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Empirical cumulative lifetime distribution function is often required for selecting lifetime distribution. When some test items are censored from testing before failure, this function needs to be estimated, often via the approach of discrete nonparametric maximum likelihood estimation (DN-MLE). In this approach, this empirical function is expressed as a discrete set of failure-probability estimates. Kaplan and Meier used this approach and obtained a product-limit estimate for the survivor function, in terms exclusively of the hazard probabilities, and the equivalent failure-probability estimates. They cleverly expressed the likelihood function as the product of terms each of which involves only one hazard probability ease of derivation, but the estimates for failure probabilities are complex functions of hazard probabilities. Because there are no closed-form expressions for the failure probabilities, the estimates have been calculated numerically. More importantly, it has been difficult to study the behavior of the failure probability estimates, e.g., the standard errors, particularly when the sample size is not very large. This paper first derives closed-form expressions for the failure probabilities. For the special case of no censoring, the DN-MLE estimates for the failure probabilities are in closed forms and have an obvious, intuitive interpretation. However, the Kaplan–Meier failure-probability estimates for cases involving censored data defy interpretation and intuition. This paper then develops a simple algorithm that not only produces these estimates but also provides a clear, intuitive justification for the estimates. We prove that the algorithm indeed produces the DN-MLE estimates and demonstrate numerically their equivalence to the Kaplan–Meier-based estimates. We also provide an alternative algorithm.

This paper is concerned with testing for the equality of failure rates in a competing risks model with two risks. Three testing procedures are investigated, namely Score test, Likelihood Ratio test and Wald test. Wald test has been considered to be the most powerful in multivariate linear regression analysis.¹ However in our application, Wald test is the most efficient one when the failure rate of the second failure type is strictly smaller than the first failure type, otherwise the Score or the Likelihood Ratio test is preferred. This phenomenon is illustrated by data from a mechanical-switch life test.² The results has been extended to a k-competing risks model. A simulation study is also given to examine the performance of the three tests.

Pancreatic cancer is the fourth leading cause of cancer deaths in the United States with five-year survival rates less than 5% due to rare detection in early stages. Identification of genes that are directly correlated to pancreatic cancer survival is crucial for pancreatic cancer diagnostics and treatment. However, no existing GWAS or transcriptome studies are available for addressing this problem. We apply lasso penalized Cox regression to a transcriptome study to identify genes that are directly related to pancreatic cancer survival. This method is capable of handling the right censoring effect of survival times and the ultrahigh dimensionality of genetic data. A cyclic coordinate descent algorithm is employed to rapidly select the most relevant genes and eliminate the irrelevant ones. Twelve genes have been identified and verified to be directly correlated to pancreatic cancer survival time and can be used for the prediction of future patient's survival.

We propose a novel screening method targeting genotype interactions associated with disease risks. The proposed method extends the maximum entropy conditional probability model to address disease occurrences over time. Continuous occurrence times are grouped into intervals. The model estimates the conditional distribution over the disease occurrence intervals given individual genotypes by maximizing the corresponding entropy subject to constraints linking genotype interactions to time intervals. The EM algorithm is employed to handle observations with uncertainty, for which the disease occurrence is censored. Stepwise greedy search is proposed to screen a large number of candidate constraints. The minimum description length is employed to select the optimal set of constraints. Extensive simulations show that five or so quantile-dependent intervals are sufficient to categorize disease outcomes into different risk groups. Performance depends on sample size, number of genotypes, and minor allele frequencies. The proposed method outperforms the likelihood ratio test, Lasso, and a previous maximum entropy method with only binary (disease occurrence, non-occurrence) outcomes. Finally, a GWAS study for type 1 diabetes patients is used to illustrate our method. Novel one-genotype and two-genotype interactions associated with neuropathy are identified.

In this chapter, we give a selective review of the nonparametric modeling methods using Cox's type of models in survival analysis. We first introduce Cox's model (Cox 1972) and then study its variants in the direction of smoothing. The model fitting, variable selection, and hypothesis testing problems are addressed. A number of topics worthy of further study are given throughout this chapter.

Recent interest in the application of microarray technology focuses on relating gene expression profiles to censored survival outcome such as patients' overall survival time or time to cancer relapse. Due to the high-dimensional nature of the gene expression data, regularization becomes an effective approach for such analyses. In this chapter, we review several aspects of the recent development of penalized regression models for censored survival data with high-dimensional covariates, e.g. gene expressions. We first discuss the Cox proportional hazards model (Cox 1972) as the primary example and then the accelerated failure time model (Kalbfleisch and Prentice 2002) for further consideration.