# Regression and simple SEM: identification and model comparison

I have two problems I am trying to solve. I am using MPLUS, but I have very general questions for which MPLUS knowledge will not be necessary. Terminology: y = DV, x = IV

(1) Simple regression (N=1300)

I want to compare two linear regressions. In the first model, a metric DV is predicted by 14 IVs as well as 2 covariates (age, sex). In the second model, regression coefficients of the 14 IVs are constrained to be equal, only age and sex are freely estimated.

In MPLUS, a simplified example would look like this ("on" marks regressions, and the brackets set regression weights equal):

Model I:

y on x1;
y on x2;
y on x3;
...
y on sex age;


Model II:

y on x1 (1);
y on x2 (1);
y on x3 (1);
...
y on sex age;


The first regression has 0 DF. Therefor, fit cannot be assessed, which in turns makes model comparison impossible.

How do I get the first model identified so I can compare the two models? Does a linear regression always have 0 DF? I thought about perhaps constraining two of the fourteen regression weights that are very close to each other equal - is that a valid or common approach? For instance, x2 and x14 have a similar nonsignificant regression coefficient estimate of .050 which I could constraint equal to identify the model. If I do that, I get a chi-square of 0.000 for that model with 1 DF -- can that be correct? The constrained model has a chi-square of 398 with 13 DF.

My next question would be how to compare these models. MPLUS outputs AIC, BIC, RMSEA, SMRM, CFI/TLI, and I understand how to interpret them generally. However, am I allowed to interpret them for a simple regression? Am I allowed to interpret them in a model that is not identified?

And as for the model test, is it correct that I simply use the chi-square values in the two models, build the difference, multiply it by 2 and look up the value in a chi-square table for the difference between the degrees of freedom of the models?

(2) SEM (N=3500)

The second problem is somewhat similar, but involves more complex models. I am new to structural equation modeling, but assume these models to be SEMs (?).

Model I consists of 9 regressions with metric DVs, each regression has the same 6 predictors. The model allows for all variables to be correlated (also y with y and x with x). The DV is a the second measurement point of y (yt2), and the regressions control for the first measurement point of (yt1). So, we're really predicting changes in y.

Model I estimates all regressions freely, while model II constrains each x to affect all y in the same way. Again, a simplified example:

Model I:

y1t2 on x1 x2 x3 y1t1;
y2t2 on x1 x2 x3 y2t2;
...
y9t2 on x1 x2 x3 y9t2;


Model II:

y1t1 on x1 x2 x3 y1t2 (1 2 3 4);
y2t1 on x1 x2 x3 y2t2 (1 2 3 5);
...
y9t1 on x1 x2 x3 y9t2 (1 2 3 6);


Is it valid to compare these two models in the way mentioned above (chi-square difference test)? Can fit indices be interpreted normally?

Model I has a chi-square of 197.293 and 72 DF, model II 617.376 and 120 DF, so that would be 120-72 DF, 2*(617.376-197.293) --> p<.001 ?

Regarding your first question, part 1:

Linear regression is "just-identified" in SEM. This is also called "fully-saturated."

A more simple example with 2 IVs and 1 DV gives:

3 variances and 3 covariances in the covariance matrix. This is your DF for SEM = 6.

Your regression includes 2 regression beta coefficients, 2 IV variances, 1 covariance between the IVs (you may or may not realize this is in the model, but it is), and 1 error variance = 6 parameters

6 DF = 6 Parameters

Unless constraints are made, regression models in SEM are always fully saturated and no assessment of model fit is possible.

Regarding Part 2:

I agree with Patrick that these are nested models and you can "test" the constraints with a chi2 test.

I'm not familiar with MPlus syntax, so I can't give you advice that is very specific. However, I can give you some general information that will hopefully be helpful. Forgive me if you already know some of what I'll be talking about -- it's not completely clear to me at what level I should give my advice.

For your first question, for any given model, your available degrees of freedom are

$$x * (x + 1) / 2$$

where x is the number of variables that you're working with. In your first example, this means that you're working with $17 * 18 / 2 = 153$ degrees of freedom. As you may already know, this number ($153$) limits the number of parameters that you can estimate.

If you were to conduct a linear regression where you allowed all your predictors (i.e., your $14$ IVs and what you call your two covariates) to predict your dependent variable, you would be estimating:

• $17$ variances ($16$ for all your predictors, $1$ for your dependent variable)
• $120$ covariances ($16 \choose 2$, since you are estimating a covariance for each pair of predictors)
• $16$ path coefficients ($1$ between each predictor and your dependent variable)

When you sum the total number of parameters you're estimating, you reach $153$, which equals your total available degrees of freedom. What this means is that any regression model is a saturated model and will perfectly reproduce the observed variance-covariance matrix.

So, I would guess what's happening in your first question is that in your model 1, you're estimating a number of parameters equal to the number of degrees of freedom that you have available. This means that any comparison between your model 1 (as currently specified) and another model (model 2) is equivalent to a comparison between a fully saturated model and your model 2.

Therefore, if you want to know whether your Model 2 has an adequate fit, you can simply interpret its fit indices on their own according to the many general guidelines that have been established (i.e., non-significant $\chi ^2$, RMSEA below .06, lower bound of the 90% CI of the RMSEA below .05, upper bound below .10, all correlation residuals below .10, etc).

As for your second question, I don't quite understand what models you're trying to fit. However, in general, the $\chi ^2$ test is valid if your less complex model (i.e., the model for which you're estimating fewer parameters) is a nested subset of the other; that is, if your less complex model is the same as your more complex model except that some of the parameters that were previously freely estimated are now constrained to some value.