jung, 2013 efa with small sample sizes

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Behavioural Processes 97 (2013) 9095

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Behavioural Processes

j ournal homepage: www.elsevier .com/ locate /behavproc

Exploratory factor analysis with small sample sizes:A comparison ofthree approaches

SunhoJung

School of Management, Kyung HeeUniversity, 1 Hoegi-dong, Dongdaemon-gu, Seoul, 130-872, Republic of Korea

a r t i c l e i n f o

Article history:

Received 23 February 2012

Received in revised form 13 August 2012Accepted 29 November 2012

Keywords:

Exploratory factor analysis

Small sample size

Regularized exploratory factor analysis

Generalized exploratory factor analysis

Unweighted least-squares

a b s t r a c t

Exploratory factor analysis (EFA) has emerged in the field ofanimal behavior as a useful tool for deter-

mining and assessing latent behavioral constructs. Because the small sample size problem often occurs in

this field, a traditional approach, unweighted least squares, has been considered the most feasible choice

for EFA. Two new approaches were recently introduced in the statistical literature as viable alternatives

to EFA when sample size is small: regularized exploratory factor analysis and generalized exploratory

factor analysis. A simulation study is conducted to evaluate the relative performance of these three

approaches in terms of factor recovery under various experimental conditions of sample size, degree

of overdetermination, and level of communality. In this study, overdetermination and sample size are

the meaningful conditions in differentiating the performance ofthe three approaches in factor recovery.

Specifically, when there are a relatively large number offactors, regularized exploratory factor analysis

tends to recover the correct factor structure better than the other two approaches. Conversely, when

few factors are retained, unweighted least squares tends to recover the factor structure better. Finally,

generalized exploratory factor analysis exhibits very poor performance in factor recovery compared to

the other approaches. This tendency is particularly prominent as sample size increases. Thus, generalized

exploratory factor analysis may not be a good alternative to EFA. Regularized exploratory factor analysis

is recommended over unweighted least squares unless small expected number offactors is ensured.

2013 Elsevier B.V. All rights reserved.

1. Introduction

Exploratory factoranalysis (EFA) is a basic tool in the behavioral

sciences. Theattractionof EFAis its ability to investigatethe nature

of unobservable behavioral constructs (often called latent vari-

ables) that account for relationships among measured variables.

Traditionally, unweighted least squares (ULS) and maximum like-

lihood(ML)havebeen two of themostpopular estimation methods

in EFA (e.g., Fabrigar et al., 1999). Despite the popularity of ULS and

ML, the development of a new factor analysis procedure continues

to be actively studied (e.g., Bentler and de Leeuw, 2011). Recently,

two new approaches have been published in the statistical litera-

ture as viable alternatives to EFA in small sample size situations:One is regularized EFA (REFA) (Jung and Lee, 2011), and the other

is generalized EFA (GEFA) (Trendafilov and Unkel, 2011).

Researchers in animal behavior have long recognized the bene-

fit of studying an underlying latent variable and the importance of

EFA. However, the small sample size problem often limits the gen-

eral applicability of EFA in animal behavior studies (e.g., Budaev,

2010). Both REFA and GEFA are shown to perform well particularly

when sample sizes are small (Jung and Lee, 2011; Trendafilov and

E-mail address: [email protected]

Unkel, 2011). Thus, use of these approaches can be valuable in ani-

mal behavior research. In fact, our review found one instance of

an application of REFA in the study of animal personality (Konecn

et al., in press). However, the new approaches may still be novel to

researchers in animal behavior. Moreover, previous research has

not compared the performance of the new approaches to that of

traditional approach. It would be useful to know under what cir-

cumstances the new approaches are preferable to the traditional

approach. The objective of this article, therefore, is to make the

new approaches more accessible to researchers in animal behav-

ior, and to evaluate the traditional and new approaches in terms of

factor recovery capability using a Monte Carlo simulation study.

The article is organized as follows. We offer a general overviewpointing to the relative strengths and weaknesses of the traditional

andnew approaches. Then, we describethedesignofthesimulation

study and report results. An empirical study is conducted as well,

to investigate factor recovery in a more realistic context. Finally,

we discuss the implications of the study and also offer recommen-

dations based on the factor recovery capability.

2. Background

In animal behavior research, small sample sizes may be rule

rather than the exception (e.g., Havstad and Olson-Rutz, 1991). The

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S. Jung / Behavioural Processes97 (2013) 9095 91

potential application of EFA in this research even involves those

cases where the number of variables exceeds the sample size (e.g.,

Liu and Gershenfeld, 2003). In general, MLis most frequently rec-

ommended for use in EFA (e.g., Fabrigar et al., 1999). However,

ML requires large sample sizes to yield reliable and stable solu-

tions (e.g., MacCallum et al., 1999). Previous research has shown

that ULS can yield high quality solutions with small sample sizes

(below 50) under ideal conditions of few factors, highly reliable

data, andhigh communalities(e.g., Preacherand MacCallum, 2002).

Although these conditions often fail in practice, ULS has been the

most feasible option for EFA in animal behavior research (Budaev,

2010). Here, only ULS is considered the traditional approach (here-

after referred to as ULSFA).

ULSFA, REFA, and GEFA are based on a common factor model.

This model partitions variance of each measured variable into

two parts: unique variance (uniqueness) due to a unique factor

and common variance (communality) due to the common factors.

Unique factors represent latent variables specific to only one mea-

sured variable, whereas common factors are shared among two

or more measured variables and represent latent variables that

can explain relationships among measured variables. Thus, factor

analysis concerns only common variance to derive the patterns of

relations between the commonfactors and the measured variables

(i.e., factor loadings). However, this requires the computation ofunique variances in some way. Broadly speaking, REFA and GEFA

differ from ULSFA in that estimation of unique variances is sep-

arated from estimation of factor loadings. In REFA, only a single

parameter needs to be determined to obtain estimates of unique

variances. GEFA first derives unique factor scores and then uses

them to estimate unique variances.

Compared to ULSFA, REFA and GEFA have several advantages.

ULSFA relies on complicated iterative algorithm, whereas REFA

and GEFA simplify the estimation of parameters in the model.

REFA assumes that the true unique variances are proportional

to their tentative estimates. That is, REFA involves estimating a

single parameter representing proportion. Unique variances are

estimated by multiplying their tentative quantities by an esti-

mate of the proportion. After unique variance estimates have beenobtained, factor loadings are then computed by a closed-form for-

mula (e.g., Jreskog, 1977). On the other hand, GEFA first derives

the common and unique factor scores for each observation. For

these latent variable scores, factor loadings and unique variances

are then found respectively using separate linear regression anal-

yses.

Another advantage results from the piece-by-piece approach

that REFA and GEFA take to estimate parameters in the model.

Data sets with small samples and many measured variables are

common in behavioral research. With many parameters relative

to sample size, iterative numerical methods would be feasible but

computationally demanding. The implementation of these meth-

ods may present numerical difficulties (e.g., nonconvergence). To

deal with this problem, methodologists have advocated subdivid-ing the parameters into several subsets (e.g., Bollen, 1996).

However, compared to ULSFA, REFA and GEFA also have several

limitations.For EFA, an overall model fit index can be used in deter-

mininghowwellthe model fits a sampleas a whole. ULSFAprovides

indices to assess the models goodness of fit and also permits sta-

tistical significance testing (Browne, 1984). However, no overall fit

indices are available for these two new approaches. Another limi-

tation is the absence of software programs for implementation of

the new approaches. The lack of statistical software could make

applied researchers less motivated to understand and implement

the new approaches. In contrast, ULSFA are readily implemented

and available in many software programs, including SPSS, SAS,

Mplus (Muthn and Muthn, 2006), CEFA (Browne et al., 1998),

FACTOR (Lorenzo-Seva and Ferrando, 2006).

Beyond the theoretical characteristics among the three

approaches we have discussed, theutility of a factoranalysis proce-

dure depends heavily on its ability to recover the underlying factor

structure, which is required for good interpretation and inference.

In thenext section,we present a simulation studythat evaluatesthe

performance of REFA and GEFA relative to ULSFA in terms of factor

recovery. Based on our findings, we provide guidelines in the Dis-

cussion and recommendations section for choosing between the

three approaches.

3. Simulation design

In the present simulation study, we investigate the capability

of the three approaches in population factor recovery under sev-

eral experimental conditions of sample size, communality, and

overdetermination. These conditions are commonly encountered

in simulations based on exploratory factor analysis (e.g., Hogarty

et al., 2005). Prior research has shown that level of commu-

nality moderates the impact of sample size on factor recovery

(e.g., MacCallum et al., 1999). Degree of overdetermination was

shown to greatly influence the quality of factor solutions, partic-

ularly with small samples (e.g., Preacher and MacCallum, 2002).

Overdetermination is defined as the ratio of factors to measured

variables. If the number of observed variables (p) is held con-stant, varying the number of factors (f) leads to different degrees

of overdetermination. Fewer factors indicate higher degree of

overdetermination. Withp held constant (p= 20), the Monte Carlosimulations involved manipulating four experimental conditions

as follows.

1 Approach: REFA, GEFA, and ULSFA.

2 Sample Size: N=5, 10,20, 30,and 50.

3 Communality: High (0.60.7), Low (0.20.4), and Wide (0.20.8).

4 Overdetermination: High (f=3) and Low (f=7).

Using a method developedby Tuckeret al.(1969), six population

correlation matrices were generated to correspond to the six com-

binations of communality and overdetermination. Individual-levelmultivariate normal data were drawn from N(0,), where is the

population covariance matrix, using the Cholesky decomposition

(Wijsman, 1959). One hundred samples were generated for each

level of the experimental conditions resulting in 3000 samples for

each approach (5 Sample Sizes3 levels of communality2 degrees

of overdetermination100 Replications). All sample matrices were

factor analyzed using each of the three approaches. All simulations

were carried out using MATLAB (R2010a).

4. Simulation results

To assess the degree of factor pattern recovery under the

three approaches, we computed the root mean squared deviation

(denoted asg) (Velicer and Fava, 1998) as follows:

g=

trace

( )

( )

pf

1/2

. (1)

This index represents the average difference between sample

loadings andtheir correspondingpopulation loadings. In thisstudy,

thevalueofgwascomputed after the sign of loading estimates was

recovered usinga procedure suggested by Cliff (1966). Because this

value can be affected by the rotational indeterminacy, the loadings

produced by each of the three approaches were not submitted to

the rotation method.

An ANOVA test was performed using the root mean squared

deviation g as the dependent variable and the four experimental


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92 S. Jung / Behavioural Processes97 (2013) 9095

Table 1

The results of an ANOVA testfor the root mean squared deviation (g) (Sig.= significance, 2 = effect size).

Source SS DF F Sig. 2

Approach (A) 1,020,833 2 23806 0.00 0.84

Communality (C) 1205 2 28 0.00 0.01

Overdetermination (O) 25576 1 1192 0.00 0.12

Sample size (N) 56418 4 657 0.00 0.23

A*C 3044 4 35 0.00 0.02

A*O 33353 2 111 0.00 0.15

A*N 270325 8 1576 0.00 0.59C*O 1582 2 36 0.00 0.01

C*N 2711 8 15 0.00 0.01

O*N 1530 4 17 0.00 0.01

A*C*O 3005 4 35 0.00 0.02

A*C*N 6902 16 20 0.00 0.04

A*O*N 7500 8 43 0.00 0.04

C*O*N 2266 8 13 0.00 0.01

A*C*O*N 2266 16 6 0.00 0.01

Error 190,236 8910

conditions as design factors. Table 1 presents the results of the

ANOVA. As the table shows, all of the main and interaction effects

werestatistically significant.This should notbe toosurprising given

the large numberof observations (i.e.,a total of 9000 observations).

Thus,it is crucial toexaminethe effectsizeas well (e.g.,Paxton et al.,

2001). Following accepted practice, we focus on main and interac-tion effects whose sizes were at least medium (i.e.,2 greater than0.06) (Cohen, 1988).

First, approach (2 = 0.84), overdetermination (2 = 0.12), andsample size (2 =0.23) had large main effects. Here, approachwas the most important determinant of the root mean squared

deviation, followed in descending order by sample size and

overdetermination. There are likely meaningful differences in the

root mean squared deviation among the three approaches. It

appears that there was little difference in the root mean squared

deviation between REFA (7.2) and ULSFA (7.3), whereas GEFA

exhibited a much higher level of the root mean squared devia-

tion (29.9). Moreover, it is likely that the levels of the root mean

squared deviation were higher when the number of factors was

small (16.5)than when the number of factors was large (13.1). Thisresult is somewhat counterintuitive and needs some explanation.

In the study, the large number of factors (f= 7) has the larger num-

berof estimatedparameters thanthe small number of factors (f=3):

the former has 140 unknown elements, and the latter has 60. Note

that the number of unknown elements is used as denominator to

calculate the root mean squared deviation.

Second, the ANOVA test reveals two two-way interactions

worthy of examination: that between approach and sample size

(2 =0.58) and that between approach and overdetermination(2 = 0.15). Fig. 1 displays the average values of the root meansquared deviation of the three approaches across sample sizes. On

average, the loading estimates of ULSFA involved a little smaller

values ofg than those under REFA when N 10, whereas REFA

resulted in the smallest value ofg at N=5. GEFA exhibited thelargest values ofgacross all sample sizes. Fig. 1 demonstrates thepoor factor recovery of GEFA particularly for larger sample sizes. In

fact, GEFA had much higher values ofgwith N= 50 than withN= 5.Conversely, REFA and ULSFA performed better in factor recovery

for larger sample sizes. Fig. 2 displays the average values of the

root mean squared deviationof the threeapproachesunder thetwo

levels of overdetermination. When the number of factors is small

(i.e.,f= 3), ULSFA was associated with the smallest level of the root

mean squared deviation (7.3) and REFA was associated with the

second smallest (7.9). Conversely, when the number of factors is

large, REFA involved the smallest level of the root mean squared

deviation (6.6) and ULSFA had the second smallest (7.2). At both

levels of overdetermination, GEFA yielded the largest level of the

root mean squared deviation.

The above ANOVA test suggested that factor recovery of the

three approaches was distinct between the two levels of overde-

termination. To gain a greater understanding of factor recovery

capability of the three approaches under this condition, we further

investigated the overall finite-sample properties of the estimated

loadings of the three approaches across the overdetermination lev-els. Tables 2 and 3 presents the average relative biases, standard

deviations, and mean square errors of factor loading estimates

obtained from thethree approaches under the two levelsof overde-

termination. Among these properties, the mean square error (MSE)

is the average squared difference between a parameter and its esti-

mate, therebyindicatinghow faran estimate is,on average,fromits

parameter value, i.e., the smaller the mean square error, the closer

the estimate is to the parameter. Specifically, the MSE is given by

MSE(j) = E[(j j)2

] = E[(j E(j))2

] + (E(j) j)2. (2)

As shown in Eq. (2), the MSE of an estimate is the sum of its

variance and squared bias. Thus, the MSE accounts for both bias

and variability of the estimate (Mood et al., 1974).

Fig.1. Theaveragevaluesfor indexof rootmeansquareddeviation(g) oftheloading

estimates obtained from the three approaches across different sample sizes.


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Table 2

Overall finite-sample properties of the loading estimates of the three approaches under high overdetermination (f = 3) acrossdifferent sample sizes (RB= relative bias (%),

SD = standard deviation, MSE =mean square error).

GEFA ULSFA REFA

RB SD MSE RB SD MSE RB SD MSE

N= 5 90 0.91 0.91 4 0.46 0.21 8 0.44 0.20

N= 10 195 1.04 1.41 6 0.34 0.11 4 0.34 0.12

N= 20 304 1.12 2.21 9 0.25 0.06 10 0.25 0.06

N= 30 419 1.13 2.95 3 0.20 0.04 2 0.21 0.04

N= 50 608 1.10 4.54 1 0.15 0.02 2 0.17 0.03

Table 3

Overall finite-sample properties of the loading estimates of the three approaches under low overdetermination (f = 7) across different sample sizes (RB= relative bias (%),

SD = standard deviation, MSE =mean square error).

GEFA ULSFA REFA

RB SD MSE RB SD MSE RB SD MSE

N= 5 58 0.62 0.44 20 0.35 0.13 11 0.31 0.09

N=10 217 0.82 0.93 11 0.32 0.11 4 0.27 0.07

N= 20 449 0.97 1.58 25 0.26 0.07 16 0.21 0.04

N= 30 498 1.05 2.12 8 0.24 0.06 7 0.19 0.03

N= 50 486 1.14 3.14 28 0.20 0.04 16 0.15 0.02

In the study,absolutevalues of relative bias greater than 10 per-cent may be singled out as unacceptable (Lei, 2009). As shown in

Table 2, when few factors are retained, both REFA and ULSFA on

average yielded unbiased estimates of loadings across all sample

sizes whereas GEFA yielded tremendously and positively biased

loading estimates regardless of sample sizes. In addition to relative

bias, it is important to evaluatethe stabilityof the three approaches.

The stability of each approach was examined by computing the

standard deviation. REFA produced on average the standard devia-

tions of the estimates almost identical to those of ULSFA. As sample

size increased, the standard deviations of the estimates tended to

decrease across these two approaches. Conversely, GEFA was asso-

ciated with much larger standard deviations, and the degree of

Fig.2. Theaveragevaluesfor indexof rootmeansquareddeviation(g)oftheloading

estimates obtained from the three approaches acrosstwo levels of overdetermina-

tion (3 factors= high overdetermination, 7 factors= lower overdetermination).

standard deviations tended to increase with larger sample sizes.Overall,ULSFA showedthe smallest mean squareerrors of the load-

ingestimates. However, thedifferences in the mean squared errors

of the estimates were negligibly small between ULSFA and REFA.

On the other hand, GEFA clearly exhibits its inconsistency: larger

sample size does not help produce the loading estimates closer to

the true population values.

As Table 3 shows, when a large number of factors are retained,

REFA was theleast biasedof the three approaches.REFAled totoler-

able degreeof bias forestimates ofloadings acrosssamplesizes. The

estimates under REFA were consistently associated with smaller

standard deviations than those under the other approaches. On

average, REFA has the smallest mean squareerrors of factorloading

estimates across all ample sizes. In particular, this tendency of fac-

tor recoverywas more apparentin the two smallest sample sizes. Ingeneral, under low overdetermination, REFA led to more accurate

and stable solutions than the other approaches.

5. Empirical example

The simulation study presented in the previous section com-

pared the performance of the three approaches under various

experimental conditions commonly encountered in behavioral

research. Nonetheless, the range of conditions in this study may

still be limited in scope. Questions are often raised regarding the

validity of results from a Monte Carlo simulation study. In this sec-

tion, we provide concrete empirical evidence supporting results of

the simulation study.An empirical example is presented using the sensory attribute

data collected by quantitative evaluation of fluid milk using a con-

sumer panel (Chapman et al., 2001). The data set consisted of nine

milkproducts describedby eight sensory attributes (e.g., aroma, fla-

vor, texture, etc.).Twelve sensory panelists were asked to rate each

of the products on each attribute, using an intensity scale ranging

from 0 (not present) to 10 (extremely strong). Because the attribute

rating data collected from multiple panelists often reveal a lack of

agreement, the attributeratingsof the panelists are aggregated into

a singlegroup compositevalue. Thearithmeticmeanof theindivid-

ual ratings of the panelists was computed to aggregate the ratings

(see Chapman et al. (2001), pp. 1415). The aggregation of response

data helps minimize individual-level biases and errors, and thus

improves estimation accuracy (Van Bruggen et al., 2002).


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94 S. Jung / Behavioural Processes97 (2013) 9095

Table 4

Factor loadings and uniquevariancesfor thefour factors from thesensory attribute data (F= factors, = unique variances).

Attributes REFAa ULSFA

Fl F2 F3 F4 Fl F2 F3 F4

Cooked aroma 0.965 0.001 0.027 0.228 0.230 0.979 0.008 0.040 0.209 0.000

Caramel aroma 0.476 0.529 0.615 0.222 0.262 0.524 0.576 0.556 0.292 0.151

Grainy aroma 0.949 0.015 0.251 0.024 0.251 0.975 0.007 0.224 0.021 0.035

Cooked flavor 0.681 0.556 0.090 0.383 0.268 0.727 0.591 0.073 0.341 0.144

Sweet flavor 0.030 0.107 0.865 0.192 0.403 0.046 0.081 0.987 0.147 0.000

Bitter flavor 0.189 0.008 0.242 0.815 0.433 0.187 0.008 0.191 0.965 0.000

Dry texture 0.007 0.888 0.094 0.102 0.371 0.022 0.992 0.094 0.083 0.186

Lingering aftertaste 0.002 0.740 0.332 0.422 0.358 0.023 0.813 0.424 0.398 0.264

a Factornames: F1: Cooked; F2: Drying/lingering; F3: Sweet;F4: Bitter.

Kaisers criterion suggested a four factor solution. To facilitate

interpretation of the results, the factors were then orthogonally

rotated using a varimax rotation. GEFA was not considered for the

study due to its worst performance on factor recovery across all

experimental conditions. In this empirical study, REFA and ULSFA

were applied to the mean attribute ratings. If Heywood cases

(i.e., negative variance estimates) occur in the iteration process,

ULSFA set communalities to one (e.g., Dillon et al., 1987). The

results of these analyses are presented in Table 4. The varimax

rotated factor loadings produced by ULSFA were almost consis-tently larger than the corresponding REFA solutions. This was due

to the existence of zero entries in the estimated unique variances

under ULSFA. Some researchers assume that the Heywood case

variable is explained entirely by the corresponding common factor.

However, this assumption is unrealistic because of the presence of

measurement error in behavioral research. It appears that ULSFA

may overestimate factor loadings and therefore pose a more diffi-

cult interpretation problem. Results from ULSFA indicated that an

attribute of caramel aroma may cross-load (i.e., having large fac-

tor loadings on the second and third factors). Compared to ULSFA,

REFA offered greater conceptual clarity on the extracted factors.

6. Discussion and recommendations

In this article, we undertook investigations of the relative

performance of regularized exploratory factor analysis and gener-

alized exploratory factor analysis with respect to the more familiar

unweighted least squares in exploratory factor analysis using both

simulated and empirical data. Based on our simulation results, we

would provide some recommendations for the applied researcher

in animal behavior.

First, we recommend the adoption of regularized exploratory

factor analysis as a sensible alternative to generalized exploratory

factor analysis, a technique specifically designed to derive fac-

tors when the number of variables is larger than the number

of observations. As we demonstrate, regularized exploratory fac-

tor analysis performed much better than generalized exploratory

factor analysis in terms of factor recovery, under small sample sit-uations. Moreover, the loading estimates produced by generalized

exploratory factor analysis never converge to the true population

values as sample size increases(i.e., the estimates are inconsistent).

Second, if the number of expected factors is small, we rec-

ommend the use of unweighted least squares. On average, this

approachperforms better thanor as wellas regularized exploratory

factor analysis in terms of factor recovery. The relatively good per-

formanceof unweighted leastsquares in situations involving fewer

underlying factors has been previously reported in the literature

(e.g., Preacher and MacCallum, 2002).

Finally, if researchers have little confidence thatthe true num-

ber of factors is small, they should use regularized exploratory

factor analysis because it outperforms unweighted least squares in

factor recovery when there are a relatively large number of factors.

Under the same situation, all three approaches yielded biased esti-

mates of factor loadings. However, regularized exploratory factor

analysis had (acceptably) lower levels of relative bias compared to

the other two approaches. Moreover, regularized exploratory fac-

tor analysis resulted in more accurate estimates of loadings. The

high performance level of regularized exploratory factor analysis

was more prominent in very small samples.

Researchers routinely rely on factor retention criteria such as

parallel analysis (Horn, 1965) in order to determine the number of

factors to retain unless they have an a priori hypothesis about thenumberof factors to extract. Yet, no extantmethod forfactor reten-

tion decisionscan avoiderroneous conclusions.In addition, it is not

straightforward to assess the degree of under- and overdetermina-

tion of the number of factors. Other things being equal, retaining

a small number of factors is deemed preferable because retaining

too many often hinders interpretability. In most cases, however,

retainingtoo fewfactors maytend to reduce communalities, which

in turn has adverse effect on factor recovery in small samples (e.g.,

MacCallum et al., 1999). Unless researchers have tremendous con-

fidence that the correct number of factors is small, we suggest that

theyconsider regularized exploratory factor analysisa complement

or substitute forunweighted least squares under small samplesize.

Our study provides a greater understanding of the three avail-

able approaches to exploratory factor analysis with small samplesizes. We hope that this study leads researchers in animal behavior

to adoptregularized exploratory factor analysis in manysituations,

particularly those in which the researchers should expect to retain

a relative large number of factors. In many research domains, it

is often unavoidable to have data with many measured variables

and small sample size; some specific examples include behavior

genetics and neuroimaging data analysis, among others. In these

domains, use of regularized exploratory factor analysis can also be

valuable.

As doothersimulationstudies, thepresentstudyis notfreefrom

limitation. Althoughthe simulation studytook intoaccount various

experimental conditions that are frequently used in Monte Carlo

simulation studies in the domain of exploratory factor analysis, it

may be necessary to contemplate a wider range of conditions formore careful investigations of the relative performance of the three

approaches.

Appendix A. Supplementary data

Supplementary data associated with this article can be

found, in the online version, at http://dx.doi.org/10.1016/j.beproc.

2012.11.016.

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jung, 2013 efa with small sample sizes

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