In this proposal, the investigator proposes a semiparametric method for joint modeling of longitudinal responses and measurement times. The semiparametric varying-coefficient regression model postulates that the influences of some covariates vary nonparametrically with time while the effects of the remaining covariates follow certain parametric functions of time. The measurement times of responses are modeled by Cox's proportional model for conditional mean rate. The modeling by Aalen's additive model will also be studied. The investigator employs the approach of Lin and Ying (2001) to estimate the nonparametric regression coefficients. The parametric components are estimated by a weighted least squares method. More efficient smoothing techniques are used for response processes. The investigator proposes a data-driven bandwidth selection criterion based on the sum of squares of residuals. This selection method will be further studied and tested empirically through simulations. Constructions of confidence bands for the parametric components, nonparametric regression coefficient functions and their cumulatives are proposed. Goodness-of-fit test procedures are proposed to check whether some regression coefficient functions follow certain parametric forms by comparing the nonparametric estimators and the corresponding estimators of the cumulative regression coefficient functions under the null hypothesis. A preliminary simulation study demonstrates that the proposed estimation and hypothesis testing methods are promising. The investigator will study the asymptotic properties, such as consistency, weak convergence and consistency of the estimators of the asymptotic variances, for all the proposed estimators. Finite sample properties of the proposed estimation methods and goodness-of-fit tests will be evaluated through extensive numerical simulation studies. The investigator also proposes to study related problems in optimal choice of the weight function, optimal design of measurement times and informative censoring.
The investigator proposes to seek statistical models with a physical or biological basis and biologically interpretable parameters and to develop statistically efficient methods to better understand linear or nonlinear behavior of response process. The proposed semiparametric varying-coefficient regression model allows for additional flexibility to explore how covariate effects change over time and provides a base to test whether simpler models with scientific relevance hold. The proposed research when carried out would help to enrich a collection of statistical tools which have important impact on the analysis of longitudinal data. Giudelines for practical applications will be developed. The methods will be used to analyze longitudinal data in AIDS related medical research to develop more effective treatments and to other real examples in medical studies. The research of the problems proposed here will also generate many research topics at different levels suitable for graduate and undergraduate studies, therefore promotes involvement of students in current scientific research.
