The model

We need to specify the model, in terms of fixed effects and random effects. First, we expand the Fixed Effects tab and include the interaction into the model terms, by selecting both variables on the left panel and pushing them on the right.

Then we expand the Random Component tab and fill in the intercept as random effects across school (we will add random terms later on, here we start with the intercept as random for the sake of simplicity).

Results

The first table, Model Info recaps the model call (the formula used in R), the description of the model (family and link function), and some overall model index. Here we can look at the R-squared, for datail see technical details and R piecewise package implementation

Based on the R-square indexes, we can see that our error of approximation of the data decreases of 4% ( \({R^2}_{marg}=.039\) ) thanks to the fixed effects, whereas all effects together decrease our error of approximation of 37% (\({R^2}_{cond}=.378\)).

The other information in the table helps to interpret the results. In particulat, the row Direction is useful. It gives the definition of the logit that is used, including which is the reference group of the dependent variable. In the example, it indicates that we are predicting pass=1 against pass=0. Thus we know that all the independent variables positively related with the logit are positively related with the odd of passing the exam.

Random component

For this model, with only intercepts as random coefficients across schools, the random component table is pretty simply. It shows the variance of the random intercepts. It is non-zero, so we are happy.

Omnibus test

The omnibus (Wald) Chi-Squared test tests the main effects of the independent variables and their interaction. Notice that in GAMLj the continuous variables are centered to their mean by default, and thus we can interpret the linear effects of math and activity as average effects or main effects . Based on the p-value, our results seem to support an interaction and two main effects.

Parameter estimates

The same information can be extracted from the parameters estimates table.

Here we also obtain the odd ratio (exp(B)) of the effects, useful to interpret the effects in terms of rate of change in the dependent variable odd.

Plots

For generalized linear models, mixed included, a good strategy is to visualize the effects by plotting the predicted values. GAMLj plots the predicted values after transforming them back to the original scale of the dependent variable, in this case probability.

Ask for the plot in the Plots panel. Add math as the variable whose values go in the Horizontal axis and activity as separate lines.

Because activity is a continuous variable, the separated lines will show the effect of math for three interesting values of the moderator activity. The defaul in GAMLj is to show effects for the moderator set at Mean-1SD, Mean, and Mean+1SD. This default can be altered in the Covariates scaling tab.

Thus, for the average level of activity (gray line) there’s an increase of probability of passing the exam along the scores of math. The increase, however, is much stronger for students with one standard deviation above average of activities (yellow line), whereas for student with a few activities (blue line), the probability of passing the exam does not change much depending on the math score (recall the data are simulated, the interpretation is provided only as an exercise).

We can also visualize the random effects by asking them in the imput panel.

Notice that the random effects depict different curves for different schools, even though the only random effect is the intercept. This is not weird in generalized linear models. The random intercept is estimated for the logit, thus it is the intercept of the straight lines computed for predicting the logit. When the logit is transformed back to probabilities, the function relating Y to X is no longer a straight line, and its shapes changes depending also on the value of the interecpt. If we wish to appreciate how the random linear effects vary, we can ask for the Linear Predictor plot, which plots the effects in the logit scale.

As expected, the random effects are all parallel, because we allowed only the intercepts to be random.

At this point, one can expand the model by allowing also the IVs effects to vary, and evaluate the goodness of the models, comparing them, and further investigate the relationships we observed, with simple effects analysis and additional plots.