Quantile Regression for Repeated Responses Measured with Error

Many problems in biostatistics involve response variables that are difficult to measure accurately. Muscular strength in animals and humans quantified through the grip strength, and usual nutrient intake of an individual, are two examples of such variables. These studies often collect, on the same subject, multiple measurements of the response variable containing measurement error. One research interest in these studies is estimating the conditional quantile function of the latent response variable given a set of subject-specific covariates and the contaminated replicates.;Conventional quantile regression (QR) is an established technique to estimate the conditional quantile function of a response variable. However, it cannot be directly employed for estimation in the current problem because the response variable is observed with measurement error. Naively replacing the latent response variable by the subject-specific average of the contaminated replicates in a QR model could lead to serious bias in the quantile estimates. Other methods based on transformations have been proposed to estimate the conditional quantiles of a latent variable. However, these methods: i) rely on the existence of a transformation to achieve both constant variance and normality, which could be unrealistic in many problems; and ii) do not provide a method to compute quantile curves when a continuous covariate is present.;Therefore, we propose a new seminonparametric estimation approach to estimate the conditional quantile function of a latent variable given subject-specific covariates and contaminated replicates. The proposed method, described in Chapter 2, involves a location-scale shift model that allows the subject-specific random effects to follow a flexible seminonparametric distribution and accounts for measurement errors. A virtue of the proposed method is that it does not rely on the normality and constant variance assumptions on the subject-specific random effects. Moreover, the variance of the subject-specific random effects distribution is allowed to be a function of continuous covariates. We derive the asymptotic results for the proposed estimator and, through simulation studies, demonstrate that the proposed method leads to consistent estimates of the conditional quantiles of the latent response variable. To assess the practical usage of the proposed method, a grip strength data set from laboratory mice is analyzed.;In Chapter 3, we extend the location-scale shift model proposed in Chapter 2 by allowing the location and scale functions to be modeled nonparametrically. This current model handles the possible misspecification of the location and scale functions in the model proposed in Chapter 2. We illustrate the value of the proposed method by estimating the conditional quantile curves of usual sodium intake as a function of age. This application entails careful work to deal with the estimation and selection of tuning parameters in our method under a complex survey design. We handle these challenges by incorporating sampling weights into the estimation procedure and a replication method for variance estimation with the proposed method.;In Chapter 4 we investigate the performance of some adaptations of the existing estimation methods based on QR and bias-correction approaches. An advantage of these methods is that some of the model assumptions required in Chapters 2-3 can be avoided. First, we propose the simulation-extrapolation (SIMEX) and empirical SIMEX methods to mitigate the bias of the naive QR. Second, we propose two estimators based on corrected-loss functions in a QR model. SIMEX-based methods slightly reduce the bias of quantile coefficients, particularly for the intercept. The performance of the SIMEX-based estimators is highly dependent on the extrapolant being considered. The corrected-loss estimators give approximately unbiased estimates of the conditional quantiles. However, practical implementation of these estimators is challenging due to the computational issues that characterize corrected-score methods.