The more parsimonious model is, of course, the one without the additional paths. To override the preference for parsimony, therefore, you will have to show that the additional paths, as a set,

*significantly*reduce the overall model chi-square, thus improving model fit. As you move along in your careers, you may wish to adopt additional criteria, such as whether the reduction in chi-square appears substantively large in addition to being statistically significant, but for now, we'll use statistically significant change as our criterion.

You can display your results in a table, as follows:

--------------------------------------------------------------

Model....................................X2.............df....

--------------------------------------------------------------

Model w/ fewer parameters.....----............---...

Model w/ added parameters.....----............---...

--------------------------------------------------------------

Delta (change)..........................----............---...

--------------------------------------------------------------

The chi-square change score (the top chi-square minus the bottom one) can be treated like any other chi-square value and be referred to a chi-square table, with degrees of freedom equal to delta df (top df minus bottom df).

**For the version of the universities model without the three paths listed above, the chi-square is 210.98 (31 df), whereas for the model that adds the three paths, chi-square = 169.04 (28 df).**

*UPDATE, March 7, 2017:*

**UPDATE, March 11, 2012:**Xiaohui photographed the explanation I diagrammed on the board, linking number of paths in a model, goodness of fit, chi-square, and degrees of freedom. A key point was to demonstrate that if one model has a higher chi-square than another model, it will also have a higher number of degrees of freedom.**All of the green phrases go together: a model with fewer paths (which preserves a higher df) will have a poorer fit and thus a higher chi-square.**The red terms represent the opposite of the green terms, and thus**the red terms go together, as well: a model with more paths (which depletes the df) will lead to a better fit and thus a lower chi-square.**

**UPDATE, March 5, 2008:**Kristina photographed the decision-tree I drew on the board, to augment our discussion of comparative model testing. Here it is (you can click on the image to enlarge it).*And now, back to our regular programming...*

An important condition for being able to conduct comparative model tests is that the two models being compared to each other must possess the property of

**. Two models are nested if they can be converted from one to the other either by**

*nestedness**only adding parameters*to one to obtain the other, or

*only removing parameters*from one to obtain the other. By

*parameters*, we mean anything that is freely estimated in SEM (e.g., structural paths, non-directional correlations). If you start with one model and convert it to a new, second model by both

*adding and substracting*parameters from the initial model, the two models will

*not*fulfill the criteria for nestedness and thus cannot be compared via the delta chi-square test.

The following two diagrams provide examples of nested and non-nested models.

An analogous situation exists in multiple regression. You can do a delta R-square test to see, for example, if a model with predictor set A, B, C, D, and E accounts for significantly more variance in the dependent variable than does predictor set A, B, and C. ABC is contained -- that is nested -- within ABCDE, thus permitting the statistical comparison. You could not, however, test whether predictor set ABCD

**F**accounts for more variance than set ABCD

**E**, because the change in models would have required both dropping a predictor and adding one. If ABCDE was the starting point, we would have dropped E and added F.

We'll use the following article to delve more deeply into comparative model testing:

Bryant, A. L., Schulenberg, J., Bachman, J. G., O'Malley, P. M., & Johnston, L. D. (2000). Understanding the links among school misbehavior, academic achievement, and cigarette use: A national panel study of adolescents.

*Prevention Science, 1*, 71-87.