Influences on factorial accuracy : a Monte Carlo investigation

The correct numbers of factors in these 27 population correlation matrices were extracted via the principal axes factoring procedure using the true population commonalities in the diagonals.

The correct numbers of factors were rotated via the normalized Varimax procedure.

Sixteen hundred and twenty sample correlation matrices with different ratios of observations per variable were randomly generated from the 27 population correlation matrices.

These 1,620 sample correlation matrices were factored via the principal axes procedure (no iterations) using squared multiple correla tions, when the inverse of the matrix existed, as communality estimates; otherwise the absolute value of the highest off diagonal correlation was used. Factoring was carried to the point where the subsequent eigenvalue was less than or equal to zero, or to the value of n/2, where n is the number of variables, whichever was smaller.

Four separate Varimax rotations were performed to the following criteria:

A. Condition I(--rotate one less factor in each sample than in the corresponding population.

B. Condition II--rotate the same number of factors in each sample that is contained in the corresponding population.

C. Condition III--rotate one more factor in each sample than in the corresponding population.

D. Condition IV--rotate the numbers of factors as determined by the scree tests.

The congruence coefficients and the means square errors between factor loadings on the rotated population factors and factor loadings on the corresponding rotated sample factors were computed. Corresponding factors were defined as those sample factors which had the highest congruence coefficient with the population factors. If a congruence coefficient were negative then all sample loadings on that factor were reflected and the congruence coefficient and mean square error recomputed.

The dependent variables (congruence coefficients or mean square errors) for each factor were separately regressed on the independent variables (number of factors, proportion of common variance, etc.) and their interactions under each of the four conditions previously discussed.

Mean accuracy indices were compared via dependent sample t tests for the conditions of overfactoring versus underfactoring. Results indicate that far fewer observations are needed than were previously thought, and that overfactoring is superior to underfactoring; but, the issue is moot without more precise ways of deter mining the correct number of factors, and then there would be no logic to overfactoring.

When there are few factors, many variables, and a high proportion of common variance in the population, samples can be based on fewer observations than variables and still be highly replicable.