E From the menus, choose View → Analysis Properties. E In the Analysis Properties dialog box, click the Output tab.
E Select Modification indices and enter a suitable threshold in the field to its right. For this example, the threshold will remain at its default value of 4.
Chi-square = 33.215 Degrees of freedom = 3 Probability level = 0.000
Requesting modification indices with a threshold of 4 produces the following additional output:
According to the first modification index in the M.I. column, the chi-square statistic will decrease by at least 13.161 if the unique variables eps2 and eps4 are allowed to be correlated (the actual decrease may be greater). At the same time, of course, the number of degrees of freedom will drop by 1 because of the extra parameter that will have to be estimated. Since 13.161 is the largest modification index, we should consider it first and ask whether it is reasonable to think that eps2 and eps4 might be correlated.
Eps2 represents whatever pre_opp measures other than verbal ability at the pretest.
Similarly, eps4 represents whatever post_opp measures other than verbal ability at the posttest. It is plausible that some stable trait or ability other than verbal ability is measured on both administrations of the Opposites test. If so, then you would expect a positive correlation between eps2 and eps4. In fact, the expected parameter change (the number in the Par Change column) associated with the covariance between eps2 and eps4 is positive, which indicates that the covariance will probably have a positive estimate if the covariance is not fixed at 0.
It might be added that the same reasoning that suggests allowing eps2 and eps4 to be correlated applies almost as well to eps1 and eps3, whose covariance also has a fairly large modification index. For now, however, we will add only one parameter to Model A: the covariance between eps2 and eps4. We call this new model Model B.
Model B for the Olsson Data
Below is the path diagram for Model B. It can be obtained by taking the path diagram for Model A and adding a double-headed arrow connecting eps2 and eps4. This path diagram is saved in the file Ex09-b.amw.
Modification Indices (Group number 1 - Default model) Covariances: (Group number 1 - Default model)
M.I. Par Change
eps2 <--> eps4 13.161 3.249 eps2 <--> eps3 10.813 -2.822 eps1 <--> eps4 11.968 -3.228 eps1 <--> eps3 9.788 2.798
You may find your error variables already positioned at the top of the path diagram, with no room to draw the double-headed arrow. To fix the problem:
E From the menus, choose Edit → Fit to Page. Alternatively, you can:
E Draw the double-headed arrow and, if it is out of bounds, click the Resize (page with arrows) button. Amos will shrink your path diagram to fit within the page boundaries.
Results for Model B
Allowing eps2 and eps4 to be correlated results in a dramatic reduction of the chi-square statistic.
You may recall from the results of Model A that the modification index for the covariance between eps1 and eps3 was 9.788. Clearly, freeing that covariance in addition to the covariance between eps2 and eps4 covariance would not have produced an additional drop in the chi-square statistic of 9.788, since this would imply a negative chi-square statistic. Thus, a modification index represents the minimal drop in the
Chi-square = 2.684
chi-square statistic that will occur if the corresponding constraint—and only that constraint—is removed.
The following raw parameter estimates are difficult to interpret because they would have been different if the identification constraints had been different:
As expected, the covariance between eps2 and eps4 is positive. The most interesting result that appears along with the parameter estimates is the critical ratio for the effect of treatment on post_verbal. This critical ratio shows that treatment has a highly significant effect on post_verbal. We will shortly obtain a better test of the significance of this effect by modifying Model B so that this regression weight is fixed at 0. In the meantime, here are the standardized estimates and the squared multiple correlations as displayed by Amos Graphics:
Regression Weights: (Group number 1 - Default model)
Estimate S.E. C.R. P Label
post_verbal <--- pre_verbal .889 .053 16.900 ***
post_verbal <--- treatment 3.640 .477 7.625 ***
pre_syn <--- pre_verbal 1.000
pre_opp <--- pre_verbal .881 .053 16.606 ***
post_syn <--- post_verbal 1.000
post_opp <--- post_verbal .906 .053 16.948 ***
Covariances: (Group number 1 - Default model)
Estimate S.E. C.R. P Label
pre_verbal <--> treatment .467 .226 2.066 .039 eps2 <--> eps4 6.797 1.344 5.059 ***
Variances: (Group number 1 - Default model)
Estimate S.E. C.R. P Label
In this example, we are primarily concerned with testing a particular hypothesis and not so much with parameter estimation. However, even when the parameter estimates themselves are not of primary interest, it is a good idea to look at them anyway to see if they are reasonable. Here, for instance, you may not care exactly what the correlation between eps2 and eps4 is, but you would expect it to be positive. Similarly, you would be surprised to find any negative estimates for regression weights in this model. In any model, you know that variables cannot have negative variances, so a negative variance estimate would always be an unreasonable estimate. If estimates cannot pass a gross sanity check, particularly with a reasonably large sample, you have to question the correctness of the model under which they were obtained, no matter how well the model fits the data.
Model C for the Olsson Data
Now that we have a model (Model B) that we can reasonably believe is correct, let’s see how it fares if we add the constraint that post_verbal does not depend on treatment.
In other words, we will test a new model (call it Model C) that is just like Model B except that Model C specifies that post_verbal has a regression weight of 0 on treatment.