Survival data censoring

In the next section we will discuss Kaplan-Meier curves, which are used both to display the data and also to enable the calculation of summary statistics. We will then cover the logrank and Gehan-Wilcoxon tests which are simple two group comparisons for censored survival data (akin to the unpaired t-test), and then extend these ideas to incorporate centre effects and also allow the inclusion of baseline covariates. 

Kaplan and Meier (1958) introduced a methodology for estimating, from censored survival data, the probability of being event-free as a function of time. If the event is death then we are estimating the probability of surviving and the resultant plots of the estimated probability of surviving as a function of time are called either Kaplan-Meier (KM) curves or survival curves. 

Proportional Hazards Model Likelihood for Censored Survival Data... 

The proportional hazards model is used to regress censored survival data on when the values of a set of predictor variables are known for each observation. For each observation we record f j, its time of death, Wi the censoring variable which equals 1 if we observe the true death ti, and 0 when the observation is censored and all we know the true lifetime is greater than ti, and the values of the predictors Xji,..., Xip for that observation. 

In the sections that follow, terminologies and functions used to characterize survival data are first explained, followed by the application of nonlinear mixed effects modeling to the analysis of nonrandomly censored ordered categorical longitudinal data with application to analgesic trials. 

The combination of censoring and differential follow-up creates some unusual difficulties in the analysis of such data that cannot be handled properly by standard statistical methods. Thus, a different approach called survival analysis or censored survival analysis was developed for the analysis of such data. 

When analyzing survival data, summary statistics may not have the desired statistical properties, such as unbiasedness, because of possible censoring. The sample mean, for instance, is no longer an unbiased estimator of the population mean (of survival time). Thus, other methods are needed for presenting such data. An approach would be estimating the underlying true distribution. With the distribution estimated, it is then possible to estimate other quantities of interest such as median or mean. The distribution of the random variable T can be described by the usual cumulative distribution function... 

The Kaplan-Meier estimates produce a step function for each group and are plotted over the lifetime of the animals. Planned, accidentally killed, and lost animals are censored. Moribund deaths are considered to be treatment related. A graphical representation of Kaplan-Meier estimates provide excellent interpretation of survival adjusted data except in the cases where the curves cross between two or more groups. When the curves cross and change direction, no meaningful interpretation of the data can be made by any statistical method because proportional odds characteristic is totally lost over time. This would be a rare case where treatment initially produces more tumor or death and then, due to repair or other mechanisms, becomes beneficial. 

A statistical technique used to test the significance of differences between the survival curves associated with two different treatments. It is often used to analyze survival (life vs. death) data when there are censored observations (observations that are unknown because a subject has not been in the study long enough for the outcome to be observed) or to analyze the effects of different treatment procedures. ... 

In many cases an endpoint directly measures time from the point of randomisation to some well-defined event, for example time to death (survival time) in oncology or time to rash healing in Herpes Zoster. The data from such an endpoint invariably has a special feature, known as censoring. For example, suppose the times to death for a group of patients on a particular treatment in a 24 month oncology study are as follows ... 

It is this specific feature that has led to the development of special methods to deal with data of this kind. If censoring were not present then we would probably just takes logs of the patient survival times and undertake the unpaired t-test or its extension ANCOVA to compare our treatments. Note that the survival times, by definition, are always positive and frequently the distribution is positively skewed so taking logs would often be successful in recovering normality. 

Sometimes, these data are presented in a shorter table that displays only those time points at which an individual had an event or was censored, and thus the only values of time for which the probability of survival changes. It is more common, however, to see analyses of this type displayed graphically. The Kaplan-Meier estimate of the survival distribution is displayed for both groups in Figure 8.3. The survival curves displayed in the figure are termed "step functions" because of their appearance. We return to the interpretation of Figure 8.3 after we have fully specified the survival distribution function. 

In some cases, the exact time at which an event occurs is not known, but the event is known to have occurred between two recorded times. Such cases are termed interval censored. This type of censoring is present in the analgesic trial example presented later in this chapter. Survival time analysis is better suited than logistic analysis to the analysis of interval or right censored data. 

Likelihood Function. For prospective cohort data, the likelihood function is the product of individual likelihoods over aU the subjects in the cohort. Assuming that participants are cancer free at their age of entry the study (ac ), and that we censor individuals in case of death from other causes or in case they survive cancer until the end of follow-up, the individual likelihoods are given by... 

Kaplan-Meier procedure. A metliod of estimating survival probabilities from data on survival times for individuals in a cohort which can handle the case where a number of the individuals are still alive, so that their eventual survival times are censored. 

Survival analysis. The analysis of time to event data in particular, but not exclusively, where the event is death. A common feature of such data is that they are very skewed and that there are many censored values. Survival analysis is one of the single most important topics in medical statistics, although its importance to pharmaceutical statistics is, because of the nature of the trials usually run in drug development, relatively less important than the contents of standard textbooks on medical statistics might suggest. 

When we have eensored survival times data, and we relate the linear predictor to the hazard function we have the proportional hazards model. The function BayesCPH draws a random sample from the posterior distribution for the proportional hazards model. First, the function finds an approximate normal likelihood function for the proportional hazards model. The (multivariate) normal likelihood matches the mean to the maximum likelihood estimator found using iteratively reweighted least squares. Details of this are found in Myers et al. (2002) and Jennrich (1995). The covariance matrix is found that matches the curvature of the likelihood function at its maximum. The approximate normal posterior by applying the usual normal updating formulas with a normal conjugate prior. If we used this as the candidate distribution, it may be that the tails of true posterior are heavier than the candidate distribution. This would mean that the accepted values would not be a sample from the true posterior because the tails would not be adequately represented. Assuming that y is the Poisson censored response vector, time is time, and x is a vector of covariates then... 

Klein, J. P., and Moeschbeiger, M. L. (2003), Survival Analysis Techniques for Censored and Truncated Data, Second Edition, Springer, New York. 

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