Mx Output from Full Genetic Common Factor Model

PARAMETER SPECIFICATIONS
MATRIX F
             1         2         3         4
   1        13         0         0         0
   2         0        14         0         0
   3         0         0        15         0
   4         0         0         0        16
MATRIX X
             1
   1         1
   2         2
   3         3
   4         4
MATRIX Y
             1
   1         5
   2         6
   3         7
   4         8
MATRIX Z
             1
   1         9
   2        10
   3        11
   4        12

MX PARAMETER ESTIMATES
MATRIX F
             1         2         3         4
   1    46.208      .000      .000      .000
   2      .000    39.171      .000      .000
   3      .000      .000    31.522      .000
   4      .000      .000      .000    34.684
MATRIX X
             1
   1    15.088
   2    13.416
   3    13.293
   4    13.553
MATRIX Y
             1
   1     1.189
   2     5.119
   3     4.546
   4     5.230
MATRIX Z
             1
   1     4.142
   2     6.250
   3     7.146
   4     5.765

 Chi-squared fit of model >>>>>>>    46.77 
 Degrees of freedom >>>>>>>>>>>>>       56
 Probability >>>>>>>>>>>>>>>>>>>>     .806
 Akaike's Information Criterion >

Earlier in this chapter we alluded to the fact that confirmatory factor models allow one to statistically test the significance of model parameters. We can perform such a test on the present multivariate genetic model. The Mx output above shows that the shared environment factor loadings are much smaller than either the genetic or non-shared environment loadings. We can test whether these loadings are significantly different from zero by modifying slightly the Mx script to fix these parameters and then re-estimating the other model parameters. There are several possible ways in which one might modify the script to accomplish this task, but one of the easiest methods is simply to change the Y to have no free elements.

Performing this modification in the first group effectively drops all $C$ loadings from all groups because the Matrices= Group 1 statement in the second and third group equates its loadings to those in the first. Thus, the modified script represents a model in which common factors are hypothesized for genetic and non-shared environment effects to account for covariances among the observed variables, and unique effects are allowed to contribute to measurement variances. All shared environment effects are omitted from the model.

Since the modified multivariate model is a sub- or nested model of the full common factor specification, comparison of the goodness-of-fit chi-squared values provides a test of the significance of the deleted $C$ factor loadings (see Chapter ). The full model has 56 degrees of freedom and the reduced one: $2\times 8(8+1)/2 - 12 =60$ d.f. Thus, the difference chi-squared statistic for the test of $C$ loadings has $60 - 56 = 4$ degrees of freedom. As may be seen in the output fragment below, the $\chi^{2}_{60}$ of the reduced model is 51.08, and, therefore, the difference $\chi^{2}_{4}$ is $51.08 - 46.77 = 4.31$, which is non-significant at the .05 level. This non-significant chi-squared indicates that the shared environment loadings can be dropped from the multivariate genetic model without significant loss of fit; that is, the arithmetic data are not influenced by environmental effects shared by twins.

Parameter estimates from this reduced model are given below.