Linktest

Linktest is a statistical diagnostic method used in generalized linear models, most commonly after logistic regression, to assess model specification and the appropriateness of the chosen link function. The idea is to examine whether the fitted linear predictor from the model sufficiently captures the relationship between predictors and the outcome, or whether there are nonlinearities or interactions that the model has failed to incorporate.

In practice, after fitting a model, the method uses the linear predictor η = Xβ (the predicted logit in a logistic model). A second regression is then run with the outcome as the dependent variable and the linear predictor η and its square η^2 as regressors. The key statistic is the coefficient on η^2: if this squared term is statistically significant, it suggests potential misspecification, such as nonlinear effects of predictors, omitted interactions, or an incorrect link function. If the η^2 coefficient is not significant, the model is considered adequately specified with respect to these aspects, though this is not a guarantee of perfect fit.

Linktest is implemented in various statistical software packages, sometimes under specific commands (for example, as a diagnostic after logistic or probit modeling). It should be interpreted alongside other diagnostics, such as calibration tests, goodness-of-fit measures, and ROC analysis. Because the test relies on sample size and model structure, it may yield misleading signals in small samples or highly complex models, making it prudent to use multiple diagnostic tools when evaluating model specification.