Appendix 9. On the indeterminacy of the general factor when based on two factors.
In this appendix, we examine whether the rotation toward a general factor was an artifact related
the particular constraints we imposed to solve the indeterminacy involved in a Schmid-Leiman
rotation when based on two factors. This rotation involves three steps. First, the extracted
factors are rotated toward oblique (correlated) simple structure. Second, a so-called higher-order
factor is extracted from the factor correlation matrix. Third, the higher-order factor is regressed
onto the variables through matrix transformation.
When the original solution consists of two factors, the identification of the higher-order factor
(step 2) is underdetermined because it involves estimating two loadings based on one factor
correlation (the two genetic factors correlated at r = .45 based on the Geomin rotation). The
common solution to this problem, which we applied, is to fix the two loadings to be identical at
[sqrt(r), sqrt(r)], where r equals the factor correlation. This way, one only has to estimate a
single loading such that the solution becomes determined. However, a number of other possible
values, ranging from [1,r] to [r,1], can also solve the higher-order factor. Below (Table 1) we
display the corresponding general factor loadings when based on these two extreme solutions to
the higher-order factor indeterminacy (i.e., [1,r] to [r,1]). Even when based on these two most
extreme solutions to the interminacy, all but one loading on the general factor were quite large.
Table 1. Loadings on the general genetic factor based on the two extreme solutions.
General genetic factor
Diagnosis
Based on (1, r)
Based on (r, 1)
Schizophrenia
0.81
0.45
Schizoaffective
0.87
0.33
Bipolar
0.57
0.32
Depression
0.34
0.42
Anxiety
0.29
0.54
ADHD
0.13
0.55
Drug
0.41
0.89
Alcohol
0.32
0.69
Crime
0.29
0.64
Note. Loadings equal to or greater than .30 are bolded.
Note, however, that these two loading patterns are based on the two most extreme solutions to
the indeterminacy. To explore how similar our solution was to the entire range of all possible
values that could solve the indeterminacy, we compared solutions using the factor congruence
coefficient, which is a measure of factor loading similarity that can be interpreted as a correlation
coefficient, that is, it ranges from -1 to 1. We computed this index between the general factor
based on our solution of fixing the two higher-order loadings to be equal at [sqrt(r), sqrt(r)], and
all the other possible values the general factor could take on, ranging from [r,1] to [sqrt(r),
sqrt(r)] to [1,r]. As can be seen below, the factor similarity between our general factor solution
and all other possible general factor solutions remained very high (mean congruence coefficient
= .98; SD = .02; min = .94). In other words, no matter how this indeterminacy was resolved, the
general factor appeared about the same.
In addition, we computed the factor congruence coefficient between the general factor based on
the [sqrt(r), sqrt(r)] solution, and the general factor based on three factors (as displayed in
Appendix 7). The advantage of the latter is that it did not suffer from an indeterminacy problem
because it estimated three higher-order factors based on three factor correlations. The
congruence coefficient equaled .96. In other words, the general factor based on a constrained
solution is highly similar to a general factor based on an unconstrained solution. Overall, we
view these analyses as evidence that the general factor is not an artifact related to the specific
constraints of our rotation.