Rotation to a Partially Specified Target Matrix in Exploratory Factor Analysis: How Many Targets?

Abstract

The purpose of this study was to explore the influence of the number of targets specified on the quality of exploratory factor analysis solutions with a complex underlying structure and incomplete substantive measurement theory. Three Monte Carlo studies were performed based on the ratio of the number of observed variables to the number of underlying factors. Within each study, communality, sample size, and the number of targets were manipulated. Outcomes included a measure of congruence and a measure of variability with regard to the rotated pattern matrix. The magnitude of the main effect for the influence of the number of targets on congruence and variability ranged from moderate to large. The magnitude of the interaction between the number of targets and level of communality ranged from small to moderate with regard to congruence and variability. Consistent with theoretical expectations, the minimum number of targets to specify to be reasonably assured of obtaining an accurate solution varied across study conditions.

Keywords:

• communality
• exploratory structural equation modeling
• Monte Carlo
• simulation
• overdetermination
• sample size

Notes

¹From this point forward all factors were assumed to be standardized.

²When Θ is squared, the diagonal values also can be interpreted as unique variances.

³MacCallum et al. (1999) also cited results from a vast literature based mainly on previous Monte Carlo experiments. This literature is not reviewed in this article for textual parsimony and because extant reviews are available (e.g., MacCallum et al., 1999).

⁴The original true common factors were orthogonal but consistent with MacCallum et al. (1999) population values taken following oblique rotation (direct quartimin rotation) to reflect common practice (Fabrigar, Wegener, MacCallum, & Strahan, 1999).

⁵In Study 3, due to problems with rotation identification, this procedure was altered slightly on occasion. For example, when h = low and t = r (i.e., 7 targets per column) specifying the lowest six pattern coefficients as targets (along with the highest pattern coefficient) in each column led to a poorly identified rotated solution. To remedy this problem, one of the six lowest pattern coefficients was respecified as a nontarget in each column and another small pattern coefficient in each column was identified as a suitable replacement. Each B is available on request.

^aThe r − 1 condition was dropped in Study 3 as detailed in Footnote 6.

⁶In Study 3, the r – 1 condition was dropped due to persistent convergence problems when N ≤ 100 and h = wide or low. For example, when N = 60 and h = wide, 0 of 1,000 requested replications were completed despite rotation identification of the population values.

⁷Consistent with Asparouhov and Muthén (2009), small values (in this case, ≤ |.08|) were counted as approximately zero.

⁸The full ANOVA table for each study and outcome is available on request from the first author.