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The National Alzheimer's Coordinating Center Uniform Data Set includes test results from a battery of cognitive exams. Motivated by the need to model the cognitive ability of low‐performing patients we create a composite score from ten tests and propose to model this score using a partially linear quantile regression model for longitudinal studies with non‐ignorable dropouts. Quantile regression allows for modeling non‐central tendencies. The partially linear model accommodates nonlinear relationships between some of the covariates and cognitive ability. The data set includes patients that leave the study prior to the conclusion. Ignoring such dropouts will result in biased estimates if the probability of dropout depends on the response. To handle this challenge, we propose a weighted quantile regression estimator where the weights are inversely proportional to the estimated probability a subject remains in the study. We prove that this weighted estimator is a consistent and efficient estimator of both linear and nonlinear effects.