Motivated by investigating the relationship between progesterone and the days in a menstrual cycle in a longitudinal study, we propose a multi-kink quantile regression model for longitudinal data analysis. It relaxes the linearity condition and assumes different regression forms in different regions of the domain of the threshold covariate. In this paper, we first propose a multi-kink quantile regression for longitudinal data. Two estimation procedures are proposed to estimate the regression coefficients and the kink points locations: one is a computationally efficient profile estimator under the working independence framework while the other one considers the within-subject correlations by using the unbiased generalized estimation equation approach. The selection consistency of the number of kink points and the asymptotic normality of two proposed estimators are established. Secondly, we construct a rank score test based on partial subgradients for the existence of kink effect in longitudinal studies. Both the null distribution and the local alternative distribution of the test statistic have been derived. Simulation studies show that the proposed methods have excellent finite sample performance. In the application to the longitudinal progesterone data, we identify two kink points in the progesterone curves over different quantiles and observe that the progesterone level remains stable before the day of ovulation, then increases quickly in five to six days after ovulation and then changes to stable again or even drops slightly

翻译：通过在纵向研究中调查月度和月经周期中的天数之间的关系,我们提出一个用于纵向数据分析的多金四分位回归模型。它放松了线性条件,并在门槛共变区的不同区域采取了不同的回归形式。在本文中,我们首先提出长度数据多金四分位回归。提出两个估算程序来估计回归系数和近点位置:一个是工作独立框架下的计算高效剖面估计器,另一个是使用公正通用估计方程式法来考虑主题内部相关性。它放松了线性条件,并在门槛共差区域的不同区域采取了不同的回归形式。在长度研究中,我们首先提出一个基于部分次偏差的位回归回归率和近点回归。测试统计的无效分布和当地替代分布都得到了推算。模拟研究表明,拟议的方法在使用不偏差的主体内,使用公正的普遍估计方位方位方位公式的方法;在使用近半年期数据之前,我们再次确定两个直位分数的分数,然后在正值前的半年中,我们再确定两个直位数,然后在正位曲线前,然后再测量两个直到正点。