Equality Constraints

This week, we'll learn about equality constraints. An equality constraint tells the SEM computer program that, in reaching its solution, it must provide the identical unstandardized coefficient for all parameters within a set that has been designated for equality (even when equality constraints have been imposed, standardized coefficients may not be exactly the same within the constrained set).

Implementation of equality constraints in Mplus is illustrated at this site (see section 3.0).*

Suppose that equality constraints are placed on the structural paths from two predictor constructs to an outcome construct. In the absence of constraints, one predictor might, for example, take on an unstandardized coefficient of .40 and the other predictor might take on a value of .30. Because of the constraints, however, the values must be identical, so both paths might take on a value of .35 (I don't know that the solution must always "split the difference," but it's probably a reasonable way to think about it).

Constraining the two (or more) values to equality, of necessity, harms model fit; the mathematically optimal MLE paths in the above example would have been .40 and .30. Giving the two paths the identical .35 is thus suboptimal mathematically, but it provides greater parsimony because we can say that a single value (.35) works reasonably well for both paths.

Equality constraints involve comparative model-testing and delta-chi square tests, as we've seen before. In this new context, what we do is run the model twice, once without constraints and once with (this is considered "nested"). If the delta-chi square test (with delta df) is significant, we say the constraints significantly harm model fit and we ditch them. If the rise in chi-square due to the constraints is not significant, we then retain the constraints in the name of parsimony.

Equality constraints have at least four purposes, as far as I can tell:

1. Theory/hypothesis testing. The following example, from one of my older articles, involves trying to test if three adolescent suicidal behaviors -- thoughts, communication, and attempts -- are three gradations on the same underlying dimension or are more qualitatively different. We reasoned that, if the behaviors were gradations along the same dimension, then each psychosocial predictor should relate equivalently to the three suicidal behaviors. If, on the other hand, the suicidal behaviors were qualitatively different, then a given predictor might relate significantly to one of them, but not all three.

2. Playing the "Devil's Advocate."

At first glance, one path may look like it is significantly larger than another. However, this proposition must be tested by constraining the paths in question to equality and conducting a delta chi-square test. The following articles illustrate the process.

Foster, T. D., Froyen, L. C., Skibbe, L. E., Bowles, R. P., & Decker, K. B. (2016). Fathers’ and mothers’ home learning environments and children’s early academic outcomes. Reading and Writing, 29, 1845–1863.

Breckler, S.J. (1990). Applications of covariance structure modeling in psychology: Cause for concern? Psychological Bulletin, 107, 260-273.

3. Longitudinal/panel models.

We will go over several examples, including some from:

Farrell, A.D. (1994) Structural equation modeling with longitudinal data: Strategies for examining group differences and reciprocal relationships. Journal of Consulting and Clinical Psychology, 62, 477-487.

As the title notes, this article is also good for studying multiple-group modeling...

4. Multiple-group analyses. Here's a graphic I made to illustrate the use of equality constraints in this context. (AMOS uses letters to denote paths constrained to equality, whereas Mplus uses numbers.)


*Designation of equality is done with the AMOS program via letters. For a given parameter, you can go to the "Object Properties" box and, for the parameter value, you can pre-specify a letter such as A, B, C, etc. Any two (or more) parameters to which you assign an A will all become constrained to take on the same unstandardized value; any two (or more) parameters to which you assign a B will be become constrained to take on identical unstandardized values, etc. Only parameters with the same letter will take on identical values. In other words, the uniform coefficient taken on by the set of A constraints will (almost certainly) be different from the uniform coefficient taken on by the B set.