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."

We will look at the example (near Figure 3) in the following article, which we can access online via the TTU Library.

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.

"Equivalent Models" Problem

We'll next consider what is known as the "equivalent models" problem, and other cautions and limitations of SEM. In doing so, we will look at two articles that are available via the TTU library's website:

MacCallum, R.C., Wegener, D.T., Uchino, B.N., & Fabrigar, L.R. (1993). The problem of equivalent models in applications of covariance structure analysis. Psychological Bulletin, 114, 185-199.

(We'll also have a song about the equivalent models problem, entitled, "Your Model's Only One.")

Tomarken, A.J., & Waller, N.G. (2003). Potential problems with "well fitting" models. Journal of Abnormal Psychology, 112, 578-598.

In addition, regarding one of MacCallum et al.'s suggestions, here's an article that incorporates participation in a randomized experimental program (yes/no) into an SEM model.

Florsheim, P., Burrow-Sánchez, J., Minami, T., McArthur, L. & Heavin, S. (2012). The Young Parenthood Program: Supporting positive paternal engagement through co-parenting counseling. American Journal of Public Health, 102, 1886-1892.

Stating Hypotheses

Today, I'd like to cover interpretive clarity in writing about your hypotheses and results. Many beginning writers on SEM simply restate the numerical information from their output, tables, and figures, without providing substantive interpretations.

Earl Babbie's textbook, The Practice of Social Research (2007, 11th ed.) contains a guest essay by Riley E. Dunlap, entitled "Hints for Stating Hypotheses" (p. 47). Here is an excerpt of what I believe is the key advice:

The key is to word the hypothesis carefully so that the prediction it makes is quite clear to you as well as others. If you use age, note that saying "Age is related to attitudes toward women's liberation" does not say precisely how you think the two are related... You have two options:"

1. "Age is related to attitudes toward women's liberation, with younger adults being more supportive than older adults"...

2. "Age is negatively related to support for women's liberation"...

As Dunlap demonstrates, these two statements of the hypothesis let the reader know with specificity which people are expected to hold which type of attitudes (I have added the color and bold emphases above).

Results should be described similarly -- not just that a standardized regression path between Construct A and Construct B was .46, but that (given the positively signed relationship) the more respondents do whatever is embodied in Construct A, the more they also do what is embodied in Construct B.

One of my SEM-based publications from several years ago, which is accessible on TTU computers via Google Scholar, can serve as a guide.

Thomas, G., Reifman, A., Barnes, G.M., & Farrell, M.P. (2000). Delayed onset of drunkenness as a protective factor for adolescent alcohol misuse and sexual risk-taking: A longitudinal study. Deviant Behavior, 21, 181-210.

(Added April 29, 2015): This webpage explains the distinction between a hypothesis (when you have a prediction) and a research question (when you don't).