Repeated letters increase the ambiguity of strings:
Evidence from identification, priming and samedifferent tasks

Iliyana V. Trifonovaa,b and James S. AdelmanaaUniversity of Warwick, bUniversity of York

Corresponding author:
Iliyana V. Trifonova
Department of Psychology
University of YorkYork
YO10 5DD
United Kingdom.
iliyana.v.trifonova@gmail.com

Abstract
Letters are often repeated in words in many languages. The present work explored themechanisms underlying processing of repeated and unique letters in strings across threeexperimental paradigms. In a 2AFC perceptual identification task, the insertion but not thedeletion of a letter was harder to detect when it was repeated than when it was unique (Exp. 1).
In a masked primed same-different task, deletion primes produced the same priming effectregardless of deletion type (repeated, unique; Exp. 2), but insertion primes were more effective
when the additional inserted letter created a repetition than when it did not (Exp. 3). In a samedifferent perceptual identification task, foils created by modifying a repetition, by either
repeating the wrong letter or substituting a repeated letter, were harder to reject than foils
created by modifying unique letters (Exp. 4). Thus, repetition effects were task-dependent.
Since considering representations alone would suggest repetition effects would always occur ornever occur, this indicates the importance of modelling task-specific processes. The similarity
calculations embedded in the Overlap Model (Gomez et al., 2008) appeared to always predict a
repetition effect, but its decision rule for the task of Experiment 1 allowed it to predict theasymmetry between insertions and deletions. In the Letters in Time and Retinotopic Space(LTRS; Adelman, 2011) model, repetition effects arise only from briefly presented stimuli as
their perception is incomplete. It was therefore consistent with Experiments 2-4 but required atask-specific response bias to account for the insertion-deletion asymmetry of Experiment 1.

Keywords: visual word recognition; repeated letters; masked priming; orthographic processing;computational modelling; letter processing2

1. Introduction
Processing visual information in alphabetic languages requires the successful identification of
constituents within multiple hierarchical levels. The identification of a word in a sentence relies
upon the recognition of the letters and their order within the word unit. The encoding of letter
identities and their position has been the focus of much orthographic processing research. This
research has investigated how the visual system encodes letter information and discriminates
between several word candidates before selecting (the correct) one from a set of representations.
The results of this research effort have led to the development of letter and word processing
theories that have been explicitly formulated in the architecture of computational models ofvisual word recognition tasks. The mechanisms of letter identity and position encoding are
implemented in the encoding schemes of these models and determine both the set of consideredcandidates and the difficulty of the process of selection.
Key findings about how letter information is processed when letters are embedded in strings
include: the activation of abstract letter representations is not affected by differences in font,case and size (e.g. Bowers, Vigliocco, & Haan, 1998; Kinoshita & Kaplan, 2008); letter strings
are processed in a specialized manner that is different from the one associated with otherdomains such as symbol strings and there is a processing advantage for initial positions(Aschenbrenner, Balota, Weigand, Scaltritti & Besner, 2017; Jordan, Thomas, Patching, &Scott-Brown, 2003; Scaltritti & Balota, 2013; Scaltritti, Dufau, & Grainger, 2018; Schoonbaert& Grainger, 2004; Tydgat & Grainger, 2009); and the encoding of letter information is initially
imprecise and there is a high degree of positional uncertainty in early word recognition stages(e.g. Kinoshita & Norris, 2009; Lupker, Perea, & Davis, 2008; Peressotti & Grainger, 1999;
Schoonbaert & Grainger, 2004; Van Assche & Grainger, 2006; Welvaert, Farioli, & Grainger,2008). Less clear is how letter information is processed when the same letter is present morethan once in one string; evidence has been mixed as to the existence of effects of repeatedletters in word identification. Such cases are however important because words with letter
repetitions are more common than those without in many languages including English, Frenchand Dutch (Trifonova & Adelman, 2019).
Recently, Trifonova and Adelman (2019) have provided evidence that the presence of repeatedletters in words delays their visual processing. This was demonstrated with a regression
approach on megastudy data of two key visual word recognition tasks: lexical decision andword naming. In these tasks, a letter string is presented on a computer screen and a response3

time is measured from the onset of the presentation. In lexical decision, participants indicate
whether the letter string forms a genuine word or not by pressing one of two corresponding
keys, whereas in word naming, they read the word aloud. Trifonova and Adelman found an
inhibitory effect of letter repetition in three different languages, English, Dutch and French. The
effect was most robust for cases in which the repeated letters appeared in close proximity but
not in immediate adjacency.
These findings are reminiscent of several previously reported perceptual phenomena suggesting
interference in cases of repeated visual information. These include: the repeated-letter
inferiority effect (Bjork & Murray, 1977; Egeth & Santee, 1981) describing poor performance incases of repeated target letters in tachistoscopic letter identification studies; the homogeneity
effect (Mozer, 1989) demonstrating lower accuracy in identifying the number of target letters incases of repetitions; the redundancy masking effect (Sayim, & Taylor, 2019) describing asimilar effect of perceived smaller numbers of letters in peripheral vision; the repetition
blindness effect, reporting lower accuracy in detecting several instances of the same word or
letter type (Kanwisher, 1987; 1991; Luo & Caramazza, 1996). Regardless of the persistent
traces of a repeated letter phenomena reported in various cognitive processing studies, previous
factorial studies investigating orthographic processing and effects of repeated letters on theprocessing of whole letter strings have provided mixed evidence. It is, therefore, still unclear
whether the repetition of a letter embedded in a letter string could change the processing
difficulty and affect reading.
The goal of this work was to provide further evidence required for the better understanding of
the processing of repeated letters embedded in letter strings, as they would appear in words. A
more detailed exploration with factorial approach would allow a comparison with the evidence
for repeated letter effect from the regression approach on large datasets (Trifonova & Adelman,2019), as well as with previous inconsistent findings from the orthographic processing literature
related to the same research question. An overview of these findings will be shortly presented.
This will be followed by a summary of the theoretical assumptions and modelling of repeatedletters in visual word recognition. The second section will present a more detailed description of
two computational models of visual word recognition: the Letters in Time and Retinotopic
Space model (LTRS: Adelman, 2011) and the Overlap Model (Gomez, Ratcliff, & Perea, 2008)
and their assumptions regarding the processing of repeated letters. This will be followed by thepresentation of four experiments employing several orthographic processing methodologies.

4

The results of the experiments will be accompanied by predictions from LTRS and the Overlap
Model and discussed in the context of both frameworks.
Previous findings
Unlike Trifonova and Adelman’s (2019) regression approach, it has been more common toapproach the investigation of letter position and identity encoding in strings using factorial
experiments. In this approach, experiments are designed with a controlled selection of stimuli,
and the effect of interest is explored with experimental manipulations and comparison between
contrasting conditions. The experimental paradigms in visual word recognition studies,
especially those focused on early perceptual processing, often involve the presentation of somebrief perceptual event, such as the target letter string in perceptual identification tasks (e.g.,
Gomez et al., 2008), or a prime in a masked-primed lexical decision task (Forster, Davis,
Schoknecht, & Carter, 1987). In the masked-priming paradigm, participants are typicallyunaware of the presence of the prime that is briefly presented after a mask (######) and prior to
the target. Nevertheless, response times are often faster when the prime is related in form to thetarget (e.g., desihn-DESIGN) in comparison to when it is not (voctal-DESIGN) (e.g., Adelman
et al., 2014). The difference between the two conditions, the priming effect, is usuallyinterpreted as reflecting the degree to which the orthographic codes between the related prime
and the target overlap. That is, higher similarities between prime and target are expected to
produce more (facilitatory) priming.
Masked-priming lexical decision studies investigating repeated letters in the past have taken
advantage of previous findings that primes formed by disrupting the precise letter order, forexample by deleting letters from the target base word (relative position subset or deletion
primes, e.g., blcn-BALCON), still produce form priming effects (e.g., Peressotti & Grainger,
1999), contrary to the predictions of a strict position-specific encoding scheme (McClelland &Rumelhart, 1981). Primes formed by deleting a repeated or a unique letter from a French baseword (e.g., balace vs. balnce for target BALANCE) produced the same form priming effect aseach other when compared with a control unrelated six-letter word-like prime (e.g., fodiruBALANCE; Schoonbaert & Grainger, 2004). These results were, however, inconsistent with themain effect of target type reported in the same study: Targets with repeated letters were moredifficult to process. In a different series of experiments, again no effects of repeated letters werereported with primes in which one or two additional repeated or unique letters were inserted(insertion primes: e.g., jusstice, juastice, jussstice, jurqstice - JUSTICE), rather than deleted
from the target (van Assche & Grainger, 2006). These insertion primes produced the same5

priming effect as each other, relative to unrelated primes and did not differ significantly fromidentity primes (justice). However, the fact that no priming cost was reported with the insertionof one or two letters in the prime suggests that the experiments of van Assche and Grainger(2006) could have been underpowered. Evidence for larger cost with increased number of letter
insertions was later provided by Welvaert, Farioli, and Grainger (2008). Furthermore, Adelman
et al. (2014) showed that primes with one redundant doubled identity (e.g., deshhign - DESIGN)
were less disruptive to the target recognition than two redundant letter identities difference (e.g.,
desaxign - DESIGN). The results with the masked-priming lexical decision task are therefore
inconclusive as to whether repeated letter effects could be observed, and whether such effects
could depend on factors such as adjacency or number of repeated letters.
Indeed, Norris, Kinoshita, and van Casteren (2010) argued that repeated letter effects should be
more robustly observed when the repeated letters are adjacent. They also suggested that themasked-priming lexical decision task might not be appropriate for the investigation of theseeffects and argued that the (masked-priming) same-different task might be a more appropriate
tool for small sublexical effects, as it is insensitive to lexical factors and the priming effect inthis task is suggested to reflect only orthographic processes (Kinoshita & Norris, 2009; Norris& Kinoshita, 2008; Norris et al., 2010). The latter argument, however, was recently challenged
by evidence in the literature suggesting a partial phonological contribution (Lupker, Nakayama,& Perea, 2015; Lupker, Nakayama, & Yoshihara, 2018). Nevertheless, priming effects are often
larger in the same-different task than the lexical decisions task, so there may be more power insuch studies. In the same-different task, a lowercase letter string serves as a temporary reference
presented for about 1s simultaneously and above a mask of symbols (#######). This event is
succeeded by the presentation of a prime that appears at the place of the mask (in lowercase),followed by the target (in uppercase). The task of the participants is to indicate whether thereference and the target are the same or not, disregarding their case difference. Norris et al.
(2010) provided evidence that repeated letter effects could be observed with the masked
priming methodology, particularly when combined with the same-different task. They showed
greater priming by primes such as uueer compared to ulger for the target UNDER, as well as byanex for ANNEX compared to eupt for ERUPT. They took this evidence as a support of amechanism of “leakage” of a letter identity to nearby positions due to early perceptual
uncertainty. However, the authors did not argue that such effects could be generalized to
nonadjacent repetitions.

6

Evidence with factorial designs for the presence of repeated letter effects in cases of
nonadjacent repetitions as well as adjacent repetitions, however, has been provided by Gomez etal. (2008), who used a two-alternative forced choice perceptual identification task. Participants
had to identify a briefly presented five-letter nonword target by choosing between the correct
option and a foil option. Higher similarities between target and foil were expected to produce
lower accuracy. Accuracy was extremely low when the target contained letter repetition in bothadjacent and nonadjacent cases. Accuracy was significantly higher when the foils and not thetarget contained a letter repetition and intermediate when both the target and the foils had arepeated letter. The authors speculated that participants had a bias towards choosing a string
without repetition and suggested that the repeated letter effect might be due to repeated letters
being difficult to detect at early perceptual stages. However, Gomez et al. did not proceed to
examine this effect in more detail.
Taken together, masked-priming paradigm studies in the past have provided some evidence fordifferential processing between unique and repeated letters for adjacent repetitions (Adelman etal., 2014; Norris et al., 2010), but not for nonadjacent cases (Schoonbaert & Grainger, 2004;
van Assche & Grainger, 2006). In addition, the evidence also raises methodological concerns
regarding the sensitivity of the masked-primed lexical decision task to effects operating on aletter level, such as the repeated letter effects and an advantage of the masked-primed samedifferent task for detecting such processes. Repeated letter effects in cases of nonadjacent
repetitions have been observed on word naming and lexical decision response times withregression approach on megastudy data (Trifonova & Adelman, 2019) and with a factorial
approach in a lexical decision (Schoonbaert & Grainger, 2004) and perceptual identification
experiments (Gomez et al., 2008). Further evidence in this area seems to be required to resolve
the inconsistent findings and to establish how repeated letters in nonadjacent positions affect theprocessing of the whole string unit, how their processing differs from that of corresponding
unique letters.

The present experiments
The present work investigated the effects of repeated letters in more detail with a factorial
approach. We aimed to further understand how the previous inconsistencies in repetition effects
might arise from differences in task demands, task sensitivity, and orthographic context. We
focused on the under-studied non-adjacent repetitions, using exclusively nonword stimuli to7limit the influence of lexical effects. Across our four experiments, we used three tasks believed
to be sensitive to early perceptual processing and enhanced our power with larger than usual
participants numbers. To interpret the results and compare models, we also modeled thesestimulus relationships and (where possible) particular tasks with several models of visual word
recognition.
The first of our four experiments used the two-alternative forced choice perceptual
identification task. This experiment focused on testing how the visual system keeps track of thenumber of occurrences of letter identities. Experiments 2 and 3 employed the masked primed
same-different task as a comparator for Experiment 1 with a task less susceptible to response
biases. The first three experiments included conditions with asymmetric orthographic
relationships, in which one of the stimuli has either a deleted or an additional inserted letter, andthe deletion or insertion may be a unique or repeated letter. Experiment 4 continued theinvestigation of the effect with new conditions in which the orthographic pair comprises ofstimuli of the same length, and the manipulations include letter replacements. The task inExperiment 4 was a hybrid between the previous two paradigms, a same-different perceptual
identification task (without priming). This further explored the processing of strings withrepeated letters and tests whether the results we obtain generalize to different orthographic
contexts and another paradigm.

Previous modelling
Theories of visual word recognition, like the evidence in the literature, have been inconsistent in
the way they treat repeated letters. An example of such inconsistency comes from successors of
the Interactive Activation (IA) Model (McClelland & Rumelhart, 1981) such as open bigram
models (Grainger, Granier, Farioli, van Assche, & van Heuven, 2006; Grainger & van Heuven,2003) and the Spatial Coding Model (SCM; Davis, 2010) and its predecessor SOLAR (Davis,
1999). These models were designed to overcome the limitations of the absolute position
encoding scheme of their predecessor and to explain masked-priming data reporting strong
priming effects from form-related primes with different cases of absolute letter order violations,and they have not been applied to tasks in which a briefly presented string is to be identified1.

1

Moreover, although the McClelland and Rumelhart (1981) model was often applied to theidentification of letters in nonwords, and a mechanism was described for deciding between two
different word alternatives, these mechanisms do not readily generalize to decisions between two

8

The commonality between these interactive activation models is the assumption that the processof selecting the right candidate for among words in the vocabulary (lexical selection) is
mediated by mechanisms (implemented with the same equations as McClelland and Rumelhart)
in which input information leads to activation of consistent candidates which further compete
for selection, while inconsistent candidates are inhibited early on. The consistent candidates
that are activated may include some that are only partially consistent with the input information,and they are only partially activated. These models therefore assume that the matching between
an input string and the candidates is graded, and the match is affected by the amount of both theconsistent and inconsistent information. One of the major differences between these models istheir encoding schemes.
With respect to detecting letter repetitions, the Interactive Activation Model (McClelland &Rumelhart, 1981) is insensitive, as the letter identity information operates independently foreach letter position. The open bigram model (Grainger & van Heuven, 2003) is sensitive torepetitions. This model encodes letter position and identity through an intermediate set ofrepesentations (open bigrams) formed by two-letter cobinations with a correct relative left-toright order. As the word representations are represented by the unique set of open bigrams andduplications of letters lead to duplications of open bigrams, letter repetitions can affect thepredictions of the model via the calculation of the matching between the input and thecandidates. The Spatial Coding Model (Davis, 2010) uses letter representations for identities
and letter positions are represented by spatial patterns. The same letter representations can beused for the same letter in different positions. However, the model has an explicit mechanismthat deals with letter repetitions to ensure the model effectively treats repeated letters in thesame way as different ones. Therefore, this model is not sensitive to letter repetitions (seeTrifonova & Adelman, 2019, for a more detailed discussion of these models and their relevanceto repeated letter effects). It is unclear how these models should be extended to deal withidentification of briefly presented nonwords or priming to nonwords (in tasks other than lexicaldecision) given the centrality of the lexical matching process, but it is possible to examine thenumerical match that would drive activation for analogous situations in the priming of words.
Such “match scores” are commonly examined to understand the predictions of these types ofmodels, although the exact predictions can be influenced by other aspects of the aforementioned
lexical matching process.

nonword alternatives, because (like the successor models) decisions are based on activity in thepermanent lexical representations.

9Another model designed to overcome the limitations of the position specific encoding scheme is
the Overlap Model (Gomez et al., 2008). This model implements positional uncertainty byrepresenting letter identities in a brief letter string as normal distributions along a letter position
dimension. The distributions are centered on the letter positions and their standard deviations
are free parameters with the initial position typically having a smaller standard deviation (lessnoise) than the other positions. The model asseses the similarity between two letter strings bysummating the areas of overlap under the curves of the brief string that match a comparison
string whose representation is uniformly spread within the relevant position window (i.e., ±0.5
from the center). The two-alternative forced-choice perceptual identification task isimplemented by first applying a power function to the overlap scores so-calculated, and thenapplying the Luce (1959) choice rule to the resulting values. Gomez et al. demonstrated that themodel could account for effects such as letter transpositions, insertions, deletions and indeed
repetitions, as in these cases a consistent letter identity spills over, or leaks to nearby positions
and still contributes for a higher overlap score relative to inconsistent letter identity, despite its
incorrect position. Gomez et al. also (in an appendix) ordinally compared overlap scores fromthe model to priming effects from a handful of studies, much as match scores from interactive
activation based models as have been, but these investigations did not extend to studies of letter
repetitions.
An alternative theory of orthographic processing that is also not based on the IA framework andthat has predicted visual processing differences between repeated and unique letters was
proposed by Adelman (2011). In his model, Letters in Time and Retinotopic Space (LTRS), theprocess of word identification and both the observed masked priming and perceptual(tachistoscopic) identification effects are explained by the means of gradual accumulation andavailability of visual information in time. Unlike the IA models, priming effects do not result
from a graded match that weighs the consistent and inconsistent information and activates thetarget to a proportionate degree. Rather, in LTRS, inconsistent information is treated as clear
evidence that a stimulus is not a particular target.
According to LTRS (Adelman, 2011), during perceptual processing of the prime, several
potential targets can be temporarily consistent with the prime, which causes lexical processingof those targets. When any inconsistency is detected between the prime and a potential target,that target no longer undergoes further lexical processing. Perceptual processing of the prime
continues (until the prime offset) so that any remaining potential targets can be evaluated. It is
the timing of the detection of inconsistency (or timing of the detection of target onset if no10

inconsistency is detected) that determines the size of the priming effect that target would
receive. In LTRS this timing is probabilistic, not deterministic.
Bigger discrepancies from the target give more opportunities for earlier detection ofinconsistency and therefore lead to smaller priming effects. Differences in cases of repeatedletters are predicted in some conditions: While a unique excess letter is immediately discrepant
whenever it is perceived, a repeated letter cannot be perceived to be in excess – and hence
discrepant – without further information, either about its twin, or (for non-adjacent repetitions)
its local context of other letters. Adelman (2011) showed that LTRS accounts for the repeatedletter results in tachistoscopic identification of Gomez et al. (2008), and in primed lexicaldecision of Schoonbaert and Grainger (2004) and Norris et al. (2010) and argued that for Van
Assche and Grainger’s (2006) primed lexical decision experiments, the observed null results
were within the predicted range, given the power and precision of those experiments and thesmall average effects predicted by LTRS for these conditions.
Present modelling
The data from the present experiments will be fitted with LTRS (Adelman, 2011), and theOverlap Model (Gomez et al., 2008). The models will be avaluated across several paradigms –two-alternative forced choice perceptual identification, masked priming (in a same-differenttask), and perceptual identification (again in a same-different task). Additional implementationsof the models for these tasks will be presented where necessary. In addition to the simulations
with LTRS and the Overlap Model, successors of the Interactive Activation model (McClelland& Rumelhart, 1981) will also be evaluated with match scores. The match scores calculations
will be obtained with Davis’s Match Score calculator(http://www.pc.rhul.ac.uk/staff/c.davis/utilities/matchcalc/index.htm) for the SOLAR model(Davis, 1999), as well as for two versions of the overlap bigram scheme: open bigram (Grainger& van Heuven, 2003), and the overlap open bigram (Grainger et al., 2006). The predictions of
all models will be presented along with the results of each experiment. In addition, thosepredictions and the key findings will be summarized in a table (Table 14) and will be further
discussed in the General Discussion section.
In the next section, we will present a more formal description of LTRS (Adelman, 2011) and theOverlap Model (Gomez et al., 2008) and their application to the tasks used in this work. This
will be followed by the presentation of the four new experiments.

112. Models
2.1. LTRSIn the below, we explain the Letters in Time and Retinotopic Space model (LTRS: Adelman,2011) explicating several aspects that were implicit in the original paper, including explicit
formulae for the derivation of numerical predictions and demonstration of the use of explicit
listing of possible perceptual states to derive qualitative parameter-free predictions (in thosecases where the model makes such predictions).
LTRS is a model in which different elements of perceptual information arrive at random times
to form discrete perceptual states. Each stimulus can lead to some perceptual states but notothers. Elements of information about different letters arrive independently, so the sequence ofperceptual states is not entirely predictable, and different states are passed through on different
presentations of the same stimulus. The various intermediate states in which not all perceptual
information has arrived are ambiguous between several possible stimuli.
When a stimulus presentation is brief and post-masked, perception may be terminated in one ofthese intermediate ambiguous states. If the task is to identify the briefly presented stimulus fromtwo options, accuracy on a single trial depends on whether perception terminates in a state thatis ambiguous between the two response options: If it is ambiguous, then guessing must occur. If
it is unambiguous, then the information identifies the correct option.
Figure 1 summarizes the core processes of the model for a presentation of the word “top” in thecontext of a two-alternative forced-choice between “top” and “cop”. After an initial attentional
delay, processing starts at (0,0,0); each dimension represents the processing that has occurred
for a particular letter position in the presented string. When no processing has occurred, any
string would be compatible with the perceptual constraints, which are summarized as “*”. If thepresentation ceased while perceptual processing was in this state, the participant would clearlyneed to guess, because the percept is consistent with the incorrect foil option as well as thecorrect target option, indicated in the diagram by the dashed border. Otherwise, the next
perceptual event could occur for any of the three letter dimensions, moving to (1,0,0), (0,1,0) or(0,0,1); each corresponds to the relevant letter being identified without any currently useful
positional information. There are two types of perceptual event for each letter dimension (0→1,12which provides letter identity and minimal positional information; 1→2, which provides moredetailed positional information) so perceptual processing is complete if and when (2,2,2) isreached. For the “top”-“cop” trial, of course, guessing must occur if the first letter has not been
perceived, and an accurate response can be made if the first letter has been perceived (subject tothe effects of masking and motor error).
Figure 2 provides a similar summary for a trial in which the stimulus presented is “pop” and theoptions are “pop” and “cop”. All calculations regarding the perception of each position areunchanged from the preceding example (assuming unchanged parameters). The consequences,
however, differ because the first letter is now a “p” that is the same identity as the common
third position of the two options, and not the “t” that occurs only in the target. As such, theperceptual constraints at (1,0,0) can be summarized as “*p*” – a “p” with no (currently) useable
positional information. In other words, this “p” could be the “p” in final position of “cop” (whenin fact it is a first “p” of “pop”) because to decide that this “p” is not the final “p”, it would need
to be known that there is another “p”, or that there is an “o” after this “p”, or that this “p” is at
the beginning – any of these additional pieces of information would disambiguate the situation.
Therefore, a guess must occur for this state in the “pop”-“cop” trial, where a correct response
was possible for the analogous state in the preceding “top”-“cop” example. For these examples,
this is the only state whose implications differ between “top”-“cop” and “pop”-“cop” (although
the situation is more complicated for internal repetitions). Since the probabilities of analogous
states (those with matching co-ordinates) are the same for the two types of trial, the probabilityof an accurate response must be higher in the “top”-“cop” trial than the “pop”-“cop” trial.
Note that this is a qualitative prediction that does not depend on the details of the timings,
probabilities, or possible movement between perceptual states. The prediction occurs because(i) for two trials of the same stimulus length, any possible perceptual state for the first trial has
an analogous state for the second trial with an identical probability; (ii) there exists a state thatis ambiguous between options for the second trial, but the analogous state is unambiguous forthe first trial; and (iii) there is no state for which the converse is true. The states for the trials at
hand meet these conditions because the first letter has been perceived and cannot be part of“cop” but can be part of “top” in the “top”-“cop” trial, but can be mistaken for the final “p” of“cop” in the “pop”-“cop” trial, because it is possible to partially perceive the stimulus in a waythat there is no way to determine the position of this letter.
Similar considerations are relevant for trials in which the brief stimulus is a prime, rather than
the target of an identification decision, but it is not the final state that is relevant, but rather the13

time spent in ambiguous states: that is, perceptual states of the prime that cannot be
distinguished from the target. Lexical access occurs for words while they are consistent with thecurrent perceptual state (what has been perceived so far), so the target is primed an amountequal to the time for which the prime percept is ambiguous with the target. (A similar process
may occur for nonwords for which a representation has been established, but not nonwords ingeneral.) Therefore, more priming occurs for prime-target combinations (in, e.g., lexicaldecision) that are less accurate as target-foil combinations in identification (with the brief primeof the priming trial being analogous to the brief target of the identification trial).
LTRS thus explains graded similarity-like effects with brief stimuli without positing a gradedsimilarity-like computation as part of the cognitive processes involved. Instead, the core
premise of LTRS is that such phenomena reflect the (probabilistic) incompleteness of thepercept when the stimulus is brief, with potentially relevant information simply being missing.
As such, what is important is how soon relevant information – such as an inconsistency between
foil and target or between prime and target – can be perceived, that is, how easily an
inconsistency is detected. Additional inconsistencies make it easier overall to detect that aninconsistency exists, but different information has different time courses, so the number ofinconsistencies is not strictly relevant, and no single number can summarize difficulty at
different time points (indeed, the ordering is not always preserved). This contrasts with othermodels that attribute these phenomena to a system that tolerates inconsistences and/or mistrusts
the percept and have an underlying match or distance score. Although a mechanism must exist
for expert readers to tolerate typographical and spelling errors, LTRS was designed to explain
identification and priming phenomena without invoking this mechanism.
In the following, we detail for the original Adelman (2011) LTRS model (1) the perceptualstates: their probability distributions over time, and how the perceptual constraints follow fromeach perceptual state; (2) calculations for predicted accuracy for the two-alternative forcedchoice identification task; (3) calculations for predicted response time priming for tasks withbriefly presented primes; and (4) the role of stochastic dominance in qualitative predictions of
the model. Further elaborations of the model will be discussed later for Experiment 1 – (§3.3.1)
where an alternative guessing rule is considered – and Experiment 4 – (§4.3.1) where the samedifferent identification task requires a new decision rule.
2.1.1. Perceptual processing and inference common to all paradigms

14Each letter in a briefly presented string can be in one of three perceptual states: unperceived;
perceived with relative location information; or perceived with full location information. On
each trial, each letter starts in the unperceived state (state 0) and at times determined by arandom distribution can pass to a partially perceived state (state 1) and at times determined byanother random distribution from that state to a fully perceived state (state 2). These processes
occur independently, so at any time during the stimulus, or at stimulus offset, any combinationof different states of different letters is possible. However, each letter’s state proceeds in theincreasing direction (except for a possible post-mask effect). The perceptual state of the lettersin an n-letter long stimulus at time t therefore can be characterized as a vector: x(t) = (x1, x2, , xn)
where the xi can take values 0 (no information), 1 (partial information) or 2 (full information).
The perceptual information that can be extracted from the stimulus s in perceptual state x iscomplied into perceptual constraints ψ that can be summarized in a textual pattern format. We
justify the value of ψ for a given x and s based on the information contained in an intermediate
representation y (though the intermediate representation need never be computed to derivepredictions); example values of y can be seen in Tables 1 and 2, and their interpretation will
now be described.
The perceptual information that is derived so far from a particular stimulus s in a particular
perceptual state is a set of perceived letters (those letters with state xi > 0) each of which istagged with some location information (whose values are only meaningful in relation to thelocations of other letters), plus always available are the left and right edge co-ordinates (lmin andrmax) of the entire stimulus. This information is a vector y where the i indices from x are notavailable, and the letters have indices j that are arbitrary and not related to letter order. A letteryj corresponding to state 1 has the information (zj, cj) where zj is the letter’s identity, and cj is ahorizontal co-ordinate somewhere inside that letter; such a location is insufficient to determine
if this letter is adjacent to another letter, or is first or last in the string. This is, however,
sufficient information to determine the order of two perceived letters. A letter yi corresponding
to state 2 has the information (zj, lj, rj) where the zj is the letter’s identity (same as state 1), lj is
its left edge co-ordinate and rj is its right edge co-ordinate; this location information is sufficient
to determine (in combination with lmin and rmax that are always known) if the letter is first or last,and if the letter is adjacent to another letter, if the edges of that letter are also available.

15

Figure 1. Diagram of all possible perceptual states (and all possible transitions between them) in LTRS when the stimulus s = “top”, and theconsequences for such an identification trial with the foil “cop”. A similar diagram for a repeated-letter stimulus (“pop”) is given in Figure 2.

Figure 2. Diagram of all possible perceptual states (and all possible transitions between them) in LTRS when the stimulus s = “pop”, and theconsequences for such an identification trial with the foil “cop”. A similar diagram for a non-repeated-letter stimulus (“top”) is given in Figure 1.

17

Table 1. Enumeration of possible perceptual states x, example corresponding perceptual information y, and corresponding textual constraint
patterns ψ for a stimulus s = “top” whose edges are at 0.4 and 4: “t” has width 1; “o” has width 1.2, and “p” has width 1.4. On any given trial, at
most seven of these states are reached before post-masking. Table 2 shows a similar trial in which s = “pop”.
State x =Information y: The numbers and letter order here areConstraint
Could be “cop”? Possible Next States(x1, x2, x3) examples only and would vary from trial to trial
pattern ψ
0, 0, 0{ lmin = 0.4, rmax = 4 }*Y(0,0,1) OR (0,1,0) OR (1,0,0)0, 0, 1{ (p, 3.1), lmin = 0.4, rmax = 4 }*p*
Y(0,0,2) OR (0,1,1) OR (1,0,1)
0, 0, 2{ (p, 2.6, 4), lmin = 0.4, rmax = 4 }*pY(0,1,2) OR (1,0,2)
0, 1, 0{ (o, 1.9), lmin = 0.4, rmax = 4 }*o*Y(0,1,1) OR (0,2,0) OR (1,1,0)
0, 1, 1{ (p, 3.1), (o, 1.9), lmin = 0.4, rmax = 4 }*o*p*Y(0,1,2) OR (0,2,1) OR (1,1,1)
0, 1, 2{ (o, 1.9), (p, 2.6, 4), lmin = 0.4, rmax = 4 }*o*pY(0,2,2) OR (1,1,2)
0, 2, 0{ (o, 1.4, 2.6), lmin = 0.4, rmax = 4 }+o+Y(0,2,1) OR (1,2,0)
0, 2, 1{ (p, 3.1), (o, 1.4, 2.6), lmin = 0.4, rmax = 4 }+o*p*Y(0,2,2) OR (1,2,1)
0, 2, 2{ (p, 2.6, 4), (o, 1.4, 2.6), lmin = 0.4, rmax = 4 }+op*
Y(1,2,2)
1, 0, 0{ (t, 0.8), lmin = 0.4, rmax = 4 }*t*N(1,0,1) OR (1,1,0) OR (2,0,0)
1, 0, 1
{ (t, 0.8), (p, 3.1), lmin = 0.4, rmax = 4 }*t*p*N(1,0,2) OR (1,1,1) OR (2,0,1)
1, 0, 2{ (t, 0.8), (p, 2.6,4), lmin = 0.4, rmax = 4 }*t*pN(1,1,2) OR (2,0,2)
1, 1, 0{ (t, 0.8), (o, 1.9), lmin = 0.4, rmax = 4 }*t*o*N(1,1,1) OR (1,2,0) OR (2,1,0)
1, 1, 1
{ (t, 0.8), (p, 3.1), (o, 1.9), lmin = 0.4, rmax = 4 }*t*o*p*N(1,1,2) OR (1,2,1) OR (2,1,1)
1, 1, 2{ (t, 0.8), (p, 2.6, 4), (o, 1.9), lmin = 0.4, rmax = 4 }*t*o*pN(1,2,2) OR (2,1,2)
1, 2, 0{ (t, 0.8), (o, 1.4, 2.6), lmin = 0.4, rmax = 4 }*t*o+N(1,2,1) OR (2,2,0)
1, 2, 1
{ (t, 0.8), (p, 3.1), (o, 1.4, 2.6), lmin = 0.4, rmax = 4 }*t*o*p*N(1,2,2) OR (2,2,1)
1, 2, 2{ (t, 0.8), (p, 2.6, 4), (o, 1.4, 2.6), lmin = 0.4, rmax = 4 }*t*op
N(2,2,2)2, 0, 0{ (t, 0.4, 1.4), lmin = 0.4, rmax = 4 }t*
N(2,0,1) OR (2,1,0)
2, 0, 1
{ (t, 0.4, 1.4), (p, 3.1), lmin = 0.4, rmax = 4 }t*p*N(2,0,2) OR (2,1,1)
2, 0, 2{ (t, 0.4, 1.4), (p, 2.6, 4), lmin = 0.4, rmax = 4 }t+pN(2,1,2)2, 1, 0{ (t, 0.4, 1.4), (o, 1.9), lmin = 0.4, rmax = 4 }t*o*N(2,1,1) OR (2,2,0)
2, 1, 1
{ (t, 0.4, 1.4), (p, 3.1), (o, 1.9), lmin = 0.4, rmax = 4 }t*o*p*
N(2,1,2) OR (2,2,1)
2, 1, 2{ (t, 0.4, 1.4), (p, 2.6, 4), (o, 1.9), lmin = 0.4, rmax = 4 }t*o*pN(2,2,2)2, 2, 0{ (t, 0.4, 1.4), (o, 1.4, 2.6), lmin = 0.4, rmax = 4 }to+N(2,2,1)2, 2, 1
{ (t, 0.4, 1.4), (p, 3.1), (o, 1.4, 2.6), lmin = 0.4, rmax = 4 }to*p*N(2,2,2)2, 2, 2{ (t, 0.4, 1.4), (p, 2.6, 4), (o, 1.4, 2.6), lmin = 0.4, rmax = 4 } top
N
Note – In y, the ordering of letter elements is arbitrary (and ignored), and the precise numerical values vary randomly and with the properties of thephysical display, but this information is abstracted away to produce location and font invariances in the constraints. The possible ψ constraints inthe final column constitute C(“top”) the constraint list consistent with “top”.

Table 2. Enumeration of possible perceptual states x, example corresponding perceptual information y, and corresponding textual constraint
patterns ψ for a stimulus s = “pop” whose edges are at 0 and 4: “o” has width 1.2, and both “p”s have width 1.4. On any given trial, at most sevenof these states are reached before post-masking. Table 1 shows a similar trial in which s = “top”.
State x =Information y: The numbers and letter order here areConstraint
Could be “cop”?
Possible Next States(x1, x2, x3) examples only and would vary from trial to trial
pattern ψ
0, 0, 0{ lmin = 0, rmax = 4 }*Y(0,0,1) OR (0,1,0) OR (1,0,0)0, 0, 1{ (p, 3.1), lmin = 0, rmax = 4 }*p*
Y(0,0,2) OR (0,1,1) OR (1,0,1)
0, 0, 2{ (p, 2.6, 4), lmin = 0, rmax = 4 }*pY(0,1,2) OR (1,0,2)
0, 1, 0{ (o, 1.9), lmin = 0, rmax = 4 }*o*Y(0,1,1) OR (0,2,0) OR (1,1,0)
0, 1, 1{ (p, 3.1), (o, 1.9), lmin = 0, rmax = 4 }*o*p*Y(0,1,2) OR (0,2,1) OR (1,1,1)
0, 1, 2{ (o, 1.9), (p, 2.6, 4), lmin = 0, rmax = 4 }*o*pY(0,2,2) OR (1,1,2)
0, 2, 0{ (o, 1.4, 2.6), lmin = 0, rmax = 4 }+o+Y(0,2,1) OR (1,2,0)
0, 2, 1{ (p, 3.1), (o, 1.4, 2.6), lmin = 0, rmax = 4 }+o*p*Y(0,2,2) OR (1,2,1)
0, 2, 2{ (p, 2.6, 4), (o, 1.4, 2.6), lmin = 0, rmax = 4 }+op*
Y(1,2,2)
1, 0, 0{ (p, 0.8), lmin = 0, rmax = 4 }*p*
Y(1,0,1) OR (1,1,0) OR (2,0,0)
1, 0, 1{ (p, 0.8), (p, 3.1), lmin = 0, rmax = 4 }*p*p*N(1,0,2) OR (1,1,1) OR (2,0,1)
1, 0, 2{ (p, 0.8), (p, 2.6,4), lmin = 0, rmax = 4 }*p*pN(1,1,2) OR (2,0,2)
1, 1, 0{ (p, 0.8), (o, 1.9), lmin = 0, rmax = 4 }*p*o*N(1,1,1) OR (1,2,0) OR (2,1,0)
1, 1, 1{ (p, 0.8), (p, 3.1), (o, 1.9), lmin = 0, rmax = 4 }*p*o*p*N(1,1,2) OR (1,2,1) OR (2,1,1)
1, 1, 2{ (p, 0.8), (p, 2.6, 4), (o, 1.9), lmin = 0, rmax = 4 }*p*o*pN(1,2,2) OR (2,1,2)
1, 2, 0{ (p, 0.8), (o, 1.4, 2.6), lmin = 0, rmax = 4 }*p*o+N(1,2,1) OR (2,2,0)
1, 2, 1{ (p, 0.8), (p, 3.1), (o, 1.4, 2.6), lmin = 0, rmax = 4 }*p*o*p*N(1,2,2) OR (2,2,1)
1, 2, 2{ (p, 0.8), (p, 2.6, 4), (o, 1.4, 2.6), lmin = 0, rmax = 4 }*p*op
N(2,2,2)2, 0, 0{ (p, 0, 1.4), lmin = 0, rmax = 4 }p*N(2,0,1) OR (2,1,0)
2, 0, 1{ (p, 0, 1.4), (p, 3.1), lmin = 0, rmax = 4 }p*p*N(2,0,2) OR (2,1,1)
2, 0, 2{ (p, 0, 1.4), (p, 2.6, 4), lmin = 0, rmax = 4 }p+pN(2,1,2)2, 1, 0{ (p, 0, 1.4), (o, 1.9), lmin = 0, rmax = 4 }p*o*N(2,1,1) OR (2,2,0)
2, 1, 1{ (p, 0, 1.4), (p, 3.1), (o, 1.9), lmin = 0, rmax = 4 }p*o*p*
N(2,1,2) OR (2,2,1)
2, 1, 2{ (p, 0, 1.4), (p, 2.6, 4), (o, 1.9), lmin = 0, rmax = 4 }p*o*pN(2,2,2)2, 2, 0{ (p, 0, 1.4), (o, 1.4, 2.6), lmin = 0, rmax = 4 }po+N(2,2,1)2, 2, 1{ (p, 0, 1.4), (p, 3.1), (o, 1.4, 2.6), lmin = 0, rmax = 4 }po*p*N(2,2,2)2, 2, 2{ (p, 0, 1.4), (p, 2.6, 4), (o, 1.4, 2.6), lmin = 0, rmax = 4 } pop
N
Note – In y, the ordering of letter elements is arbitrary (and ignored), and the precise numerical values vary randomly and with the properties of thephysical display, but this information is abstracted away to produce location and font invariances in the constraints. The possible ψ constraints inthe final column constitute C(“pop”) the constraint list consistent with “pop”.
19The useful perceptual information present is more conveniently summarized as the textual
pattern ψ, which can include wildcards. Two wildcard characters are useful to express
constraints in LTRS’s perceptual information: We use + to indicate that at least one letter isneeded between the preceding and following item for a match to be made and * to indicate that
letters may be present or absent between the preceding and following item. In this format, awhole pattern match is implied, and the beginning and end must match. Thus, go matches g*oand *g*o but not g+o; gro and grio match all three; and agro matches *g*o but not g*o or g+o;and grob matches none of them. Table 1 illustrates the relationship between x, y and ψ for theexample stimulus “top”, adding y examples to the information of Figure 1; and Table 2
illustrates the same for the example “pop” of Figure 2.
For a perceptual state x(t) at a time t during perception of a stimulus s, the perceptual system
has available perceptual information y(x(t); s) from which it can compute the perceptual
constraints ψ by a procedure equivalent to that in Table 3.
Table 3. Pseudocode algorithm to summarize y constraints as a textual pattern, ψ.
Identify the left-most unprocessed letter yj with the lowest left-edge lj or internal position cj
If lj is unavailable or lj ≠ lmin (i.e, is not at the stimulus left-edge), append “*” to pattern
Append the current letter identity zj to pattern.
Mark the current letter yj as processed.
Keep a temporary record of the most recently processed letter p = j.
While elements of y remain unprocessed:
Update j by identifying the left-most unprocessed letter yj – that with the lowest lj or cj
If the previous letter’s right-edge rp or the current letter’s left edge lj is unavailable, then append “*” topatternIf both rp and lj are available and the current letter is not adjacent to the previous letter, i.e., rp ≠ lj, thenappend “+” to pattern
Append the current letter identity zj to pattern
Mark current letter yj as processed
Record the most recently processed letter p = j.
Loop
If the right edge of the last letter rj is unavailable or rj ≠ rmax (i.e., it is not at the right edge of the stimulus), thenappend “*” to pattern

The constraints ψ ultimately follow from the following: If a letter is in state 1 or 2, its identity is
known. If a letter is in state 2, it is also known whether it is the initial or final letter. If two
letters are both in either state 1 or 2, their order is additionally known. If two letters are both instate 2, then whether they are adjacent is also known (non-adjacency can also be implied by anintervening letter).
When we are calculating the predictions of the model, it is unnecessary to do computations thatinvolve y. We can work out ψ directly from x – through the procedure in Table 4 – and thissimplifies away from the details of positional information (cj, lj, rj, lmin, rmax), as the result is thesame as would be achieved via obtaining y and applying Table 3’s procedure.
Table 4. Pseudocode algorithm to summarize perceptual constraints arising from a given x and s
as a textual pattern, ψ.
If x1 < 2 (the first position does not have full positional information), append “*” to pattern
If x1 > 0 (the first position has been perceived), append s1 (its identity) to pattern
If x1 = 2, let a = 1
Let i = 2; Do:
If xi = 2 and a = 1 and xi-1 ≠ 2, append “+” to pattern
If xi = 0 and xi-1 > 0, append “*” to pattern
If xi > 0 append si to pattern and let a = 0
If xi = 2, let a = 1
Increment i
While i ≤ n
If xn = 1, append “*” to pattern

For any given vocabulary item or response option w, only a finite set of sets of perceptual
constraint patterns can be reached from the 3n possible x states; this set of consistent patterns
C(w) can be computed by iterating over the 3n possible x states. The perceptual informationfrom a stimulus s at a given time t, y(t; s) on any trial can be compared to the set C(w) to
determine whether the percept is consistent with the vocabulary item or response option w. The
Boolean values, v(w; y) = 1 if ψ(y) ∈ C(w), 0 otherwise, are central to the LTRS conception of
how perceptual information is used: Incomplete perceived perceptual information is comparedto the incomplete perceptual information that could possibly be perceived from a stimulus, andmodus tollens applies to determine inconsistency. That is: If the stimulus s were the option w,
21

then the percept ψ would be one of the C(w); if it is not, then s cannot be w. So, for the example
stimulus “top”, C(“top”) is exactly the elements of the third column of Table 1. When deciding
if a percept might be “top”, it is compared to the possible percepts of “top”, C(“top”).
To produce numerical predictions, specification of the probabilistic time course of the percept x(in Equations 2-4) makes it possible to compute (by summing the probability of all consistentcases) the probability that the percept from a stimulus s at time t will be consistent with avocabulary item or response option w, that is:𝑃(𝑉(𝑤) = 1; 𝑠, 𝑡) = ∑𝒙: 𝜓(𝒙; 𝑠)∈ 𝐶(𝑤)(1)

𝑃(𝑿(𝑡) = 𝒙) .

a value used to compute all LTRS predictions. That is, the probability that the percept x is
consistent with the item w under consideration (that v(w) is 1) is the probability that the percept
is in one of the states that produces a constraint pattern ψ that is among the constraint patterns
consistent with the considered item w (C(w)). Since these states are mutually exclusive, the total
probability of consistency is the sum of the probabilities of these consistent states (those states x
whose corresponding pattern ψ is in C(w)).
Consider our examples in Table 1 and 2 of targets “top” and “pop” with respect to the foil“cop”. The difference in probability of ambiguity comes from the different terms involved in
the sum: 𝒙: 𝜓(𝒙; 𝑠) ∈ 𝐶(𝑤) for the two different targets. For the trial with the target s = “pop”
and foil w = “cop”, the sum is based on the first ten x values in Table 2. In contrast, for thetarget s = “top” with the same foil, only the first nine of these are included from Table 1. Solong as the probability of the x = (1,0,0) state (the tenth row) is greater than 0, P(V(cop) = 1) –the probability the foil “cop” is compatible with the percept – is higher for “pop”-“cop” than it
is for “top”-“cop”. Percepts arising from “pop” are more probably ambiguous than those arisingfrom “top,” and this does not depend on the specifics of the timing or sequencing of theperceptual states, nor any parameters (so long as the same parameters are used in both cases).
The above aspects of the model serve to make qualitative predictions in many (but not all) casesof interest (so long as it is assumed certain probabilities will be greater than 0, which is true for
finite parameter values). To make specific quantitative predictions, the probabilities ofperceptual states over the time course of perception need to be specified. The time course of thepercept x has two components, an initial random perceptual delay t0 (in milliseconds) affecting
all components equally on the same trial, and the otherwise independent subsequent transitions
from 0 to 1 to 2 in the individual elements xi. The transitions from 0 to 1 have a constant hazard
βi|n dependent on the letter position i and stimulus length n; when time is expressed in22

milliseconds, the β values have a natural unit of kHz (letters/ms). The transitions from 1 to 2
have a constant hazard λβi|n proportional to that for the first transition (λ is a ratio of kHz/kHz).
The specification of constant hazard defines an exponential distribution, so it can be shown that:
𝑃(𝑋𝑖 = 0 | 𝑇0 = 𝑡0 ; 𝑡) = 𝑒 −𝛽𝑖|𝑛 (𝑡−𝑡0 ) ;𝑃(𝑋𝑖 = 2 | 𝑇0 = 𝑡0 ; 𝑡) = 1 −1(𝑒 −𝛽𝑖|𝑛 (𝑡−𝑡0 ) − 𝑒 −𝜆𝛽𝑖|𝑛 (𝑡−𝑡0 ) ) − 𝑒 −𝛽𝑖|𝑛 (𝑡−𝑡0 )λ−1𝑃(𝑋𝑖 = 2 | 𝑇0 = 𝑡0 ; 𝑡) = 1 − (𝛽𝑖|𝑛 (𝑡 − 𝑡0 ) + 1)𝑒 −𝛽𝑖|𝑛 (𝑡−𝑡0 )if 𝜆 ≠ 1;

if 𝜆 = 1; and𝑃(𝑋𝑖 = 1 | 𝑇0 = 𝑡0 ; 𝑡) = 1 − 𝑃(𝑋𝑖 = 0 | 𝑇0 = 𝑡0 ; 𝑡) − 𝑃(𝑋𝑖 = 2 | 𝑇0 = 𝑡0 ; 𝑡).

(2)These processes are independent, so
𝑃(𝑿 = 𝒙 | 𝑇0 = 𝑡0 ; 𝑡) = ∏ 𝑃(𝑋𝑖 = 𝑥𝑖 | 𝑇0 = 𝑡0 ; 𝑡) .
𝑖

(3)

The distribution of T0 is N(α, σ) (with α and σ normally in milliseconds) so the non-conditional
probabilities for V = 0 and (complementarily) V = 1 are computed by numerical integration of𝑃(𝑉(𝑤) = 0; 𝑠, 𝑡) = ∫∑𝒙: 𝜓(𝒙;𝑠)∉ 𝐶(𝑤)𝑑𝑃(𝑇0 = 𝑡0 )∏ 𝑃(𝑋𝑖 = 𝑥𝑖 | 𝑇0 = 𝑡0 ; 𝑡) 𝑑𝑡0
𝑑𝑡0
𝑖

(4)

where the dP/dt0 is normal density function.

2.1.2. Two-alternative forced choice perceptual identification
In the task of Experiment 1, a stimulus s is presented for a time t then masked, and correct
option s and a foil option f are presented as a two-alternative forced choice, so if only one v is 1,
then that option should be chosen, if possible. Clearly, v(s) is always 1 because the percept isalways consistent with the presented stimulus. Therefore, accuracy only depends on theprobability of v(f) being 0 after the post-mask – that is, how probably the percept rules out the
foil by providing constraints inconsistent with the foil.
The effect of the post-mask can be to cause a loss of information for a letter. Such a transition
back to 0 can occur for each xi independently with probability φ. The P(Xi = 1) and P(Xi = 2) ofEquation 2 are then adjusted to masked versions of these probabilities by multiplication by (1 -23

φ) and the lost probability is added to P(Xi = 0). These probabilities for the masked case can beinserted into Equations 3 and 4 to calculate 𝑃(𝑉 = 0) specifically for the masked case.
So, if after the effect of the post-mask, v(f) = 0, then responding is correct, unless a premature
guess occurs with probability ε. If v(f) = 1, then a guess must be made. Accuracy is therefore
computed thus:P(no guess) = (1 − ε)𝑃(𝑉(𝑓) = 0; 𝑠, 𝑡, masked)
P(accurate) = P(no guess) + gc(s, f)P(guess) =(1 − ε)𝑃(𝑉(𝑓) = 0; 𝑠, 𝑡, masked) + 𝑔𝑐 (𝑠, 𝑓)(1 − (1 − ε)𝑃(𝑉(𝑓) = 0; 𝑠, 𝑡, masked))(5)

where gc(s, f) is the probability of a correct guess.
In most previous applications of LTRS to this task, trials in which w1 was the stimulus and w2
was the foil formed the same condition as trials in which w2 was the stimulus and w1 was thefoil, so that the probability of responding correctly from a guess averaged to gc = 0.5 within acondition. Some forms of bias could occur but would not affect the overall accuracy of theconditions: A bias towards, for instance, the more frequent of an option pair would average out
between the trials in which that response was correct and those in which it was incorrect.
Moreover, there is no way one could be biased towards, for instance, the “transposed-letter”
option in a pair such as swan-sawn, because this is a reciprocal relationship within the pair. (Ahand bias was fitted in some cases, but the fitting for the current experiments did not require thislevel of detail.)This version of the model was the one we first fitted to Experiment 1. We also describe below
in the information on modelling individual experiments (i) a modified version with a different
guessing rule for Experiment 1; and (ii) a similar model for the same-different task ofExperiment 4.

2.1.3. Masked priming
In priming paradigms, a briefly presented prime s is presented before a task must be performed
on a target w that has an established representation. In prior applications of LTRS, w was aword with an established lexical entry in the lexical decision, and no priming was predicted for“unseen” nonwords. In the present Experiments 2 and 3, a representation w0 is established for a24reference nonword stimulus presented clearly at the beginning of the trial. The task is a samedifferent judgment between w0 and w, in which s typically has a priming effect on trials where
the correct answer is same. (This difference between seen [same] and unseen [different] targets
is analogous to the word-nonword distinction in lexical decision.) This generalization is simple
in LTRS because it does not specify how other aspects of lexical processing and the actual
lexical decision occur, only how much time is later “saved” due to prime processing.
In LTRS, priming is caused by the prime going through incompletely perceived states that areconsistent with the target, so a non-identical prime percept can be ambiguous with the target;that is v(w) can be 1 during the prime even if the prime is not identical to the target. Priming is
not produced by only the ambiguity at the offset of the prime. Rather, potential identities for theprime are undergoing processing when their v is 1, and the time spent on this processing of thetarget during the prime is saved on later processing of the target, affecting response times.
Prime processing is assumed to continue after prime offset for a period lasting ω (omega)
during which no changes occur to the prime percept.
Priming as access for a target w from prime s on a trial with prime duration l is therefore𝑙𝛾=∫𝑣(𝑤; 𝒚(𝑡; 𝑠)) 𝑑𝑡 + 𝜔𝑣(𝑤; 𝒚(𝑙; 𝑠))𝑡=𝑡0

Mean priming as access for a fixed 𝑡0 is therefore𝑙𝐸(Γ | 𝑠, 𝑤, 𝑙, 𝑡0 ) = ∫𝑃(𝑉(𝑤) = 1; 𝑠, 𝑡) 𝑑𝑡 + 𝜔𝑃(𝑉(𝑤) = 1; 𝑠, 𝑙)

𝑡=𝑡0

The dependency on 𝑡0 is removed by observing that priming must always be empirically𝑡0measured relative to another condition, and ∫𝑡=0𝑣(𝑤) 𝑑𝑡 does not depend on s. The predicted

mean relative priming of a target w by a prime s1 relative to its control s0 is therefore just the𝑡0
difference between the mean priming for each of the two primes (after adding ∫𝑡=0
𝑃(𝑉(𝑤) =1; 𝑠, 𝑡) 𝑑𝑡 to each):𝐸(Γ | 𝑠1 , 𝑤, 𝑙) − 𝐸(Γ | 𝑠0 , 𝑤, 𝑙)𝑙= ∫ 𝑃(𝑉(𝑤) = 1; 𝑠1 , 𝑡) 𝑑𝑡 + 𝜔𝑃(𝑉(𝑤) = 1; 𝑠1 , 𝑙)𝑡=0𝑙

− (∫ 𝑃(𝑉(𝑤) = 1; 𝑠0 , 𝑡) 𝑑𝑡 + 𝜔𝑃(𝑉(𝑤) = 1; 𝑠0 , 𝑙))
𝑡=0

25

(6)

whose integrals do not depend on t0 and can be evaluated by numerical integration.
Explicit modelling of any final judgment regarding stimuli that are not presented briefly isbeyond the scope of LTRS; only the priming effect of the briefly presented stimulus ismodelled.

2.1.4. Identifying stochastic dominance and stochastic equivalence to establish parameter-free
LTRS predictions
It is possible in several cases to establish qualitative parameter-free predictions of LTRS bypairing (or grouping) analogous states that have the same probability across two (or more)conditions.
For the same parameters and number of letters in the briefly presented stimulus, the probabilityof each possible state x does not depend on the properties of the stimuli in the experiment. For
each condition, we can examine the v for the relevant stimulus-probe pairing (target-foil[remembering v is always 1 between target and target] or prime-target) for each x.
If for all x, the two vs (one for each condition) are equal, then stochastic equivalence implies
that the accuracy or priming must be the same between the two conditions, the predictedaccuracy will be the same in two conditions, regardless of parameters. For example, in an
identification task, if the stimulus is cat and the foil conditions are coo and cob: if the a or t isperceived, both coo and cob are inconsistent with the stimulus; otherwise, both are consistent.
If instead there are differences at some x states, and these differences are all in the same
direction (among states that are different, one condition always has a v of 1), then the moreoften ambiguous condition will be less accurate or produce more priming, regardless ofparameters. For example, if the stimulus is cat and the foil conditions are act and hot, then ifeither the c or a is perceived, then hot is inconsistent, but act may or may not be consistent,
depending on the other information perceived. The cases when act is inconsistent involve both cand a being perceived, the precise position of c (initial) being perceived, or the precise positionof both a and t being perceived (a before t does not suffice, but at is informative). In none of26

these cases is hot consistent. Thus, regardless of parameters, hot is more accurately rejected
than act.
Parameter-free qualitative predictions do not always exist, but they do for Experiments 1-3 that
follow, as well as for two of the simple effects (and hence the one corresponding main effect)
and the interaction of Experiment 4. A detailed enumeration of these cases is given in
Appendixes C and D.

2.2. The Overlap Model
The Overlap Model (Gomez et al., 2008; Ratcliff, 1981) is a model whose defining component
is a process to calculate an overlap score between briefly perceived strings and more stably
represented strings that can be entered into a decisional model. Overlap is calculated on thebasis the positional representation of letters in strings can be treated as containing uncertainly as
to the location of a letter that is indexed by a probability distribution. For strings that are nolonger visible (i.e., briefly presented then masked) the uncertainty of an individual letter inposition i is indexed by the normal distribution N(i, si2) where si is a position-dependent
standard deviation parameter. By contrast, for a clearly presented test alternative, there is lessuncertainty, and whatever uncertainty exists is contained within a region of width 1 centered on
the position i, i.e., (i - ½, i + ½).
The calculation of overlap between two strings, T and S, is described by these authors as being:
5

𝑖+½

∑∫
𝑖=1 𝑖 − ½

𝑖+½

𝑓𝑇 (𝑥)𝑑𝑥 ∫𝑖−½

for 5-letter strings, alongside a diagram like Figure 3.

27

𝑓𝑆 (𝑥)𝑑𝑥(7)

Figure 3. Typical graphical representation of the calculation of overlap for a response option“piles” when the briefly presented stimulus is “plies”.
In all applications, fT is a density representing the visible string (target or foil) at test in thelower part of the Figure, and fS is a density representing the uncertainly positioned brief
stimulus in the upper part of the Figure. The diagram shows an fT for each position, and an fS ineach position, and the overlap is the sum of the shaded regions of the upper portion of the
diagram. It is not explicit in Equation 7 – but it is clear in the diagram and the authors’ code –that it is meant that in each term of the sum it is fT for the letter in slot i and fS for the letter (ifsuch exists) of the same identity. Indeed, the code involves two nested loops whose computation
is more explicitly expressed using a double sum:
5

5

𝑖+½

𝑂(𝑇, 𝑆) = ∑ ∑ 𝐼(𝑖, 𝑗) ∫𝑖=1 𝑗=1𝑖−½𝑖+½

𝑓𝑇,𝑖 (𝑥)𝑑𝑥 ∫𝑖−½

𝑓𝑆,𝑗 (𝑥)𝑑𝑥

where I(i, j) is 1 if the identity of letter i in T matches the identity of letter j in S, and 0
otherwise; fT,i is explicitly the density of letter i in string T; and fS,j is explicitly the density ofletter j in string S.

28

(8)

Figure 4. Expanded graphical representation of the calculation of the overlap for a response
option “piles” when the briefly presented stimulus is “plies” (i.e., the response option is atransposed-letter foil). Each row represents a position (and identity) in the briefly presented
stimulus, each column represents a position in the response option it is being compared with.
The left curved areas represent the positional uncertainty from the briefly presented stimulus.
The right rectangles represent the lack of uncertainty from the clearly visible response option,
and are present (of value 1) if the letter identities match between the stimulus and response
option. Elements occupying the same row and column of the left and right arrays are multiplied
before summation. Since the right elements are always 1 or 0, then in effect, the right arrayshows which elements of the left are added together to calculate the overlap.
The calculation can be understood more clearly if we consider each of the five normal
distributions for each of the briefly presented letters (fS,j) separately, on the left hand side in eachof the rows of Figure 4. Each is split into five possible regions of overlap. On the right handside of the diagram, the identity matching is combined with the distributions of the clearly
displayed option (I(i, j).fT,i). The areas on the left are multipled by the corresponding areas on
the right, and their sum is the overlap. Each area on the right is either one or zero, indicating29

whether or not the identities match; in the model code, this is exactly how the overlap is in factcalculated, i.e:
5

5

𝑖+½

𝑂(𝑇, 𝑆) = ∑ ∑ ∫𝑖=1 𝑗=1 𝑖 − ½

𝑓𝑆,𝑗 (𝑥)𝑑𝑥 𝐼(𝑖, 𝑗).

This representation also clearly shows how overlap calculations are made when a letter isrepeated in either the briefly presented stimulus (Figure 5) or the clearly seen response option(Figure 6). In both cases, the repeated letter provides additional overlap because there is both
overlap within the matching position and overlap from “leakage” of the normally distributed
representation centred on one position into another position.

Figure 5. Expanded graphical representation of the calculation of the overlap for a response
option “piles” when the briefly presented stimulus is “pipes”. The interpretation of the diagram
is as for Figure 4. Both instances of the repeated “p” in the stimulus contribute to the overlap,
even though there is only one instance in the response option.

30

Figure 6. Expanded graphical representation of the calculation of the overlap for a response
option “pipes” when the briefly presented stimulus is “piles”. The interpretation of the diagram
is as for Figure 4. The single instance of “p” in the stimulus makes two contributions to theoverlap, as there is leakage in its positional representation to overlap with both instances of “p”
in the response option.
For response alternatives that involve a deletion or insertion of a letter, resulting in an option
with a different length to the stimulus, a further assumption is required. For strings of similar
lengths, these comparisons involving a stretching process that is described via diagram by
Gomez et al. (2008) and made explicit in code provided by Gomez (2020): The stimulus string
and the test string are put on a common scale by adjusting the test string T to probe the region
from 0.5 to m + 0.5, where m is the number of letters in S, while maintaining the total area of
each fT,j.
𝑛

im
+½
n

𝑚

𝑂(𝑇, 𝑆) = ∑ ∑ 𝐼(𝑖, 𝑗) ∫(i−1)m
𝑖=1 𝑗=1n31+½

𝑓𝑆,𝑗 (𝑥)𝑑𝑥

where there are n letters in T and m letters in S. fT has been stretched and 𝑓𝑆,𝑗 has not, so thenormal distributions continue to have their original means j and variances sj2. This is illustrated
in Figure 7, where a six-letter alternative is squeezed into the space of five letters (between 0.5
and 5.5), and the five-letter stimulus distributions are similarly partitioned into six regions
instead of five.

Figure 7. Expanded graphical representation of the calculation of the overlap for a response
option “pliers” when the briefly presented stimulus is “plies”. The interpretation of the diagram
is as for Figure 4. When the lengths of stimulus and response option do not match a stretchingor shrinking process occurs. In this example, while the number of columns is increased to 6 tomatch the number of letters, the numerical labels of the axis are still linked to the number ofletters in the stimulus.
2.2.1. Constraint of standard deviation parameters
Although Gomez et al. (2008) treated the sj parameters controlling the width of the standard
deviation as free parameters in their original fitting of their experiments, they went on to also
examine the fits with a constraining formula:32

𝑠𝑗 = 𝑑. (1 − 𝑒 −(𝑖−0.5)/𝑟 )where d is an asymptote parameter for the largest standard deviation (for a letter infinitely to theright) and r is a rate parameter controlling how quickly the asymptote is reached (and how much
lower the lowest standard deviation – for the initial letter – is than the asymptote). Gomez(2020) indicates that this version of the model – with fewer parmeters – should be considered
the standard version.
2.2.2. Two-alternative forced choice perceptual identification
For a two-alternative forced choice task, between two possible test strings T1 and T2, given abriefly presented S, Gomez et al. (2008) modeled the decision as being made using a Choice
Rule based on the overlap values raised to a power:
𝑃(choose 𝑇𝑘 ) ∝ 𝑂(𝑇𝑘 , 𝑆)𝑎𝑘

(9)

Although a single a = ak was fitted regardless of the identity of the stimuli in designs without
repeated letters by Gomez et al. (2008), different values of ak were used for Ts with and without
letter repetitions. A lower value was fitted for options that had repeated letters, which Gomez et
al. interpreted as reflecting a bias against these options.

2.2.3.

Masked priming

For masked primed lexical decision, Gomez et al. (2008) suggest it may be reasonable toexamine the overlap scores O(T, S) with S as the prime (presented briefly and not available at
response) and T as the target of the lexical decision, and they illustrate this as a means to obtainordinal predictions for experiments comparing different prime types with the same target stimuli(using an alternative means of dealing with comparing strings of different lengths). It is notclear how this should be generalized to comparisons between targets of different type (e.g.,different lengths, containing repetitions or not) because O(T, T) is not the same across different
types of stimuli (unlike other forms of match score where self-similarity is 1).

3.

Experiment 1: Perceptual Identification with Insertions and Deletions

In this experiment, the two-forced-choice perceptual identification task was used for theinvestigation of repeated letter effects. A previous study using the same task for exploring such
33

effects was the study of Gomez et al. (2008). What sets the two studies apart is the criticalmanipulation used in the two studies, evident in the relationship between the target and the foil(the wrong choice). In the present study, the target and the foil were never the same length. The
foil was formed by either deleting or inserting a letter. In their Experiment 4, Gomez et al.
compared strings with the same length that included replacements and transpositions. They had
three main conditions in which the repeated letters were either in the target, or in the foil or in
both. The repeated letters in their conditions also appeared within different distances. The
present study, on the other hand, focuses on a non-adjacent repetition in which the repeatedletters are always separated by one letter. This choice was motivated by the stable inhibitory
pattern for repeated letters within that distance demonstrated in Trifonova and Adelman (2019),
as well as by the necessity for more evidence due to the gap in the orthographic processing
research literature. The purpose of the experiment was to test whether the number of
nonadjacent repeated letters could be determined in early perceptual stages. If the perceptual
system is not able to detect or keep track of the number of the identities, such processing
limitations could explain the inhibitory effect reported in the study of Trifonova and Adelman.
Another important aim of this experiment was to provide evidence regarding the cause of the
low accuracy results for the repeated letter targets of Experiment 4 in Gomez et al. (2008). One
possible explanation for their result could be a bias toward choosing targets with no repetition
due to some unnaturalness of targets with repeated letters. Such an explanation does not imply
any effects due to the presence of repeated letters per se. Another explanation, however, could
be that the presence of repeated letters raises the level of processing difficulty of those targets.
To test the reason of the results reported by Gomez et al. the comparison was made between
trial type (repeat vs unique), rather than target type, as in the case of Gomez et al. As therepeated letters were not only in the target, but also in the foil, it could be tested whether any
effect of repetition could be attributed to bias of choosing a string with no letter repetition.
Choosing a deletion foil (the string with no repeated letters) will lead to lower accuracy inrepeated trial condition with eight-letter targets. However, choosing seven-letter target (thestring with no repeated letters) more often than the insertion foil (with repeated letter) will leadto a higher accuracy in the repeated trial condition for seven-letter target. Therefore, if any
effect of repetition could be attributed to bias of choosing a string with no letter repetition, theaccuracy of repeated letter trials should be lower only for eight-letter targets and not for sevenletter targets.

34

Like the study of Gomez et al., (2008), nonword stimuli were used for the intended
manipulation. Unlike their study’s five-letter stumili, however, the present experiment used
seven- and eight-letter stimuli. The greater length was chosen for two reasons. First, longer
lengths allow for variation in the positions in which the repetitions occur, thus making the items
less predictable and more variable. Second, repetitions are more likely to be observed in longer
words. Therefore, the processes involved in possible observed effects would be more likely togeneralize to processing of longer items with repeated letters.
3.2. Method
3.2.3.

Participants

The data of 96 undergraduate students from the University of Warwick were included in theanalysis. All reported English as their native language. They took part in the experiment inexchange for course credit. Another 24 participants did not perform reliably above chance level(accuracy was less than 56%) and were dropped from the analyses. In addition, the data of one
participant were not retrieved succesfully due to machine failure and three other participants
were excess to the number that would equate counterbalancing lists. The total number of tested
participants was 124.

3.2.4.

DesignThe major variable of interest was the presence or absence of a repetition in a trial, or letter type(repeated, unique). In addition, there were two different levels of target length (seven letters,
eight letters). In each trial, the target and the foil were never the same length. The criticalcomparison was the one between repeated or unique letter type. When the targets were eight
letters long, the critical manipulation was done in the target, which either contained repeatedletters (DRARTIEN) or not (DRALTIEN). In this case, the foil was seven letters long anddiffered from the target in deletion of the critical letter (DRATIEN). When the targets wereseven letters long (DRATIEN), and the roles of the stimuli were reversed, the repeated letterswere either present or not in the foil (here DRARTIEN and DRALTIEN served as foils) and so
foils differed from the target by insertion of the critical letter. Although targets and foils had
different lengths, which could lead to some strategic performamce, this difference was notapparent between these two lengths, especially when nonword targets were very briefly
35displayed and sandwiched between a longer mask. Furthermore, this length difference wasconstant for both critical conditions and could not explain any possible repeated letter effects.
This design afforded testing of how participants would respond to both insertions and deletionsof letters that were either already present or not in the target. The purpose of this contrast was totest the ability of the participants to detect the number of letters that shared the same identity ina brief presentation of pronounceable letter strings. To avoid the influence of possible
confounds, we created 32 different lists, counterbalancing factors such as position of the correct
response, occurrence of the repetition, and critical letter. These will be further explained in thenext section. An illustration of all the possible conditions can be seen in Figure 8.

36Figure 8. Conditions in Experiment 1

3.2.5.

Stimuli

The materials2 consisted of 352 families of 8 nonword items, which were constructed so thatthey were pronounceable and did not deviate substantially from the orthotactics andphonotactics of English. Each of the base pseudowords of the families was aneightletternonword that contained a repetition of a letter. The repeated letters occurred equal
times (88) in positions 2 and 4; 3 and 5; 4 and 6; 5 and 7. Half of the repeated letters in eachposition condition were consonants and the other half were vowels. The initial and final letterswere never repeated. For each of the families, 7 additional derivatives were constructed so thateach family of items had 8 different members in it. The second version of each item wasconstructed by replacing the repeated letters with other repeated letters from the same
corresponding consonant-vowel class (e.g., OLELUVAN-OBEBUVAN). The third version of
the items were derived by replacing the first occurrence of the repeated letter from the first item
version with the repeated letter from the second item version (e.g., OBELUVAN). The fourth
item from the family was constructed in a similar way as the third one, the only difference beingthat the second occurrence of the repeated letter of the first version was the one that was
replaced by the repeated letter of the second type (e.g., OLEBUVAN). Thus, the first four
versions of the items contained two different eight-letter nonwords with repeated letters(repeated condition) and two different nonwords with no repeated letters (unique condition). Allfour versions were used to keep the design symmetrical as well as to counterbalance possible
statistical regularities effects such as letter and bigram frequencies.
The next four versions of the item families represented one-letter-deletion derivatives of theeight-letter nonwords. They were constructed by deleting one of the letters in the critical
positions where a repetition occurred in the repeated condition. The fifth version of the items
were derived by deleting the first occurrence of the repeated letter of the first type
(OLELUVAN-OELUVAN). The sixth item was derived by deleting the second occurrence ofthe first type (OLELUVAN-OLEUVAN). The seventh and the eighth versions were constructedin the same way as the previous two versions, but the first and the second occurrences of thesecond repeated letter type were deleted (OBEBUVAN-OEBUVAN; OBEBUVANOBEUVAN). The initial 352 base items were constructed so that their deletion derivatives
remained pronounceable.

2

Research data file including stimuli, data, analysis, DMDX scripts, simulation files of all experiments isavailable to download at: http://adelmanlab.org/repeated-expts/Research%20Data.zip

Each of the eight versions of an item family served as a target in a perceptual identification twoalternative forced-choice task, in which participants had to choose the correct response from atarget and a foil. Each of those versions also served as a foil. There were two possible foils foreach target. When the target was one of the four eight-letter targets, the two possible foils werethe corresponding seven-letter versions with omission of one of the two letters in critical
positions. For example, the two possible foils for OLELUVAN were OELUVAN and
OLEUVAN. When the target was one of the seven-letter item versions, the two possible foils
were the corresponding repeated or unique eight-letter versions. Thus, when the length of thetarget was seven letters, the foils had an additional inserted letter. This was the letter in the same
position that was omitted to form the seven-letter target. Thus, the two possible foils for
OLEUVAN were OLELUVAN and OLEBUVAN.
Each participant saw only one of the eight versions of the items as a target with one of the two
possible foils. The position of the correct response and their corresponding left and right buttons
was carefully counterbalanced between the lists, so that the same target-foil pair appears in both
possible left-right configurations. These manipulations led to the construction of 32 different
counterbalancing lists. In addition, within each list, the correct responses were equal times on
the left and on the right for each of the contrasting conditions.
3.2.6.

ProcedureThis and the following experiments were conducted on a 17” CRT Sony Trinitron CPD-G220
monitor with the refresh rate set to 10ms. The DMDX software (Forster & Forster, 2003) wasused for the presentation of the stimuli and for data collection. All stimuli were presented inblack on a white background. Each trial began with a 10-symbol mask (##########), presented
in the center of the screen for 900 ms in a Courier New font, size 23. The mask was followed by
the target nonword, presented for 110 ms in upper Courier New font, size 20. After thepresentation of the target, the screen remained blank for 10 ms after which the mask appeared
again in the place of the target simultaneously with two choices which were displayed below themask. The choices were in Courier New font, size 23 and were presented to the left and to theright of the middle of the screen. Participants had to perform a two-alternative forced-choicetask. One of the alternatives was the target itself and represented the correct choice and thewrong alternative was one of the two possible foils of that target. Participants were asked toindicate which of the two alternatives was displayed on the screen by pressing either the left
shift key for the alternative on the left or the right shift key for the alternative on the right. They
39

were instructed to be as accurate as possible and had up to 2000 ms to respond. Accuracy
feedback was given after each trial. In addition, participants’ current percentages of correct
responses were displayed after the completion of every 44 trials. Participants were encouraged
to constantly try to improve their performance as much as possible and were given a break inthe middle of the experiment.

3.3. Results
The accuracy results per condition can be seen in Figure 9. A generalized linear mixed-effects
model with binomial distribution was fitted for the accuracy analyses of all the trials in theexperiment. The model contained letter type (repeated/unique), target length (seven/eight) andtheir interaction as fixed effects and by-subjects and by-items intercepts and by-subjects slopes
for letter type, target length and their interaction as well as by-items slopes for target length as
random effects. The results revealed significant main effect of letter type, χ2(1) = 34.029, p <.001; significant main effect of target length χ2(1) = 28.258, p < .001; and a significant
interaction between the two factors, χ2(1) = 33.259, p < .001. Participants were significantly lessaccurate when the targets were eight-letters long than when the targets were seven-letters long.
Pairwise comparisons revealed that the repeated letter condition was significantly harder thanthe unique letter condition only for the trials with the shorter seven-letter targets with a letter
insertion in the foils, χ2(1) = 67.07, p < .001. The difference between the repeated and uniqueletter conditions was not significant for the longer eight-letter trials with deletion in the foils, χ2
< 1.

40

Figure 9. Percentage accuracy in Experiment 1 per letter condition (repeated vs. unique) for
seven-letter targets with letter insertion in foils and for eight-letter targets with letter deletion in
foils: Empirical data (from left, panel 1); LTRS prediction (panel 2); and LTRS prediction withbias (panel 3); Overlap predictions (panel 4), Overlap without bias predictions (panel 5).

3.4. Modelling
3.4.3.

LTRS

Parameter-free predictions of LTRS
Insertion foils For both types of insertion foil, the discrepancy in the foil is only detectable (vonly equals 0) when the adjacency of the letters surrounding the omission is perceived; adetailed enumeration of states is given in Appendix C's Table C1. The probability of thisoccurring does not depend on the identity of the foil because these correspond to the same
abstract perceptual states x, so these conditions are predicted to be equivalent.
Deletion foils For some of the possible perceptual states x in which the critical letter isperceived, whether or not the percept is consistent with the foil, v(f), differs by foil condition;
the detailed enumeration of states is given in Appendix C’s Table C2, and can be summarized
as follows. In the unique case, the critical unique letter is always inconsistent with the foil, but
in the repeated case, the critical repeated letter is not always distinguishable from its twin. Thisdifference occurs when four conditions are all met: (i) the critical letter that differs between the
two targets has been perceived; (ii) its twin has not been perceived; (iii) the central letter41between the repetitions has not been perceived; and (iv) the adjacency of the critical letter andits non-central neighbour has not been perceived (either because the non-central neighbour has
not been perceived at all or because at least one of these two letters has not reached the state offull information). There are no cases where the unique condition can produce consistency where
the repeated condition does not. As such, accuracy is predicted to be higher in the uniquecondition.
These predictions were not borne out in the data, so we re-examined one aspect of the model.
Biasing the guess in LTRS
When considering the misfit of the original LTRS to the data of Experiment 1, we observed that
our response options do not conform to a reciprocal relationship: If participants preferred tochoose an option with a repeated letter, this would affect our conditions differently: Responses
in the repeated-insert condition would gain accuracy at the expense of accuracy in the repeateddelete condition, while retaining the pattern of average accuracy for the two repeated conditions
being greater than the average accuracy for the two unique conditions.
We added such a bias to LTRS as follows. While the probability of a correct guess g remained
0.5 on trials where neither option contained a repetition, a biased guess was made when one ofthe options contained a repetition, with the probability of choosing the option with the repeatedletter being 0.5 + (n-r/R)b/n where b is a parameter controlling the size of repeated letter biases,r is the number of excess repeated letters (in the sense of: by how much the length n exceeds thenumber of unique letter identities) and R is the number of letter identities for which there is arepetition. In the present experiment, this was always 0.5 + 7b/8, and this is the probability of acorrect guess on repeated-deletion-foil conditions, and the probability of a correct guess on
repeated-insertion-foil conditions was 0.5 - 7b/8.
Numerical predictions of LTRS
Figure 9 includes numerical predictions of LTRS both with and without the adjustment to bias.
Our initial attempts to optimize model parameters indicated that this data set did not sufficiently
constrain the parameters, as some took on extreme values without substantially improving thefit of the model. Therefore, we fixed some parameters to be similar to previous fits, andconstrained others to be equal. In particular, within each length, non-initial letter rates were set
to be equal. The final parameters are listed in Appendix A. As can be seen in Figure 9, the42

original model predicted repeated letter effects only when the repetition was in the target but
captured the observed data pattern once a repetition bias was implemented.

3.4.4.

Overlap Model

An Overlap Model (Gomez et al., 2008) simulation of Experiment 1 was run after optimizationof the model’s parameters: the asymptote and rate parameters (controlling the standard
deviations for each of the letter positions in the stimuli), scaling exponent parameters forchoosing the unique letter item (a) and the repeated letter item (b). A separate simulation was
also run where only one exponent was fitted (a) and so there was no implemented bias in themodel for repeated letters. The parameters for both simulations are listed in Appendix B. As
can be observed in Figure 9, the model captured the repeated letter effect for the shorter targets
in both simulations.
Notably, even without separate parameters for the unique and repeated cases, the repetition
effect for trials with eight-letter targets (that could contain a repetition) and seven-letter foils
was correctly predicted to be noticeably smaller than the repetition effect with seven-lettertargets and eight-letter foils (that could contain a repetition). In both cases, the overlap between
the foil and the target increases when a repetition is present, because the critical letter in thelonger item contributes to the overlap only if it is a repetition. As one would expect, theincreased similarity between foil and target tends to increase the chances that an incorrect foil
response will occur. However, the model’s decision is also influenced by the overlap between
the target and itself (but not the overlap between the foil and itself). Critically, unlike manyother similarity schemes, self-overlaps can vary. In the foil-insertions conditions with the sevenletter targets, the target-target overlap is unaffected by the repetition that occurs in the foil –does not vary – so accuracy is just controlled by foil-target overlap, allowing for a strong
repetition effect coming from the foil. In the foil-deletion conditions with the eight-lettertargets, the target-target overlap is affected and is higher when the target contains a repetition.
So, when the repetition deletion condition is compared with the unique deletion condition, boththe foil-target overlap and the target-target overlap going into the decision rule are higher,
resulting in only a mild decrease in accuracy.
In its original form with a bias affecting unique and repeated options differently, the model
fitted the data slightly better, with the exponent parameter being slightly higher for repeated43than unique items, which is the opposite direction of bias to the original application to Gomez etal.’s (2008) Experiment 4. Thus, the Overlap Model tended to agree with LTRS that participants
were biased towards the repeated-letter options, though this was not as critical to the Overlap
Model’s fit as LTRS’s.3
3.4.5.

Open bigram models and SOLAR

The predictions of several successors of the Interactive Activation Model (McClelland &Rumelhart, 1981) were also explored by calculating match scores for the nonword pairs in themain 4 experimental conditions in Experiment 1 (these also apply to the main contrasts inExperiment 2 and Experiment 3). The match scores calculations were obtained with Davis’s
Match Score calculator (http://www.pc.rhul.ac.uk/staff/c.davis/utilities/matchcalc/index.htm)
for the SOLAR model (Davis, 1999), as well as for two versions of the overlap bigram scheme:
open bigram (Grainger & van Heuven, 2003), and the overlap open bigram (Grainger et al.,2006). As can be seen in Table 5, SOLAR predicts no difference between the repeated andunique conditions. The open bigram models, however, predict stronger similarities in the cases
including repeated letters than the cases with no repetition in both target length conditions. In
addition, all models generally predict stronger similarity between the two strings when thecomparison word is the shorter stimulus and the longer stimulus is compared to the shorter one,than in the opposite case, suggesting an asymmetric relationship between the strings in the pair.

Table 5. Mean match score calculations by target length and letter type for SOLAR (Davis,
1999), Open Bigram (Grainger & van Heuven, 2003), Overlap Open Bigram (Grainger et al.,2006)
TargetLengthModel

SOLAR
Open Bigram
Overlap
Open Bigram

7

Repeated
DRATIENDRARTIEN
0.97
0.9
0.978

8

UniqueDRATIENDRALTIEN
0.97
0.85
0.883

3

Repeated
DRARTIENDRATIEN
0.873
0.845
0.78

UniqueDRALTIENDRATIEN
0.873
0.71
0.753

Conversely, when we fitted the new version of LTRS with bias to Gomez et al.’s (2008) Experiments,
LTRS agreed that there was a slight bias against the repeated-letter items. The improvement in fit
compared to the original LTRS was, however, fairly small.

44

3.5. Discussion
The results of Experiment 1 showed that participants performed significantly worse when an
additional inserted letter in the foil was already present in the target, therefore producing a letter
repetition, then when the inserted letter was different from any letters in the target. These results
suggest that letter numerosity might be difficult to process at early stages of orthographic
processing. A bias toward choosing a string with no repeated letters was not observed in these
conditions. Such a bias was offered as an explanation by Gomez et al. (2008) for the observed
lower accuracy for targets with repeated letters and foils with no repeated letters in a twoalternative forced-choice perceptual identification task. Two separate simulations with theOverlap Model showed that the results of Experiment 1 could be captured by the model withand without an implemented bias. The results of Experiment 1 also indicated lower accuracy forthe eight-letterg targets than for the seven-letter targets, suggesting that the longer targets were
harder to perceive. In the trials, in which the foil was missing a letter from the target, theaccuracy was not influenced by the type of the missing letter, repeated or unique. The differencein the pattern of the repeated letter effect in the two target length conditions signals fororthographic processing asymmetries between insertions and deletions string pairs. LTRS
(Adelman, 2011) predicted a difference between the two letter type conditions only when thecritical letter was present in the target. This prediction represented the opposite pattern of theobserved data and was based on the original implementation of the model for forced-choice
perceptual identification task with two alternatives. The presence of the repeated letters effectonly for the shorter targets and the inconsistency with the predictions of LTRS could be
explained by generally higher processing difficulty for items with repeated letters that could inturn cause a strategic preference for the repeated item as a form of overcompensation. Thishypothesis was tested in a subsequent simulation with a version of LTRS augmented with thisbias. This model captured the observed data pattern with the inclusion of a slight preference forthe repeated letter item when the model guessed because the percept was ambiguous. The
susceptibility to bias-based explanations is a feature of the task (not peculiar to LTRS) andshows that this task is not ideal for comparisons where contrast of interest is not symmetricalbetween response options.

45

4.

Experiment 2: Same-Different Task with Deletion Primes

The next two experiments followed up on the evidence from Experiment 1 that distingushingrepeated letter identities is perceptually more demanding than distinguishing corresponding
different letter identities. The results of Experiment 1 demonstrated effects of repeated letterswith a two-alternative forced-choice perceptual identification task with a manipulation of letter
insertion in the foil. This result suggested that the two choices were perceived as more similar in
the repeated letter condition than in the unique letter condition. As already discussed, it ispossible that the different results with deletions and insertions in the foils could be attributed toadjusted strategic performance that compensates for processing limitatation in cases of repeatedletters.
Experiments 2 and 3 aimed to further explore the repeated letter effects with deletion andinsertions with a paradigm that is less susceptible to conscious strategies. They employed themasked priming paradigm combined with the same-different task. We used the same-different
task for several reasons. First, it could be combined with the masked-priming paradigm andprovide priming results that are not affected by strategic effects and biases. Furthermore, unlike
the lexical decision task in which the priming effects are usually restricted to word targets,robust masked form priming effects occur with the same-different task for both word andnonword targets (Kinoshita & Norris, 2009). This ensures the compatibility of the task with the
nonword stimuli used in Experiment 1 and affords for a systematic exploration of the effectacross the different paradigms. Finally, it has been demonstrated that the masked-priming samedifferent task is sensitive to small orthographic manipulations, such as effects of repetition, butthese effects were demonstrated only with adjacent conditions (Norris et al., 2010). The focus of
the present work was to establish whether there was a more general mechanism involved in adifferential processing between two nonadjacent repeated identities and two different identities
in a letter string. The aim was to establish whether the previous repeated letter findings
observed with the masked-priming same-different task could be extended to nonadjacent
repetitions in the context of the insertion and deletion manipulations used in the perceptual
identification task in Experiment 1.
The purpose of Experiment 2 was to test whether a repeated letter effect, analogous to the one
observed in Experiment 1, could also be demonstrated with the masked-priming paradigm and adeletion manipulation in the prime. Experiment 2 explored whether there was a higher
orthographic similarity between a deletion prime and target with repeated letters than between a46

deletion prime and target with no repeated letters. The effect of identity primes on the two target
types was also tested to make sure that any deletion priming differences could not be attributed
to one of the target types being more prone to priming in general.
4.2. Method
4.2.3.

Participants

Sixty-four native English speakers took part in the experiment for a small payment and were
included in the analyses. Two other participants were excluded due to low accuracy scores(correct on less than 70% of the trials).
4.2.4.

DesignAs the task was to determine whether two letter strings, a reference and a target, were same ordifferent, there were two levels of trial types that occurred equal times: same and different. In
addition, the design of the experiment contained three more factors: 2 x target types (repeatedletters, unique letters), x 2 prime relatedness (related, control) x 2 prime length (7 letters, 8
letters) x 2 trial type (same, different). In addition, the different trials had two types of reference
type (repeated, unique). The four different prime types comprised of related identity anddeletion primes and their corresponding control unrelated conditions. Each participant saw only
one version of an item in only one of the possible conditions but was presented with all thedifferent conditions in the design. There were 32 different counterbalancing lists. Examples of
the conditions in same trials could be seen in Figure 10.

47

Figure 10. Conditions in Experiment 2, Same trials.

4.2.5.

Stimuli

Stimuli were re-used from Experiment 1. To minimize the counterbalancing list conditions,from each item family only one of the two eight-letter-long versions containing repetition andonly one not containing repetition were selected as targets for both the same and different trials.
They also served as references in the same trials. In the same trials, the eight-letter related prime
was identical to the target and the reference. The seven-letter related prime was the same as theidentity with omission of a single letter. The omitted letter was the different one between therepeated letter family version (e.g. OBEBUVAN) and the nonrepeated letter one (e.g.
OBELUVAN), so the relationship between the seven-letter related prime (e.g. obeuvan) and thetwo target types was the same: The prime contained seven out of eight letters of the target. The
control primes were constructed by pairing each family with another family so that bothfamilies shared no more than three common letters. Where it was not possible to pair families sothat control primes shared two or fewer letters with their target, one letter was changed in theprime to meet this constraint. After those manipulations, the control primes contained no morethan two common letters with the targets. The seven-letter control primes were constructed bydeleting one common letter from the eight-letter unrelated primes. Where there was a choice,48

the letter whose omission preserved pronounceability was chosen. The seven-letter unrelated
primes had no more than one common letter with the target.
The previously unselected eight-letter versions of an item family served as references for thedifferent trials. There were two different reference types. The reference either contained arepetition for which the second repeated letter version from a family was selected (e.g., abebulit
from the ARERULIT family) or not, for which the second unique letter version from the family
was selected (e.g., olebuvan from the OLELUVAN family). The two different reference types
occurred equal times in the different trials for each target type, so the outcome could not be
determined only be the presence or absence of a repetition in the reference. For half of the items
the repeated letter reference was selected and for the other half the unique letter reference wasselected. A zero-contingency scenario was adopted for the different trials. In this scenario, thecorrect response cannot be predicted by the relationship between the reference and the prime, as
in both trial type conditions, the prime is related to the reference. In the different trials, theidentity primes were the same as the reference, the seven-letter related prime was more related
to the reference than to the target (e.g., apoplecy-apolecy-ARORLECY) and the two control
prime conditions were the same as the control primes in the same trials and were neither related
to the reference nor to the target.
4.2.6.

Procedure

Each trial began with a presentation of a 10-symbol (##########)mask in the center of thescreen and the reference above it. These were presented for 1 s after which the reference
disappeared, and the prime replaced the mask. The prime was displayed for 60 ms and it was
followed by the target. The target stayed on the screen until response with a 2-s timeout. The
reference and the prime were presented in lowercase and the target was presented in uppercase.
Courier New font was used, sizes 12.5 for the reference and the prime, and 20 for the target.
Participants were instructed to respond as quickly and as accurately as possible whether the pairof nonwords they saw on the screen were the same or different by pressing one of twocorresponding buttons (left shift for different, right shift for same). They were instructed todisregard the difference in the case. The presence of the primes was not mentioned.

4.3. Results49

4.3.3.

Same Trials

Response Time. Prior to the response time analyses, trials with incorrect responses (8.7%) and
response times faster than 150 ms and slower than 1500 ms were removed (1.3% of the correct
trials). Mean response times and error rates by condition are displayed in Table 6. A linear
mixed-effects model was fitted with target type (unique/repeated), prime relatedness(related/control), prime length (7 letters/ 8 letters) and their interaction as fixed effects and bysubjects and by-items intercepts as random factors. The random slopes were excluded from themodel as it failed to converge.

Table 6. Mean Response Times (ms), Error Rates (%) by Condition and LTRS priming
predictions for Experiment 2, Same Trials

Prime Length78Deletion/Control

Identity/Control

Target Type Unique

Repeated

UniqueRepeated

Prime
Related735(8.0)742(8.2)

739(8.3)

743(7.9)

Control

763(9.9)

767(8.8)760(9.3)

764(8.9)

Priming28(1.9)25(0.6)

21(1)

21(1)

LTRS

51

51

63

63

The effect of prime relatedness was significant, χ2(1) = 35.153, p < .001. The effect of target
type was not significant χ2(1) = 1.859, p = .173. The effect of prime length and all theinteractions were not significant, χ2 <1. These results suggested that participants were notsignificantly delayed in the repeated letters target condition. Тhey were also equally primed by
the identity 8-letter primes and by a deletion 7-letter primes and the priming effect did not differ
across target type conditions.

50

Accuracy. For the accuracy analyses, a generalized linear mixed-effects model with binomial
distribution was fitted with the same structure as the model for the response time analyses. The
results revealed a main effect of prime relatedness, χ2(1) = 4.89, p = .027. Participants produced
significantly more errors when the primes were unrelated than when the primes were related.
No other results were significant, all χ2 <1.

4.3.4.

Different Trials

Response Time. Prior to the response time analyses, trials with incorrect responses (20.2%) and
response times faster than 150 ms and slower than 1500 ms were removed (1.7% of the correct
trials). Response times for the different reference and target type conditions are displayed inTable 74. A linear mixed-effects model was fitted with target type (unique/repeated), prime
relatedness (related/control) and prime length (7 letters/ 8 letters), reference type (repeated/unique) and their interaction as fixed effects and by-subjects and by-items intercepts as random
factors. The results revealed a significant main effect of target type, χ2(1) = 510.344, p < .001;
reference type, χ2(1) = 486.748, p < .001; and significant interaction between target type andreference type, χ2(1) = 17.869, p < .001. The effect of prime length approached significance,
χ2(1) = 3.01, p < .083.
As can be seen in Table 7, participants were fastest in the condition in which both the reference
and the target contained a repetition. In this condition, the reference was two-letter different
from the target, as the repeated letter identity was replaced with a different one (dlaltienDRARTIEN). Separate contrasts between this condition and the conditions with one letter
difference between the reference and the target confirmed a significant delay, χ2(1) = 129.43, p< .001; χ2(1) = 178.86, p < .001; for contrasts with the repeated-unique (dlaltien-DLARTIEN)
and unique-repeated (olebuvan-OBEBUVAN) reference type-target type conditions,
respectedly. Crucially, the difference between the latter two conditions was also significantly
different, χ2(1) = 7.52, p = .006. This result suggested that when there was a letter repetition
only either in the reference or in the target, the repetition is less resource-demanding if its
representation has been established (presented as the reference) rather than when the repeatedletter string does not have an established representation and is perceived for the first time(presented as a nonword target in a different trial). The remaining pairwise contrasts confirmed

4

A more detailed table for the different trials in Experiment 2 is included in the supplementary material.

51

that the most difficult reference type-target type condition, which contained letter transpositionand no repeated letters (olebuvan-OBELUVAN) was significantly slower than the other three
conditions, χ2(1) = 305.27; 52.955; 24.098, all p < .001.

Table 7. Response Times (ms) and Error Rates (%; in parentheses) by Reference and Target
Type in Different Trials, Experiment 2
Reference

Target

RT/Error Rate

RT

RT

mean

prime 7 prime 8

Repeated (apoplecy)

Unique (AROPLECY)

775 (20)783

766

Unique (olebuvan)

Repeated (OBEBUVAN)

808 (22)812

805

Unique (olebuvan)

Unique (OBELUVAN)

877 (34)874

879

Repeated (apoplecy)

Repeated (ARORLECY)

686 (06)

688

684

Accuracy. For the accuracy analyses, a generalized linear mixed-effects model with binomial
distribution was fitted with target type, reference type and their interaction as fixed effects andby-subjects and by-item intercepts as random effects. The prime relatedness and length were
excluded from the model as it failed to converge even after dropping the random slopes. The
results revealed a main effect of target type, χ2(1) = 304.852, p < .001, a main effect ofreference type, χ2(1) = 211.578, p < .001, and a significant interaction between the two factors,
χ2(1) = 57.099, p < .001. A pairwise contrast between the repeated-unique and unique-repeatedreference type-target type conditions revealed that the 2% difference was significant, χ2(1) =4.749, p = .029. Accuracy was significantly lower when the repetition was in the target and notin the reference than vice versa.

4.4. Modelling
4.4.3.

LTRS

Parameter-free predictions of LTRSFor Experiment 2, the analysis for deletion primes is the same as that for the insertion-foil
conditions of Experiment 1 (when the briefly presented stimulus is 7 letters long; Table C1 of52

Appendix C) with the briefly presented prime in Experiment 2 taking the role of the briefly
presented target in Experiment 1.
That is, the two different targets are not affected differently because the inconsistency is in theabsence of a letter. This is detected when the adjacency of the two surrounding letters is
detected, regardless of the identity of the letter in the target. Therefore, priming is predicted tobe identical for repeated and unique conditions, and this is a categorical prediction of the model,
not dependent on parameters.
Likewise, identity priming cannot differ for targets of the same length, as no discrepancies canoccur (but β parameters can depend on the length).
Numerical predictions of LTRS
It would be possible to choose LTRS parameters that produce any positive amount of primingfrom seven-letter deletion primes (e.g., 26ms), and any positive amount of identity priming foreight-letter stimuli (e.g., 21ms) if no restriction were placed on the β parameters.
Instead, for purposes of illustration, Table 6 includes the numerical LTRS priming predictions
that come from the parameters that Adelman (2011) optimized for word priming in the lexicaldecision task. The model predicted identical priming effects for the repeated and uniqueconditions. This prediction captured the observed lack of difference between the repeated andunique conditions. The predicted bigger priming effect for the identity eight-letter primes thanfor the deletion seven-letter primes, however, was not observed in the data.

4.4.4.

Overlap Model

Overlap scores between the targets and each of the prime types were calculated using theasymptote and rate parameters suggested by Gomez (2020) to control the standard deviations.
As can be seen in Table 8, the identity value was higher for a repeated letter target, suggestingthat this target type overlaps more with itself than the unique letter target type, due to therepeated letter. The related deletion prime also had a higher overlap scores for the repeatedletter targets than for the unique letter targets. As the values for the identity values were not thesame for both conditions, and they were not fixed to 1, the interpretation of the predictions was
not straightforward. These predictions, however seemed at odds with the observed lack of53

difference between the repeated and unique cases in both the identity and the deletion prime
conditions.
Table 8. Overlap scores for Experiment 2
TargetRepeated
UniqueIdentity
8-letter
2.86
2.66

Control
8-letter
0.21
0.21

Related
7-letter
2.23
2.08

Control
7-letter
0.1
0.1

4.5. Discussion
The results of Experiment 2 showed similar deletion and identity priming effects for the
repeated and unique target types. These results did not appear to be consistent with thepredictions of the Overlap Model (Gomez et al., 2008) and also disagreed with the higher matchscores for the repeated conditions than the unique given by open bigram schemes. The lack ofdifference between the repeated and unique conditions was predicted by LTRS and wasconsistent with previous studies that have investigated the same effect with such subset primesin the lexical decision task (Schoonbaert & Grainger, 2004). Furthermore, there was no effectof target type in the response time analysis in the same trials, suggesting that both types oftargets took the same time to process, therefore showing no effect of repeated letters. The
different trials, however, provided some important results regarding repeated letters. The
evidence suggests that the repeated letters are more difficult to process when the string in which
they are embedded does not have an established representation than when the presence of arepetition is anticipated.

5.

Experiment 3: Same-Different Task with Insertion Primes

Experiment 3 tested whether a repeated letter effect could be established if the relationshipbetween the prime and the target was the reversed case of the one in Experiment 2. The longer
stimulus with the additional letter was the prime (the brief event), and the shorter stimulus
served as the target. Experiment 3 explored whether a possible masked priming effect withrepeated letters was asymmetrical, and whether it could be affected by the anticipation of therepetition as well as its processing time. We tested whether the orthographic similarity between
54

insertion primes and targets could be affected by the status of the inserted letter (repeated vs
unique). This experiment differs from previous studies with similar manipulation by the task(same-different rather than lexical decision) and the lexicality of the targets (nonwords rather
than words). The aim was to establish whether the insertion prime with repeated letters would
produce stronger priming effect than insertion primes with no repeated letters. This expectation
was generated by the results of Experiment 1, suggesting that foils with an inserted letter
already present in the target were more similar to targets than foils with an inserted letter thatwas not present in the target. It was also caused by the results of the different trials inExperiment 2, suggesting that repeated letters might be more difficult to process when theirpresence is not anticipated. As in the previous two experiments, the repetitions of letters in theinsertion primes were nonadjacent with one intervening letter between the two repeated ones.
5.2. Method
5.2.3.

Participants

Seventy-five native English speakers took part in the experiment for a small payment. The last
three participants were added to complete the counterbalancing after three were dropped fromthe sample due to low accuracy scores (correct on less than 75% of the trials), leaving data from
72 for analysis.
5.2.4.

Design and Stimuli

The same set of items as those in Experiment 2 were used. However, this time the seven-letter
items served as targets and references and were primed by the eight-letter items. The references
and the primes were identical for the same and different trial types. There were three different
prime types: A related prime, containing an insertion of a letter already present in the target(e.g., obebuvan-OBEUVAN); a related prime, containing an insertion of letter not alreadypresent in the target (e.g., obeluvan-OBEUVAN); a control prime (e.g., mactrido-OBEUVAN).
For the different trials, the alternative seven-letter version was chosen from the family set as atarget. It differed by the same trial target and the reference by only one letter (e.g., obeuvanmactrido-OLEUVAN). Examples of the conditions in same trials could be seen in Figure 11.
The different trials differed from the same trials only by the target (e.g., OLEUVAN).

55

Figure 11. Conditions in Experiment 3, Same trials.

5.2.5.

Procedure

The procedure was identical to the one in Experiment 2.
5.3. Results
5.3.3.

Same Trials

Response Time. Prior to the response time analyses, trials with response times faster than 150
ms and slower than 1500 ms (0.47%) and incorrect responses (7.43%) were removed. Mean
response times and error rates by condition are displayed in Table 9. A linear mixed-effects
model was fitted with a prime type (unrelated/ related repeat/ related unique) as a fixed effectand by-subjects and by-items intercepts and slopes for prime type as random factors (the fullmodel). The effect of prime type was significant, χ2(2) = 83.402, p < .001. A post-hoc pairwise
comparison between the three conditions revealed that the difference between the repeated letter
related prime conditions and the unrelated prime condition was significant, χ2(1) = 80.001, p <.001, as was the difference between the unique letter prime and the unrelated prime, χ2(1) =47.829, p < .001. The difference between the two related primes was also significant, χ2(1) =7.52, p = .006, with participants responding significantly faster in the related repeat condition
than in the related unique condition.

56

Table 9. Mean Response Times (ms) and LTRS priming predictions for Experiment 3, Same
trialsResponse Times

Priming (ms)LTRS

Prime Type
Control

656

Related repeat

618

38

32

Related unique

627

28

27

Accuracy. The error rates for the control, related repeat and related unique conditions were 9%,7.1% and 6.6%, respectively. For the accuracy analyses, a generalized linear mixed-effects
model with binomial distribution was fitted with prime type as a fixed effect and by-subjects
and by-items intercepts and slopes for prime type as random effects (the full model). The effectof prime type was significant, χ2(2) = 9.621, p = .008. Post-hoc tests revealed that the difference
between the repeated letter related prime and the unrelated prime was significantly different,
χ2(1) = 7.847, p = .005, as was the difference between the related unique prime and theunrelated prime, χ2(1) = 6.292, p = .012, but not the difference between the two related prime
type conditions (χ2 <1). People produced significantly fewer errors in the two related prime
conditions than in the unrelated prime condition.

5.3.4.

Different Trials

Response Time. Prior to the response time analyses, trials with response times faster than 150
ms and slower than 1500 ms (0.70%) and incorrect responses (11.21%) were removed. The
mean response times in the different trials for the control, related repeat and related uniqueconditions were 676 ms, 682 ms, and 679 ms, respectively. A linear mixed-effects model was
fitted with prime type (unrelated/ related repeat/ related unique) as a fixed effect and by-subjects
and by-items intercepts and by-items slopes for prime type as random factors (the by-subjects
slope for prime type was dropped due to converge failure). The effect of prime type was not
significant, χ2(2) = 2.259, p = .323.

57

Accuracy. In the different trials, the error rates for the control, related repeat and related uniqueconditions were 9.9%, 12.8% and 11.7%, respectively. For the accuracy analyses, a generalized
linear mixed-effects model with binomial distribution was fitted with prime type as a fixed
effect and by-subjects and by-items intercepts and slopes for prime type as random effects (fullmodel). The effect of prime type was significant, χ2(2) = 12.896, p = .002. Post-hoc pairwise
contrasts revealed significant difference between the repeated letter related prime and theunrelated prime, χ2(1) = 12.879, p < .001, and between the unique related and unrelated prime,χ2(1) = 4.213, p = .040. The difference between the two related conditions did not reachsignificance, χ2(1) = 2.819, p = .093. People produced significantly more errors in the related
prime conditions than in the unrelated prime conditions.
5.4. Modelling
5.3.1. LTRSParameter-free predictions of LTRSFor Experiment 3, the analysis for insertion primes is the same as that for the deletion-foil
conditions of Experiment 1 (when the briefly presented stimulus is 8 letters long; Table C2 ofAppendix C) with the briefly presented prime in Experiment 3 taking the role of the briefly
presented target in Experiment 1.
That is, in some of the cases where the unique inserted letter is perceived and causes aninconsistency, the corresponding case for the repeated unique letter is ambiguous (as it could be
the other instance of the same identity; These are described in §3.3.1) and does not cause aninconsistency. There are no cases where the repeated insertion causes an inconsistency and theunique insertion does not. Therefore, priming is predicted to be greater for the repeatedcondition than the unique condition, and this is a categorical prediction of the model, notdependent on parameters.
Numerical predictions of LTRS
It would be possible to choose LTRS parameters that produce any positive priming predictions
where the repeated insertion produces greater priming than the unique insertion (i.e., fit thepriming summary in Table 9 perfectly).

58

Instead, for purposes of illustration, Table 9 includes the numerical LTRS priming predictions
that come from the parameters that Adelman (2011) optimized for word priming in the lexicaldecision task. Unlike Experiment 2, the model’s predictions for Experiment 3 was bigger
priming effect for the repeated condition than unique conditions. This prediction matched theobserved data qualitatively. LTRS predicted a smaller effect in the case of the unique letter
insertion as the cessation of priming in this case was affected by perceiving one wrong letter
identity (the inserted letter), while in the case of the repeated letter insertion, the detection of theinserted letter with consistent identity as well as additional contextual information is required toterminate the accumulation of priming.
5.3.2. Overlap Model
As for Experiment 2, we evaluated the predictions of the Overlap Model (Gomez et al., 2008)
by calculating overlap scores between the target and the primes in Experiment 3. Again, weused the asymptote and rate parameters suggested by Gomez (2020) to set the standard
deviations for the model. As can be seen in Table 10, the overlap was higher between therepeated letter insertion prime and the target than between the unique letter insertion prime andthe target.
Table 10. Overlap scores for Experiment 3
Repeated
8-letter
Overlap
2.77
score

Unique8-letter
2.60

Control
8-letter
0.24

5.5. Discussion
The most important finding in Experiment 3 was the stronger priming effect produced in therepeated letter condition than in the unique letter condition. To the best of our knowledge, this is
the first reported case of nonadjacent repeated letter effects obtained with the masked priming
paradigm. The results showed that a repeated letter effect could be obtained with an insertion
manipulation in the prime when the repetition was not anticipated and had a limited processing
time. The effect was qualitatively captured in the predictions of LTRS (Adelman, 2011). It wasalso in accordance with the predictions of the Overlap Model (Gomez et al., 2008) and as can beseen in Table 5, the results were in line with the match scores of open bigram models (Grainger
et al., 2006; Grainger, van Heuven, 2003), but not with SOLAR (Davis, 1999).
59

6.

Experiment 4: Perceptual Identification in a Same-Different Task

The following experiment continued investigating the repeated letter effects with a task that wasa hybrid between the previous two tasks. As in the case of Experiment 1, the task in Experiment
4 was also perceptual identification. However, the presentation of the target was followed byonly one option, which was either the same as the target or a different one, thus resembling thesame-different task in Experiment 2 and Experiment 3. The perceptual identification samedifferent task was selected for two reasons. First, to minimize the role of decision processes thatinvolve two alternatives. Second, to minimize top-down effects which might be present in a taskwith a reference stimulus and might influence early perceptual processes. The aim of thisexperiment was to further explore processes underlying the encoding of letter position and letter
identity with a different paradigm and to provide further information of how processing
difficulty might differ depending on whether a string contains repeated letter identities or not.
This time, the contrasted conditions always included stimuli of the same length, rather than ofpairs with missing or additional letter, as in the previous experiments. The purpose was to test
whether the visual system discriminates equally well between two items, target and foil,
depending on whether the target had repeated letters or not, in two different new scenarios with
two new foil types. The observed data was compared to the predictions of LTRS (Adelman,2011) and the Overlap Model (Gomez et al, 2008), as well as to match scores from open bigram(Grainger et al., 2006; Grainger, van Heuven, 2003) and spatial coding schemes (SOLAR;
Davis, 1999).

6.2. Method
6.2.3.

Participants

The analyses included the data from 96 participants. They were all native English speakers who
took part in the experiment for course credit. The data from another 24 participants were excluded
as they did not perform reliably above chance level (correct on less than 56% of the trials). Thoseparticipants were replaced to equalize the observations per counterbalancing list. The total
number of participants was 120.

60

6.2.4.

DesignDue to the nature of the task, the design included two different trial types: same and different.
In the different trials, there were two foil types: substitution and wrong repetition. In the firstcase, the foil contained two substituted letter identities with preserved positional and identity
information in the nonsubstituted letters (unique target-foil pair: abcdefgh-abijefgh; repeatedtarget-foil pair: adcdefgh-adijefgh). In the second case, there were no new inconsistent letter
identities in the foil. However, there were positional and numerosity violations resulting fromdeletion of one letter and insertion of a different consistent one (unique: abcdefgh-acbcefgh;
repeated: adcdefgh-acdcefgh). In addition, the design included two presentation typeconditions: normal and enhanced. In the enhanced condition, the critical letter was presented
briefly (20 ms) in its position before the presentation of the whole target string. The purpose
was to test whether any possible resolution of temporal perceptual ambiguity could interact withthe type of the critical letter (unique or repeated). Examples of the conditions could be seen in
Figure 12 for same trials and Figure 13 for different trials.

Figure 12. Conditions in Experiment 4, Same trials. Examples with two different items.

6.2.5.

Stimuli61

A subset of 256 family items from the materials of Experiment 1-3 were used to form two
different target types: targets with repeated letters (OLELUVAN) and targets with unique lettersonly (OBELUVAN). All targets had eight letters. As in the previous experiments, the repeatedletters in the repeated target condition were always separated by one unique letter. In half of theitems, the repeated letter was a consonant and in the other half it was a vowel. In addition, eachof the consonant-vowel half was further divided into two groups, so that in half of those items
the unique letter target was formed by changing the first occurrence of the repeated letter (as in
OLELUVAN-OBELUVAN), while in the other half the change was in the second occurrence(ABEBULIT-ABERULIT). In each of the consonant-vowel and occurrence subgroups, therepeated letters appeared equal times in critical position pairs (one replaced and one
nonreplaced) 2 and 4; 3 and 5; 4 and 6; 5 and 7.

Figure 13. Conditions in Experiment 4, Different trials.

As the task required a comparison between two strings (one-alternative-forced choice; are theysame or different?), different alternatives (foils) were also created for the different trial
condition. These were also two different types, foils with repeated letters (wrong repetition
foils) and foils with unique letters only (substitution foils). The wrong repetition foil was
created by first deleting the repeated or unique letter in the critical replacement position(OLELUVAN and OBELUVAN will become OELUVAN) and then inserting a copy of theintervening letter in the other critical position. Thus, the wrong repetition foils for the targets
62

OLELUVAN and OBELUVAN are: OELEUVAN and OEBEUVAN, respectively (see Figure13). The items with a second occurrence replacement, such as ABEBULIT-ABERULIT will
form AEBEULIT and AEREULIT as wrong repetition foils. The substitution foil was formedby substituting two of the target letters with other two unique letters. The substituted letterswere always the intervening letter (between the two critical positions) and the unchanged
occurrence of the repeated letter that is common between the repeated letter type target andunique letter type target (e.g. the L in position 4 in OLELUVAN-OBELUVAN). The
corresponding unique letter substitution foils for the repeated letter target OLELUVAN is
OLYMUVAN and for the unique letter target OBELUVAN the substitution foil is
OBYMUVAN.
In the enhanced presentation condition, the letter in the critical replaced position was briefly
presented before the target. Thus, in the repeated letter target condition, the presentation of thetarget OLELUVAN was preceded by the short presentation of letter L in position 2, while in theunique letter target condition, the target OBELUVAN was preceded by the short presentation ofletter B in position 2. The enhanced letters were identical for the different trial condition.
6.2.6.

Procedure

The procedure of Experiment 4 resembled the one of Experiment 1 with a few differences thatreflected the number of the alternatives (one, instead of two), the enhancement conditions, and aslightly longer presentation of the targets (120 ms, instead of 110 ms). The target duration was
longer due to the increased difficulty in Experiment 4, in which all the targets had eight letters.
The same apparatus and software were used as in the previous experiments. Each trial beganwith a 10-symbol mask (##########), presented in the center of the screen for 900 ms in a
Courier New font, size 23. In the normal presentation condition, the mask was followed by thetarget nonword, presented for 120 ms in upper Courier New font, size 20. In the enhanced
presentation condition, the presentation of the target was preceded by 20 ms presentation of theenhanced critical letter. After the presentation of the target, the screen remained blank for 10 ms
after which the mask appeared again in the place of the target simultaneously with one
comparator which was displayed below the mask. The comparator was in Courier New font,size 23. Participants had to perform a forced-choice same-different task. They had to decide
whether the target and the comparator were the same or different and were asked to respond bypressing one of two corresponding keys. In half of the counterbalancing lists, the left shift was
the key for different and the right shift was the key for same, and in the other half the keys were63

switched5. They were instructed to be as accurate as possible and had up to 2000 ms to respond.
Accuracy feedback was given after each trial. Participants were encouraged to constantly try to
improve their performance as much as possible and were given a break in the middle of theexperiment.

6.3. Results
6.3.3.

Same Trials

The accuracy results per condition for same trials can be seen in Table 11. A generalized linear
mixed-effects model with binomial distribution was fitted for the accuracy analyses with target
type (repeated/unique), presentation (enhanced, normal) and key allocation (right-same or leftsame) and all their interactions as fixed effects and by-subjects and by-items intercepts andslopes for target type (repeated/unique), presentation and their interaction as random effects.
The results revealed significant effects of target type, χ2(1) = 8.347, p = .004, and presentation,
χ2(1) = 4.845, p = .028. No other results were significant, χ2(1) < 1. Overall, in the same trials,
participants were significantly more accurate when the target contained repeated letters. They
were also significantly facilitated by the presentation of the single letter enhancement.
Table 11. Mean Accuracy (%) by Condition in Same Trials, Experiment 4

Presentation
Target type

NormalEnhanced

Repeated

78.2

79.4

Unique75.7

77.4

6.3.4.

Different Trials

5

Data were first collected with counterbalancing lists in which the left key was different, and the right
key was same. Subsequently, the keys were counterbalanced to make sure that the data pattern was
not affected by key allocation through effects such as differential guessing for the two positions. As thepattern remained broadly the same, here we report the combined results with key allocation included
as a fixed effect. Tables with results from the separate experiments (4a and 4b) are included in the
supplementary material.

64

The accuracy results per condition for different trials can be seen in Table 12. A generalized linear
mixed-effects model with binomial distribution was fitted with target type (repeated/unique),presentation (enhanced, normal), foil type (repeated/unique), and their interactions as well as key
allocation as fixed effects, and the by-subjects and by-items intercepts and slopes for target type,presentation and foil type as random effects. The interactions between the factors were notincluded as random factors as the full model failed to converge. The effects of target type and foil
type were significant, χ2(1) = 63.434, p < .001, χ2(1) = 328.123, p < .001 respectively, as was theirinteraction χ2(1) = 14.815, p < .001. Pairwise comparisons between the repeated and unique target
type conditions revealed a significant difference in both wrong repetition and substitution foil
type conditions, χ2(1) = 74.436, p < .001; χ2(1) = 13.092, p < .001. Participants were significantly
less accurate when they had to reject a foil for the repeated target type than for the unique target
type. The interaction between foil type and presentation was also significant, χ2(1) = 3.849, p =.050, as was the interaction between foil type and key allocation, χ2(1) = 3.858, p = .050. The
three-way interaction between key allocation, presentation type and foil type was not significant,χ2(1) = 2.334, p = .127. There were no other significant results.

Table 12. Mean Accuracy (%) by Condition in Different Trials, Experiment 4. Examples ofrepeated and unique letter targets with corresponding substitution and wrong repetition foils:
OLELUVAN - OLYMUVAN/OELEUVAN; OBELUVAN - OBYMUVAN/OEBEUVAN

Foil type
Unique (substitution)

Repeated (wrong repetition)
Presentation

Target type

NormalEnhanced

NormalEnhanced

Repeated

60.5

64.3

31.0

29.8

Unique

66.5

66.3

42.9

41.7

6.4. Modelling
6.4.3.

LTRS

Adapting LTRS to same-different decisions

65

In Experiment 4, a brief stimulus s is presented, before a mask and a clearly presented probe w
for a same-different judgement between the two stimuli. This does not correspond to any
previous application of LTRS, but there is a straightforward extension, which we will describe.
On same trials v(w) will always be 1 whereas on different trials v(w) could be 0 or 1. Thus a v of
0 is unambiguous evidence for “different” so responding is accurate when v is 0 unless apremature guess is made with probability ε. In contrast, a v of 1 is ambiguous, unless all xi are 2,
in which case the percept is complete and consistent with the comparator; this means a “same”
response must be correct, and this response will be made unless a premature guess has occurred.
Otherwise, a guess is made, which is a “same” response with probability gs. Thus, accuracy on
different trials is computed:
P(no guess | different trial) = (1 − ε)𝑃(𝑉(𝑤) = 0; 𝑠, 𝑡, masked)
P(accurate | different trial) =P(no guess | different trial) + P(the guess is different)P(guess | different trial) =(1 − ε)𝑃(𝑉(𝑤) = 0; 𝑠, 𝑡, masked) + (1 − 𝑔𝑠 )(1 − (1 − ε)𝑃(𝑉(𝑤) = 0; 𝑠, 𝑡, masked))(10)

and accuracy on same trials is computed thus:P(no guess | same) = (1 − ε) ∏𝑖 𝑃(𝑋𝑖 = 2; 𝑡)P(accurate | same trial) =P(no guess | same trial) + P(the guess is same)P(guess | same trial) =(1 − ε) ∏ 𝑃(𝑋𝑖 = 2; 𝑡) + 𝑔𝑠 (1 − (1 − ε) ∏ 𝑃(𝑋𝑖 = 2; 𝑡))
𝑖

𝑖

(11)

Provable properties of LTRS predictions
A detailed enumeration of the x states for the different trials is given in Appendix D along withan explanation of how these can (or cannot) derive predictions for the simple effects andinteraction for these trials (ignoring the enhancement manipulation). These show that thepredictions of LTRS are that: There is a simple effect of target repetition to create more guesses
and hence more errors for both foil conditions (because these probabilities must be positive) andthe interaction is such that the target repetition effect is greater when there is also a repetition in
the foil. Some details follow.
Repetition in the target The source of the effect of repetitions in the target is that the uniquetarget receives accurate responses in cases of x that perceives an identity that does not appear in
the foils (i.e., a unique identity) and where in the corresponding case for the repeated target, a66

single twinned letter is perceived without other inconsistency information being unavailablecausing a guess. In other words, either of two letters in the repeated target potentially match one
letter in the foil whereas the same letter in the foil is matched by only one letter of the uniquetarget.
Repetition in the foil We cannot state a parameter-free prediction of the model for the effect offoil repetition because while there are several cases of x where the construction of the repetition
foil reduces accuracy, there are also a few cases where the reordering of letters in the repeatedletter foil increases accuracy.
When the letter between the critical letter and its (potential) twin is perceived, this produces aninconsistency in the unique foil but not necessarily the repetition foil, as this letter is present(indeed, present twice) in the repetition foil but not (at all) in the unique foil. There are manysuch cases that give an accuracy advantage to the unique foil conditions.
There are, however, a limited number of cases that provide a countermanding influence (give an
accuracy disadvantage to the unique foil conditions). These involve the critical letter and its
outside flanker being perceived fully and the potential twin and the interior letter not being
perceived at all. It is presently unclear whether it is possible to find parameters where thesecases outweigh the cases with the opposite effect, but numerical fits are presented below. In any
case, according to LTRS, the effect of foil repetition here has nothing to do with repetition perse, and everything to do with letter identities and letter orderings being preserved.
Interaction There is a substring of the target, involving the middle letter and the twin or
potential twin of the critical letter, that is only present in the foil when both target and foil haverepeated letters (e.g., target: OLELUVAN, foil: OELEUVAN have the shared bigram “EL” but
OLELUVAN-OLYMUVAN, OBELUVAN-OEBEUVAN and OBELUVAN-OBYMUVAN do
not contain any such shared bigrams in the letter triple that is manipulated). This provides anadditional opportunity for confusion and reduces accuracy predicted by LTRS for this cell of thedesign only. This creates an interaction involving the two repetition effects compatible with thedata, but again, LTRS is not attributing this effect to repetition per se.
Numerical Predictions of LTRS
Mindful of the risks of overfitting, we again fixed some parameters and constrained others to beequal. The parameters that were fixed mostly control the effect of stimulus duration, which was
not manipulated in this experiment, and those that were constrained were for different positions,67

which we are averaging over anyway, so these restrictions are not important for representing
how the model performs for these data. For the simulation of Experiment 4, the implementation
for perceptual identification task in LTRS was changed so that it reflects the same-different
nature of the task in this experiment, and the enhancement manipulation was ignored. Unlike
the original implementation of the identification task with two-alternative forced choice, where
the decision is determined mostly by comparing the compatibility of the two alternatives withthe percept (and the correct option is always compatible), here the decision is made bycomparison between the percept and the comparator. Two changes were made. First, on same
trials, the perfect match between target and comparator is positively determined if (and only if)
all perceptual elements are available. Second, for trials where neither a positive determinationof ‘same’ or ‘different’ is available, then the bias parameter is treated as the probability of a‘same’ response. This implementation, therefore, is not able to capture any effects in the same
trials as it will produce identical results in all same trial conditions when the perceptual
parameters are fixed. We examined predictions for different trials6 after optimizing letter rates(subject to constraints)7 and ‘same’-response bias parameters on the basis of all trials. As can beobserved in Figure 14, LTRS managed to capture the data pattern, specifically, the lower
accuracy in cases of repeated targets especially when the target was compared with the repeatedletter foil type.
6.4.4.

Overlap Model

To adapt the Overlap Model for the task with a single comparator string, the overlap for the“missing” alternative was replaced with a parameter t that operated as a form of soft threshold.
That is:𝑃(𝑠𝑎𝑚𝑒) =𝑂(𝑇𝑘 , 𝑆)𝑎𝑘
𝑂(𝑇𝑘 , 𝑆)𝑎𝑘 + 𝑡(12)

If the overlap between target and reference, raised to the relevant exponent, is equal to thethreshold, then the model is guessing between “same” and “different” responses, with greater
overlap increasing the probability of a “same” response.

6

A stronger ‘same’-response bias for trials with repeated letters in the comparator would suffice toaccommodate the ‘same’ trials without disrupting the model’s ability to make the other predictions
correctly (it would enhance the foil-type difference) but is not interesting.
7Here, interior letter rate parameters were fixed to be equal, and parameters not optimized werechosen arbitrarily. The full parameter set is given in the appendix.

68

A simulation was run with the Overlap Model after optimization of the following parameters:
scaling exponent for a unique condition (a), scaling exponent for a repeated condition (b), theasymptote and rate parameters (controlling the standard deviations for the letter positions), anda threshold value. The optimized parameters are listed in appendix B. As can be seen in Figure
14, the Overlap Model managed to match the observed data pattern quite well and predicted thepresence of repetition effects, with bigger magnitude in the condition with wrong repetition
foils.

Figure 14. Mean accuracy by target type and foil type in different trials, Experiment 4.
Empirical data (left), LTRS accuracy predictions (middle), and Overlap accuracy predictions(right). Examples of repeated and unique letter targets with corresponding substitution andwrong repetition foils: OLELUVAN-OLYMUVAN/OELEUVAN; OBELUVAN OBYMUVAN/OEBEUVAN
5.3.3. Open bigram models and SOLAR
As in the insertion and deletion conditions in Experiments 1-3, we explored the predictions of
the SOLAR model (Davis, 1999), as well as the two versions of the overlap bigram models(Grainger et al., 2006; Grainger & van Heuven, 2003) by calculating match scores for thesubstitution and wrong repetition foils in Experiment 4, separately for each target type. As canbe seen in Table 13, SOLAR again predicted no difference between the repeated and the uniqueconditions for each of the foil types. In addition, the difference between the similarity of both
foil types with the target was predicted to be small. The open bigram models, on the contrary,
predicted stronger similarities between the repeated letter targets and the two foil types thanbetween the unique letter target and the same foil types. In addition, this difference was69

particularly large in the cases of the wrong repetition foils, which were also predicted to begenerally much more similar to both target types than the corresponding substitution foils. The
open bigram schemes were in accordance with the observed data pattern.

Table 13. Mean match score calculations by target type and foil type in Experiment 4 for
SOLAR (Davis, 1999), Open Bigram (Grainger & van Heuven, 2003), Overlap Open Bigram(Grainger et al., 2006)
Foil Type
Target Type

Model
SOLAR
Open Bigram
Overlap
Open Bigram

Substitution(OLYMUVAN / OBYMUVAN)

Wrong Repetition(OELEUVAN / OEBEUVAN)

Repeated

UniqueRepeated

Unique(OLELUVAN)(OBELUVAN)(OLELUVAN)(OBELUVAN)

0.8
0.52
0.54

0.8
0.46
0.53

0.85
0.88
0.79

0.85
0.68
0.73

6.5. Discussion
The results of Experiment 4 provided strong evidence that the presence of a repetition changes
the mechanisms underlying the processing of a letter string. It also demonstrates that the results
are task dependent. In the same trials, participants were facilitated when the target contained arepetition and the number of possible identities was one less than the condition in which the target
contained eight unique identities. The results in the different trials were in the opposite direction
and the repeated letter conditions were significantly harder than the unique conditions. The factthat both foil types were harder to reject when compared to repeated letter targets than uniqueletter ones suggests that the presence of repeated letters makes the string difficult to distinguishand generally more similar to many input candidates. This means that the result of Experiment 1
that there was no cost for repeated letters in briefly presented targets does not generalize. The
pattern of the results in the different trials was captured in the predictions of LTRS (Adelman,2011) and the Overlap Model (Gomez et al., 2008) and were also in accordance with repetitionsensitive open bigram schemes (Grainger et al., 2006; Grainger & van Heuven, 2003). The results
disagreed with the predictions of SOLAR (Davis, 1999).

70

7.

General Discussion

The results from these experiments provide strong evidence that the mechanisms underlying
processing of repeated letters in a letter string are different than those of unique letters.
Repetition effects were observed in three paradigms contrasting nonadjacent repetition of letterswith unique letters: a two-alternative forced-choice perceptual identification task, a masked
primed same-different task, and a same-different perceptual identification task. The results alsosuggested that the pattern of the effect was sensitive to methodological idiosyncrasies andshould be interpreted in the context of the experimental task. Table 14 summarizes the key
experimental results and the predictions of models discussed below.

71

Table 14. Summary of the data pattern in Experiments 1-4 and corresponding predictions of visual word recognition models
Experiment

Data Pattern

LTRS

Overlap

Open Bigram

SOLAR

E1: 2AFC
Perceptual Identification

Insert in Foil:
Repeat < Unique

As data (model withrepetition bias)

As data

Lower accuracy (<) suggests higher
similarity with the target

Delete in Foil:
Repeat = UniquePredicts difference inaccuracy for both insert
and delete:
Repeat < UniquePredicts no difference
in both insert anddelete:Repeat = Unique

E2: Masked-priming SameDifferent(same trials)

Delete in Prime:
Repeat = Unique

As data

Predicts difference inpriming:
Repeat > Unique*Predicts difference inpriming:
Repeat > Unique

As data

Insert in Prime:
Repeat > Unique

As dataAs dataAs data

Predicts no difference:
Repeat = Unique

Substitute in Foil:
Repeat < Unique

As dataAs dataAs data

Predicts no difference
for both:Repeat = Unique

Larger priming (>) suggests higher
similarity with the target
E3: Masked-priming SameDifferent(same trials)
Larger priming (>) suggests higher
similarity with the target
E4: Same-Different
Perceptual Identification(different trials)
Lower accuracy (<) suggests higher
similarity with the target

Wrong Repeat in Foil:
Repeat < Unique

Note – *Although match scores were higher for primes containing repetition than primes containing unique letters, the overlap of the targets with themselves was
also higher for targets with repeated letters than targets with only unique letters, leading to difficulty in interpreting the predicted results.

Experiment 1 used a two-alternative forced-choice perceptual identification paradigm. The foils
had an additional or missing letter relative to the target, and the inserted or deleted letter could
be a repetition of another target letter identity or a unique identity. A letter repetition effect wasobserved such that foils with an inserted repetition were more likely to be incorrectly chosen
than foils with an inserted unique letter. This suggests a difficulty with identifying thenumerosity of letters in the briefly presented target string: Participants chose foils with thewrong number of a particular letter more than they chose foils with a wrong letter. There was no
similar robust effect for comparisons of targets with and without repetitions. In contrast, thebasic mechanisms of LTRS (Adelman, 2011) predict that ambiguity is created when the briefly
presented targets have repeated letters relative to unique letters: A letter identity present in thetarget but not the foil is more easily perceived that the numerosity of the repetition or otherdisambiguating positional information. At a verbal level of description, this agrees with ourcharacterization of the data: Numerosity detection is difficult, so decisions where one of theoptions contains a repetition are difficult.
With the original decision rule of LTRS, though, the model produces a repetition effect in the
foil-deletion trials, not the foil-insertion trials, because those trials whose foils omit a repeatedletter have targets that contain the difficult-to-perceive repetition. However, participants could
try to counteract this difficulty by using a different decision rule that disproportionately chooses
the repeated letter option when such an option is available and the percept is ambiguous.
Additional modelling with LTRS with this extra response bias mechanism produced theobserved pattern of results.
Gomez et al. (2008) had also found it necessary to include a response bias involving repeatedletters in their modelling of two-alternative forced-choice data with the Overlap Model, but theyrequired the opposite bias to avoid choosing the option with the repeated letter. Such a bias
would increase accuracy for foils that include repetitions, as these would be rejected more often,
but in the data, these were numerically the least accurate trials. Nevertheless, the Overlap Model
was successful in modelling the outcome of the present Experiment 1, regardless of whether the
extra parameter for bias was used or not.
The results of Experiment 1 were more consistent with the predictions of open bigram models(Grainger et al., 2006; Grainger & van Heuven, 2003) which predict higher similarity in theconditions with repeated letters than conditions with unique letter only, than SOLAR, which
predicts that no repetition effect should be observed as the two letter conditions are equallysimilar (Davis, 1999). This experiment provided evidence that repeated letter identities are notalways processed as unique identities, therefore challenging the encoding scheme of SOLAR.
The same insertion and deletion manipulations were used to explore repeated letter effects witha masked priming methodology in the next two experiments. The more implicit measurement ofconfusability in these paradigms – participants have limited awareness of, and no explicit taskon, the prime – means that the results can be considered more directly, without consideration of
response biases.
If items that differ only in numerosity of a letter are more confusable than those differing in a
letter identity, then it would be expected that primes with a letter deleted relative to the target
would be more effective if that deletion changed only the numerosity of a letter – that is, deleted
a repeated letter. No such effect was found (on “same” responses) in our Experiment 2 with amasked primed same-different task; this was consistent with the findings of Schoonbaert andGrainger (2004) with a similar manipulation in the masked primed lexical decision task.
If detecting the numerosity of a letter in a briefly presented string posed a particular difficulty,
then it would be expected that primes with a letter inserted relative to the target would be more
effective if that insertion changed only the numerosity of a letter – that is, the insertion created arepeated letter. In Experiment 3, this effect was found. Given the implicit nature of the priming
manipulation, we expect that this insertion-deletion asymmetry in repetition effects – observed
without relevant response biases – is the one that naturally arises from orthographic processing.
We believe the finding of Experiment 3 to be novel: Van Assche and Grainger’s (2006)previous investigation of primes with insertions did not find a difference between repeated andunique insertions. However, Van Assche and Grainger used masked primed lexical decision,
which Norris et al. (2010) have argued is less sensitive than the masked primed same-different
task that we used.
If the presence of a repetition effect in Experiment 3 but not Experiment 2 could imply theeffect will only emerge when the manipulation of repetition affects the briefly presented string,but the opposite pattern was seen in Expeirment 1. There, the effect emerged only when the
manipulation affected the foil, and the options presented effectively required a decision on letter
numerosity. The apparent inconsistency might therefore be attributed to properties of the task.
This was reflected in our modelling with LTRS. While for Experiment 1, LTRS needed a taskspecific process to accommodate the findings, LTRS naturally makes the correct prediction forExperiments 2 and 3. A repetition effect occurs in LTRS only for the insertion primes and not74

the deletion primes, because the insertion creates an inconsistent piece of information that will
require more processing to detect in the repeated case, but the deletion leaves behind the same
inconsistency regardless of repetition. The same reasoning – that it is the presence ofinconsistent information, not the absence of consistent information, that controls priming –applies to comparisons of replacement (zudge-JUDGE) and insertion (zjudge-JUDGE) primes:
LTRS performance is largely controlled by inconsistent (z) information not consistent (j)information, and so these types of primes should be equivalent, as Lupker, Spinelli and Davis(2020) have found.
The representational scheme in the Overlap Model allows for there to be more similarity inrepresentations involving letter repetitions than those without, because representations of lettersleak into nearby positions, allowing the letters in repetitions to contribute to the overlap morethan once. This would account for the findings of a repetition effect in Experiment 3, but not theabsence of one in Experiment 2, though there is some ambiguity here, because for the betweentarget comparison, the targets had different maximum similarities, and the model does not offer
an explicit account of this kind of priming task.
As before, representations based on open bigrams gave higher similarities for the stimulus pairs
involving repetitions, suggesting a repetition effect should be observed, and SOLAR
representations treated repetitions as irrelevant, suggesting none should be observed. Whileneither of these types of models predicted the strict asymmetry in repetition effects, all suggestthat there is an asymmetry between insertions and deletions, which may contribute to thedifferent effect of repetitions on insertions compared to deletions.
While Schoonbaert and Grainger’s (2004) experiments and Trifonova and Adelman’s (2019)
megastudy analyses found that lexical decisions to words were slower for items with letter
repetitions, in Experiment 2’s same-different task with nonwords, latencies to respond “same”
were not affected by letter repetition. The difference between the prior studies and the present
Experiment 2 may be due to task-specific processes linked to lexical decision and the samedifferent task, or to differences in the type of stimuli. For instance, if the repetition effect on
clearly presented stimuli is based on how well those stimuli have been learned – which could be
affected by perceptual difficulty due to repetition – then the effect would be limited to words.
Experiment 4, which examined the accuracy of identification of briefly presented strings with asame-different decision, also did not find an inhibitory effect of repeated letters on “same”
trials; indeed, the effect was that “same” received more accurate responses when a repetition75

was present. This effect was anticipated by the Overlap Model, which allows for repeated lettersto contribute more than once to overlap calculations. By contrast, LTRS would again need toinclude a response bias for the repeated letter items to account for this finding.
In the “different” trials of Experiment 4, though, less accurate responding was seen for targets
involving repetitions. This effect was stronger when the foil contained a repetition that was notin the target (and removed the original repetition if present, so unique: OBELUVANOEBEUVAN; repeated: OLELUVAN-OELEUVAN) than it was when the foil replaced twoletters (including a repeated one if present, so unique: OBELUVAN-OBYMUVAN; repeated:
OLELUVAN-OLYMUVAN). The pattern of performance on different trials was anticipated by
LTRS and Overlap Model, and by the open bigram representational schemes. Nevertheless, thisis not a universal prediction. In edit distance schemes of similarity, both foils with repetition are
distant by one deletion and one insertion; and foils with two substitutions are distant by those
two substitutions. Moreover, as before, the SOLAR scheme is generally insensitive torepetition.
Across our experiments, it appears that unique identities in letter strings are more easily
processed accurately than repetitions, and strings with repetitions are therefore treated as moresimilar to comparison strings. While the duplication may offer an advantage in detecting thepresence of the repeated letter identity, identifying the numerosity of repeated identity seems tointroduce processing obstacles that do not occur when the same number of distinct identities
need to be processed. Future research is needed to establish whether the observed effects are
specific to letter strings or also generalize to other domains, such as digit strings, as well.
The evidence provided in the present study suggests that there are processing limitations in theencoding of repeated letter information in strings. These results are in accordance with findingsof tachistoscopic letter identification studies (Bjork & Murray, 1977; Egeth & Santee, 1981)
demonstrating interference in cases of processing two identical letters in a visual array, thestudy of Mozer (1989) demonstrating lower accuracy in evaluation of numerosity of letters incases of repetitions, and resemble the perceptual phenomenon of repetition blindness(Kanwisher 1987; 1991; Luo & Caramazza, 1996).
Taken together, the evidence provided by the experiments here was broadly consistent withLTRS (Adelman, 2011) and its predictions that letter repetition in strings requires additional
processing steps for disambiguation. It was also generally in line with models that could
account for repetition effects due to mechanisms in which letter position is weakly represented,
76

such as the Overlap Model (Gomez et. al, 2008) and open bigram models (Grainger et al, 2006;
Grainger & van Heuven, 2003). The data, however, also presented some challenge to all
aforementioned modelling frameworks. The observed repetition effects are at odds with
schemes, such as the spatial coding, that treat unique letters in the same way as repeated ones
(SCM, SOLAR; Davis, 1999; 2010). The results are also not entirely consistent with open
bigram schemes which would suggest that there should be difference between the repeated andunique conditions in both insertion and deletion cases. Although the predictions of both LTRS
and the Overlap Model generally captured the data pattern well, LTRS required an additional
implementation of repetition bias to fit the data of Experiment 1 and the match scores forExperiment 2 of the Overlap Model could be interpreted as prediction for repetition effect inboth identity and deletion primes where there was none.
This study extends previous findings of perceptual effects of repetition and demonstrates
repeated letter effects when letters are embedded in strings. The results would be interpreted inmany frameworks as suggesting that string stimuli containing repeated letters bear close
resemblance to more input candidates than corresponding unique letters strings. The special
properties of letter repetitions highlight to us, however, that assumptions regarding therelationship between letter identity and letter position – and indeed the concept of resemblance– require critical consideration. Since letter identities do not always have a one-to-one
correspondence to letter positions, it is not necessarily natural to consider the letter location as
an integral part of the same component representation as the letter identity.
Most models of orthographic processing treat information about letter identity and information
about letter position as part of an integrated orthographic code that becomes stronger and/or
clearer over time. The influence of both identity and position on lexical processing increases
over time due to a single integration (e.g., activation) process that continually compares a stored
orthographic code to a perceived orthographic code. In these models, the influence of identity
may be stronger than that of position. This means that when time is short, we may not have thepower to experimentally detect a small influence of position but may have the power to detect
stronger influence of identity. However, the sense here in which positional information can be“slower” than identity is only illusory: The positional effect is theoretically present but
undetectable.
However, it seems perfectly feasible and natural to ask whether the information about positionsof letters – or even numerosities of letters – becomes available to lexical identification processes
later than information about the presence of letter identities. In LTRS (Adelman, 2011),
77

positional information is available later in part directly and in part indirectly because the orderof two letters requires both to have been perceived. It is by no means the only conceivablemodel in which different perceptual dimensions of letter strings become available at different
times. We believe exploring the variety of possible models of this type and the corresponding
empirical questions are likely to be more productive than positing increasingly complex matchscore calculations as underpinning orthographic processing.
Allowing for different perceptual information to become available for consideration at different
times is critically a possible mechanism to dissociate salience from importance. Salient things
are generally perceived sooner, but they are not necessarily the most relevant to a perceptual
decision. Therefore, it seems possible that information perceived late could be given any
amount of weight (i.e, more or less than information perceived early) in deciding the identity ofa word. This is not possible if resemblance drives both apparent perceptual speed and lexicalmatching. As a matter of rhetoric (and mathematical simplification) LTRS (Adelman, 2011)
totally dissociates salience and importance by keeping importance constant – all negative
information is critically important – but this is not a necessary part of this dissociation: LTRS’s
relatives in categorization (Lamberts, 2000, 2002) parameterize decisional weight separatelyfrom salience.
Placing the present results alongside related ones concerning delayed processing for words withrepeated letters (Schoonbaert & Grainger, 2004; Trifonova & Adelman, 2019) suggests thatboth are linked to issues of salience of positional information for repeated letters in perception.
We anticipate that it will prove important to conceptually distinguish the concept of salience asperceptual speed from the concepts of decisional weight and attention. Progress in this direction
may depend on models becoming more general – in terms of tasks – as new paradigms attempt
to probe these issues more directly.

78

Acknowledgments
This work was supported by the Leverhulme Trust (Project Grant RPG-2013-408) and an Early
Career Fellowship awarded to the first author by the Institute of Advanced Study, University of
Warwick, United Kingdom.

79

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83

Appendix A. LTRS (Adelman, 2011) Parameters
Table A1. LTRS parameters for Experiment 1, original implementation

Parameter asdinitial8 other8 initial7 other7 eephilambda

20

10

0.118

0.01

1

Value

0.004

0.026

0.025

0.01

Table A2. LTRS parameters for Experiment 1, implementation with repeated letter bias

Parameter asdinitial8 other8 initial7 other7 eephilambda rl

20

10

0.278

1

Value

0.003

0.027

0.027

0.01

0.01

0.082

final

ee

phi

lambda ps

0.01

0.03

Table A3. LTRS parameters for Experiment 4

Parameter asdinitial

38

20

0.87237 0.00553 0.00001 0.01

Value

middle0.783

Appendix B. Overlap Model (Gomez et al., 2008) Parameters
Table B1. Overlap parameters for Experiment 1
ParameterabdrOverlap with bias

Overlap without bias

6.776
6.803
1.007
1.011

6.708
1.007
1.072

Table B2. Overlap parameters for Experiment 4

Parameter

a

b

d

r

threshold

Value

7.784

7.717

1.284

1.599

3442.776

Appendix C.
Tabulation of states to analyse conditions of Experiments 1-3
Table C1. Map from perceptual states x to consistency status v for briefly presented 7-letter stimuli (insertion-in-foil conditions for Experiment 1;
deletion-prime conditions in Experiment 2).
x constraint

v unique-f

v repeat-fv(f) (foil: Exp. 1)or v(w) (target: Exp. 2)

Unique example
s: dratienf or w: DRALTIEN

Repeat example
s: dratienf or w: DRARTIEN

i = 3 (letter a); i + 1 = 4 (letter t)
xi = 2; xi+1 = 2
0
0
at
at
xi = 2; xi+1 = 1
1
1
a*t
a*t
xi = 2; xi+1 = 0
1
1
a*/a+
a*/a+
xi = 1; xi+1 = 2
1
1
a*t
a*t
xi = 1; xi+1 = 1
1
1
a*t
a*t
xi = 1; xi+1 = 0
1
1
a*
a*
xi = 0; xi+1 = 2
1
1
*t/+t
*t/+t
xi = 0; xi+1 = 1
1
1
*t
*t
xi = 0; xi+1 = 0
1
1
*/+
*/+t
Note – i refers to the position before the missing letter and i + 1 refers to the position after the deletion. There is no representation of the irrelevant
initial dr or final ien letters; the r is irrelevant because there is nothing about it that can affect the match: its presence, postposition to d, preposition
to a, and relative location to the non-adjacent letters are all consistent with both the w options. A slash (/) means “or”.

Table C2. Map from perceptual states x to consistency status v for briefly presented 8-letter stimuli (deletion-in-foil conditions for Experiment 1;
insertion-prime conditions in Experiment 3)
x constraint

v unique-sv repeat-s

Unique example
Repeat example
s: draltiens: drartienf or w: DRATIEN
f or w: DRATIEN
v(f) (foil: Exp. 1)or v(w) (target: Exp. 3)
i = 4 (l or r); j = 3 (a); k = 2 (r); h = 5 (t)xi > 0; xk > 000
Contains l (and r)
Contains r twice
xh = 2; xi = 0; xj = 200*a+t/r*a+t*a+t/r*a+t
xh = 2; xi > 0; xj > 0; xk = 000*alt/*a*lt/*a*l*t*art/*a*rt*/*a*r*t*
xh = 1; xi > 0; xj > 0; xk = 000*a*l*t/*al*t*a*r*t*/*ar*t*
xh = 2; xi = 2; xj = 0; xk = 0
0
0
*lt
*rt
xh = 2; xi = 1; xj = 0; xk = 0
0
1
*l*t
*r*t
xh < 2; xi > 0; xj = 0; xk = 001*l*/*l*t*r*/*r*t
xh = 2; xi = 0; xj = 1
1
1*a*t/r*a*t*a*t/r*a*t
xh = 1; xi = 0; xj > 0; xk > 0
1
1
ra*t/r*a*t
ra*t/r*a*t
xh = 1; xi = 0; xj > 0; xk = 0
1
1*a*t/+a*t*a*t/+a*t
xh = 1; xi = 0; xj = 0; xk > 0
1
1
r*t
r*t
xh = 0; xi = 0; xj > 0; xk > 0
1
1
ra*/r*a*ra*/r*a*xh = 0; xi = 0; xj > 0; xk = 0
1
1*a*/+a**a*/+a*xh = 0; xi = 0; xj = 0; xk > 0
1
1
r*/r+
r/r+
xh > 0; xi = 0; xj = 0; xk = 0
1
1
*t
*t
xh = 0; xi = 0; xj = 0; xk = 0
1
1
*/+
*/+
Note – i refers to the position in f or s of the deleted letter; k refers to the position of the repetition (or where the repetition would be); and j refers
to the intermediate position; and h refers to the position adjacent to i that is not j. There is no representation of the irrelevant d or ien letters in theexamples. A slash (/) means “or”.
All the cases where the two v columns differ, the unique column has the advantage (is the one in which the inconsistency is detectable), and so theunique foil has greater probability of inconsistency, regardless of the exact probabilities of those rows. Therefore, the model categorically predicts(regardless of parameters or stimulus duration) that responses will be more accurate for these foils (if guessing is equally accurate).

87Appendix D.
Tabulation of states to analyse conditions of Experiment 4
Table D1. Map from perceptual states x to consistency status v for briefly presented stimuli and their foils on “different” trials in Experiment 4.
x constraint

Repetition in target(e.g., OLELUVAN)
Rep. in foil
No foil rep.

No repetition in target(e.g, OBELUVAN)
Rep. in foil
No foil rep.

fx

OELEUVAN

OLYMUVANOEBEUVAN

OBYMUVAN

xi > 0; xk > 0
xk = 2; xl = 2
xh = 2; xi = 2; xj > 0; xk = 0

0
0
0

0
0
0

0
0
0

0
0
0

N
N
N

xh < 2; xi = 2; xj = 1; xk = 0
xh = 2; xi = 2; xj = 0; xk = 0
xh < 2; xi = 2; xj = 0; xk = 0; xl < 2
xi = 1; xj = 2; xk = 0; xl = 2
xi = 1; xj = 2; xk = 0; xl < 2
xi = 1; xj = 1; xk = 0
xi = 1; xj = 0; xk = 0
xh = 2; xi = 0; xj = 2; xk = 2; xl < 2

1
0
1
0
1
1
1
0

0
1
1
0
0
0
1
0

1
0
1
0
1
1
1
0

0
1
1
0
0
0
1
0

F
f
A
N
F
F
A
N

xh < 2; xi = 0; xj = 2; xk = 2; xl < 2
xh = 2; xi = 0; xj = 2; xk = 1
xh < 2; xi = 0; xj = 2; xk = 1
xh = 2; xi = 0; xj = 2; xk = 0; xl = 2
xh = 2; xi = 0; xj = 2; xk = 0; xl < 2
xi = 0; xj = 1; xk = 2; xl < 2

1
0
1
0
1
1

0
0
0
0
0
0

0
0
0
0
1
0

0
0
0
0
0
0

I
N
I
N
F
I

xi = 0; xj = 1; xk = 1
xh = 2; xi = 0; xj = 1; xk = 0
xh < 2; xi = 0; xj > 0; xk = 0

1
1
1

0
0
0

0
1
1

0
0
0

I
F
F

xi = 0; xj = 0; xk = 1; xl = 2
1
1
0
0
T
xi = 0; xj = 0; xk > 0; xl < 2
1
1
0
0
T
xi = 0; xj = 0; xk = 0
1
1
1
1
A
Note – i refers to position of the critical (potentially repeated) letter that appears in all foils. j refers to the “interior” position of the latter adjacent to i that isrepeated in repetition foils and deleted in substitution foils. k refers to the other (potentially repeated) position adjacent to j. l refers to the other “flanker” position
adjacent to k, and h refers to the other “flanker” position adjacent to i. In the “fx” (“effects”) column: A = a guess is made in all conditions; F = a guess is made in
conditions with repetition in the foil; f = a guess is made in conditions without repetition in the foil; T = a guess is made in conditions with repetition in the target;
N = a guess is not made in any condition; I = a guess is made only in the condition where both target and foil contain a repetition.

Table D1 delineates the relationship between possible states x and consistency statuses v in eachof the four conditions of Experiment 4. Using the categorization of these states in the final
column, we can delineate the probability of a guess, which is linearly related to the probabilityof an error, as follows:𝑃(guess | 𝑇𝑅𝑒𝑝 & 𝐹𝑅𝑒𝑝) = 𝑃(𝐴) + 𝑃(𝑇) + 𝑃(𝐹) + 𝑃(𝐼)
𝑃(guess | 𝑇𝑅𝑒𝑝 & 𝐹𝑈𝑛𝑖𝑞) = 𝑃(𝐴) + 𝑃(𝑇) + 𝑃(𝑓)𝑃(guess | 𝑇𝑈𝑛𝑖𝑞 & 𝐹𝑅𝑒𝑝) = 𝑃(𝐴) + 𝑃(𝐹)𝑃(guess | 𝑇𝑈𝑛𝑖𝑞 & 𝐹𝑈𝑛𝑖𝑞) = 𝑃(𝐴) + 𝑃(𝑓)because A, T, F, f, I, and N are mutually exclusive, exhaustive options.
The predicted simple effects on guessing are therefore:
Target repetition effect when foil is repeated =𝑃(guess | 𝑇𝑅𝑒𝑝 & 𝐹𝑅𝑒𝑝) − 𝑃(guess | 𝑇𝑈𝑛𝑖𝑞 & 𝐹𝑅𝑒𝑝) = 𝑃(𝑇) + 𝑃(𝐼)
Target repetition effect when foil is unique =𝑃(guess | 𝑇𝑅𝑒𝑝 & 𝐹𝑈𝑛𝑖𝑞) − 𝑃(guess | 𝑇𝑈𝑛𝑖𝑞 & 𝐹𝑈𝑛𝑖𝑞) = 𝑃(𝑇)
Foil repetition effect when target is repeated =𝑃(guess | 𝑇𝑅𝑒𝑝 & 𝐹𝑅𝑒𝑝) − 𝑃(guess | 𝑇𝑅𝑒𝑝 & 𝐹𝑈𝑛𝑖𝑞) = 𝑃(𝐹) − 𝑃(𝑓) + 𝑃(𝐼)
Foil repetition effect when target is unique =𝑃(guess | 𝑇𝑈𝑛𝑖𝑞 & 𝐹𝑅𝑒𝑝) − 𝑃(guess | 𝑇𝑈𝑛𝑖𝑞 & 𝐹𝑈𝑛𝑖𝑞) = 𝑃(𝐹) − 𝑃(𝑓)By subtracting either pair of simple effects (the first two or the last two) it can be seen that there
is a predicted super-additive interaction of the two forms of repetition of magnitude P(I).
Both foil repetition effects contain the term P(F) – P(f) because in the F rows, the unique foil
has the advantage (will not cause a guess) but in the f row, the repeated foil has the advantage.
The probabilities of these rows depend on parameters and the stimulus duration. We therefore
cannot show that P(F) > P(f) for all possible parameter values, so we cannot state a parameterfree prediction of the model for the effect of foil repetition.
The F rows consist of cases where the E (letter between critical letter and its twin) is perceived
without further relevant disambiguation, because this only appears in the repeated target. The f

rows consist of the cases that, in the example in the table, when the initial OE/OB/OL letters areperceived to be adjacent and the subsequent two letters are not perceived at all – OE in therepetition foil is inconsistent with either OB or OL. In other words: these involve the criticalletter and its outside flanker being perceived fully and the potential twin and the interior letternot being perceived at all.

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Supplementary material

Repeated letters increase the ambiguity of strings: Evidence from identification, priming and same-different tasks
Iliyana V. Trifonovaa,b and James S. AdelmanaaUniversity of Warwick, bUniversity of York

Results: Experiment 2, Different Trials
Table S1. Mean Reaction Times (ms) and Error Rates (%; in parentheses) by Condition in Experiment 2, Different Trials

Prime Length

Target7

8

Deletion/Control

Identity/Control

UniqueRepeated

UniqueRepeated

Type
UniqueRepeated

UniqueRepeated

UniqueRepeated

UniqueRepeated

Related870 (36)784 (19)815 (22)682 (5)876 (35)772 (22)807 (23)691 (5)Control

878 (34)782 (17)809 (22)694 (6)883 (33)761 (20)802 (23)676 (6)

Reference
Type
Prime

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Results: Experiment 4aSame Trials
The effect of target type was significant, χ2(1) = 4.554, p = .033.
The effect of presentation was not significant, χ2(1) = 1.509, p = .219, nor was the interaction
between the two factors, χ2(1) < 1.
Table S2. Mean Accuracy (%) by Condition in Same Trials, Experiment 4a

Presentation
Target type

NormalEnhanced

Repeated

78.8

79.6

Unique75.7

77.0

Different Trials
The effect of target type was significant, χ2(1) = 33.178, p < .001.
The effect of foil type was highly significant, χ2(1) = 148.391, p < .001.
The interaction between target type and foil type was significant, χ2(1) = 5.691, p = .017, as was
the interaction between presentation and foil type, χ2(1) = 6.63, p = .01.
The three-way interaction between foil type, target type and presentation was marginallysignificant, χ2(1) = 3.012, p = .083.
Table S3. Mean Accuracy (%) by Condition in Different Trials, Experiment 4a

Foil Type
Repeated (Wrong repetition)

Unique (Substitution)

Presentation
Target type

NormalEnhanced

NormalEnhanced

Repeated

32.6

29.6

56.8

63.4

Unique

43.0

41.8

64.5

64.8

Results: Experiment 4bSame Trials
The effect of target type was not significant, χ2(1) = 2.141, p = .143. Although there was a trend
for higher accuracy in the repeated target type condition, it was less pronounced than inExperiment 4a.
The effect of presentation type was approaching significance, χ2(1) = 3.002, p = .083. Theinteraction between the two factors was not significant, χ2(1) < 1.
Table S4. Mean Accuracy (%) by Condition in Same Trials, Experiment 4b

Presentation
Target type

NormalEnhanced

Repeated

77.5

79.3

Unique75.8

77.9

Different Trials
Consistent with the results of Experiment 4a, the effects of target type and foil type weresignificant, χ2(1) = 35.684, p < .001, χ2(1) = 239.361, p < .001, as was their interaction χ2(1) =8.675, p = .003.
The interaction between foil type and presentation was, however, not significant, χ2(1) <1, nor
was the three-way interaction between foil type, target type and presentation type, χ2(1) <1.
Table S5. Mean Accuracy (%) by Condition in Different Trials, Experiment 4b

Foil Type
Repeated

UniquePresentation

Target type

NormalEnhanced

NormalEnhanced

Repeated

29.4

29.9

64.2

65.1

Unique

42.8

41.7

68.5

67.7

94