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Motivated by investigating the relationship between progesterone and the daysin a menstrual cycle in a longitudinal study, we propose a multi-kink quantileregression model for longitudinal data analysis. It relaxes the linearitycondition and assumes different regression forms in different regions of thedomain of the threshold covariate. In this paper, we first propose a multi-kinkquantile regression for longitudinal data. Two estimation procedures areproposed to estimate the regression coefficients and the kink points locations:one is a computationally efficient profile estimator under the workingindependence framework while the other one considers the within-subjectcorrelations by using the unbiased generalized estimation equation approach.The selection consistency of the number of kink points and the asymptoticnormality of two proposed estimators are established. Secondly, we construct arank score test based on partial subgradients for the existence of kink effectin longitudinal studies. Both the null distribution and the local alternativedistribution of the test statistic have been derived. Simulation studies showthat the proposed methods have excellent finite sample performance. In theapplication to the longitudinal progesterone data, we identify two kink pointsin the progesterone curves over different quantiles and observe that theprogesterone level remains stable before the day of ovulation, then increasesquickly in five to six days after ovulation and then changes to stable again oreven drops slightly

Multi-Kink Quantile Regression for Longitudinal Data with Application to the Progesterone Data Analysis