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Kasper and ÜnlüAssumptions of factor analytic approachesdensity0.0 0.2 0.4 0.6 0.8true distributionestimated distributionnormal distribution−4 −2 0 2 4factor scoreFIGURE 8 | Distributions of the estimated (blue curve) and true (green curve) factor scores on the fourth factor under the exploratory factor analysismodel for sample size n = 1,000, factor space dimension k = 4, and true skewness µ 3 = −2. The unit normal distribution is plotted as a reference (red curve).Table 4 | Discrepancy averages and standard deviations ¯D and s, respectively.n k Model Latent variable distributionNormal Slightly skewed Strongly skewed¯D s ¯D s ¯D s200 4 PCA a 0.143 0.156 0.143 0.156 0.158 0.173EFA b 0.129 0.142 0.124 0.136 0.141 0.154PAA c 0.128 0.139 0.124 0.136 0.137 0.150600 4 PCA 0.076 0.087 0.075 0.086 0.091 0.106EFA 0.063 0.072 0.062 0.072 0.080 0.095PAA 0.066 0.075 0.064 0.074 0.082 0.096200 8 PCA 0.165 0.169 0.162 0.166 0.172 0.176EFA 0.154 0.157 0.152 0.155 0.156 0.159PAA 0.135 0.138 0.134 0.138 0.143 0.146600 8 PCA 0.119 0.123 0.118 0.123 0.125 0.130EFA 0.106 0.112 0.107 0.112 0.115 0.120PAA 0.097 0.101 0.095 0.099 0.102 0.105aPCA, principal component analysis; b EFA, exploratory factor analysis; c PAA, principal axis analysis.factor analytic procedures, even though they are performed withmetric scale indicator variables, seem not to be appropriate forestimating properties of ability in the “non-normal case.” Significanceof this result on sensitivity of factor score estimationto the nature of the latent distribution has been discussed forthe PISA study, which is an international survey with impacton education policy making and the education system in Germany(see Sections 1 and 3.1). In addition to that discussion,the classical factor analytic approaches have been examined inmore detail on PIRLS large scale assessment data, corroboratingthe results that we have obtained from the simulation study(see Section 6).A primary aim of our work is to develop some basic understandingfor how and to what extent the results of classical factoranalyses (in the present paper, PCA, EFA, and PAA) may be affectedby a non-normal latent factor score distribution. This has to be distinguishedfrom non-normality in the manifest variables, whichhas been largely studied in the literature on the factor analysis ofitems (cf. Section 3.2). In this respect, regarding the investigationof non-normal factors, the present paper is novel. However, thisis important, since it is not difficult to conceive of the possibilitythat latent variables may be skewed. Interestingly, moreoverwe have seen that a purely computational dimensionality reductionmethod can perform surprisingly well, as compared to theFrontiers in Psychology | Quantitative Psychology and Measurement March 2013 | Volume 4 | Article 109 | 137
Kasper and ÜnlüAssumptions of factor analytic approachesABCD∣∣ ∣∣FIGURE 9 | Distributions of the absolute differences ∣ˆl i;xy − l xy as a“function” of factor model and skewness of the latent abilitydistribution. Factor models are principal component analysis (PCA, orPC), exploratory factor analysis (EFA, or EA), and principal axis analysis(PAA, or PA). Numbers 1, 2, and 3 stand for normal, slightly skewed,and strongly skewed population latent ability values, respectively. Thepanels are for the different sample sizes and numbers of underlyingfactors.results obtained based on latent variable models. This observationmay possibly be coined a general research program: whether genuinestatistical approaches (originally based on variables without ameasurement error) can work well, perhaps under specific restrictionsto be explored, when latent variables are basically postulated,seemingly more closely matching the purpose of analysis.www.frontiersin.org March 2013 | Volume 4 | Article 109 | 138
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Table of Contents05 Is Data Cleanin
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OsborneAssumptions and data cleanin
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ORIGINAL RESEARCH ARTICLEpublished:
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Hoekstra et al.Why assumptions are
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García-PérezStatistical conclusio
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TressoldiPower replication unreliab
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NimonStatistical assumptionsand Del
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