Sweating the Small Stuff: Does data cleaning and testing ... - Frontiers

Kasper and ÜnlüAssumptions of factor analytic approachesWe decided to analyze released items of the PIRLS 2006 study(IEA, 2007) to have an empirical basis for the selection of skewnessvalues for ω(= ν). We used a data set of dichotomously scoredresponses of 7,899 German students to 125 test items. Figure 1displays the distribution of the PIRLS items’ (empirical) skewnessvalues. 6We decided to simulate under three conditions for the distributionsof ω. Under the first condition, ω m (m = 1, . . ., k) arenormal with µ 1m = 0, µ 2m = 1, µ 3m = 0, and µ 4m = 3. Under thesecond condition, ω m (m = 1, . . ., k) are slightly skewed withµ 1m = 0, µ 2m = 1, µ 3m = −0.20, and µ 4m = 3. Under the thirdcondition, ω m (m = 1, . . ., k) are strongly skewed with µ 1m = 0,µ 2m = 1, µ 3m = −2, and µ 4m = 9. The error terms were assumedto be unit normal, that is, we specified µ 1h = 0, µ 2h = 1, µ 3h = 0,and µ 4m = 3 for ω h (h = k + 1, . . ., k + p). Skewness and kurtosisof any z i under each of the three conditions were computedusing Mattson’s method (Section 4.1). The values are reported inTables 1 and 2 for the four and eight dimensional factor spaces,respectively.Under the slightly skewed distribution condition, the theoreticalvalues of skewness for the manifest variables range between−0.060 and −0.005, a condition that captured approximately 20%of the considered PIRLS test items. Under the strongly skewed distributioncondition, the theoretical values of skewness lie between−0.599 and −0.047, a condition that covered circa 30% of thePIRLS items (cf. Figure 1). Based on these theoretical skewnessand kurtosis statistics, we can see to what extent under these modelspecifications the distributions of the manifest variables deviatefrom the normal distribution.How to generate variates ω i (i = 1, . . ., k + p) such that theypossess predetermined moments µ 1i , µ 2i , µ 3i , and µ 4i ? To simulatevalues for ω i with predetermined moments, we used thegeneralized lambda distribution (Ramberg et al., 1979)ω i = λ 1 + uλ 3− (1 − u) λ 4λ 2,6 All figures of this paper were produced using the R statistical computing environment(R Development Core Team, 2011; www.r-project.org). The source files arefreely available from the authors.where u is uniform (0, 1), λ 1 is a location parameter, λ 2 a scaleparameter, and λ 3 and λ 4 are shape parameters. To realize thedesired distribution conditions for the simulation study (normal,slightly skewed, strongly skewed) using this general distributionits parameters λ 1 , λ 2 , λ 3 , and λ 4 had to be specified accordingly.Ramberg et al. (1979) tabulate the required values for the λ parametersfor different values of µ. In particular, for a (more orless) normal distribution with µ 1 = 0, µ 2 = 1, µ 3 = 0, and µ 4 = 3the corresponding values are λ 1 = 0, λ 2 = 0.197, λ 3 = 0.135, andλ 4 = 0.135. For a slightly skewed distribution with µ 1 = 0, µ 2 = 1,µ 3 = −0.20, and µ 4 = 3, the values are λ 1 = 0.237, λ 2 = 0.193,λ 3 = 0.167, and λ 4 = 0.107. For a strongly skewed distributionwith µ 1 = 0, µ 2 = 1, µ 3 = −2, and µ 4 = 9, the parameter valuesare given by λ 1 = 0.993,λ 2 = −0.108·10 −2 ,λ 3 = −0.108·10 −2 ,andλ 4 = −0.041·10 −3 .Remark. Indeed, various distributions are possible (see Mattson,1997); however, the generalized lambda distribution provesto be special. It performs very well in comparison to other distributions,when theoretical moments calculated according to theMattson formulae are compared to their corresponding empiricalmoments computed from data simulated under a factormodel (based on that distribution). For details, see Reinartz et al.(2002). These authors have also studied the effects of the useof different (pseudo) random number generators for realizingthe uniform distribution in such a comparison study. Out ofthree compared random number generators – RANUNI fromSAS, URAND from PRELIS, and RANDOM from Mathematica– the generator RANUNI performed relatively well or better.In this paper, we used the SAS program for our simulationstudy. 7Besides the number of factors and the distributions of thelatent variables, sample size was varied. In the small samplecase, every z i consisted of n = 200 observations, and in thelarge sample case z i contained n = 600 observations. Table 3summarizes the design of the simulation study. Overall there7 For the factor analyses in this paper, we used the SAS program and its PROC FAC-TOR implementation of the methods PCA, EFA, and PAA. More precisely, variationof the PROC FACTOR statements, run in their default settings, yields the performedprocedures PCA, EFA, and PAA (e.g., EFA if METHOD = ML).0.3density0.20.10.0−6−4−2−1.5−1−0.500.512skewness valueFIGURE 1 | Distribution of the skewness values for the 125 PIRLS test items.www.frontiersin.org March 2013 | Volume 4 | Article 109 | 128