Adaptive Model Testing For Longitudinal Generalized Linear Models

Longitudinal data combines the information of cross section and time series,which is wide?ly concerned in many fields,including biology,medicine,finance and economics.However,with the rapid development of scientific and technological means,people can often collect large-scale Longitudinal data sets,and the expansion of the dimension of data brings great difficulties and challenges to the model checking of longitudinal data.In the classical scenarios with independen-t identically distribution data,a dimension-reduction model-adaptive test procedure is proposed for parametric single-index models to solve the problem of typical curse of dimensionality.In this paper,we construct a dimension reduction adaptive-to-model test for longitudinalgeneralized linear model.It has the feature that fully uses the dimension reduction structure under the null hypothesis and automatically adapt to the general model structure under alternative hypothesis.This model adaptation strategy can avoid the curse of dimensionality in the sense that the test behaves like a classical test with only one covariate.The convergence rate of the test statistic to its weak limit can be very accelerated compared to the classical local smoothing methods and the sensitivity to alternative models can be enhanced.Simulation studies in balanced and unbalanced data are conducted to examine the finitesample performance of the proposed method.The purpose is threefold.In the first example,for the null model which is linear,we compare the proposed test denoted as T_n with the testproposed by Wang and Qu(2009)and the test S~?in(3-3).In the second and third example,wecheck the usefulness of the dimension reduction technique to overcome the curse of dimension?ality by comparing our test with the test in(3-3).The proposed test also have model adaptation property compared with the test in(3-4).Finally,a real data analysis is shown for illustration.We now apply the proposed testto the primary biliary cirrhosis(PBC)data from a double-blinded randomized trail conducted by the Mayo Clinic.With the proposed test,we obtain that the test statistic T_n takes value of4.174 and the p-value is 0.Therefore,it indicates that a marginal linear regression model is not plausible for predicting the response.To give a more formal check for the result,we use the nonparametric regression model with smoothing spline and marginal linear regression model with the GEE estimation to analyze the data set.Then we use out-of-sample testing to compare different models~^forecasting accuracy and it also shows that a marginal linear regression model is not plausible for predicting the response.