Psychological Methods
1999, Vol.4. No. 3.272-299

Copyright 1999 by (he American Psychological Association, Inc.
1082-989X/99/S3.00

Evaluating the Use of Exploratory Factor Analysis in
Psychological Research
Leandre R. Fabrigar

Duane T. Wegener

Queen's University

Purdue University

Robert C. MacCallum

Erin J. Strahan

Ohio State University

Queen's University

Despite the widespread use of exploratory factor analysis in psychological research,
researchers often make questionable decisions when conducting these analyses.
This article reviews the major design and analytical decisions that must be madewhen conducting a factor analysis and notes that each of these decisions has
important consequences for the obtained results. Recommendations that have been
made in the methodological literature are discussed. Analyses of 3 existing empirical data sets are used to illustrate how questionable decisions in conducting
factor analyses can yield problematic results. The article presents a survey of 2
prominent journals that suggests that researchers routinely conduct analyses using
such questionable methods. The implications of these practices for psychologicalresearch are discussed, and the reasons for current practices are reviewed.

Since its initial development nearly a century ago(Spearman, 1904, 1927), exploratory factor analysis(EFA) has been one of the most widely used statisticalprocedures in psychological research. Despite thislong history and wide application, the use of factor
analysis in psychological research has often been
criticized. Some critics have raised concerns about
fundamental limitations of factor analysis for contributing to theory development (e.g., Gould, 1981; Hills,
1977; Overall, 1964). For instance, Armstrong (1967),
in an article entitled "Derivation of theory by meansof factor analysis or Tom Swift and his electric factor
analysis machine," argued that factor analysis had
limited utility for aiding in the development of theory,
because it could not be relied on to provide meaningful insights into data.1 He attempted to demonstrate
this point by creating artificial data with a known
structure and then ostensibly showing that EFA failed
to accurately represent the structure. 2 Other critics
have not challenged the fundamental utility of EFA
but have instead criticized the manner in which it is

Leandre R. Fabrigar and Erin J. Strahan, Department of
Psychology, Queen's University, Kingston, Ontario,
Canada; Duane T. Wegener, Department of Psychology,
Purdue University; Robert C. MacCallum, Department of
Psychology, Ohio State University.
Erin J. Strahan is now at Department of Psychology, University of Waterloo, Waterloo, Ontario, Canada.
This research was supported by a grant from the Social
Sciences and Humanities Research Council of Canada and
Grant PO1-MH/DA56826 from the National Institutes of
Health. We thank Stephen G. West for his comments on an
earlier version of this article.
Correspondence concerning the article should be addressed to Leandre R. Fabrigar, Department of Psychology,
Queen's University, Kingston, Ontario, K7L 3N6, Canada.
Electronic mail may be sent to fabrigar@psyc.queensu.ca.

1

The name Tom Swift refers to a character in a popularseries of juvenile science fiction novels published in the
1960s. In each novel, Tom Swift makes use of futuristic
devices with near-miraculous powers. Armstrong's reference to this character in the title of his article highlighted
what he regarded as the naive belief by researchers in fundamental utility of EFA.
2
It is important to note that Armstrong (1967) was not
the first person to examine the effectiveness of EFA procedures using data sets with a known underlying structure(e.g., see Thurstone, 1947; Cattell & Dickman, 1962; Cattell& Sullivan, 1962; Cattell & Jaspers, 1967). Interestingly, inthese other cases, the authors concluded that appropriate
EFA procedures were reasonably effective in revealing the
known underlying structure of the data.

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USE OF FACTOR ANALYSIS

sometimes applied (e.g., Comrey, 1978; Ford, MacCallum, & Tail, 1986; Gorsuch, 1983; McNemar,
1951; Skinner, 1980). In this article, we primarily
address the latter issue. That is, we explore the manner in which factor analysis is applied in psychological research and evaluate the soundness of current
practices. However, we contend that these two issues
are intertwined. The utility of factor analysis fortheory development is dependent on the manner inwhich it is implemented (see Cattell, 1978; Comrey,1978). Furthermore, we suggest that some critics who
have questioned the fundamental value of factor
analysis have not been sufficiently sensitive to thisrelationship.
We begin our discussion by reviewing some of the
major methodological decisions that researchers must
make when conducting a factor analysis. Next, we
illustrate with published data sets how poor choices
when making these decisions can substantially distort
the results. We then turn our attention to the extent towhich current use of factor analysis reflects soundpractice-. We conclude with discussions of the implications of current factor analytic practices for psychological theory and reasons for the prevalence of certain practices.

Methodological Issues in the Implementation of
Factor Analysis
Perhaps more than any other commonly used statistical method, EFA requires a researcher to make anumber of important decisions with respect to how theanalysis is performed (see Finch & West. 1997). Specifically, there are at least five major methodologicalissues that a researcher should consider when conducting a factor analysis. First, he or she must decide
what variables to include in the study and the size andnature of the sample on which the study will be based.
Second, a researcher must determine if EFA is themost appropriate form of analysis given the goals of
the research project. Third, assuming that EFA is appropriate, a specific procedure to fit the model to thedata must be selected. Fourth, the researcher must
decide how many factors should be included in themodel. Finally, it is usually necessary for a researcher
to select a method for rotating the initial factor analytic solution to a final solution that can be more
readily interpreted. Each of these decisions can haveimportant consequences for the results obtained (seeArmstrong & Soelberg, 1968; Cattell, 1978; Comrey,
1978; Ford et al., 1986; MacCallum, 1983; MacCal-

273

lum, Widaman, Zhang, & Hong, 1999; Velicer &Fava, 1998; Weiss, 1976). To the extent that a researcher makes poor decisions, the analysis is more
likely to provide misleading results. However, researchers often appear to be unaware of the issues
involved in these decisions.

Decisions in Conducting an EFA
Study design. As with any statistical procedure,
the utility of the results obtained in EFA is in largepart determined by the soundness of the design of thestudy from which the data are collected. Within thecontext of EFA, one design issue that is especially
important is what measured variables to include in the
study (Cattell, 1978). If a researcher inadequately
samples measured variables from the domain of interest, he or she may fail to uncover important common factors. Conversely, if measured variables irrelevant to the domain of interest are included, thenspurious common factors might emerge or true common factors might be obscured. Therefore, researchers should carefully define their domain of interestand specify sound guidelines for the selection of measured variables.
Research suggests that EFA procedures provide
more accurate results when each common factor isrepresented by multiple measured variables in theanalysis (i.e., when common factors are "overdetermined"; MacCallum et al., 1999; see also Velicer &Fava, 1998). Methodologists have recommended thatat least three to five measured variables representing
each common factor be included in a study (MacCallum et al., 1999; Velicer & Fava, 1998). Thus, when
designing studies for which EFA is likely to be used,a researcher should consider the nature and number ofcommon factors he or she expects might emerge. The
total number of measured variables included should
be at least 3 to 5 times the number of expected common factors, and the selected variables should include
multiple variables likely to be influenced by each of
the common factors. Alternatively, in cases in which
there is little or no basis to anticipate the number andnature of common factors, a researcher should attempt
to delineate as comprehensively as possible the population of measured variables for the domain of interest. He or she should then include in the study asample of these measured variables that is as large as
feasible (see Cattell, 1978).
Sound selection of measured variables also requiresconsideration of psychometric properties of measures.
When EFA is conducted on measured variables with

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FABRIGAR, WEGENER, MAcCALLUM, AND STRAHAN

low communalities (i.e., variables for which the common factors explain little variance), substantial distortion in results can occur (MacCallum et al., 1999;
Velicer & Fava, 1998). There are a number of reasons
why communalities for measured variables might below. One obvious reason is low reliability. As explained later, variance due to random error cannot, bydefinition, be explained by common factors. Becauseof this, variables with low reliability will have low
communalities and thus should be avoided. A second
reason why a variable might have a low communality
is that the variable is unrelated to the domain of interest and thus shares little in common with othermeasured variables in that domain. Therefore, to theextent such information is available, a researcher
should consider the validity (e.g., face validity, convergent validity) of measured variables when selecting items to include in the analysis.
A second important design decision is the selectionof the sample. A researcher must determine how large
the sample should be and how that sample will beselected from the population of interest. Methodologists have proposed a host of rough guidelines for
estimating an adequate sample size for an EFA. Mostof these guidelines involve determining sample size
based on the number of measured variables includedin the analysis—with more measured variables requiring larger sample sizes. Sometimes such guidelines
also specify a minimum necessary sample size regardless of the number of measured variables.
Unfortunately, there are serious drawbacks to suchguidelines. One problem is that these recommendations vary dramatically. For instance, Gorsuch (1983)
suggested a ratio of 5 participants per measured variable and that the sample size never be less than 100.
In contrast, Nunnally (1978) and Everitt (1975)
proposed ratios of 10 to 1. More important, recent
research has suggested that such guidelines are notsufficiently sensitive to a variety of important characteristics of the data (MacCallum et al., 1999;
Velicer & Fava, 1998). The primary limitation of suchguidelines is that adequate sample size is not a function of the number of measured variables per se but isinstead influenced by the extent to which factors are
overdetermined and the level of the communalities ofthe measured variables. When each common factor is
overdetermined (i.e., at least three or four measured
variables represent each common factor) and the communalities are high (i.e., an average of .70 or higher),accurate estimates of population parameters can beobtained with samples as small as 100 (MacCallum etal., 1999). However, under more moderate conditions
a sample size of at least 200 might be needed; whenthese conditions are poor it is possible that samples aslarge as 400 to 800 might not be sufficient.
It is worth noting that obtaining parameter estimates that closely approximate population values isonly one criterion a researcher might consider when
determining sample size. In some situations, additional concerns might also play a role. Most notably,
when EFA involves the testing of formal hypotheses
regarding model fit or parameter estimates (as issometimes done in maximum likelihood [ML] EFA),
statistical power might also be considered. A researcher could specify a hypothesis of interest, a desired level of power, and an assumed population value
for model fit. The sample size necessary to achievethese objectives can then be calculated (see MacCallum, Browne, & Sugawara, 1996).
Researchers should also consider the nature of thesample on which the study is based. Psychologists
often select samples based on convenience. In manycases, this practice will not pose a problem. However,if the sample is considerably more homogeneous thanthe population on the common factors, this can lead torestriction of range in the measures, thereby attenuating correlations among variables. This attenuation can
result in falsely low estimates of factor loadings andcorrelations among factors (see Comrey & Lee, 1992;
Gorsuch, 1983; Tucker & MacCallum, 1997). Additionally, selection biases related to a single measured
variable in the analysis can also distort results (Tucker& MacCallum, 1997). For these reasons, researchers
should consider the nature of the measured variables
they are investigating and the manner in which theirsample is selected. Overly homogeneous samples andsamples whose selection is related to measured variables in the analysis should be avoided. Thus, when
there is a substantial basis to expect that conveniencesamples will not be appropriate, a researcher should
consider obtaining a sample representative of the
population of interest. Alternatively, a researcher
might wish to select a sample to maximize varianceon measured variables relevant to the constructs ofinterest and minimize variance on measured variables
irrelevant to the constructs of interest (see Cattell,1978).
Determining whether EFA is appropriate. The
primary purpose of EFA is to arrive at a more parsimonious conceptual understanding of a set of measured variables by determining the number and natureof common factors needed to account for the pattern

USE OF FACTOR ANALYSISof correlations among the measured variables. That is,
EFA is used when a researcher wishes to identify a setof latent constructs underlying a battery of measured
variables. Before using EFA. a researcher shouldcarefuily consider if this is a goal of the research
projec.
In reaching this decision, it is important to recognize that the goal of identifying latent constructs (i.e.,understanding the structure of correlations among
measured variables) is different from that of data reduction. Data reduction involves taking scores on alarge set of measured variables and reducing them toscores on a smaller set of composite variables thatretain as much information from the original variables
as possible. Data reduction does not attempt to model
the structure of correlations among the original variables. This distinction is important, because different
methods have been designed to achieve these two
objectives. If the goal is to arrive at a parsimonious
representation of the associations among measured
variables, EFA can be an appropriate form of analysis.
If the goal is data reduction, principal components
analysis (PCA) is more appropriate. Many researchers
mistakenly believe that PCA is a type of EFA when infact these procedures are different statistical methods
designed to achieve different objectives (for a discussion of the distinction, see Bentler & Kano, 1990;
Bookstein, 1990; Gorsuch, 1990: Loehlin, 1990;
McArdie, 1990; Mulaik, 1990; Rozeboom. 1990;
Schonemann, 1990; Steiger, 1990; Velicer & Jackson,
1990a, 1990b; Widaman, 1990).
EFA is based on the common factor model (Thurstone, 1947). This model postulates that each measured variable in a battery of measured variables is alinear function of one or more common factors andone unique factor. Common factors are unobservablelatent variables that influence more than one measured variable in a battery and are presumed to account for the correlations (covariances) among themeasured variables (i.e., two measured variables are
assumed to be correlated, because they are influenced
by one ;>r more of the same common factors). Uniquefactors are latent variables that influence only one
measured variable in a battery and do not account forcorrelations among measured variables. Unique factors are assumed to have two components: a specificfactor component (i.e., systematic latent factors that
influence only one measured variable) and an error ofmeasurement component (i.e., unreliability in a measured variable). The goal of the common factor model
is to understand the structure of correlations among275

measured variables by estimating the pattern of relations between the common factor(s) and each of themeasured variables (i.e., as indexed by factor loadings).
In contrast, PCA does not differentiate between
common and unique variance. Rather, this approachdefines each measured variable as a linear function ofprincipal components, with no separate representationof unique variance. Mathematically, these principal
components can be defined as linear composites of theoriginal measured variables and thus contain bothcommon and unique variance. Therefore, principal
components are not latent variables, and because ofthis it is not conceptually correct to equate them with
common factors (although in practice researchers often do so). Furthermore, whereas the goal of common
factor analysis is to explain correlations among measured variables, the goal of PCA is to account forvariance in the measured variables. That is, the objective of PCA is to determine the linear combinationsof the measured variables that retain as much information from the original measured variables as possible. Thus, although PCA is often referred to andused as a method of factor analysis, it is not factor
analysis at all.
Nevertheless, some methodologists have argued
that PCA is a reasonable substitute for analyses ofcommon factors and might even be superior (e.g., see
Velicer & Jackson, 1990a, 1990b). They noted that
PCA is computationally simpler than common factor
analysis and therefore requires less computer memory
and processing time (e.g., Velicer & Jackson, 1990a).
They also argued that the two approaches generally
produce very similar results (e.g.. Arrindell & van derEnde, 1985; Velicer, 1977; Velicer & Jackson, 1990a;
Velicer, Peacock, & Jackson, 1982). Also, they notedthat common factor analysis procedures sometimes
produce "Heywood cases" (e.g., Velicer & Jackson,
1990a)—situations in which a communality for ameasured variable (i.e., the proportion of variance inthe measured variable accounted for by the common
factors) is estimated to be at 1 or greater than 1. Because it is impossible to account for more than 100%of the variance in a variable, such an estimate is potentially problematic. Finally, advocates of PCA notethat it is determinate, whereas the common factor
model is not (e.g., Steiger, 1979, 1990; Velicer &Jackson, 1990a). That is, it is possible to compute anindividual person's score on a principal component,
whereas it is not possible to do so for a common
factor.

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FABRIGAR. WEGENER, MAcCALLUM, AND STRAHAN

There are, however, reasons to question these arguments. Advances in the speed and memory capabilities of computers have made the advantage of thecomputational simplicity of PCA trivial. Also, although PC A and EFA do often produce similar results, there are some contexts in which this is not thecase (e.g., see Bentler & Kano, 1990; Borgatta,
Kercher, & Stull, 1986; Gorsuch, 1988, 1990; Hubbard & Allen, 1987; McArdle, 1990; Snook & Gorsuch, 1989; Tucker, Koopman, & Linn, 1969; Widaman, 1990, 1993). Differences in results are most
likely when communalities are low (e.g., .40) andthere are a modest number of measured variables[e.g., three) per factor (Widaman, 1993). Furthermore,when the data correspond to assumptions of the common factor model, EFA produces more accurate results than PCA (e.g., McArdle, 1990; Snook & Gorsuch, 1989; Tucker et al., 1969; Widaman, 1990,
1993). In contrast, when the data are relatively consistent with the assumptions of PCA (e.g., little
unique variance present in measured variables), extraction of common factors does as well as extractionof principal components (e.g., see Gorsuch, 1990;
McArdle, 1990; Velicer et a!., 1982).
The occasional occurrence of Heywood cases
should also not be regarded as a flaw of common
factor analysis. Heywood cases often indicate that a
misspecified model has been fit to the data or that thedata violate assumptions of the common factor model(van Driel, 1978). Thus, Heywood cases can havediagnostic value (Velicer & Jackson, 1990a). In contrast, because such cases do not occur in PCA, suchproblems are not solved but simply go unnoticed; McArdle, 1990).3
Indeterminacy of individual factor scores in thecommon factor model need not be a problem. In mostapplications of EFA, this issue is irrelevant in that theobjective is to identify common factors that accountror the structure of the correlations among the measured variables. This goal does not require the computation of factor scores but rather only factor loadings and factor intercorrelations (McArdle, 1990). In
those situations where factor scores might be of interest, researchers usually obtain scores for the purpose of assessing correlations of factors with othervariables, or using factors as independent variables ordependent variables in regression models. However,the development of structural equation modelingmakes it unnecessary to estimate factor scores to obtain such information. One can use structural equation
modeling to specify a model with factors as corre-lates, predictors, or consequences of other variables
and obtain estimates of the relevant parameters without ever estimating factor scores. Additionally, recent
developments have provided useful nonstructural
equation modeling procedures for estimating the correlations between factors and other variables without
the need to compute factor scores (Gorsuch, 1997).
Finally, advocates of common factor analysis notethat it has certain advantages over PCA when the goal
is to identify latent constructs. They argue that mostmeasures used in psychological research contain somerandom error. Because EFA procedures reflect a recognition of this fact, whereas PCA does not, the common factor model is a more realistic model of thestructure or correlations (e.g., see Bentler & Kano,
1990; Gorsuch, 1973; Loehlin, 1990). Additionally,
the common factor model is testable, whereas the
PCA model is not (e.g., see Bentler & Kano, 1990;
McArdle, 1990). The common factor model specifies
certain hypotheses about the data. Thus, it can be fit todata and the model rejected if the fit is poor. In contrast, because PCA does not involve a specific hypothesis to be tested, it does not provide information
on which one could base a decision to reject the"model."Therefore, there are clear conceptual distinctions
between PCA and EFA. Although these approaches
often produce similar results, this is not true in certain
contexts. When the goal of the analysis is to identify
latent constructs underlying measured variables, it ismore sensible to use EFA than PCA (see Cattell,
1978; Gorsuch, 1983; McDonald, 1985; Mulaik,1972).
Assuming that identifying latent variables that account for the correlations among measured variables
is the goal of the research project, a researcher mustthen decide if an exploratory or confirmatory approach will be used. Both EFA and confirmatory factor analysis (CFA) are based on the common factor
model, and both seek to represent the structure ofcorrelations among measured variables using a rela-

1

Furthermore, there might be some cases in which a
Heywood case would not represent a misspecified modeland might not be problematic. For example, circumstancesin which population parameters are close to logical boundaries (e.g., if true communalities are close to 1.0) could
produce Heywood cases simply because of expected variability in parameter estimates resulting from sampling error(van Driel, 1978).

USE OF FACTOR ANALYSIS

lively small set of latent variables. However, EFA is
primarily a data-driven approach. No a priori numberof common factors is specified and few restrictions
are placed on the patterns of relations between thecommon factors and the measured variables (i.e., thefactor loadings). EFA provides procedures for determining an appropriate number of factors and the pattern of factor loadings primarily from the data. In
contrast, CFA requires a researcher to specify a specific number of factors as well as to specify the pattern of zero and nonzero loadings of the measured
variables on the common factors. Alternatively, a researcher might specify a priori a small set of competing models postulating differing numbers of factors,
different patterns of factor loadings, or both.
In situations in which a researcher has relatively
little theoretical or empirical basis to make strong
assumptions about how many common factors exist or
what specific measured variables these common factors are likely to influence, EFA is probably a moresensible approach than CFA. EFA is likely to be moredesirable in these situations, because the number ofplausible alternative models might be so large that it
would be impractical to specify and test each one in
CFA. Additionally, when a strong basis does not exist
for identifying a single model or a few specific competing models, it is quite possible that a researcher
might fail to identify a number of plausible models.
Therefore, in these contexts, the data-driven approachof EFA seems advisable.
However, when there is sufficient theoretical andempirical basis for a researcher to specify the modelor small subset of models that is the most plausible,
CFA is likely to be a better approach. This is because
CFA allows for focused testing of specific hypotheses
about the data (e.g., see Finch & West, 1997; Wegener & Fabrigar, in press). Also, the a priori nature of
CFA makes it less likely that a researcher will capitalize on chance characteristics in the data.
It is also often useful to use EFA and CFA in conjunction with one another. An EFA can be conductedin an initial study to provide a basis for specifying a
CFA model in a subsequent study. Alternatively, if
the sample size in a single study is sufficiently large,
the sample could be randomly split in half. An EFA
could then be conducted on one half of the data providing the basis for specifying a CFA model that canbe fit to the other half of the data.
Choice of model-fitting procedure. If EFA is themost appropriate form of analysis, it is then necessary
to decide what procedure will be used to fit the com-277

mon factor model to the data. A number of modelfitting methods (i.e., factor-extraction procedures) areavailable. Most widely used among these are ML,
principal factors with prior estimation of communalities, and iterative principal factors. These are different
methods for estimating the parameters (factor loadings and unique variances) of the same model, thecommon factor model.4
Although these procedures fit the same model, eachmethod does have certain advantages and disadvantages. The primary advantage of ML is that it allows
for the computation of a wide range of indexes of thegoodness of fit of the model. ML also permits statistical significance testing of factor loadings and correlations among factors and the computation of confidence intervals for these parameters (Cudeck &O'Dell, 1994). The primary limitation of ML estimation is its assumption of multivariate normality. When
this assumption is severely violated, this procedure
can produce distorted results (Curran, West, & Finch,
1996; Hu, Bentler, & Kano, 1992). On the other hand,principal factors methods (both iterated and noniterated) have the advantage of entailing no distributional
assumptions. Principal factors are also less likely than
ML to produce improper solutions (i.e., a solution
with a Heywood case or a solution that fails to converge on a final set of parameter estimates; Finch &West, 1997). The major drawback of the principal
factor methods is that they provide a much more limited range of goodness-of-fit indexes and generally do
not allow for computation of confidence intervals andsignificance tests. Regardless of these differences,when the common factor model holds reasonably wellin the population and severe violations of distributional assumptions are not present, solutions provided
by these methods are usually very similar.
Selecting the number of factors. Determining how
many factors to include in the model requires theresearcher to balance the need for parsimony (i.e., amodel with relatively few common factors) against
the need for plausibility (i.e., a model with a sufficientnumber of common factors to adequately account for

4
Although the common factor model makes a further
conceptual distinction by decomposing unique variance into
specific variance and error variance, methods of fitting thecommon factor model allow for estimates of the amount of
unique variance only and do not provide separate estimatesof specific variance and error variance.

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FABRIGAR, WEGENER, MAcCALLUM, AND STRAHAN

the correlations among measured variables). In otherwords, the goal of the researcher is to determine thenumber of "major" factors underlying a battery ofmeasures. Importantly, errors in selection of the number of factors in a model can have a substantial effecton the results obtained (e.g., Comrey. 1978; Fava &Velicer, 1992; Levonian & Comrey, 1966; Wood, Tataryn, & Gorsuch, 1996).
Traditionally, methodologists have regarded specifying too few factors in a model (i.e., underfactoring)as a much more severe error than specifying too manyfactors (i.e., overfactoring; see Cattell, 1978; Rummel, 1970; Thurstone, 1947). Empirical research has
generally supported this notion. When too few factors
are included in a model, substantial error is likely(Fava & Velicer, 1992; Wood et al., 1996). Measured
variables that load on factors not included in themodel can falsely load on factors included in themodel, and poor estimates of the factor loadings canhe obtained for measured variables that do actuallyload on the factors included in the model (Wood et al.,1996). Such distortions can result in rotated solutions
in which two common factors are combined into asingle common factor (thereby obscuring the true faci.or structure) and in solutions with complex patterns
of factor loadings that are difficult to interpret (seeComrey. 1978).
Empirical research suggests that overfactoring introduces much less error to factor loading estimatesihan underfactoring (Fava & Velicer. 1992; Wood et..;!., 1996). Such models often result in rotated solui ions in which the major factors are accurately represented and the additional factors have no measured
variables that load substantially on them or have onlya single measured variable that loads substantially oneach additional factor. Nonetheless, overfactoringshould be avoided (see Comrey, 1978; Comrey &Lee, 1992). Solutions with too many factors might
prompt a researcher to postulate the existence of cons tracts with little theoretical value and thereby develop unnecessarily complex theories. Additionally,
overfactoring can accentuate poor decisions made atother steps in a factor analysis. For example, PCA
tends to produce loadings that are larger than factor
loadings (see Snook & Gorsuch, 1989; Widaman.
1993). Thus, solutions using this approach can sometimes make minor components appear to be major
components (Wood et al., 1996).
Given these consequences, it is not surprising thatan extensive methodological literature has developed
exploring the issue of determining the optimal numberof factors. A number of procedures for answering thisquestion have been proposed. Perhaps the best knownof these procedures is the Kaiser criterion of computing the eigenvalues for the correlation matrix to determine how many of these eigenvalues are greaterthan 1 (for discussion of the nature of eigenvalues, see
Gorsuch, 1983). This number is then used as the number of factors. Although this procedure is appealing
for its simplicity and objectivity, the approach hassignificant problems. First, it is often misapplied byreferring to the eigenvalues of the correlation matrix
with communality estimates in the diagonal (i.e., thereduced correlation matrix) rather than eigenvalues of
the correlation matrix with unities in the diagonal(i.e., the unreduced correlation matrix; see Guttman.
1954; Kaiser, 1960). Application of this rule to theeigenvalues of the reduced correlation matrix is anerroneous procedure (Gorsuch, 1980; Horn, 1969).
Second, as with any mechanical rule, the procedure
can to some extent be arbitrary. For instance, it is notreally meaningful to claim that a common factor withan eigenvalue of 1.01 is a "major" factor whereas acommon factor with an eigenvalue of 0.99 is not.
Finally, in numerous studies involving both principal
components and common factors, this procedure has
been demonstrated to lead to substantial overfactoringand occasionally to underfactoring (Cattell & Jaspers,
1967; Cattell & Vogelmann, 1977; Hakstian, Rogers.
& Cattell, 1982; Linn, 1968; Tucker et al., 1969;
Zwick & Velicer, 1982, 1986). In fact, we know of nostudy of this rule that shows it to work well.
Another widely known approach for determining
the number of factors is the "scree test" (Cattell, 1966;
Cattell & Jaspers, 1967). In this procedure, the eigenvalues of the correlation matrix (or the reduced correlation matrix) are computed and then plotted in order of descending values. This graph is then examined
to identify the last substantial drop in the magnitudeof the eigenvalues. A model with the same number ofcommon factors as the number of eigenvalues prior tothis last substantial drop is then fit to the data. This
procedure is often used with eigenvalues from theunreduced correlation matrix. However, when thegoal is to identify common factors, it is more conceptually sensible to examine the plot of eigenvalues
from the reduced correlation matrix, because this matrix more directly corresponds to the common factor
model. Regardless of which matrix is used, this procedure has been criticized (e.g.. Kaiser. 1970) becauseof its subjectivity (i.e., there is no clear objective definition of what constitutes a "substantial" drop in mag-

USE OF FACTOR ANALYSIS
nitude). Additionally, sometimes the obtained patternof eigenvalues is ambiguous in that no clear substantial drop may be present. However, when strong common faciors are present in data, studies indicate that
this procedure functions reasonably well (Cattell &Vogelmann. 1977; Hakstian et al., 1982: Tucker et al.,1969).
A third factor-number procedure that has been investigated in the methodological literature is parallel
analysis (Horn, 1965; Humphreys & Ilgen, 1969;
Humphreys & Montanelli, 1975; Montanelli & Humphreys, 1976). This approach is based on a comparison of eigenvalues obtained from sample data to eigenvalues one would expect to obtain from completely
random data (i.e., the predicted means of eigenvalues
producec by repeated sets of random data). Suppose a
set of measured variables observed in a given sample
depends on m major common factors. Parallel analysis is based on the notion that the m largest sample
eigenvalues of the reduced correlation matrix should
be larger than the m largest expected values of eigenvalues obtained from repeated corresponding sets ofrandom data (based on the same sample size and number of variables). The eigenvalues that would be expected from random data are then compared with the
eigenvalues actually produced by the data, and amodel is specified with the same number of common
factors as real eigenvalues that are greater than the
eigenvalues expected from random data. Like otherobjective mechanical rules, this procedure can sometimes be arbitrary in that a factor just meeting thecriterion is retained, whereas a factor falling just below the criterion is ignored. This procedure is also notavailable in major statistical programs. Nonetheless,simulation research suggests parallel analysis functions fairly well (Humphreys & Montanelli, 1975).
Parallel analysis methods have also been developedfor use in components analysis and found to function
well in ;his context (Allen & Hubbard. 1986: Lautenschlager, 1989; Longman, Cota, Holden, & Fekken,
1989; Zwick & Velicer, 1986).
The methods described to this point for determining
the number of factors to retain involve analysis of
eigenvalues obtained in principal factors or PCA. Although >uch methods of factoring are widely used, theML method of factor extraction is becoming increasingly popular. The ML method has a more formal
statistical foundation than the principal factors methods and thus provides more capabilities for statisticalinference, such as significance testing and determination of confidence intervals. These capabilities allow279

a researcher to adopt a somewhat different approachto determining the optimal number of factors.
One can conceptualize the number-of-factors issue
as choosing the most appropriate model from a seriesof alternative factor analysis models that differ in theircomplexity (i.e., the number of factors). The objective
is to select a model that explains the data substantiallybetter than simpler alternative models (i.e., models
with fewer factors) but does as well or nearly as well
as more complex alternative models (i.e., models withmore factors). Because factor analysis is a special
case of structural equation modeling, many of the procedures used in structural equation modeling formodel selection can be applied to EFA. That is, theuse of ML estimation for EFA introduces a vast arrayof goodness-of-fit information that can be used todetermine the appropriate number of factors (seeBrowne & Cudeck, 1992; Hu & Bentler, 1998; Marsh,
Balla, & McDonald, 1988; Mulaik et al., 1989). These
fit measures, which are intended to assess the degree
to which a given model provides an approximation ofobserved correlations or covariances, can be applied
to the number-of-factors problem. The general procedure is to fit models with a range of numbers of factors, beginning with zero, and increasing throughsome maximally interesting number. Fit measures foreach model can then be evaluated, and a decision canbe made as to the appropriate number of factors toretain. The desired model is that which constitutes a
substantial improvement in fit over a model with one
fewer factor but for which a model with one morefactor provides little if any improvement in fit.5
One common statistic for assessing fit in ML factor
analysis solutions is the likelihood ratio statistic(Lawley, 1940). If N is sufficiently large and the distributional assumptions underlying ML estimation are
adequately satisfied, the likelihood ratio statistic approximately follows a chi-square distribution if thespecified number of factors is correct in the popula-s
Ideally, the preferred model should not just fit the data
substantially better than simpler models and as well as more
complex models. The preferred model should also fit thedata reasonably well in an absolute sense. When this is not
the case, a researcher should exercise some caution in interpreting the results. A preferred model that fits the data
poorly might do so, because the data do not correspond toassumptions of the common factor model. Alternatively, it
might suggest the existence of numerous minor common
factors.

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FABRIGAR, WEGENER, MAcCALLUM, AND STRAHAN

tion. Thus, this statistic can be used to test the null
hypothesis that the model holds exactly in the population with a given number of common factors. This
test can be applied to a series of numbers of factors,
beginning with zero and continuing until a nonsignificant test statistic is obtained, indicating that the common factor model with the corresponding number offactors is not rejected as a correct model in the population. Although this procedure is intuitively appealing in that it is based on a clear statistical rationale, it
is susceptible to serious problems. Most importantly,
it is highly influenced by sample size. When N islarge, even trivial discrepancies between the modeland the data are likely to give rise to rejection of themodel with any reasonable number of factors (Hakstian et al., 1982; M. L. Harris & Harris, 1971; Humphreys & Montanelli, 1975; MacCallum, 1990). In
situations where N is small, even large discrepancies
between the model and the data may not be statistically significant, thereby leading to underfactoring(Humphreys & Montanelli, 1975). In addition, the useof the likelihood ratio test is problematic, because thenull hypothesis of perfect fit is an unrealistic standard
in that all models are approximations of reality(Browne & Cudeck, 1992; Cudeck & Henly, 1991;
MacCallum, 1990). The realistic goal in factor analysis is to obtain a parsinomious solution that provides agood approximation to the real world; thus, the hypothesis of perfect fit is not generally of empirical interest.
Because of these problems, methodologists havedeveloped an array of alternative "descriptive" measures of fit. These developments began with the workof Tucker and Lewis (1973), who proposed a "reliability coefficient" for ML factor analysis solutions,and have continued through the present day. Our viewis that developments reviewed and illustrated by
lirowne and Cudeck (1992) provide a promising apnroach for assessing fit of these models. Browne andCudeck focused on the Root Mean Square Error of
Approximation (RMSEA) fit index (Steiger & Lind,
1980) and the Expected Cross-Validation Index
i ECVI; Browne & Cudeck, 1989) and illustrated theiruse for determining the number of factors in EFA.
RMSEA is an estimate of the discrepancy between themodel and the data per degree of freedom for themodel. It has been suggested that values less than 0.05
constitute good fit, values in the 0.05 to 0.08 range
acceptable fit, values in the 0.08 to 0.10 range marginal fit, and values greater than 0.10 poor fit (seeBrowne & Cudeck, 1992; Steiger, 1989).6 ECVI, on
the other hand, is an estimate of how well the solution

obtained from one sample will generalize to othersamples. Although no guidelines exist for interpreting
ECVI in absolute terms, it is useful for relative comparisons among alternative models. The smaller thevalue of ECVI, the better the expected crossvalidation of the model. An important property of
ECVI discussed by Cudeck and Henly (1991) andillustrated by Browne and Cudeck (1992) is that it issensitive to sample size. ECVI will tend to support theretention of simpler models (fewer factors) when N issmall and more complex models (more factors) when
N is large. One can think of ECVI as addressing thefollowing question; Based on the available data, for
how many factors can we obtain accurate and generalizable parameter estimates?
A major advantage of both these indexes is theavailability of confidence intervals. Thus, a researcher
can compare the point estimates and corresponding
confidence intervals of these fit indexes for a series ofmodels with varying numbers of factors. A model canbe selected that shows fit that is substantially better
than simpler models and fit that is comparable to thatof more complex models. Of course, the logic of theapproach illustrated by Browne and Cudeck (1992) ofusing RMSEA and ECVI to determine the number offactors in EFA can be extended to other descriptive fit
indexes. Thus, a researcher might choose to examine
additional indexes of model fit when determining thenumber of factors (for a recent evaluation of fit indexes, see Hu & Bentler, 1998).
Although the use of RMSEA, ECVI, and other descriptive measures of model fit has yet to be extensively tested within the context of determining thenumber of factors in EFA, there is a compelling logic
to this approach. Furthermore, many of these indexes
have been tested within the context of more generalcovariance structure models. Thus, users of ML factor
analysis would do well to consider the use of these fit
indexes when determining the number of factors toretain.
In summary, a number of procedures exist to determine the appropriate number of common factors.7

6

Although these guidelines for RMSEA are generally
accepted, it is of course possible that subsequent researchmight suggest modifications.
7
One method that has been found to perform well in
determining the number of principal components to retain is
the minimum average partial procedure (Velicer, 1976;
Zwick & Velicer, 1982, 1986). We do not discuss this pro-

USE OF FACTOR ANALYSIS

Some of these procedures are highly problematic (i.e.,eigenvalues greater than 1, the likelihood ratio statistic), whereas others are likely to perform reasonably
well s'. least under a range of conditions (e.g., scree
test, parallel analysis, descriptive indexes of model fit
such as RMSEA). However, even the best procedures
are not infallible. Furthermore, it is important to remember that the decision of how many factors toinclude in a model is a substantive issue as well as astatistical issue. A model that fails to produce a rotated -olution that is interpretable and theoreticallysensible has little value. Therefore, a researcher
should always consider relevant theory and previous
research when determining the appropriate number offactors to retain. It should also be noted that because
any individual data set is likely to have its own idiosyncrasies, decisions regarding the appropriate number of factors can be further improved by examining
the replicability of the decision over multiple data sets
(Catteil, 1978). When a given number of factors for abattery of measured variables is shown to be appropriate in more than one data set (e.g., sensible rules
for determining the number of factors give similar
results, rotated solutions produce similar interpretable
patterns of factor loadings), a researcher can be more
confident that the optimal number of factors has been
extracled.
Factor rotation. For any given solution with two
or more factors (or principal components), there exists
an infinite number of alternative orientations of thefactors in multidimensional space that will explain thedata equally well. This means that EFA models withmore than one factor do not have a unique solution.
Therefore, a researcher must select a single solution
from among the infinite number of equally fitting solutions.
The criterion most commonly used for selecting
among solutions in EFA is the property of simple

cedure, because it is a method for use with principal components and has not been developed for use with common
factors. Another approach to determining the number ofcomponents is to examine the comparability of componentscores across split halves of a sample (Everett, 1983). Unfortunately, formal approaches for determining the numberof common factors based on the stability of common factors
have nor been extensively developed and tested. Nonetheless, approaches based on this logic seem like a promising
direction for future research. Catteil (1978) suggested thattrue factors should replicate across samples whereas spurious factors should be unstable over samples.

281

structure (Thurstone, 1947). Thurstone proposed thatfor any given set of mathematically equivalent solutions, the solution with the best "simple structure"would generally be the most easily interpretable,
psychologically meaningful, and replicable. Thurstone used the term simple structure to refer to solutions in which each factor was defined by a subsetof measured variables that had large loadings relative to the other measured variables (i.e., high within-factor variability in loadings) and in which eachmeasured variable loaded highly on only a subsetof the common factors (i.e., low factorial complexity in defining variables). Therefore, Thurstone suggested that factors be rotated in multidimensional
space to arrive at the solution with the best simple
structure.
A number of analytic rotation methods have beendeveloped to seek simple structure, and numerous articles have been published comparing the utility ofthese various rotation procedures (e.g., Crawford &Ferguson, 1970; Dielman, Catteil, & Wagner, 1972;
Gorsuch, 1970; Hakstian, 1971; Hakstian & Boyd,1972; Hofmann, 1978). Although these rotation
methods differ in a number of respects, perhaps themost fundamental distinction that can be made is between orthogonal and oblique rotations. Orthogonalrotations constrain factors to be uncorrelated. Thougha number have been developed, varimax (Kaiser,
1958) has generally been regarded as the best orthogonal rotation and is overwhelmingly the mostwidely used orthogonal rotation in psychological research.
In contrast to orthogonal rotations, oblique rotations permit correlations among factors. One common
misconception among researchers is that oblique rotations require factors to be correlated. This is not thecase. If the solution with the best simple structure
involves orthogonal factors, a successful oblique rotation will provide estimates of the correlations
among factors that are close to zero and produce asolution that is quite similar to that produced by asuccessful orthogonal rotation (Harman, 1976). However, in situations in which the best simple structure isa solution with correlated factors, successful obliquerotations will produce solutions with correlated factors. Unlike orthogonal rotation, there is no single
method of oblique rotation that is clearly dominant inpsychological research. Several oblique rotation procedures are commonly used and have been found to
generally produce satisfactory solutions. These include direct quartimin rotation (Jennrich & Sampson,

282

FABRIGAR, WEGENER, MAcCALLUM, AND STRAHAN

1966), promax rotation (Hendrickson & White, 1964),
and Harris-Kaiser orthoblique rotation (Harris & Kaiser, 1964).8
Some researchers have indicated a preference fororthogonal rotation because of its simplicity and conceptual clarity (e.g., Nunnally, 1978). However, there
are a number of reasons to question the wisdom of thisview. First, for many constructs examined in psychology (e.g., mental abilities, personality traits, attitudes), there is substantial theoretical and empiricalbasis for expecting these constructs (or dimensions ofthese constructs) to be correlated with one another.
Therefore, oblique rotations provide a more accurate
and realistic representation of how constructs arelikely to be related to one another. Thus, from a substantive perspective, the restriction of uncorrelated
factors imposed by varimax and other orthogonal rotations is often unwarranted and can yield misleading
results. Second, because orthogonal rotations require
factors to be oriented at 90° angles from one anotherin multidimensional space (i.e., uncorrelated factors)
whereas oblique rotations allow for orientations ofless than 90° (i.e., correlated factors), orthogonal rotations are likely to produce solutions with poorer
simple structure when clusters of variables are lessthan 90° from one another in multidimensional space(i.e., when the true underlying factor structure isbased on correlated factors). Finally, oblique solutions
provide more information than orthogonal rotations.
Oblique rotations produce estimates of the correlations among common factors. Knowing the extent tovvhich factors are correlated with one another can ofren be useful in interpreting the conceptual nature ofi he common factors. Indeed, the existence of substanlial correlations among factors suggests that higher
order factors may exist. Correlation matrices of factors can in turn be analyzed to gain insight into thenumber and nature of these higher order factors andthereby further refine a researcher's understanding of
the data (see Gorsuch, 1983). Because orthogonal rotations do not provide correlations among factors, it is
impossible to determine if one or more higher orderfactors are present in the data.

Summary and Recommendations
The implementation of EFA requires a researcher
to address five major types of methodological decisions. There are a number of options available foreach decision, and the issues involved in selecting
among these options are sometimes complex. Nonetheless, these decisions are not arbitrary in thatsome options are clearly more optimal than others.
Thus, the utility of an EFA is likely to be a functionof the decisions made in the design of the study andthe implementation of the analysis. Indeed, it is interesting that some notable examples of the supposed
failure of EFA to provide valid insights into data involved poor factor analytic methodology. For instance, Armstrong's (1967) classic demonstration of
the alleged limitations of EFA used a PCA, the eigenvalues-greater-than-1 rule, and varimax rotation. In
light of the questionable nature these choices, it is not
surprising the analysis failed to uncover the true structure of the data.9 Given these facts, some clear recommendations can be made regarding how EFA
should be conducted.
/. Study design. Because EFA results are likely
to be more accurate if sensible decisions are made in
selecting measured variables and samples, it is essential that researchers carefully consider both of theseissues. Obviously, because EFA is used in situations
where there is relatively little prior theory and empirical evidence, variable selection can be difficult. Nonetheless, to the extent it is possible, researchers should
try to anticipate the number and nature of the factors
they expect to obtain and use this as a guide for selecting variables. We suggest researchers include atleast four measured variables for each common factor
they expect to emerge and perhaps as many as six
given that there is usually considerable uncertainty
about the nature of the common factors and their relations to the measured variables. Furthermore, although EFA should be based on comprehensive sam8

Unfortunately, the selection of oblique rotations offeredin major statistical packages is quite limited. SPSS offers
promax and Harris-Kaiser orthoblique rotations. Direct oblimin rotation is a family of rotations defined by different
values of the delta parameter, which governs the obliqueness of the solution. The default value for this parameter in
SPSS is 0. which corresponds to a direct quartimin rotation.
A new EFA program. CEFA (comprehensive exploratory
factor analysis; Browne. Cudeck, Tateneni. & Mels, 1998)
offers a much wider range of rotations.
9
Additionally, because the data created by Armstrong
have a number of properties that violate assumptions of thecommon factor model (e.g., nonlinear relations between factors and measured variables), his data are not really appropriate for EFA. Nonetheless, we found that if one uses more
optimal EFA procedures, the analysis produces a solutionthat provides a good representation of the underlying structure of the data, thus invalidating Armstrong's arguments
against the use of factor analysis.

USE OF FACTOR ANALYSIS

pling of measured variables in the domain of interest,
we recommend that researchers carefully consider thepsychometric properties of variables. Variables withlow reliabilities (e.g., below .70) should be avoided,
and when validity information is available, it should
be used as a basis for selecting items. With respect todetermining adequate sample size, the properties ofthe measured variables should be taken into account.
Under good conditions (communalities of .70 or
higher, four to five variables for each factor), a sample
size of 100 might well be adequate (although it isalways best to have larger sample sizes if possible).
Under conditions of moderate communalities (e.g.,.40 to .70) and moderate overdetermination of factors,
a sample of 200 or more seems advisable. Finally,
under poor conditions, no sample size may be sufficient to produce accurate estimates of the population
parameters. However, it seems likely that a sample
less than 400 will lead to distorted results.
2. Appropriateness of EFA. We urge researchers
to carefully consider if EFA is the most appropriate
form of analysis to meet their research objectives.
EFA should be used when the primary goal is to identify latent constructs and there is insufficient basis tospecify an a priori model (or small subset of models).
CFA should be used when the goal is to identify latentconstructs and a substantial basis exists to specify an
a priori model or small subset of models. PCA should
not be used as a substitute for EFA.
3. M,idel-fitting procedures. With respect to selecting one of the major methods of fitting the common factor model in EFA (i.e., principal factors, iterated principal factors, maximum likelihood), all three
are reasonable approaches with certain advantages
and disadvantages. Nonetheless, the wide range of fit
indexes available for ML EFA provides some basis
for preferring this approach. However, researchers
should rscognize that this procedure can produce misleading results when assumptions of multivariate normality are severely violated (see Curran et al., 1996;
Hu et a!., 1992). Therefore, we recommend that the
distributions of measured variables be examined prior
to conducting ML EFA. If nonnormality is severe(e.g., skew > 2; kurtosis >7; West, Finch, & Curran,1995), one of several remedies might be employed(see West et al., 1995). Measured variables could betransformed to normalize their distributions. Corrections to fit indexes and standard errors could be performed (Bentler & Dudgeon, 1996; Browne, 1984;
Satorra & Bentler, 1994). Alternatively, one mightwish to use a principal factors procedure.10283

4. Determining the number of factors. We suggest that researchers rely on multiple criteria when
deciding on the appropriate number of factors to include in a model. In the use of principal factors methods, we recommend the scree test and parallel analysis using eigenvalues from the reduced correlation
matrix. In ML factor analysis, we encourage the useof descriptive fit indexes such as RMSEA and ECVI
as discussed by Browne and Cudeck (1992) alongwith more traditional approaches such as the scree test
and parallel analysis. A sensible strategy in mostcases would be to use multiple methods to make thisdecision and then carefully examine the rotated solution for the suggested model to confirm that it is interpretable and theoretically plausible. In situations inwhich the sample size is sufficiently large, a researcher might also wish to randomly split the dataand examine the stability of the solution across thetwo halves (for a discussion of assessing factor comparability across samples, see Gorsuch, 1983). In contexts in which procedures suggest different numbersof factors or in which the procedures produce somewhat ambiguous results, the researcher should examine the subset of models that these procedures suggest
are most plausible. The rotated solutions for thesemodels can then be examined to see which model
produces the most readily interpretable and theoretically sensible pattern of results (Comrey, 1978; Fordet al., 1986; Hakstian & Muller, 1973; Hakstian et al.,
1982; C. W. Harris, 1967) and when possible whichof these solutions is most stable over different data
sets or split halves of data sets.
5. Rotation. Given the advantages of oblique rotation over orthogonal rotation, we see little justification for using orthogonal rotation as a general approach to achieving solutions with simple structure(Gorsuch, 1983). Instead, it is most sensible to firstexamine the solutions produced by one or more of thecommon methods of oblique rotation (e.g., promax,
orthoblique, or direct quartimin). If this solution in-

10
Another approach to addressing violations of normality is to create "item parcels" (i.e., composites of several
measured variables that tap on the same common factor).
Creation of such item parcels often results in measured variables with more normal distributions than the individual
items (West et al., 1995). However, such an approach willoften not be feasible in EFA, because little is known regarding which measured variables are influenced by thesame common factors, and thus it will be difficult to form
item parcels.

284

FABRIGAR, WEGENER, MAcCALLUM, AND STRAHAN

dicates that the factors are uncorrelated, then it isreasonable for the researcher to conduct a varimax
rotation and use this as the basis for interpretation(though this solution will likely be very similar to theoblique solution). On the other hand, if at least someof the factors are found to be correlated with one
another, it is most defensible to use the oblique solution as the basis for interpretation.

Illustration of Consequences of Choices
in Analysis
As noted earlier, some critics of EFA have been
insufficiently sensitive to the extent to which particular decisions about how the analysis is conducted can
influence the results. In order to illustrate some of theeffects that these decisions can have, we reanalyzed
three data sets from a well-developed psychologicalliterature on the tripartite model of attitude structure.
According to the tripartite model (Rosenberg &Hovland, 1960; Smith, 1947), attitudes have affective,
cognitive, and behavioral components. That is, feelings experienced when in the presence of an attitude
object, beliefs about the attributes of the object, andpast or present behaviors relevant to the object makeup the structure underlying overall summary evaluations (see Eagly & Chaiken, 1998; Petty, Priester, &Wegener, 1994; Wegener & Gregg, in press, for more
recent discussions). Numerous studies have investigated the hypothesized tripartite nature of attitudes(e.g., Breckler, 1984; Kothandapani, 1971; Ostrom,•969), and in particular, the affective and cognitive
components have received a great deal of attention as-.eparable aspects of attitude structure. For example,
assessments of the affect and cognition associatedwith particular attitude objects have independently
predicted attitudes in a variety of settings (e.g., Abel•on. Kinder, Peters, & Fiske, 1982; Crites, Fabrigar,& Petty, 1994), and confirmatory analyses have supported multifactor (tripartite) rather than single-factor
models of these attitude structures (Breckler, 1984).
Moreover, attitudes primarily based on affect versus
cognition predict different classes of behaviors (e.g.,
Millar & Tesser, 1986, 1989) and are differentiallyinfluenced by affective versus cognitive persuasive
appeals (e.g., Edwards, 1990; Fabrigar & Petty, 1999;
Millar & Millar, 1990). Therefore, there are compelling theoretical and empirical rationales for separating
affect and cognition in attitude structure. Because ofthis, there is a strong basis for expecting that EFA
should ideally recover separable affect, cognition, and(perhaps) behavior factors when examining relevant
data sets.
We addressed this question by reanalyzing three
data sets from this literature. First, we analyzed two
9-item matrices from Breckler (1984). In these matrices, three items each had been designed to assess
affective, cognitive, and behavioral aspects of attitudes, respectively (see Breckler, 1984, for detail on
the measures). We took the third data set from Crites
et al. (1994), in which general measures were developed to assess the affective and cognitive bases ofattitudes (three types of measures for each). In orderto take advantage of the existence of an additional
measure of affect and an additional measure of cognition specific to one of the attitude objects (snakes;
Breckler, 1984), we analyzed 8 items from Study 1 of
the Crites et al. (1994) article, in which people responded to the attitude object "snakes" (see Breckler,
1984; Crites et al., 1994, for additional detail on themeasures).
We wished to use these data sets to illustrate how
different procedures in conducting EFA analyses
could lead one to either accurately or inaccurately
identify the number and nature of the common factors
underlying these data sets. We submitted the matrices
to a PCA, examining both a varimax (orthogonal) anda direct quartimin (oblique) rotation. We also submitted the same matrices to a ML factor analysis, again
examining both the varimax and direct quartimin rotations. In order to determine the number of factors,
we examined the eigenvalues from the unreduced correlation matrices applying the eigenvalues-greaterthan-1 rule. For the ML factor analysis models, we
also examined the RMSEA and ECVI indexes ofmodel fit, as well as the scree test and parallel analysis(Montanelli & Humphreys, 1976) for the eigenvalues
from the reduced correlation matrix.
The sample sizes were 138, 105, and 164, for
Breckler (1984) Studies 1 and 2 and Crites et al.
(1994) Study 1, respectively. Though the sample sizes
are a bit smaller than one might want, the sampling
error that might occur in small samples would only
work against the observation of consistent results
across data sets. Therefore, consistency across the
samples would increase confidence in the obtained
results, despite any concerns about small samples.
The initial communality estimates (squared multiple
correlations) are somewhat variable in each study (seeTable 1). For Breckler, Study I , the estimates ranged
from .61 to .25 with M = .42. For Breckler, Study 2.
the estimates ranged from .81 to .37 with M = .62.

285

USE OF FACTOR ANALYSIS
Table 1
Squared Multiple Correlations (SMCs) for Items in
Breckler (1984; Studies I and 2) and Crites, Fabrigar,
and Petn (1994; Studv I)in applied research (see later discussion). Of course,
most variables in real data will have substantial
unique variance, and such conditions should be especially likely to lead to differences between EFA(which separately represents common and uniquevariance in the model) and PC A (which does not differentiate between common and unique variance).
Unfortunately, information about the distributional
properties of the items in the Breckler (1984) data sets
was unavailable. For the Crites et al. (1994) data,
however, the skew (-0.77 to 0.73, lAfl = 0.41) andkurtosis (-1.02 to 3.28. IA/1 = 0.79) of the items were
far smaller than the recommended thresholds forquestioning the adequacy of ML estimation methods(skew > 2, kurtosis > 7; West et al., 1995). Therefore,
across the data sets we analyzed, consistency in solutions could provide a compelling illustration of how
the various decisions can influence the results for real
data.

SMCs
Item

Study 1

Study

0.43
0.25
0.33

0.54
0.37
0.51

0.53
0.37
0.56

0.73
0.81
0.77

0.38
0.61
0.35

0.50
0.76
0.58

Breckler
AffectThurstone affect
Mood Checklist (+)
Mood Checklist (-)
Behavior
Action sequence
Distance
Thurstone behavior
Cognition
Thurstone cognition
Semantic differential
Listed thoughts

Crites et al.
AffectMultiresponse Checklist
0.82
0.78
Dichctomous Checklist
Semantic differential
0.78
0.34
Thurstone affectCognition
0.85
Multiresponse Checklist
Dichotomous Checklist
0.79
Semantic differential
0.85
0.56
Thurstone cognition

Number of Factors
Table 2 provides the eigenvalues for the reduced
and unreduced correlation matrices for each data set.
As can be seen from the eigenvalues for the unreduced correlation matrix, the eigenvalues-greaterthan-1 rule suggested two-factor and one-factor models for the Breckler (1984) data sets (Studies 1 and 2,
respectively). For the Crites et al. (1994) data, the rulesuggested a one-factor model (see Table 1). Screeplots of these values, though a bit ambiguous, would
perhaps suggest one-factor models for all three studies. However, given that the goal is to identify common factors, it is more sensible to examine the screeplots of the eigenvalues from the reduced correlation
matrices. These plots would once again perhaps sug-

For Crites et al., Study 1, the estimated ranged from.85 to 34 with M = .72. However, these averagecommunalities, especially for Study 2 of Breckler(1984) and Study 1 of Crites et al. (1994), are actuallyhigher man the initial communalities that often occur

Table 2
Eigenvalues for Breckler (1984; Studies 1 and 2) and Crites, Fabrigar, and Petty (1994;
Studv I), From the Unreduced and Reduced Correlation Matrices
Factor number
Correlation matrices
Breckler Study 1
Unreduced
ReducedBreckler Study 2
Unreduced
ReducedCrites et al. Study 1
Unreduced
Reduced

3.84
3.30

1.20
0.60

0..98
0,.39

0.85
0.20

0.65
0.02

0.47
-0.10

0.41
-0.12

0.35
-0.21

0.25
-0.27

5 .59
5 .24

0.91
0.40

0,.70
0,.34

0.50
0.05

0.42
-0.04

0.34
-0.07

0.23
-0.09

0.19
-0.13

0.13
-0.14

5 .72
5.48

0.88
0.50

0,.46

0.38
0.01

0.19
-0.05

0.17
-0.06

0.12
-0.08

-0.09
-0.09

0.07

286

FABRIGAR, WEGENER, MAcCALLUM, AND STRAHAN

gest a one-factor model for the three data sets. However, on the basis of these scree plots, one might also
argue for a four-factor model in the Breckler Study 1
data set, a three-factor model in the Breckler Study 2
data set, and two-factor model in the Crites et al. data
set.
When one examines the parallel analysis performedon the reduced correlation matrix (Montanelli &Humphreys, 1976), a one-factor solution appears lessreasonable. For Breckler (1984) Study 1, the first four
eigenvalues expected for random data (0.51, 0.34,
0.25, 0.17) fall below the observed eigenvalues fromthe reduced matrix (3.30, 0.60, 0.39, 0.20). For Breckler (1984) Study 2, three of the first four eigenvalues
expected for random data (0.60, 0.40, 0.29, 0.20) fallat or below the observed eigenvalues from the reduced correlation matrix (5.24. 0.40, 0.34, 0.05) with
Table 3
Selected Fit Indexes for Analyses of Breckler (1984;
Studies I and 2) and Crites, Fabrigar, and Petty (1994;
Study 1): RMSEAs and ECVls From ML and PACE
Factor Analyses
Number of factors
Model-fitting procedure

1

Breckler Study 1 (n = 138)
RMSEA
ML
0.15
PACE
0.13
ECVI
1.04
ML
PACE
0.93
Breckler Study 2 (n = 105)
RMSEA
ML
0.14
PACE
0.14
ECVI
ML
1.16
PACE
1.11
Crites et al. Study 1 (n = 164)
RMSEA
ML
0.23
PACE
0.23
ECVI
1.41
ML
PACE
1.38

2

3

4

0.12"
0.14

0.03"
0.08

0.00"
0.25

0.79
0.87

0.58
0.65

0.59
0.97

0.09
0.09

0.00
0.00

0.00
0.00

0.85
0.85

0.68
0.69

0.76
0.76

0.06
0.06

0.02
0.04

0.42
0.41

0.40
0.41

h'ote. Noniterated principal factoring and PACE factoring procjce similar solutions with no boundary estimates. (See Footnote
12 fora description of this analysis.) RMSEA = root mean squareeiror of approximation; ECVI = expected cross-validation index;
ML = maximum likelihood; PACE = partitioned covariance estimator.
"The solutions noted included one or two boundary estimates (0) for
unique variances.

the fourth observed eigenvalue falling far below the
fourth value for random data. For Crites et al. (1994)
Study 1, two of the four eigenvalues expected for
random data (0.42, 0.27, 0.19. 0.11) fall below theobserved eigenvalues (5.48. 0.50, 0.07, 0.01).'' Thus,
the parallel analyses question the one-factor decisionthat might come from the eigenvalues-greater-than-1
rule. They suggest a three-factor solution in one of the
Breckler data sets and a two-factor solution in the
Crites et al. data set (both as expected given past data
and theory), but they suggest a four-factor solution forStudy 1 of the Breckler (1984) data.
The goodness-of-fit indexes, however, strongly
suggest the theoretically expected factor structures(see Table 3). For the Breckler (1984) data sets.
RMSEA shows poor fit with the one-factor model andpresents substantial improvement with a three-factor
model. Moreover, overall fit becomes quite good (seeBrowne & Cudeck, 1992) when the third factor isadded. There is also very little overlap in the 90%
confidence intervals (CI) for the two-factor (RMSEA= 0.12, CI = 0.09-0.16 for Study 1; RMSEA =0.09, CI = 0.04-0.14 for Study 2) and three-factor
models (RMSEA = 0.03, CI = 0.00-0.10 for Study
1; RMSEA = 0.00, CI = 0.00-0.00 for Study 2) but
complete overlap between the three- and four-factor
models (RMSEA = 0.00, CI = 0.00-0.07 for Study
1: RMSEA = 0.00, CI = 0.00-0.00 for Study 2). The
ECVI has it lowest value (i.e., the smallest point estimate for a predicted discrepancy in a new sample)
for the three-factor model. This pattern clearly suggests the three-factor model as superior to the one- or
two-factor models. For the Crites et al. (1994) data,
the expected two-factor structure is strongly suggested. The model has quite poor fit with one factor

1' As noted earlier, there have also been parallel analyses
developed for determining the number of components toretain (e.g., Longman et al., 1989). Given our interest ofdetermining the number of factors underlying the data, however, it seems most reasonable to use the parallel analysis
method designed for determining the number of common
factors (i.e., parallel analysis with squared multiple correlations in the diagonal of the correlation matrix). Using
parallel analyses for the unreduced matrix suggests extraction of a single principle component (though such a conclusion is questioned by the parallel analysis on the reduced
matrix and by the indexes of EFA model fit). Moreover, one
would want to be cautious about accepting a single factor
model in this context given that underfactoring is often
thought to be a more severe error than overfactoring.

USE OF FACTOR ANALYSIS

and acceptable fit with two factors. Moreover, there isno overlap in the 90% CIs for the one-factor (RMSEA= 0.23, CI = 0.20-0.26) and two-factor models(RMSEA = 0.06, CI = 0.00-0.11) but nearly complete overlap between the two- and three-factor models (RMSEA = 0.02, CI = 0.00-0.10). Again, ECVI
fails to improve beyond the level obtained with thetheoretically expected number of factors.
When taken together, these analyses provide a useful illustration of how nonoptimal procedures can leada researcher to misleading conclusions regarding theappropriate number of factors. The eigenvaluesgreater-i ban-1 rule consistently suggested fewer factors than past research and theory would indicate wasmost plausible for each data set. The scree test provided somewhat ambiguous information. In contrast,
parallel analysis suggested the most plausible numberof factors for two of the three data sets, and model fit
suggested the most plausible number of factors in all
three data sets. Thus, consistent with the methodological literature, a researcher clearly would have been
better served relying on parallel analysis and model fit
than the eigenvalues-greater-than-1 rule when determining the number of factors for these data.
Orthogonal Versus Oblique Rotations
Researchers are often tempted to seek "conceptually distinct" factors by conducting varimax (orthogonal) rotations in factor analyses. It is sometimes
thought that this retention of statistically independent
factors "cleans up" and clarifies solutions, making
them easier to interpret. Unfortunately, this intuitionis exact.ly the opposite of what the methodologicalliterature suggests, and the present results bear out themethodology-based conclusions. Tables 4, 5, and 6
present PCA and ML factor analysis results (the left
and right columns) using varimax and direct quartimin rotations (the top and bottom portions of thetables). For ease of examination, loadings above 0.30
are presented in bold face. We have ordered factors
similarly for each analysis (for the Breckler data sets,with the most "affective" factors first, most "behavioral" factors second, and most "cognitive" factors
third; for the Crites et al. data, with the most "affective" factor first, and most "cognitive" factor second).
For each of the studies, it would have been unfortunate for a researcher to rely solely on a varimax
rotation. Examining only those loadings above 0.30,
there are many more cross-factor loadings (i.e., items
that load above 0.30 for more than one factor) withthe orthogonal rotation. Across the six varimax rota-

287

tions reported in Tables 4, 5, and 6, there are 28 times
that an item is found to load on more than one factor(or component). However, the six direct quartimin
rotations reported in Tables 4, 5, and 6 produce only
eight cross-factor loadings. Examining the loadings
below 0.30, the varimax rotation produced only 13%
with magnitudes (ignoring sign) of less than 0.10;
42.6% had values between 0.10 and 0.20, and 44.4%
had values greater than 0.20. Using the direct quartimin rotation, 60.8% of the loadings below 0.30 had
magnitudes equal or less than 0.10; 25.3% had values
between 0.10 and 0.20, and only 13.9% had values
greater than 0.20 (despite including the 25 loadingsthat fell below 0.30 using direct quartimin but wereabove 0.30 using varimax). Therefore, it seems quite
clear that direct quartimin (the oblique rotation) provided superior simple structure.
Besides the "cleaner" solutions provided by the oblique rotation, a researcher relying on an orthogonalrotation would also forfeit any knowledge of the existing correlations among factors. Although some ofthe varimax solutions for the current data (e.g., theML factor analysis solution for Breckler, 1984, Study
1) might have led the researcher to think in terms of"affect, behavior, and cognition," he or she might
have thought that there was little reason to think of thethree factors as correlated (given the orthogonal rotation). Yet, when the direct quartimin rotation wasused on the same data, it not only produced bettersimple structure clearly breaking along affective, behavioral, and cognitive lines, but it also revealed thatthe factors were correlated in the .4-.6 range (seeTables 4, 5, and 6).
Factor Analysis Versus Principal Components
Comparing the PCA and EFA columns also provides some useful information. As noted earlier,Widaman (1993) used simulation data to show that
PCA inflates loadings when compared with factor
analysis. Widaman found that "salient" loadings (i.e.,the highest loadings on a given factor) are higher in
PCA than in factor analysis and that such inflation isgreater when the salient loadings are more moderate
in value (e.g., 0.40 in the population) rather than high(e.g., 0.80 in the population). Because factor and component analyses lead to some differences in the patterns of loadings across factors in the current data,
direct comparisons of loadings might not always bemeaningful. Yet, an overall pattern consistent with
Widaman's (1993) analyses seems to be present.

288

FABRIGAR, WEGENER, IVUcCALLUM, AND STRAHAN
Table 4
Loadings for Principal Components and Common Factors Using Varimax and Direct
Quartimin Rotations: Breckler (1984: Study I)

Rotation
Item
Varimax
AffectThurstone affect
Mood Checklist (+)
Mood Checklist (-)
Behavior
Action sequence
Distance
Thurstone behavior
Cognition
Thurstone cognition
Semantic differential
Listed thoughts
Direct quartimin
AffectThurstone affect
Mood Checklist (+)
Mood Checklist (-)
Behavior
Action sequence
Distance
Thurstone behavior
Cognition
Thurstone cognition
Semantic differential
Listed thoughts

j

2
3
Note.

Principal components
analysis (components)

Maximum likelihood
factor analysis (factors)

1

2

3

1

2

3

0.83
0.47
0.77

0.29
0.57
-0.05

0.09
-0.19
0.34

0.99
0.43
0.46

0.10
0.15
0.23

0.11
0.13
0.16

0.25
-0.05
0.19

0.66
0.80
0.66

0.37
0.25
0.46

0.24
0.15
0.28

0.86
0.49
0.60

0.20
0.34
0.40

0.03
0.12
0.26

0.11
0.39
0.17

0.78
0.77
0.68

0.10
0.17
0.23

0.17
0.26
0.27

0.58
0.95
0.47

0.82
0.40
0.81

0.18
0.57
-0.20

-0.06
-0.35
0.26

1.08
0.43
0.45

-0.14
0.04
0.13

-0.05
0.03
0.03

0.14
-0.20
0.08

0.64
0.84
0.64

0.23
0.12
0.33

0.03
-0.01
0.10

0.97
0.47
0.56

-0.12
0.20
0.21

0.00
0.04
0.23

0.04
0.34
0.08

0.78
0.71
0.63

-0.01
-0.01
0.12

-0.01
-0.04
0.13

0.62
1.02
0.43

.60

—

Correlations among factors or components
_

.32
.18

.28

—

.51
.42

Loadings in bold are values above 0.30.

If one takes all the loadings that are above 0.30 for
at least one of the analyses (PCA or ML factor analysis) and compares the magnitude of those loadings
across methods, the PCA loadings tend to be higher.
For example, the salient loadings for Component 2 ofthe varimax rotation in Study 1 (in bold on Table 4)have values of 0.57, 0.66, 0.80, 0.66, and 0.39 (M =0.62). The corresponding loadings for the ML factor
analysis (Factor 2) are 0.15, 0.86, 0.49, 0.60, and 0.26
(M = 0.47). There are 16 such comparisons acrossthe analyses presented, and 12 of the comparisonsshow higher salient loadings for components than forfactors. Although existing work has generally exam-

ined possible inflation of salient loadings, some of theinflations in the current data are "nonsalient" loadings
from factor analyses that become large enough forsome researchers to consider them salient in defining
a component. Such inflations would pose a potentiallymajor problem, because they could change the presumed nature and interpretation of the constructs under investigation. One possible instance of such inflation is the creation of a sizeable loading of the AffectDichotomous Checklist (0.57) in the PCA solution for
Crites et al. (1994) Study 1 in which the direct quartimin rotation was used (see Table 4). The same loading (of an "affect" item on a primarily "cognitive"USE OF FACTOR ANALYSIS

289

Table 5
Loadings for Principal Components and Common Factors Using Varimax and Direct
Quartimin Rotations: Breckler (1984; Study 2)

Rotation
Item
Varimax
AffectThurstone affect
Mood Checklist (+)
Mood Checklist (-)
Behavior
Action sequence
Distance
Thurstone behavior
Cognition
Thurstone cognition
Semantic differential
Listed thoughts
Direct Quartimin
AffectThurstone affect
Mood Checklist (+)
Mood Checklist (-)
Behavior
Action sequence
Distance
Thurstone behavior
Cognition
Thurstone cognition
Semantic differential
Listed thoughts

Principal components
analysis (components)

Maximum likelihood
factor analysis (factors)

1

2

3

1

2

3

0.72
0.89
0.31

0.43
0.18
0.69

0.18
0.07
0.19

0.75
0.63
0.40

0.32
0.24
0.53

0.20
0.12
0.28

0.27
0.25
0.24

0.81
0.89
0.84

0.27
0.19
0.29

0.39
0.32
0.36

0.69
0.90
0.73

0.34
0.24
0.36

0.08
0.36
0.53

0.22
0.55
0.49

0.94
0.65
0.39

0.13
0.43
0.57

0.22
0.44
0.38

0.76
0.70
0.39

0.68
0.98
0.11

0.24
-0.11
0.72

0.03
-0.04
-0.01

0.81
0.71
0.22

0.02
-0.01
0.49

0.01
-0.05
0.08

0.01
-0.05
-0.04

0.87
1.01
0.90

0.04
-0.07
0.07

0.10
-0.09
0.04

0.72
1.10
0.79

0.10
-0.10
0.11

-0.05
0.18
0.42

-0.07
0.36
0.32

1.02
0.56
0.26

-0.06
0.24
0.51

-0.01
0.19
0.14

0.84
0.63
0.25

Correlations among factors or components
.59
.35
Note.

.56

.73
.52

.64

Loadings in bold are values above 0.30.

component) was 0.09 in the factor analysis. It is clear
to see that the direct quartimin ML solutions are substantially "cleaner" than the same rotation of the PCA
solutions.
Another notable difference between the PCA and
ML solutions concerns the identified correlations
among factors. Consistent with Widaman's (1993)
simulations, the correlations identified by oblique rotation of the PCA solution were substantially lower
than the correlations identified by the same rotation of
the ML factor analysis. It makes sense that PCAs
should generally underestimate relations among theconstructs, because random error is included in thecomponents. Because factor analyses remove random

error from the factors, the relations among factors are
more likely to approach the population values. Such
differences are especially evident in analyses of
Breckler (1984) Study 1. The correlations among
components (i.e., .32, .18, and .28) were roughly half
the magnitude of the correlations among factors (i.e.,.51, .42, and .60). If a researcher had used a PCA, heor she would have been tempted to conclude that thecomponents were largely independent. However, using an EFA, the same researcher would have realizedthat the smallest of the interfactor correlations was
actually greater than 0.4. These would certainly berather different conclusions. Similar deflations of correlations occur for the PCAs of Breckler (1984) Study

290

FABRIGAR, WEGENER, MAcCALLUM, AND STRAHAN
Table 6
Loadings for Principal Components and Common Factors Using Varimax and Direct
Quartimin Rotations: Criles, Fabrigar, and Petty (1994; Study 1)
D t t-

Item
Varimax
AffectMultiresponse Checklist
Dichotomous Checklist
Semantic differential
Thurstone affectCognition
Multiresponse Checklist
Dichotomous Checklist
Semantic differential
Thurstone cognition
Direct Quartimin
AffectMultiresponse Checklist
Dichotomous Checklist
Semantic differential
Thurstone affectCognition
Multiresponse Checklist
Dichotomous Checklist
Semantic differential
Thurstone cognition

Principal components

Maximum likelihood

audi^M* ^uiiipuiicHUV

idt-iui analysis uacioisj

1

i

1

2

0.63
0.67
0.57
0.91

0.66
0.61
0.70
0.09

0.81
0.80
0.74
0.56

0.48
0.44
0.52
0.20

0.27
0.31
0.28
0.15

0.89
0.86
0.90
0.83

0.38
0.40
0.41
0.38

0.87
0.81
0.85
0.64

0.42
0.49
0.35
0.94

0.64
0.57
0.69
-0.04

0.82
0.84
0.70
0.66

0.15
0.09
0.24
-0.09

-0.04
0.01
-0.04
-0.15

0.95
0.90
0.95
0.90

-0.04
0.02
0.01
0.11

0.98
0.89
0.94
0.66

—
.77

—

Correlations among factors or components

1
2

—
.46

Note.

Loadings in bold are values above 0.30.

2 (see Table 5) and Crites et al. (1994) Study 1 (seeTable 6). l 2
The analyses of the Breckler (1984) and Crites etal. (1994) data provide examples of how questionable
12
It is worth noting that some parameter estimation probk m s were encountered in the ML analyses of the Breckler(1984) Study 1 data set. Heywood cases were encounteredf.ir the two-, three-, and four-factor models. It is perhaps not
surprising that difficulties were encountered for one of the
f lur-factor models given that this model is almost certainly
inappropriate for the present data (i.e., the model is overfactored). One common reason for encountering such estimation problems is when a misspecified model is fit to thedata. In the case of the three-factor model for Study 1, thetheoretical plausibility of the solution and the comparabilityof this solution with the three-factor model solution forStudy 2 and the two-factor solution from the Crites et al.
( ! 994) data suggest that the existence of Heywood cases
d.ies not constitute a serious problem. Nonetheless, to examine if the results of the ML analyses had produced unu-ual results as a function of severe nonnormality in the data—(which could not be tested for the Breckler data sets), we
conducted noniterated principal factor analyses for these
data. These analyses produced results very similar to thoseof the ML analyses. Additionally, the noniterated principal
factor solutions for the four-factor models confirmed that afour-factor model was in fact inappropriate for these data.
The four-factor models (as was the case for the ML analyses) failed to produce readily interpretable or plausible solutions. No estimation problems were encountered in any of
the analyses of the Crites et al. (1994) data set, and notsurprisingly the noniterated principal factor analyses ofthese data produced solutions very similar to the ML analyses. Finally, to further test the stability of the ML solutions
across all three data sets, we conducted analyses using Partitioned Covariance Matrix Estimator Factor Analysis(PACE; Cudeck. 1991). This procedure is available in the
program CEFA (Browne et al.. 1998). It is a noniterated
common factor model-fitting procedure that permits thecomputation of the same fit indexes available in ML. The
PACE analyses produced patterns of fit indexes across themodels that were comparable to those of the ML analyses.
The solutions obtained from the PACE analyses were alsoquite similar to those of the ML analyses.

USE OF FACTOR ANALYSIS

procedures can produce results that are rather confusing and misleading. For all three studies, an EFA withan oblique rotation provides much better simple structure, more interpretable results, and more theoreticallyplausible representations of the data than a PCA withan orthogonal rotation. Moreover, the oblique rotation(especially when paired with EFA rather than PCA)
provides important information concerning the relations among extracted factors. Across the example
data sets, we see that different procedures for determining the appropriate number of factors, different
analytic methods (PCA vs. EFA), and different rotation procedures all have an effect on results and interpretations.
The Use of Factor Analysis in Current
Psychological Research

One obvious question that arises from our review
and illustration is the issue of the extent to which
researchers actually use the procedures we haveshown to produce misleading results. An answer tothis question is not readily available. Although methodologists have often commented on the use of factor
analysis in psychology (e.g., Comrey, 1978; Ford etal., 1986; McNemar, 1951; Skinner, 1980; see also
Gorsuch. 1983), most of these criticisms have been
based on subjective impressions rather than systematic reviews of published applications. One exception
is a review of EFA practices reported by Ford et al.
(1986). They systematically examined applications of
EFA published between 1975 and 1984 in the area ofindustrial-organizational psychology. They concluded that factor analytic practices within this area
were generally inconsistent with the methodologicalliterature. However, their review did not fully address
study design issues, was confined to a single area of
psychology, and covered applications published 14 ormore y;:ars ago.
Assessing Current Factor Analytic Practices
To explore the question of how factor analysis iscurrently used in psychological research, we conducted a systematic review of articles published from1991 through 1995 in the Journal of Personality andSocial Psychology (JPSP) and the Journal of Applied
Psychology (JAP). We selected these journals, because they represent two areas of psychology (personality-social psychology and industrial-organizational291

psychology) in which EFA has been widely used. We
also chose these particular journals, because both areamong the most prestigious journals in their respective areas, so the articles found in these journals
should presumably reflect methodologically rigorouswork.
In our review, we examined every article publishedin these journals during the specified time period todetermine if any of the statistical analyses reported in
the article addressed an exploratory factor analytic
question. If one or more analyses of this type wasconducted, we then examined the description of eachanalysis. Studies were coded for the ratio of measured
variables to factors, average reliability (or communality) of the measured variables, sample size, examination of common factors versus principal components,
model-fitting procedure, method for determining thenumber of factors, and rotation procedure. The resultsof this review are presented in Table 7. The first twocolumns of the table indicate the number of articles
and corresponding percentage of total number of articles falling into each category for JPSP. The thirdand fourth columns present the same information for
JAP.
There are several aspects of these results that meritcommentary. First, our review indicates that EFA
continues to be an extremely popular statistical procedure in psychological research. A total of 159 of the
883 articles published in 60 issues of JPSP over a
5-year period reported EFAs. A total of 58 of 455
articles published in 30 issues of JAP over the same
5-year period reported EFAs. Thus, the typical issueof these journals contained two or three articles using
EFA.
The first section of Table 7 shows the distributionof variable to factor ratios. These results indicate thatthe great majority of analyses were based on ratios ofat least 4; 1. Thus, in most analyses, factors were probably adequately overdetermined. However, there wasa nontrivial number of articles (i.e., about one in five)with ratios below 4:1. This finding is noteworthy,
because one condition in which PCAs produce substantially inflated estimates of factor loadings is when
the ratio is 3:1 or less (Widaman, 1993).
The next section of Table 7 shows the distributionof the average reliability of measured variables inanalyses. These results show that when researchersreported the reliability of their measured variables, thevalues were generally high (i.e.. .70 or greater). One
might conclude from this finding that the psychometric properties of variables analyzed in EFA are typi-

292

FABRIGAR, WEGENER, MAcCALLUM, AND STRAHAN

Table 7
Summary Information of Current Practices in the Use of
Factor Analysis

Variable

Journal of
Personality
and Social
Psychology

Journal ofAppliedPsychology

N

%

N

%

0.6
17.6
16.4
8.8
8.2
46.5
1.3

1
9
10
10
6
18
4

1.7
15.5
17.2
17.2
10.3
31.0
6.9

1.9
3.8
20.8
20.8
8.8
44.0

2
5
9
11
9
22

3.4
8.6
15.5
19.0
15.5
37.9

18.9
27.7
15.7
8.2
29.6

8
14
9
2
25

13.8
24.1
15.5
3.4
43.1

52.8
19.5
5.0
0.6
22.0

28
13
2
0
15

48.3
22.4
3.4
0.0
25.9

15.7
15.1
0.6
0.0
1.3
2.5
22.0
1.3
41.5

11
9
0
0
4
0
12
0
22

19.0
15.5
0.0
0.0
6.9
0.0
20.7
0.0
37.9

54.7
0.6
1.3
13.2
14.5
1.9
0.6
13.2

28
1
2
9
4
2
0
12

48.3
1.7
3.4
15.5
6.9
3.4
0.0
20.7

Ratio of variable to factors
1Less than 3:1
3:1
28
4:1
26
14
5:1
6:1
13
More than 6: 1
74
2
Unknown
Average reliability of variables
Less than .60
36.60-.6933.70-7933.80-.8914.90-1.00
Unknown70
Sample size
100 or less30
44
101-200
201-30025
301^0013More than 400
47
Type of analysis
84
Principal components
Common factors
31
8Multiple methods
1
Other
Unknown
35
Factor-component number procedure
25
Eigenvalue > 1 .0
24
Scree Test
1
Parallel analysis
0
Model fit
Theory2Interpretability
4
35
Multiple methods
Other
2
Unknown
66
Factor-component rotation
Varimax
87
1
Harris-Kaiser
2
PromaxDirect quartimin
21
No rotation
23
3
Multiple methods
1
Other
21
Unknowncally quite good. However, in both journals about
40% of analyses did not include reports of the reliability of the measured variables. A large number of
the analyses in which reliabilities were not reported
involved single-item measures, whereas those forwhich reliabilities were reported involved multi-item
measures. Because single-item variables are likely to
be considerably less reliable than multi-item variables, it seems probable that many of the analyses
reported in these journals were based on variables
with less than optimal psychometric properties. Alsoof interest is the fact that we found virtually no cases
in which authors reported the communalities of theirmeasured variables. This practice is unfortunate given
that, in some ways, the communalities are more informative than the reliabilities regarding the soundness of the EFA results. However, for 18 data sets wewere able to obtain the correlation matrices on which
the authors' factor analyses were based. This allowed
us to index the communalities by examining thesquared multiple correlations associated with eachdata set. The average communality associated witheach data set ranged from .12 to .65 with the medianof these averages being .425.
The third section of Table 7 presents the distribution of sample sizes across articles. More than a thirdof the articles across the two journals conducted EFA
based on modest-to-small sample sizes (i.e., samplesof 200 or less). A somewhat smaller number of articles used moderate sample sizes (i.e., 201 to 400),
and about a third or more used large samples (i.e.,
greater than 400). Thus, there was a substantial number of articles based on sample sizes sufficiently smallthat results could have been distorted if the measured
variables included in the analysis were less than optimal or the factors were underdetermined.
The next section of Table 7 indicates the greater
popularity of PCA relative to EFA. Approximately
half of the published applications reported using the
PCA method. This method was used despite the factthat in the vast majority of these articles, the primary
goal was to identify latent constructs underlying measured variables rather than data reduction per se. In
contrast, only about 20% of analyses used some formof EFA (with some type of principal factors analysis
accounting for three out of every four articles using
common factor analysis). In approximately a fourth of
the articles in both journals, it was impossible to determine what method was used.
The next section of Table 7 reports the distributionof articles across different procedures for determining

USE OF FACTOR ANALYSIS

the number of factors. The eigenvalue-greater-than-1
rule was the most popular single procedure in bothjournals, followed closely by the scree test. Use ofother methods of determining the number of factors
was relatively rare. A fairly sizable number of articles(i.e., about 20%) reported using multiple methods. In
the overwhelming majority of these cases, multiple
methods involved some combination of the eigenvalues-greaier-than-1 rule, scree test, or factor interpretability-a priori theory. However, it was extremelycommon (i.e., about 40% of the time) for authors to
fail to clearly indicate how they arrived at their decisions as to the number of factors to include.
The final section of Table 7 provides the distribution of articles across different methods of rotation.
Varimax rotation was clearly the most commonlyused rotation with approximately half of all factor
analyses in both journals using this procedure. The
second most popular method of rotation was direct
quartimin although this rotation was used in only
about a third to a fourth as many articles as varimax
rotation. A nontrivial number of articles (i.e., 13% to21%) failed to indicate the rotation used.
When viewed in their entirety, the results reported
in Table 7 are discouraging. A fair number of analyses
were based on studies in which at least one design
feature was marginal (i.e., low measured variable tofactor ratios, modest sample sizes, or both). A sizable
number of analyses provided no information regarding the psychometric properties of the measured variables. Trie most popular types of analysis and rotation(i.e., PC A and varimax rotation) were not optimal oreven ne;;.r-optimal choices. In the case of determining
the number of factors, the largest proportion of published articles indicated that multiple methods wereused. Such an approach seems sensible in light of themethodological literature. However, exclusive use of
the largely discredited eigenvalues-greater-than-1 rule
was the second most popular choice. Thus, it appears
that the same poor choices found to be problematic in
the methodological literature and demonstrated to
produce the misleading results in our examples are
popular among researchers. Equally disconcerting, theresults in Table 7 indicated that researchers often
failed to provide adequate information about the procedures used. In 15.7% of JPSP factor analyses and22.4% of JAP factor analyses it was impossible todetermine what procedures were used for at least twoof the three major analytic decisions. In more thanhalf of all factor analyses reviewed, information was
not provided concerning at least one of the decisions.

293

Implications of Current Practices for
Psychological Research
Given the prevalence of poor decision making byresearchers in the use of EFA, many readers might
wonder what the implications of such practices are forthe interpretations and conclusions reached in thesearticles. A precise answer to this question is not possible. Although the methodological literature suggests(and the examples presented in this article illustrate)that different choices in designing studies and selecting factor analytic procedures can produce substantial
differences in results, there are clearly many cases inwhich this does not occur. The methodological literature has delineated some of the conditions underwhich differences in procedures will produce substantial differences in results. However, understanding ofthese issues is far from complete. Furthermore, inmany cases, it is impossible to know how often these
conditions exist in actual data. One method for answering this question would be to conduct comprehensive reanalyses of published data sets. Unfortunately, such an approach is not practical. Beyond thefact that such a study would require the reanalysis ofhundreds of data sets, access to data sets is very difficult. It is relatively rare for researchers to report thecorrelation matrix on which their factor analyses are
based, and requesting access to data is often unsuccessful. 13
Nonetheless, we were able to address this question
in a somewhat informal manner by reanalyzing thosedata sets for which we had access to the correlation
matrices (a total of 18 data sets). These analyses revealed a number of cases in which extracting factors
versus components, using different procedures to determine the number of factors, or using different rotation methods produced changes in results sufficiently substantial to have altered the conclusions a

11

Interestingly, when we contacted authors to request
their data, we found that a relatively small proportion ofthem agreed to our request. In the majority of cases, authors
did not answer our request even after reminder letters weresent. In other cases, authors indicated that the data were nolonger available. Finally, in a few cases, authors indicatedthat they would send the data but did not or directly refused
to send the data at all. This last situation is particularlyinteresting given that current APA ethical guidelines specify
that authors publishing in APA journals must make theirdata available to third parties for a period of 5 years following publication.

294

FABRIGAR, WEGENER. MAcCALLUM, AND STRAHAN

researcher might have reached concerning the data.
For example, we compared the solutions produced by
ML factor analysis and PCA for models with the same
number of factors retained by the authors and using arotation comparable to that used by the authors. We
found that there were differences between PCA and
ML sufficiently substantial that a researcher could
have reached a different interpretation of the factors in
5 of the 18 data sets. That is, for 5 of these data sets,
one or more measured variables changed the factors
on which they had their primary loadings across thetwo types of analysis. Interestingly, for 4 of the data
sets, it was impossible to compare the PCA and ML
solutions. The solutions could not be compared, because the authors in their original analyses retained somany components relative to the number of measured
variables that the corresponding ML factor analysis
models would have had negative degrees of freedom.
Using the same models (i.e., common factor or
principal component) and fitting procedures used by:he authors and the same number of factors retainedby the authors, we also compared varimax rotationand direct quartimin rotation. We found that direct
quartimin rotations often produced slightly bettersimple structure than varimax rotation when the faclors were correlated, but the pattern of loadings was
substantially different (i.e., there were changes in thefactors on which one or more of the measured variables had their primary loadings) in only 1 of the data
sets.
Finally, using the eigenvalues-greater-than-1 rule,-cree test, and model fit (as indexed by RMSEA), we
also examined the issue of the appropriate number offactors. In 7 of the 18 data sets, the scree test, modelTit, or both contradicted the eigenvalues-greateri.han-1 rule. Interestingly, when we examined the results of the scree test and model fit (two of the better
procedures for determining the number of factors), we
found that the results often contradicted the decisions
made by the authors in the original articles. In 3 of thedata sets, both model fit and the scree test clearlysuggested a different number of factors than that retained by the authors. In 5 of the data sets, model fit
clearly suggested a different number of factors. In 1
data set, the scree test clearly suggested an alternative
number of factors. 14
Although this small sample of data sets cannot be
regarded as fully representative of the entire applied
EFA literature, it does suggest that differences in results due to changes in EFA procedures might bemore common than many researchers realize. Addi-

tionally, our survey indicated that many researchers
made more than one poor decision. The combinationof several poor decisions is likely to produce evenmore serious distortions. For example, it was relatively common for researchers to conduct a PCA, retain as many factors as eigenvalues greater than 1, andconduct a varimax rotation. This particular "package"of decisions is especially likely to result in poor recovery of the underlying factors. IS

General Discussion
We began our article by noting that researchers
must consider five major methodological issues when
conducting a factor analysis. We argued that themethodological literature suggests that not all options
available to researchers for each of these decisions areequally sound and that poor choices in designing thestudy and conducting the factor analysis can produce
poor results. To further illustrate this point, we reanalyzed previously published data sets and showed thatthe (mis)use of EFA can produce misleading results.
We demonstrated that the misleading results obtainedin these analyses did not really reflect a fundamental
weakness in EFA but instead demonstrated the consequences of using questionable analytic procedures.
We then went on to establish that the same questionable procedures used in our examples are in fact quiteprevalent in current empirical research using factor
analysis.
We believe that our review highlights two important points that should be brought to the attention ofresearchers using EFA and to readers of articles inwhich EFAs are reported. First, contrary to whatmany researchers probably believe, the decisions in
the design of studies and in selecting factor analytic
procedures are not arbitrary and inconsequential.
There is reason to consider some design features and

14

In those cases in which model fit clearly contradicted
either the eigenvalues-greater-than-1 rule or the original decisions made by the authors but the scree test did not, it wasalmost never the case that the scree test supported the eigenvalues-greater-than-1 rule or the original decision. Instead, in these cases, the scree test was sufficiently ambiguous that it was difficult to argue that it clearly indicated an
appropriate number of factors.
1?
It is perhaps not surprising that this particular choice ofoptions is so popular. This set of decisions are the default
options for the factor analysis procedure in SPSS.

USE OF FACTOR ANALYSIS

EFA procedures to be markedly better than others.
Second, this article suggests that the quality of EFAs
reported in psychological research is routinely quite
poor. Researchers sometimes base their analyses on
studies with less than optimal features, commonly
make questionable choices when selecting analytic
procedures, and do not provide sufficient information
for readers to make informed judgements of thesoundness of the EFA being reported.
One obvious question that arises from these observations is why factor analysis is so poorly used. There
are several possible reasons. One is that researchers
are ill-informed regarding the use of EFA. Although
there is a substantial methodological literature on
EFA, much of this literature is relatively complex andpublished in quantitatively oriented journals that mostpsychologists are unlikely to read on a regular basis.
With a few exceptions (e.g., Finch & West, 1997),
nontechnical, concise, and up-to-date reviews of thisliterature have generally not been available to researchers. Furthermore, most researchers probably receive little formal training in EFA. Although it iscommon for graduate programs to require their students to take courses on the use of analysis of variance(ANOVA), it is much rarer for this to be the case forEFA (-,ee Aiken, West, Sechrest, & Reno, 1990).
Many graduate programs do not offer courses thatcover factor analysis or devote only 1 week or 2 tothis topic as part of a broad survey course covering
different multivariate statistical procedures. Therefore, it is not surprising that many researchers are
relatively uninformed regarding the implications ofstudy design features and choosing different factor
analytic procedures.
A second major reason why factor analyses arepoorly conducted is simple tradition. There is a strong
tendency for researchers to conduct analyses in amanner that is similar to what has been done before.
Researchers do so because (a) they wish for their results to be directly comparable to past studies, (b) theynaively believe that procedures must be reasonable if
so many people have used them in the past, or (c) they
feel (perhaps correctly) that the surest way to avoid
difficulties in the peer review process is to do what
has been done before.
Finally, another reason for the poor use of EFA has
to do w ith the statistical software currently popular in
psychological research. Such programs are likely toexert a tremendous influence on the way analyses are
conducted. For example, many researchers probably
follow the default options of their programs, because

295

they believe that these options would not be the defaults unless they were the most acceptable methods
currently available. Additionally, if a particular procedure is not offered in these programs, it is unlikelythat a researcher will or can even be expected to use
it. Unfortunately, the factor analytic procedures offered in the major statistical programs are far from
ideal (e.g., see MacCallum, 1983; Wood et al., 1996).
Given the inadequacies of these programs, it is not
surprising that factor analyses conducted in psychological research are often far from optimal.
Although it is relatively easy to understand why
EFA is often misused, it is more difficult to formulate
how such practices might be changed. It seems unlikely that psychology departments will be able orwilling to invest the resources necessary to substantially upgrade their statistics courses. However, methodologists could accept a greater responsibility foreducating the research community regarding the useof EFA and other statistical procedures. This would
require them not only to write highly technical papers
targeted at the quantitative methodology community(which are undeniably important) but also to write
less technical papers clearly explaining the practical
implications of this methodological research. Furthermore, editors must be willing to publish these articles
in nonquantitative journals that are likely to be readby researchers. The use of EFA might also be improved by editors of journals adopting higher standards for the manner in which factor analyses are
conducted and reported. At the very least, researchers
should be required to report what procedures theyused when conducting an EFA. Researchers should
also be expected to offer a brief rationale for theirdesign decisions and choices of EFA procedures. Finally, developers and users of EFA should more actively pressure the manufacturers of the major statistical programs to improve their products. Taken
together, these initiatives would be a good start toward improving the use of EFA in psychological research.

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Received September 24, 1997
Revision received February 3, 1999
Accepted February 10, 1999 •