Survival Analysis: An Exact Method for Rare Events

Conventional asymptotic methods for survival analysis work well when sample sizes are at least moderately sufficient. When dealing with small sample sizes or rare events, the results from these methods have the potential to be inaccurate or misleading. To handle such data, an exact method is proposed and compared against two other methods: 1) the Cox proportional hazards model and 2) stratified logistic regression for discrete survival analysis data. Background and Motivation Survival analysis models are used for data in which the interest is time until a specified event. The event of interest could be heart attack, onset of disease, failure of a mechanical system, cancellation of a subscription service, employee termination, etc. Survival analysis models are used in a variety of applications, including biostatistics, social sciences, economics, and engineering. One important distinguishing characteristic of survival analysis models is the presence of censoring. Censoring arises when the event time is unknown, or if it is not known whether or not the event occurred. Survival analysis has been widely studied and applied across many areas of the biomedical and social sciences. There are many excellent text books containing more details about the application of survival analysis models. For example, Singer and Willett (2003) cover both discrete and continuous time models in Applied Longitudinal Data analysis; while Tutz and Schmid (2018) focus on discrete time models in their text Modeling Discrete Time-to-Event Data. Kleinbaum and Klein (2012) offer a self-study approach in Survival Analysis: A Self-Learning Text. As with other modeling approaches in statistics, some of the conventional methods that are used for survival analysis are hampered when sample sizes or event rates are small. As a result, there has been significant attention paid to alternatives for small or sparse samples. For example, the asymptotic log-rank test requires the hazards to be proportional, while the generalized Wilcoxon test is a good alternative when the proportional hazards assumption 1 is not met. Both tests require that groups have similar sample sizes, but are still valid in unequal follow-up situations. In situations where the group sizes are markedly different and follow-up is equal, complete permutation tests are available, including the early generalized Wilcoxon test by Gehan (1965), and the modified Savage test (Schemper, 1984). Heinze, Gnant, Schemper (2003) propose two exact tests that are suitable for use in cases where: group sizes are different, there are a low number of events, and groups have unequal follow-up times. Based on simulations, the exact test (conditional on follow-up) has been shown to be the best choice when follow-up is unequal. When follow-up is equal, the exact complete permutation has a slight advantage in terms of performance. These tests are designed to detect differences in survival curves, but do not provide estimates for the hazard rate. Wang, Lagakos, and Gray propose two types of permutation tests for accelerated failure time (AFT) models when sample sizes are small (2010); as interval estimates tend to perform poorly for small samples. The methods proposed rely on first imputing the survival and censoring times, after which the the permutation methods are applied. Based on a simulation study, the two methods have good Type I error and power properties. Additionally, the confidence intervals constructed under the AFT model have better coverage than the approach of Jin and others (2003) when used with small samples. Samuelsen (2003) proposes a method where survival data is analyzed using a conditional logistic model. So-called Type I and Type II censoring are considered, where type I censoring refers to an experiment that ends after a pre-specified number of years, and type II censoring refers to an experiment that ends after a pre-specified number of events have occurred. The results of his simulation study show that the type I censored data yields conservative coverage, and lower power than score and likelihood ratio tests; while the type II censored data also yields conservative coverage, although not as conservative as the type I results. Power was demonstrably better for the exact test than the Wald test. Additionally, coverage and power were evaluated with one covariate in the model. Rabinowitz, Betensky, and Tsiatis (2000) present an approach to survival analysis using a