The analysis of the of the model residuals allows the verification of its assumptions. The main goal of the analysis in Cox regression is the localization of outliers and the study of hazard proportionality. Typically, in regression models residuals are calculated as the differences of the observed and predicted values of the dependent variable. However, in the case of censored values such a method of determining the residuals is not appropriate. In the program we can analyze residuals described as: Martingale, deviance, and Schoenfeld. The residuals can be drawn with respect to time or independent variables.

Hazard proportionality assumption

A number of graphical methods for evaluating the goodness of fit of the proportional hazard model have been created (Lee and Wang 2003\cite{lee_wang}). The most widely used are the methods based on the model residuals. As in the case of other graphical methods of evaluating hazard proportionality this one is a subjective method. For the assumption of proportional hazard to be fulfilled, the residuals should not form any pattern with respect to time but should be randomly distributed around value 0.

• Martingale – the residuals can be interpreted as a difference in time between the observed number of failure events and their number predicted by the model. The value of the expected residuals is 0 but they have a diagonal distribution which makes it more difficult to interpret the graph (they are in the range of to 1).
• Deviance – similarly to martingale, asymptotically they obtain value 0 but are distributed symmetrically around zero with standard deviation equal to 1 when the model is appropriate. The deviance value is positive when the studied object survives for a shorter period of time than the one expected on the basis of the model, and negative when that period is longer. The analysis of those residuals is used in the study of the proportionality of the hazard but it is mainly a tool for identifying outliers. In the residuals report those of them which are further than 3 standard deviations away from 0 are marked in red.
• Schoenfeld – the residuals are calculated separately for each independent variable and only defined for complete observations. For each independent variable the sum of Shoenfeld residuals and their expected value is 0. An advantage of presenting the residuals with respect to time for each variable is the possibility of identifying a variable which does not fulfill, in the model, the assumption of hazard proportionality. That is the variable for which the graph of the residuals forms a systematic pattern (usually the studied area is the linear dependence of the residuals on time). An even distribution of points with respect to value 0 shows the lack of dependence of the residuals on time, i.e. the fulfillment of the assumption of hazard proportionality by a given variable in the model.

If the assumption of hazard proportionality is not fulfilled for any of the variables in Cox model, one possible solution is to make Cox's analyses separately for each level of that variable.