Model specification test in a semiparametric regression model for longitudinal data

We propose a model specification test for whether or not a postulated parametric model (null hypothesis) fits longitudinal data as well as a semiparametric model (alternative hypothesis) does. In the semiparametric model, we suppose that a baseline function of time is modeled nonparametrically, while the longitudinal covariate effect is assumed to be a parametric linear model. The existing kernel regression based likelihood ratio tests suffer from computing the likelihood function in the alternative hypothesis, because a specific parametric alternative is not desired. To circumvent this difficulty, we calibrate the semiparametric model to a regression model containing only the parametric parameters, and investigate the quadratic inference function in the calibrated model. The proposed approach yields an asymptotically unbiased parametric regression estimator without undersmoothing the baseline function. This provides us a simple and powerful test statistic that asymptotically follows a central chi-squared distribution with fixed degrees of freedom under the null hypothesis. Simulation studies show that the proposed test is able to identify the true parametric regression model consistently. We have also applied this test to real data and confirmed that the baseline function can be captured by a conjectured parametric form sufficiently well.