.01 .01 .00 -5. M-ratio .03 -.08 -.52 *** .58 *** -1 3 2 Table 1 . Correlation table of the parameters from the model simulation. Note: ***<.001 1 3 3 Experiment 1: Explicit speed-accuracy instructions affect static but not dynamic measures 1 3 4 of confidence 1 3 5
Next, we tested these model predictions in an experiment with human participants. We recruited 1 3 6 36 human participants who performed a task that has been widely used in the study of evidence 1 3 7 accumulation models: discrimination of the net motion direction in dynamic random dot displays 21 . 1 3 8
Participants were asked to decide whether a subset of dots was moving coherently towards the left or the 1 3 9
right side of the screen (See Figure 2A) . The percentage of dots that coherently moved towards the left or 1 4 0 right side of the screen (controlling decision difficulty) was held constant throughout the experiment at 1 4 1 20%. After their choice, and a blank screen, participants indicated their level of confidence using a 1 4 2 continuous slider. Critically, in each block, participants either received the instruction to focus on 1 4 3 accuracy ("try to decide as accurate as possible"), or to focus on speed ("try to decide as fast as 1 4 4 possible"). Consistent with the instructions, participants were faster in the speed condition than in the 1 4 5 accuracy condition, M speed = 727ms versus M accuracy = 1014ms, t(35) = 4.47, p < .001, and numerically 1 4 6 more accurate in the accuracy condition than in the speed condition, M accurate = 75.6% vs M speed = 73.8%, 1 4 7 t(35) = 1.63, p = .111. Participants were also more confident in the accuracy condition than in the speed 1 4 8 condition, M accuracy = 70 versus M speed = 67, t(35) = 3.57, p = .001 (See Figure 2D ). To shed further light on the underlying cognitive processes, we fitted these data using the 1 6 4 evidence accumulation model described in Figure 1A . The basic architecture of our model was a DDM, in 1 6 5 which noisy perceptual evidence accumulates over time until a decision boundary is reached. Afterwards, 1 6 6 evidence continued to accumulate for a specified amount of time 19 . In addition to drift rate, decision 1 6 7 boundary and non-decision time, our model featured a free parameter controlling the strength of the post-1 6 8 decision evidence accumulation (v-ratio, reflecting the ratio between post-decision drift rate and drift rate) 1 6 9
and two further parameters controlling the mapping from p(correct) onto the confidence scale (see 1 7 0
Methods). Generally, our model fitted the data well, as it captured the distributional properties of both 1 7 1 reaction times and decision confidence (see Figure 2C ). As a first sanity check, we confirmed that 1 7 2 decision boundaries were indeed different between the two instruction conditions, M speed = 1.40 versus 1 7 3 M accuracy = 1.77, t(35) = 4.60, p < .001, suggesting that participants changed their decision boundaries as 1 7 4
instructed. Also non-decision time tended to be a bit shorter in the speed condition compared to the 1 7 5 accuracy condition, M speed = 309ms versus M accuracy = 390ms, t(35) = 3.19, p = .003. Drift rates did not 1 7 6 differ between both instruction conditions, p = .368. There was a small but significant difference between 1 7 7 the two instruction conditions in the two additional parameters controlling the idiosyncratic mapping 1 7 8 between p(correct) and the confidence scale, reflecting that in the accuracy condition confidence 1 7 9
judgments were slightly higher, t(35) = 2.506, p = .017, and less variable, t(35) = 2.206, p = .034, 1 8 0 compared to the speed condition. 1 8 1
We next focused on metacognitive accuracy in both conditions (see Figure 2B ). In line with the 1 8 2 model simulations, our data showed that M-ratio was significantly affected by the speed-accuracy tradeoff 1 8 3
instructions, M speed = 0.84 versus M accuracy = 0.66, t(35) = 2.26, p = .030. Moreover, apart from these 1 8 4
between-condition differences we also observed significant correlations between M-ratio and decision 1 8 5 boundary both in the accuracy condition, r(34) = -.36, p = .030, and in the speed condition, r(34) = -.53, p 1 8 6 < .001.Consistent with the notion that metacognitive accuracy should not be affected by differences in 1 8 7 decision boundary, v-ratio did not differ between both instruction conditions, p = .938. 1 8 8 Experiment 2: Spontaneous differences in response caution relate to static but not dynamic 1 8 9 measures of metacognitive accuracy 1 9 0
Although Experiment 1 provides direct evidence that changes in decision boundary affect M-1 9 1 ratio, it remains unclear to what extent this is also an issue in experiments without speed stress. Notably, 1 9 2 in many metacognition experiments, participants do not receive the instruction to respond as fast as 1 9 3
possible. Nevertheless, it remains possible that participants implicitly decide on a certain level of response 1 9 4 caution. For example, a participant who is eager to finish the experiment quickly might adopt a lower 1 9 5
decision boundary compared to a participant who is determined to perform the experiment as accurate as 1 9 6 possible, thus leading to a natural across-subject variation in decision boundaries. To examine this 1 9 7 possibility, in Experiment 2 we analyzed data from an experiment in which participants (N = 63) did not 1 9 8 receive any specific instructions concerning speed or accuracy. Participants decided which of two boxes 1 9 9
contained more dots, and afterwards indicated their level of confidence on a continuous scale (see Figure  2 0 0 3A). The same evidence accumulation model as before was used to fit these data, and again this model 2 0 1 captured both reaction times and decision confidence distributions ( Figure 3B ). Consistent with our 2 0 2 model simulations, model fits showed a positive correlation between M-ratio and v-ratio, r(61) = .21, p = 2 0 3
.092, although this correlation was not statistically significant ( Figure 3C ). However, we again observed 2 0 4 that M-ratio correlated negatively with the fitted decision boundary, r(61) = -.34, p = .006, whereas v-2 0 5 ratio did not, r(61) = -.19, p = .129. Note that participants did not receive any instructions concerning speed or accuracy. B. Distribution of 2 1 1 reaction times and confidence for Experiment 2, using the same conventions as in Figure 2 . C. The data 2 1 2
of Experiment 2 showed a non-significant positive relation between M-ratio and v-ratio (r=.21 M-ratio and v-ratio (r=.38 ) and a significant negative 2 1 9
correlation between M-ratio and decision boundary (r=-.18) but not between v-ratio and decision 2 2 0 boundary (r= -.04). 2 2 1 2 2 2 Experiment 3: Replication in an independent dataset 2 2 3
To assess the robustness of our findings, in Experiment 3 we aimed to replicate our analysis in an 2 2 4 independent dataset with high experimental power. To achieve this, we searched the confidence database 2 2 5 32 for studies with high power (N > 100) in which a 2CRT task was performed with separate confidence 2 2 6 ratings given on a continuous scale. Moreover, because our fitting procedure was not designed for 2 2 7 multiple levels of difficulty, we focused on studies with a single difficulty level. We identified one study 2 2 8 that satisfied all these constraint ( Figure 3D ; Prieto, Reyes & Silva, under review). Their task was highly 2 2 9 similar to the one reported above, but their high experimental power (N=204) assured a very sensitive 2 3 0 analysis of our claims. Consistent with the previous analysis, model fits on this independent dataset 2 3 1 showed a positive and statistically significant correlation between M-ratio and v-ratio, r(202) = .38, p < 2 3 2
.001, suggesting that both variables capture shared variance reflecting metacognitive accuracy (see Figure  2 3 3 3F). We again observed that M-ratio correlated negatively with the fitted decision boundary, r(202) = -2 3 4
.18, p = .009, whereas no relation with decision bound was found for v-ratio, r(202) = .04, p = .535. 2 3 5
Metacognitive accuracy is a quickly emerging field in recent years. Crucial to its study is a 2 3 7 method to objectively quantify the extent to which participants are able to detect their own mistakes, 2 3 8 regardless of decision strategy. We here report that a commonly used static measure of metacognitive 2 3 9
accuracy (M-ratio) highly depends on the decision boundary -reflecting decision strategy -that is set for 2 4 0 decision making. This was the case in simulation results, in an experiment explicitly manipulating the 2 4 1 tradeoff between speed and accuracy, and in two datasets in which participants received no instructions 2 4 2 concerning speed or accuracy. We propose an alternative, dynamic, measure of metacognitive accuracy 2 4 3
(v-ratio) that does not depend on decision boundary. 2 4 4 2 4 5
Caution is warranted with static measures of metacognition 2 4 6
The most important consequence of the current findings is that researchers should be cautious 2 4 7 when interpreting static measures of metacognitive accuracy, such as M-ratio. In the following, we will 2 4 8
discuss several examples where our finding might have important implications. In the last decade there 2 4 9
has been quite some work investigating to what extent the metacognitive evaluation of choices is a 2 5 0 domain-general process or not. These studies often require participants to perform different kinds of tasks, 2 5 1 and then examine correlations in accuracy and in metacognitive accuracy between these tasks 3,11-14,33 . For 2 5 2 example, Mazancieux and colleagues 11 asked participants to perform an episodic memory task, a 2 5 3 semantic memory task, a visual perception task and a working memory task. In each task, participants 2 5 4 rated their level of confidence after a decision. The results showed that whereas correlations between 2 5 5 accuracy on these different tasks were limited, there was substantial covariance in metacognitive accuracy 2 5 6 across these domains. Because in this study participants received no time limit to respond, it remains 2 5 7 unclear whether this finding can be interpreted as evidence for a domain-general metacognitive monitor, 2 5 8 or instead a domain-general response caution which caused these measures to correlate. Another popular 2 5 9
area of investigation has been to unravel the neural signatures supporting metacognitive accuracy 13,14,34-2 6 0 36 . For example, McCurdy et al. observed that both visual and memory metacognitive accuracy correlated 2 6 1 with precuneus volume, potentially pointing towards a role of precuneus in both types of metacognition.
It remains unclear, however, to what extent differences in response caution might be responsible for this 2 6 3 association. Although differences in response caution are usually found to be related to pre-SMA and 2 6 4 anterior cingulate 24,25 , there is some suggestive evidence linking precuneus to response caution 37 . 2 6 5 Therefore, it is important that future studies on neural correlates of metacognition rule out the possibility 2 6 6 that their findings are caused by response caution. Finally, our study has important consequences for 2 6 7 investigations into differences in metacognitive accuracy between specific, e.g. clinical, groups. For 2 6 8 example, Folke and colleagues 17 reported that M-ratio was reduced in a group of bilinguals compared to 2 6 9 a matched group of monolinguals. Interestingly, they also observed that on average bilinguals had shorter 2 7 0 reaction times than monolinguals, but this effect was unrelated to the group difference in M-ratio. 2 7 1
Because these authors did not formally model their data using evidence accumulation models, however, it 2 7 2 remains unclear whether this RT difference results from a difference in boundary, and if so to what extent 2 7 3 this explains the difference in M-ratio between both groups that was observed. In a similar vein, 2 7 4
individual differences in M-ratio have been linked to psychiatric symptom dimensions, and more 2 7 5 specifically to a symptom dimension related to depression and anxiety 5 . At the same time, it is also 2 7 6 known that individual differences in response caution are related to a personality trait known as need for 2 7 7 closure 38 . Given that need for closure is, in turn, related to anxiety and depression 39 , it remains a 2 7 8 possibility that M-ratio is only indirectly related to these psychiatric symptoms via response caution. 2 7 9 2 8 0
The potential of dynamic measures of metacognition 2 8 1
In order to control for potential influences of response caution on measures of metacognitive 2 8 2 accuracy, one approach could be to estimate the decision boundary and examine whether the relation 2 8 3
between metacognitive accuracy and the variable of interest remains when controlling for decision 2 8 4 boundary (e.g., using mediation analysis). However, a more direct approach would be to estimate 2 8 5 metacognitive accuracy in a dynamic framework, thus taking into account differences in response caution. 2 8 6
In the current work, we proposed v-ratio (reflecting the ratio between post-decision drift rate and drift 2 8 7 rate) as such a dynamic measure of metacognitive accuracy (following the observation that post-decision 2 8 8 drift rate indexes how accurate confidence judgments are 19,20 ). In both simulations and empirical data, we 2 8 9
observed a positive relation between v-ratio and M-ratio, suggesting they capture shared variance. 2 9 0
Critically, v-ratio was not correlated with decision boundary, suggesting it is not affected by differences 2 9 1 in response caution. Thus, our dynamic measure of metacognition holds promise as a novel approach to 2 9 2 quantify metacognitive accuracy while taking into account the dynamics of decision making. 2 9 3
In our approach we allowed the drift rate and the post-decision drift rate to dissociate. This 2 9 4
proposal is in line with the view of metacognition as a second-order process whereby dissociations 2 9 5 between confidence and accuracy might arise because of noise or bias at each level 40-42 . However, when 2 9 6
formulating post-decision drift rate as a continuation of evidence accumulation, it remains underspecified 2 9 7 which evidence the post-decision accumulation process is exactly based on. It has been suggested that 2 9 8 participants can accumulate evidence that was still in the processing pipeline (e.g. in a sensory buffer) 2 9 9 even after a choice was made 30,43 . However, it is not very likely that this is the only explanation, 3 0 0 particularly in tasks without much speed stress. One other likely possibility, is that during the post-3 0 1 decision process, participants resample the stimulus from short-term memory 44 . Because memory is 3 0 2 subject to decay, dissociations between the post-decision drift rate and the drift rate can arise. Other 3 0 3 sources of discrepancy might be contradictory information quickly dissipating from memory 45 which 3 0 4
should lower metacognitive accuracy, or better assessment of encoding strength with more time 46 which 3 0 5
should increase metacognitive accuracy. 3 0 6
To sum up, we provided evidence from simulations and empirical data that a common static 3 0 7 measure of metacognition, M-ratio, is confounded with response caution. We proposed an alternative 3 0 8 measure of metacognition based on a dynamic framework, v-ratio, which is insensitive to variations in 3 0 9
caution, and may thus be suitable to study how metacognitive accuracy varies across subjects and 3 1 0
conditions. 3 1 1
Computational model 3 1 3
Data were simulated for 100 observers with 500 trials each. For each simulated observer, we 3 1 5 randomly selected a value for the drift rate (uniform distribution between 0 and 2.5), for the decision 3 1 6 boundary (uniform distribution between .5 and 3), for the non-decision time (uniform distribution 3 1 7
between .2 and .6) and for the v-ratio (uniform distribution between 0 and 1.5; see below for details). To 3 1 8 estimate meta-d', data is needed for both of the possible stimuli (i.e., to estimate bias); therefore, for half 3 1 9
of the trials we multiplied the drift rate by -1. Finally, we fixed the values for starting point (z = .5), 3 2 0 within-trial noise (ߪ = 1) and post-decision processing time (1s).
Fitting procedure 3 2 2
We coded an extension of the drift diffusion model (DDM) that simultaneously fitted choices, 3 2 3 reaction times and decision confidence. The standard DDM is a popular variant of sequential sampling 3 2 4 models of two-choice tasks. We used a random walk approximation, coded in the rcpp R package to 3 2 5
increase speed 47 , in which we assumed that noisy sensory evidence started at z*a; 0 and a are the lower 3 2 6
and upper boundaries, respectively, and z quantifies bias in the starting point (z = .5 means no bias). At 3 2 7 each time interval ߬ a displacement Δ in the integrated evidence occurred according to the formula shown 3 2 8
in equation (1) 
Evidence accumulation strength is controlled by v, representing the drift rate, and within-trial 3 3 0 variability, σ , was fixed to 1. The random walk process continued until the accumulated evidence crossed 3 3 1 either 0 or a. After boundary crossing, the evidence continued to accumulate for a duration depending on 3 3 2 the participant-specific median confidence reaction time. Importantly, consistent with the signal detection 3 3 3
theoretical notion that primary and secondary evidence can dissociate, we allowed for dissociations 3 3 4 between the drift rate governing the choice and the post-decision drift rate. For compatibility with the M-3 3 5 ratio framework, we quantified metacognitive accuracy as the ratio between post-decision drift rate and 3 3 6
drift rate, as shown in equation (2):