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The overall objective of this research proposal is semi-parametric inference of the Cox proportional hazards (PH) regression model for a survival function Pr(X > x I Z = z) = S(x I z) = S 0 (x)ezbeta , where X is a time-to-event variable. which is subject to interval censoring, Z represents the covariates, S0 is a baseline survival function, and beta represents the regression coefficients. The main objective of our research is to develop asymptotic inferences of the generalized maximum likelihood estimators (GMLE) of beta and S(. I z). A critical limitation with GMLE under interval censoring is that it is computationally feasible only for a small data set. We therefore propose to also investigate asymptotic properties of a computationally simpler alternative to GMLE, namely two-stage estimators (TSE) of beta and S(. I z) obtained by a two-stage modified Newton-Raphson algorithm involving data grouping. In the four years of our research, we have implemented a foolproof algorithm for obtaining TSE, proved consistency and established asymptotic normality for both GMLE and TSE under both discrete and continuous distributional assumptions, and proposed new diagnostic method for PH assumption. Also, we have successfully applied our asymptotic Cox regression methodology to the analysis of a large-scale, long-term breast cancer relapse follow-up study. Our results will be useful to data analysis of breast cancer relapse follow-up studies, chemoprevention intervention trials and genetic studies on familial aggregation of breast cancer and related cancers.

The overall objective of this research proposal is semi-parametric inference of the Cox proportional hazards (PH) regression model for a survival function Pr(X > x/Z=z) = S(x/z) = S.(x) ex beta, where X is a timetoevent variable, which is subject to interval censoring, Z represents the covariates, So is a baseline survival function, and Z represents the regression coefficients. One objective of our research is to develop asymptotic inferences of the generalized maximum likelihood estimators (GMLE) of beta and S(.

The overall objective of our research proposal is nonparametric inference of the joint survival function S(x1, ..., xd) =Pr(X1 > x1, ..., Xd > xd) of d (>= 2) correlated time-to-event variables X1, ..., Xd, each of which is subject to interval censoring. The standard estimator of S is the generalized maximum likelihood estimator (GMLE) S. However, S cannot be expressed in a closed-form expression and its statistical properties have not been studied in the multivariate case. The technical objectives of this pioneer methodological research proposal are to develop asymptotic generalized maximal likelihood (GML) inference of S and to derive efficient computational algorithms for the GML procedure. In our fourth and final year of research, we have implemented a computer software for asymptotic inference of GMLE p of the correlation coefficient p between a pair of the X' variables. When the censoring distribution is continuous, we have numerically established p is not asymptotically normal and we have implemented a bootstrap method for obtaining interval estimator of p. Thus in our four years of research, we have successfully completed the tasks we proposed. The results will be useful to breast cancer researchers pursuing chemoprevention intervention trials involving multiple surrogate endpoint biomarkers, and genetic epidemiologists conducting studies on familial aggregation of breast cancer and related cancers.

Interval-censored (IC) data are encountered in three areas of breast cancer research. The most common application is in clinical relapse follow-up studies in which the study endpoint is disease-free survival. When a patient relapses, it is usually known that the relapse takes place between two follow-up visits, and the exact time to relapse is unknown. In statistics, we say relapse time is interval censored. Interval censoring is also encountered in breast cancer registry studies in which information on family history of cancer is updated periodically. The Strang Breast Surveillance Program for women at increased risk for breast cancer, for instance, has enlisted over 800 women with complete pedigree information which is verified and updated continuously. Family history data such as age at diagnosis of a specific cancer, or a benign but risk-conferring condition, are obtained from each registrant at each update. Time to a cancer event, and definitely time to first detection of a benign condition, are at best known to fall in the time interval between the last update and age at diagnosis. A third but increasingly important area of application of interval censoring is in breast cancer chemoprevention experiments or prevention trials, which involve the observation of one or more surrogate endpoint biomarkers (SEB) over time. The scientific question of interest here is the estimation of time for the SEB to reach a target value, and time from cessation of intake of a chemopreventive agent to the loss of its protective effect. Unfortunately, the exact values of both these time variables are known only to lie in between two successive assay inspection times.

The overall objective of this research proposal is semi-parametric inference of the Cox regression model for a survival function Pr(X > x/Z=z)=S(x/z)= So(x)ez beta, where X is subject to interval censoring, Z represents the covariates, So is a baseline survival function, and beta represents the regression coefficients. One objective of our research is to develop asymptotic inference of the generalized maximum likelihood estimator (GMLE) of the regression coefficients beta and S(./z).

The overall aim of our research proposal is the statistical nonparametric inference of the redistribution-to-the-center estimator (RTCH) and the generalized maximum likelihood estimator (GMLE) for the survival function of a time-to- event variable that is subject to interval censoring. The RTCH, which is proposed by us, has a closed-form expression and is equal to the GMLE under a homogeneous condition. The GMLH is the standard optimal estimator in survival analysis. However, it cannot be expressed in a closed-form expression, and asymptotic distribution theory for it has been limited. From the asymptotic study of the RTCE, we have gained important insight into proofs of asymptotic properties of the GMTE. In the past four years we have established consistency, asymptotic normality and asymptotic efficiency of the GMLE under a variety of conditions. In addition, we have derived an asymptotic nonparametric two-sample distance test procedure for comparing two populations. Finally, we have begun investigating the asymptotic inference of Cox regression model for interval-censored data by establishing consistency and asymptotic normality of the %GMLE of the regression parameters under some finite assumptions. These preliminary results are being applied to a breast cancer prognostic relapse follow-up study.

The overall aim of our research proposal is the statistical inference of nonparametric estimates, the restribution-to-the-inside estimator (RTIE) and the generalized maximum likelihood estimator (GMLE),for the survival function of a time-to-event variable that is subject to interval censoring. The RTIE, which is proposed by us, has a closed-form expression and is equal to the GMLE under a homogeneous condition. The GMLE is the standard optimal procedure in survival analysis. However, no closed-form expression for the GMLE is available, and asymptotic distribution theory for it has been limited. Our research efforts in the third year have focused on the asymptotic inference of the GMLE under conditions more general than the discrete distribution assumption that we previously imposed on the censoring variables. Additionally, we have derived an asymptotic nonparametric the sample test procedure for comparing two populations. Finally, we have begun investigating the asymptotic inference of Cox regression model for interval-censored data by establishing consistency of the GMLE of the model parameters under finite assumptions on both the survival and censoring distributions.

The overall aim of our research proposal is the statistical inference of nonparametric estimates, the redistribution-to-the-inside estimator (RTIE) and the generalized maximum likelihood estimator (GMLE), for the survival function, where the survival time is subject to interval censoring. The GMLE is the standard optimal procedure in survival analysis. However, a closed form expression for this estimator has not been derived, and the asymptotic distribution theory for it has been very little known. In our original proposal, we created the RTIE, which has a closed form expression, and which has been shown by us to be the GMLE under certain conditions. Working on the theory of the RTIE has provided us with important clues to the asymptotic theory concerning tile GMLE. Our research efforts in the second year have focused on attacking the asymptotic distribution of the GMLE under the assumption that the censoring random vector is discrete. Under such an assumption, we have successfully established the asymptotic properties of the GMLE as well as those of the RTIE.

The overall aim of our research proposal is the statistical investigation of a non-parametric estimate which we call redistribution-to-the-inside estimator (RTIE) for the survival function s(t) = Pr(X > t), where the survival time X is subject to interval censoring. Our research efforts in the first year have been focused on two aspects of RTIE for interval censored data satisfying the condition that for any pair f censoring intervals, either they are disjoint, or one is a subset of the other (DI condition). First, we have completed the implementation of a computer program oded in the C language to carry out the RTIE estimation procedure, including a Kaplan-Meier type plotting program for RTIE written in the S+ language. Second, we derive the uniform almost sure limit (strong consistency) of RTIE for any arbitrary S and any arbitrary censoring distribution function C under the DI model.

A poly-t density is a density which is proportional to a product of at least two t-like factors, each of which is of a certain form where d is a positive number, micron (underlined) is an arbitrary location vector and M (underlines) is a symmetric semi-positive definite scale matrix. In general, M (underlines) is a function of d. Such a density arises, for example, in the Bayesian analysis of a linear model with a normal error term, independent normal priors on the linear parameters and inverted-gamma priors on the variance components. A theorem about the asymptotic normality of the density as a subset of the individual d's tend to infinity is proved under very general conditions. A corollary specifically related to the Bayesian linear regression model with two variance components. The Tiao-Zellner expansion for approximating the particular poly-t form involving two proper multivariate t factors is extended to the case of two arbitrary t-like factors.