Estimating the product‐moment correlation in samples with censoring on both variables

Bivariate samples may be subject to censoring of both random variables. For example, for two toxins measured in batches of wheat grain, there may be specific detection limits. Alternatively, censoring may be incomplete over a certain domain, with the probability of detection depending on the toxin level. In either case, data are not missing at random, and the missing data pattern bears some information on the parameters of the underlying model (informative missingness), which can be exploited for a fully efficient analysis. Estimation (after suitable data transformation) of the correlation in such samples is the subject of the present paper. We consider several estimators. The first is based on the tetrachoric correlation. It is simple to compute, but does not exploit the full information. The other two estimators exploit all information and use full maximum likelihood, but involve heavier computations. The one assumes fixed detection limits, while the other involves a logistic model for the probability of detection. For a real data set, a logistic model for the probability of detection fitted markedly better than a model with fixed detection limits, suggesting that censoring is not complete.     

A mixture model for occupational exposure mean testing with a limit of detection

Information from detectable exposure measurements randomly sampled from a left-truncated log-normal distribution may be used to evaluate the distribution of nondetectable values that fall below an analytic limit of detection. If the proportion of nondetects is larger than expected under log normality, alternative models to account for these unobserved data should be considered. We discuss one such model that incorporates a mixture of true zero exposures and a log-normal distribution with possible left censoring, previously considered in a different context by Moulton and Halsey (1995, Biometrics 51, 1570-1578). A particular relationship is demonstrated between maximum likelihood parameter estimates based on this mixture model and those assuming either left-truncated or left-censored data. These results emphasize the need for caution when choosing a model to fit data involving nondetectable values. A one-sided likelihood ratio test for comparing mean exposure under the mixture model to an occupational exposure limit is then developed and evaluated via simulations. An example demonstrates the potential impact of specifying an incorrect model for the nondetectable values.     

Survival analysis in clinical trials: past developments and future directions

The field of survival analysis emerged in the 20th century and experienced tremendous growth during the latter half of the century. The developments in this field that have had the most profound impact on clinical trials are the Kaplan-Meier (1958, Journal of the American Statistical Association 53, 457-481) method for estimating the survival function, the log-rank statistic (Mantel, 1966, Cancer Chemotherapy Report 50, 163-170) for comparing two survival distributions, and the Cox (1972, Journal of the Royal Statistical Society, Series B 34, 187-220) proportional hazards model for quantifying the effects of covariates on the survival time. The counting-process martingale theory pioneered by Aalen (1975, Statistical inference for a family of counting processes, Ph.D. dissertation, University of California, Berkeley) provides a unified framework for studying the small- and large-sample properties of survival analysis statistics. Significant progress has been achieved and further developments are expected in many other areas, including the accelerated failure time model, multivariate failure time data, interval-censored data, dependent censoring, dynamic treatment regimes and causal inference, joint modeling of failure time and longitudinal data, and Baysian methods.     

Impact of nonignorable coarsening on Bayesian inference

The coarse data model of Heitjan and Rubin (1991) generalizes the missing data model of Rubin (1976) to cover other forms of incompleteness such as censoring and grouping. The model has 2 components: an ideal data model describing the distribution of the quantity of interest and a coarsening mechanism that describes a distribution over degrees of coarsening given the ideal data. The coarsening mechanism is said to be nonignorable when the degree of coarsening depends on an incompletely observed ideal outcome, in which case failure to properly account for it can spoil inferences. A theme in recent research is to measure sensitivity to nonignorability by evaluating the effect of a small departure from ignorability on the maximum likelihood estimate (MLE) of a parameter of the ideal data model. One such construct is the "index of local sensitivity to nonignorability" (ISNI) (Troxel and others, 2004), which is the derivative of the MLE with respect to a nonignorability parameter evaluated at the ignorable model. In this paper, we adapt ISNI to Bayesian modeling by instead defining it as the derivative of the posterior expectation. We propose the application of ISNI as a first step in judging the robustness of a Bayesian analysis to nonignorable coarsening. We derive formulas for a range of models and apply the method to evaluate sensitivity to nonignorable coarsening in 2 real data examples, one involving missing CD4 counts in an HIV trial and the other involving potentially informatively censored relapse times in a leukemia trial.