Partial Confirmatory Factor Analysis: Described and Illustrated on the NEO–PI–R

Abstract

Exploratory factor analytic (EFA) studies frequently conclude with the recommendation that future research should attempt to confirm the exploratively identified factor model solution via confirmatory factor analysis (CFA). In this article, I describe partial CFA (PCFA) as a technique to help researchers justify the recommendation of testing via CFA an EFA-derived model. Based on a Revised NEO Personality Inventory (Costa & McCrae, 1992) facet correlation matrix, the Five-factor model was examined via PCFA and was found not to be appropriate for testing via CFA. I conclude that researchers should supplement an EFA with a PCFA if there is an interest in eventually attempting to confirm the exploratively derived factor model via CFA.

Acknowledgment

Gilles Gignac is now Director of Research & Development at Genos Pty Ltd., Melbourne, Victoria, Australia.

Notes

¹ I acknowledge that there are limitations to the generalizability of the often-cited model-fit cutoff rules (Marsh, Hau, & Wen, 2004). Further, close-fit indexes are, in effect, single-unit representations of either the residual correlation/covariance matrix or the difference between the observed correlation/covariance matrix and the residual correlation/covariance matrix. For these reasons, exclusive or “blind” reliance on close-fit values in the evaluation of model acceptability should not be condoned. Instead, some detailed examination of the residual correlation matrix and the normalized covariance matrix should also be undertaken (see Gignac, 2007, for more details).

² The NFI was illustrated for didactic purposes, as it has been superseded by the CFI (Bentler, 1990).

³ See the “Wheaton” Amos example data set to learn how to input a correlation matrix into SPSS for analysis. Note as well that a PCFA does not have to be performed on an inputted correlation matrix. Raw data can be used as well. If a correlation matrix is intended to be factor analyzed within SPSS, the analysis must be performed via a syntax file with the FACTOR MATRIX = IN(COR*) command. In this study, I tested the FFM via PCFA with the following SPSS syntax:

• FACTOR MATRIX = IN(COR*)

• /VARIABLES N1 N2 N3 N4 N5 N6 E1 E2 E3 E4

• E5 E6 O1 O2 O3 O4 O5 O6 A1 A2 A3 A4 A5 A6

• C1 C2 C3 C4 C5 C6

• /MISSING LISTWISE

• /ANALYSIS N1 N2 N3 N4 N5 N6 E1 E2 E3 E4

• E5 E6 O1 O2 O3 O4 O5 O6 A1 A2 A3 A4 A5

• A6 C1 C2 C3 C4 C5 C6

• /PRINT INITIAL KMO REPR EXTRACTION

• /FORMAT SORT

• /CRITERIA FACTORS(5) ITERATE(25)

• /EXTRACTION ML

• /ROTATION VARIMAX.

⁴ SPSS calculates the percentage of variance accounted for by a factor by dividing the sum of the squared loadings by the number of variables included in the factor analysis. However, such a procedure is conceptually more congruent with a component analysis, that is, a data reduction technique that is based on the total variance of the variables. In contrast, a common factor analysis (such as MLE) is based only on common variance. Consequently, it may be argued to be more reasonable to divide the sum of the squared loadings associated with a factor by the amount of common variance associated with the variables within the correlation matrix. Note that this argument is equally applicable to standard EFA as it is to PCFA. In line with this reasoning, the total common variance associated with the 30 × 30 correlation matrix was estimated by summing the total communality (i.e., the R ² associated with regressing each facet onto the remaining 29 facets) associated with each variable across all 30 variables included in the correlation matrix. Based on such an analysis, the total common variance associated with the correlation matrix was calculated to be 14.08. Thus, the sum of squared loadings for each factor was divided by 14.08 to estimate the percentage of common variance accounted for by a factor (symbolized %S² _C in Table 2) rather than 30, which would be an estimate of the total variance accounted for by a factor (symbolized by %S² _T in Table 2) Within SPSS, this information is obtained by summing the values in the “Initial” column of the “Communalities” table that is produced by default in the MLE factor analysis.