Interval mapping of quantitative trait loci for time-to-event data with the proportional hazards mixture cure model

Interval mapping using normal mixture models has been an important tool for analyzing quantitative traits in experimental organisms. When the primary phenotype is time-to-event, it is natural to use survival models such as Cox's proportional hazards model instead of normal mixtures to model the phenotype distribution. An extra challenge for modeling time-to-event data is that the underlying population may consist of susceptible and nonsusceptible subjects. In this article, we propose a semiparametric proportional hazards mixture cure model which allows missing covariates. We discuss applications to quantitative trait loci (QTL) mapping when the primary trait is time-to-event from a population of mixed susceptibility. This model can be used to characterize QTL effects on both susceptibility and time-to-event distribution, and to estimate QTL location. The model can naturally incorporate covariate effects of other risk factors. Maximum likelihood estimates for the parameters in the model as well as their corresponding variance estimates can be obtained numerically using an EM-type algorithm. The proposed methods are assessed by simulations under practical settings and illustrated using a real data set containing survival times of mice after infection with Listeria monocytogenes. An extension to multiple intervals is also discussed.     

Two-level proportional hazards models

We extend the proportional hazards model to a two-level model with a random intercept term and random coefficients. The parameters in the multilevel model are estimated by a combination of EM and Newton-Raphson algorithms. Even for samples of 50 groups, this method produces estimators of the fixed effects coefficients that are approximately unbiased and normally distributed. Two different methods, observed information and profile likelihood information, will be used to estimate the standard errors. This work is motivated by the goal of understanding the determinants of contraceptive use among Nepalese women in the Chitwan Valley Family Study (Axinn, Barber, and Ghimire, 1997). We utilize a two-level hazard model to examine how education and access to education for children covary with the initiation of permanent contraceptive use.     

A proportional hazards cure model for the analysis of time to event with frequently unidentifiable causes

We propose a semiparametric method for the analysis of masked-cause failure data that are also subject to a cure. We present estimators for the failure time distribution, the cure rate, and the covariate effect on each of these, assuming a proportional hazards cure model for the time to event of interest and we use the expectation-maximization algorithm to conduct the likelihood maximization. The method is applied to data from a breast cancer clinical trial.     

Estimation in the cox proportional hazards model with left-truncated and interval-censored data

We show that the nonparametric maximum likelihood estimate (NPMLE) of the regression coefficient from the joint likelihood (of the regression coefficient and the baseline survival) works well for the Cox proportional hazards model with left-truncated and interval-censored data, but the NPMLE may underestimate the baseline survival. Two alternatives are also considered: first, the marginal likelihood approach by extending Satten (1996, Biometrika 83, 355-370) to truncated data, where the baseline distribution is eliminated as a nuisance parameter; and second, the monotone maximum likelihood estimate that maximizes the joint likelihood by assuming that the baseline distribution has a nondecreasing hazard function, which was originally proposed to overcome the underestimation of the survival from the NPMLE for left-truncated data without covariates (Tsai, 1988, Biometrika 75, 319-324). The bootstrap is proposed to draw inference. Simulations were conducted to assess their performance. The methods are applied to the Massachusetts Health Care Panel Study data set to compare the probabilities of losing functional independence for male and female seniors.     

Adaptive Lasso for Cox's proportional hazards model

We investigate the variable selection problem for Cox's proportionalhazards model, and propose a unified model selection and estimationprocedure with desired theoretical properties and computationalconvenience. The new method is based on a penalized log partiallikelihood with the adaptively weighted L₁ penalty on regressioncoefficients, providing what we call the adaptive Lasso estimator.The method incorporates different penalties for different coefficients:unimportant variables receive larger penalties than importantones, so that important variables tend to be retained in theselection process, whereas unimportant variables are more likelyto be dropped. Theoretical properties, such as consistency andrate of convergence of the estimator, are studied. We also showthat, with proper choice of regularization parameters, the proposedestimator has the oracle properties. The convex optimizationnature of the method leads to an efficient algorithm. Both simulatedand real examples show that the method performs competitively.     

Tests of proportional hazards and proportional odds models for grouped survival data

In this paper, we derive score test statistics to discriminate between proportional hazards and proportional odds models for grouped survival data. These models are embedded within a power family transformation in order to obtain the score tests. In simple cases, some small-sample results are obtained for the score statistics using Monte Carlo simulations. Score statistics have distributions well approximated by the chi-squared distribution. Real examples illustrate the proposed tests.     

Analysis of capture-recapture data with a Rasch-type model allowing for conditional dependence and multidimensionality

In this article, we show that, if subjects are assumed to be homogeneous within a finite set of latent classes, the basic restrictions of the Rasch model (conditional independence and unidimensionality) can be relaxed in a flexible way by simply adding appropriate columns to a basic design matrix. When discrete covariates are available so that subjects may be classified into strata, we show how a joint modeling approach can achieve greater parsimony. Parameter estimates may be obtained by maximizing the conditional likelihood (given the total number of captures) with a combined use of the EM and Fisher scoring algorithms. We also discuss a technique for obtaining confidence intervals for the size of the population under study based on the profile likelihood.     

Nonparametric estimation in a cure model with random cure times

Acute respiratory distress syndrome (ARDS) is a life-threatening acute condition that sometimes follows pneumonia or surgery. Patients who recover and leave the hospital are considered to have been cured at the time they leave the hospital. These data differ from typical data in which cure is a possibility: death times are not observed for patients who are cured and cure times are observed and vary among patients. Here we apply a competing risks model to these data and show it to be equivalent to a mixture model, the more common approach for cure data. Further, we derive an estimator for the variance of the cumulative incidence function from the competing risks model, and thus for the cure rate, based on elementary calculations. We compare our variance estimator to Gray's (1988, Annals of Statistics 16, 1140-1154) estimator, which is based on counting process theory. We find our estimator to be slightly more accurate in small samples. We apply these results to data from an ARDS clinical trial.     

Maximum likelihood methods for cure rate models with missing covariates

We propose maximum likelihood methods for parameter estimation for a novel class of semiparametric survival models with a cure fraction, in which the covariates are allowed to be missing. We allow the covariates to be either categorical or continuous and specify a parametric distribution for the covariates that is written as a sequence of one-dimensional conditional distributions. We propose a novel EM algorithm for maximum likelihood estimation and derive standard errors by using Louis's formula (Louis, 1982, Journal of the Royal Statistical Society, Series B 44, 226-233). Computational techniques using the Monte Carlo EM algorithm are discussed and implemented. A real data set involving a melanoma cancer clinical trial is examined in detail to demonstrate the methodology.     

Cox proportional hazards models with left truncation and time‐varying coefficient: Application of age at event as outcome in cohort studies

When analyzing time‐to‐event cohort data, two different ways of choosing a time scale have been discussed in the literature: time‐on‐study or age at onset of disease. One advantage of choosing the latter is interpretability of the hazard ratio as a function of age. To handle the analysis of age at onset in a principled manner, we present an analysis of the Cox Proportional Hazards model with time‐varying coefficient for left‐truncated and right‐censored data. In the analysis of Northern Manhattan Study (NOMAS) with age at onset of stroke as outcome, we demonstrate that well‐established risk factors may be important only around a certain age span and less established risk factors can have a strong effect in a certain age span.     

Analysis of current status data with missing covariates

Statistical inference based on right-censored data for the proportional hazards (PH) model with missing covariates has received considerable attention, but interval-censored or current status data with missing covariates has not yet been investigated. Our study is partly motivated by the analysis of fracture data from the 2005 National Health Interview Survey Original Database in Taiwan, where the occurrence of fractures was interval censored and the covariate osteoporosis was not reported for all residents. We assume that the data are realized from a PH model. A semiparametric maximum likelihood estimate implemented by a hybrid algorithm is proposed to analyze current status data with missing covariates. A comparison of the performance of our method with full-cohort analysis, complete-case analysis, and surrogate analysis is made via simulation with moderate sample sizes. The fracture data are then analyzed.