Results

The log-likelihood of the baseline model (M₀) is 96.124. This represents the fit of the model when including only the constant. Initially every child is predicted to belong to the category in which most observed cases fell. In this example there were 39 children who had display rule understanding and only 31 who did not. Therefore, of the two available options it is better to predict that all children had display rule understanding because this results in a greater number of correct predictions. Overall, the model correctly classifies 55.7% of children (since 55.7% of the children actually had display rule understanding).

In M₁, false-belief understanding (fb) is added to the model as a predictor. As such, a child is now classified as having display rule understanding based on whether they passed or failed the false-belief task. The output above shows summary statistics about the new model. The overall fit of the new model is assessed using the Deviance. Remember that large values of the deviance statistic indicate poorly fitting statistical models.

If fb has improved the fit of the model then the value of deviance should be less than the value when only the constant was included (because lower values of deviance indicate better fit). When only the constant was included, deviance = 96.124, but now fb has been included this value has been reduced to 70.042. This reduction tells us that the model is better at predicting display rule understanding than it was before fb was added. We can assess the significance of the change in a model by taking the deviance of the new model and subtracting the deviance of the baseline model from it. The value of the model chi-square statistic ( ) works on this principle and is, therefore, equal to deviance with fb included minus the value of deviance when only the constant was in the model ( 96.124 − 70.042 = 26.083). This value has a chi-square distribution. In this example, the value is significant at the p < .001 level and so we can say that overall the model predicts display rule understanding significantly better than with fb included than with only the constant included.

The Coefficients table tells us the estimates for the coefficients for the predictors included in the M₁ model (namely, fb and the constant). The coefficient represents the change in the logit of the outcome variable associated with a one-unit change in the predictor variable. The logit of the outcome is the natural logarithm of the odds of Y occurring.

The Wald statistic has a chi-square distribution and tells us whether the b coefficient for that predictor is significantly different from zero. If the coefficient is significantly different from zero then we can assume that the predictor is making a significant contribution to the prediction of the outcome (Y). For these data it seems to indicate that false-belief understanding is a significant predictor of display rule understanding (note the significance of the Wald statistic is less than .001).

We can also look at the Model Summary table to see if the other predictors (age and the interaction effect) add predictive value to the model. First, we have M₃, which reduced the Deviance by another 2.3, compared to M₂. We can use this result ( ) combined with its degrees of freedom (68-67 = 1), obtain the p-value of 0.131 that is listed in the table. The decrease in Deviance is therefore not significant, which indicates that age does not improve our model predictions to the extent that it’s worth including in the model. Here we also see BIC shining. The BIC is similar to the Deviance, in that a lower number indicates better model fit, but the BIC has some extra heavy penalty for model complexity (i.e., the number of predictors). That is a nice feature because in model selection we generally want to have a built-in Occam’s razor, where we want to predict as well as possible, while using as few predictors as possible. Since M₃ has an additional predictor, while not improving the predictions much, the BIC for M₃ is actually higher than that of M₂ - bad news for M₃! The same goes for M₄, where adding the interaction does not improve model fit significantly (  (1) = 0.123, p = .726), and where the BIC is even higher than that of M₃.

Assuming we are content that the model is accurate and that false-belief understanding has some substantive significance, then we could conclude that false-belief understanding is the single best predictor of display rule understanding. Furthermore, age and the interaction of age and false-belief understanding did not significantly predict display rule understanding. This conclusion is fine in itself, but to be sure that the model is a good one, it is important to examine the residuals, which brings us nicely onto the next task.