The Nature of Survival Data

Survival data is any data that deals with time until the occurrence of any well-defined event such as:


• death from a specific cause
  
• the onset of disease
  
• readmission e.g. of a patient in whom disease had been in remission
  
• recovery of platelet count after bone marrow transplantation.

A common feature of survival data is the presence of censoring and truncation observations. Event times are said to be censored when subjects withdraw from the study (i.e. subjects survived to a certain time beyond which their status is unknown), or the study is completed before the end point is reached.

Methods of Analysis

Data that have censored or truncated observations are said to be incomplete, and the analysis of such data requires special techniques. Survival analysis, therefore, is a set of statistical methods for studying survival data. There are basically three methods that could be used in the analysis of survival data, namely fully parametric, semi-parametric and non-parametric.

Fully Parametric methods assume the knowledge of the distributions of the survival times (e.g.: Log-logistic, Exponential, Weibull, Gompertz). Non-parametric models make no assumptions of the survival times (e.g.: the Kaplan and Meier estimators). The semi-parametric models assume a parametric form for the effects of the explanatory variables but make no assumptions as per the distributions of the survival times.

Time Independent and Dependent Covariates

Covariates whose values do not change over time are said to be time independent, whereas covariates whose values are subject to change with time are called time-dependent covariates. Time-dependent variables can be used to model the effects of subjects transferring from one treatment group to another. Consider the hip fracture study of the Centre (MCHP). Some of the variables in this study could be modeled as time-dependent where interest is in determining the effects of surgery delay on survival or mortality.

The Cox Proportional Hazard (PH) Model

The most widely used model in the analysis of survival data is the Cox proportional hazard model. This is an example of semi-parametric models and can be used in analyzing time independent variables. When time independent covariates are modeled, the only assumption PH makes is that the effect of each covariate is the same at all time points. This implies that the validity of the results based on PH model depends on the non-violation of this assumption. This thus calls for testing for the non-violation of the assumption in any analysis that we do.

The test for the non-violation of PH assumption can either be done formally or graphically. The formal test is done by interacting each covariate of interest with the log(time) variable. Using log(time) instead of time variable ensures that there is no numerical overflow. A non-significant coefficient in the interaction covariate means that the PH assumption is satisfied.

In addition to the formal tests above, it is recommended to also graphically examine the residuals, which can be done using the Schoenfeld and/or weighted Schoenfeld residuals. The weighted Schoenfeld residuals method is preferred if there are tied observations. It is also good to formally carry out a linear-trend test on the residuals. This is done by examining the Spearman and Pearson correlation coefficients. A non-significant p-value for the correlation coefficients means there is no trend.

The Cox Non-Proportional Hazard Model

The Cox non-proportional hazard model is an extension of the PH model. This model (non-PH) is used in analyzing time-dependent covariates. For this model, there is no need to carry out any tests for the violation of the model, as the model makes no assumptions with respect to covariate change over time.

SAS

All the models described above can easily be done in SAS e.g. for the fully parametric models use PROC LIFEREG, for the non-parametric use PROC LIFETEST and for the semi-parametric (PH models) use PROC PHREG.