A Model Implied Instrumental Variable Approach to Exploratory Factor Analysis (MIIV-EFA)

Abstract

Spearman (Am J Psychol 15(1):201–293, 1904. https://doi.org/10.2307/1412107) marks the birth of factor analysis. Many articles and books have extended his landmark paper in permitting multiple factors and determining the number of factors, developing ideas about simple structure and factor rotation, and distinguishing between confirmatory and exploratory factor analysis (CFA and EFA). We propose a new model implied instrumental variable (MIIV) approach to EFA that allows intercepts for the measurement equations, correlated common factors, correlated errors, standard errors of factor loadings and measurement intercepts, overidentification tests of equations, and a procedure for determining the number of factors. We also permit simpler structures by removing nonsignificant loadings. Simulations of factor analysis models with and without cross-loadings demonstrate the impressive performance of the MIIV-EFA procedure in recovering the correct number of factors and in recovering the primary and secondary loadings. For example, in nearly all replications MIIV-EFA finds the correct number of factors when N is 100 or more. Even the primary and secondary loadings of the most complex models were recovered when the sample sizes were at least 500. We discuss limitations and future research areas. Two appendices describe alternative MIIV-EFA algorithms and the sensitivity of the algorithm to cross-loadings.

Appendices

Appendices

1.1 Appendix 1: Alternative Algorithms

The algorithm we report in the text was not the first one we developed. Because it may be instructive to others who seek to develop instrumental variable EFA algorithms, we give a brief description of two other algorithms that we tried before settling on the one in the main text. We also discuss their shortcomings.

Version 1

Our earliest version of the algorithm has several steps that we summarize below:

• (1) Regress each variable on all remaining variables and calculate the R$^{\textrm{2}}$ values.

• (2) Start with a one-factor model, using the variable with the highest R$^{\textrm{2}}$ value as the scaling indicator.

• (3) Use MIIV-2SLS estimator (the MIIVsem package) to estimate the model.

• (4) Find variables with significant Sargan test statistics. If more than one variable has significant test statistics, create a new factor. If none or one variable is significant, the algorithm stops and prints the current model as the final model.

• (5) Load the variables with significant Sargan test statistics exclusively on the new factor, again use the variable with the highest R$^{\textrm{2}}$ as the scaling indicator. Repeat steps 3–5 until no more than one variable has a significant Sargan test statistic.

Fig. 8

Sensitivity for cross-loading variables.

It is not hard to see that this version is quite simple compared to the current algorithm. But with its simplicity comes limitations. One is that the variable with the highest R$^{\textrm{2}}$ might not always be the most appropriate. For example, if the data generating model is a multiple factor model with one variable cross-loaded on all factors, that variable is likely to have a very large R$^{\textrm{2}}$ value but questionable as a scaling indicator. Another limitation of this algorithm is that it only considers Sargan test statistics when checking for ‘problematic variables’ but not factor loadings. This means the algorithm does not remove nonsignificant factor loadings. Finally, this algorithm assumes factor complexity of one so that no cross-loadings are allowed, which is not very realistic for many empirical examples. It is for these reasons that we did not stay with this algorithm.

Version 2

The second version of the algorithm improved version one, and it is very close to the retained algorithm that we reported in the main text. It shares almost all steps, except that it does not include cross loadings. In fact, when no cross-loadings are allowed, this version usually ends up creating extra factors to explain these omitted cross-loading relations.

1.2 Appendix 2: Sensitivity of Cross-Loading Variables

Figure A1 shows the sensitivity for cross-loading variables—the first row is the sensitivity on the primary factor, the second row the secondary loadings on cross-loading factor(s), and the third row the overall sensitivity.

As represented in Fig. 5, sensitivity for cross-loading variables on their main factors is still high for all three significance levels at a sample size of 100. However, the overall sensitivity dropped at a sample size of 100 across all four simulations because of a lower sensitivity for cross-loadings variables on their corresponding secondary factor(s). This means that it is not that MIIVefa cannot recover any cross-loadings at small sample sizes, but that not all cross-loadings    are always correctly recovered. When dealing with empirical data, the finding for cross-loadings can be aided by substantive background knowledge that simulations do not have. It is worth mentioning that when the sample size is large, MIIVefa works very well at even recovering cross-loadings in addition to primary loadings. Note that the seeming decrease in sensitivity on secondary loadings at N$=$100 is due to the falsely high sensitivity at N$=$60. At N$=$60, as shown in Figs. 2, 3, and 4, the data structure recovery rates and overall sensitivity are both very low. However, when we look at sensitivity for a specific path/relation, it does not account for how accurately the rest of the paths were recovered. For example, at N$=$60 for simulation 5, we could have recovered a model that is very far away from the true DGM and the cross-loading variable is loaded on all factors, which would have resulted in a 100% sensitivity for that specific cross-loading variable. When the sample size increases to 100, again as shown in Figs. 2, 3, and 4, the data structure recovery rates, especially for primary loadings, and sensitivity increase as well, but not too much for secondary loadings on cross-loaded factors. This means that the algorithm is more likely to recover a model that is closer to the true DGM, although it is not uncommon for cross loading variables to be only recovered on the primary factor.