Use of Pseudo-observations in the Goodness-of-Fit Test for Gray's Time-Varying Coefficients Model

Survival analysis has been used to estimate underlying survival or failure probabilities and to estimate the effects of covariates on survival times. The Cox proportional hazards regression model is the most commonly used approach. However, in practical situations, the assumption of proportional hazards (PH) is often violated. The assumption does not hold, for example, in the presence of the time-varying effect of a covariate. Several methods have been proposed to estimate this time-varying effect via a time-varying coefficient. The Gray time-varying coefficients model (TVC) is an extension of the Cox PH model that employs penalized spline functions to estimate time-varying coefficients. Currently, there is no method available to assess the overall goodness-of-fit for the Gray TVC model. In this study, we propose a method based on pseudo-observations. By using pseudo-observations, we are able to calculate residuals for all individuals at all time points. This avoids concerns with the presence of censoring and allows us to apply the residual plots used in general linear regression models to assess the overall goodness of fit for censored survival regression models. Perme and Andersen used the pseudo-observations method to assess the fit for the Cox PH model. We extend their method to assess the fit for the Gray TVC model and illustrate how we applied this approach to assess the fit for a model that predicts posttransplant survival probability among children who were under the age of 12 years, had end-stage liver disease, and underwent liver transplantation between January 2005 and June 2010.The method has significant public health impact. The Cox PH model is the most cited regression method in medical research. When data violate the PH assumption, The Gray TVC model or an alternative should be used in order to obtain unbiased estimates on survival function and give correct inference on the relationship between potential covariates and survival. The proposed goodness-of-fit test offers a tool to investigate how well the model fits the data. If results show a lack of fit, further modification for the model is necessary in order to obtain more accurate estimates.