Samprit Chatterjee - Handbook of Regression Analysis With Applications in R

The ‐tests and ‐test of Section 1.3.3are special cases of a general formulation that is useful for comparing certain classes of models. It might be the case that a simpler version of a candidate model (a subset model) might be adequate to fit the data. For example, consider taking a sample of college students and determining their college grade point average ( ), Scholastic Aptitude Test (SAT) evidence‐based reading and writing score ( ), and SAT math score ( ). The full regression model to fit to these data is

Instead of considering reading and math scores separately, we could consider whether can be predicted by one variable: total SAT score, which is the sum of and . This subset model is

with . This equality condition is called a linear restriction, because it defines a linear condition on the parameters of the regression model (that is, it only involves additions, subtractions, and equalities of coefficients and constants).

The question about whether the total SAT score is sufficient to predict grade point average can be stated using a hypothesis test about this linear restriction. As always, the null hypothesis gets the benefit of the doubt; in this case, that is the simpler restricted (subset) model that the sum of and is adequate, since it says that only one predictor is needed, rather than two. The alternative hypothesis is the unrestricted full model (with no conditions on ). That is,

versus

These hypotheses are tested using a partial ‐test. The ‐statistic has the form

(2.1)

where is the sample size, is the number of predictors in the full model, and is the difference between the number of parameters in the full model and the number of parameters in the subset model. This statistic is compared to an distribution on degrees of freedom. So, for example, for this GPA/SAT example, and , so the observed ‐statistic would be compared to an distribution on degrees of freedom. Some statistical packages allow specification of the full and subset models and will calculate the ‐test, but others do not, and the statistic has to be calculated manually based on the fits of the two models.

An alternative form for the ‐test above might make clearer what is going on here:

That is, if the strength of the fit of the full model (measured by ) isn't much larger than that of the subset model, the ‐statistic is small, and we do not reject the subset model; if, on the other hand, the difference in values is large (implying that the fit of the full model is noticeably stronger), we do reject the subset model in favor of the full model.