Firstly, IPCW KM estimators using each of the models described in Section 3.1.3 are fitted to the Liver Registration data set. Figures 3.1, 3.2 and 3.3, compare the IPCW KM estimators using Cox and Weibull models for censoring to the standard KM estimator of the marginal survival function. We see that all the plots in Figures 3.1, 3.2 and 3.3 give similar IPCW KM estimators that do not deviate greatly from the standard KM estimator. This suggests that the potentially informative censoring in the Liver Registration data set has little effect on the estimate of the survival function. This does not agree with the estimates of the survival function found in Chapter 2, which suggested that even a small amount of dependence between T and C would result in a fairly large change in the estimate of the survival function.
One possible reason why the IPCW KM estimator does not vary greatly from the standard KM estimator is that the dependence between T and C is not completely due
Figure 3.1: Plots comparing IPCW KM estimators with unweighted KM estimators, using Cox Model 1 and Weibull Model 1 for censoring respectively
Figure 3.2: Plots comparing IPCW KM estimators with unweighted KM estimators, using Cox Model 2 and Weibull Model 2 for censoring respectively
Figure 3.3: Plots comparing IPCW KM estimators with unweighted KM estimators, using Cox Model 3 and Weibull Model 3 for censoring respectively
to shared prognostic factors included in the model for time to censoring. There could be residual dependence caused by unmeasured prognostic factors. Scharfstein and Robins (2002) and Rotnitzky et al. (2007) developed methods that allow the effect of residual de- pendence on an estimator that assumes sequential ignorability of censoring to be assessed. This is covered in more detail in Section 3.7. Unfortunately, the estimator considered in Section 3.7 is not the IPCW KM estimator presented in Section 3.1.1, so the effect of possible residual dependence on the IPCW KM estimate of the survival function cannot be assessed.
However, this analysis using the IPCW KM estimator is fairly simplistic and does not allow for adjustment for significant covariates for time to failure. Therefore we fit IPCW Cox models for time to failure to the Liver Registration data set. These allow us to assess the effect of informative censoring on individual parameter estimates and also the estimated survival function for individuals in the data set.
Several IPCW Cox models for time to death are fitted to the data set. The same baseline covariates will be included in all the models for time for failure. These are primary liver disease category, ethnicity, age, UKELD score at time of registration, serum sodium at time of registration and INR at time of registration. However, different models are used for to time to censoring and the corresponding IPCW estimates for each model are presented, along with the unweighted estimates obtained by fitting the standard Cox model. The models for time to censoring that are used were discussed in Section 3.1.3.
obtained by fitting all these models using Cox models for censoring and Weibull models for censoring using both stabilised and unstabilised weights.
Figure 3.4: Point estimates and 95% confidence intervals for parameters in time to failure model, for unweighted Cox model and IPCW Cox model using Cox models 1, 2 and 3 for time to censoring respectively. All the weights used in IPCW estimates are stabilised.
We find that the IPCW estimates using stabilised weights, which are shown in Figure 3.4, the point estimates are slightly different from the standard point estimates, but gen- erally significant covariates do not become non-significant or vice versa. This is with the exception of some of the estimates for the Chinese level of ethnicity. Under the standard Cox model, this parameter estimate has wide bounds as there are only a small number of individuals with this ethnicity in the data set. However the use of weights here is anal- ogous to the use of sampling weights. This means that the number of observations with this ethnicity is being increased so there is less uncertainty about this parameter estimate. However, we see that for the IPCW estimates that use unstabilised weights, which can be seen in Figure 3.5, there are more changes from the standard estimates. Several different levels of the categorical variables that are significant under the standard model, become non-significant. However, these changes are likely to be caused by the heavy censoring in the data set making some of these unstabilised weights quite large.
Figure 3.5: Point estimates and 95% confidence intervals for parameters in time to failure model, for unweighted Cox model and IPCW Cox model using Cox models 1, 2 and 3 for time to censoring respectively. All the weights used in IPCW estimates are unstabilised.
Figure 3.6: Point estimates and 95% confidence intervals for parameters in time to failure model, for unweighted Cox model and IPCW Cox model using Weibull models 1, 2 and 3 for time to censoring respectively. All the weights used in IPCW estimates are stabilised.
Figure 3.7: Point estimates and 95% confidence intervals for parameters in time to failure model, for unweighted Cox model and IPCW Cox model using Weibull models 1, 2 and 3 for time to censoring respectively. All the weights used in IPCW estimates are unstabilised.
For the covariates and factor levels that remain significant when using an IPCW Cox model, we will examine the changes the estimated hazard ratios. There is a slight decrease in the point estimates of the hazard ratio for patients with metabolic liver disease when using an IPCW Cox model. This suggests that the standard Cox model slightly overesti- mates the hazard ratio for these patients. The hazard ratios for age, UKELD score, serum sodium and INR all also remain significant when using an IPCW Cox model. However, there is very little difference between the point estimates from the standard Cox model and the point estimates from the IPCW Cox models.
The IPCW estimates using Weibull models for time to censoring, with both stabilised and unstabilised weights, can be seen in Figures 3.6 and 3.7 respectively. The results in these two figures are very similar, suggesting that when using a Weibull proportionals hazards model for time to censoring it does not matter whether stabilised or unstabilised weights are used. The changes from the standard estimates are also similar to those observed in Figure 3.5, with some levels of categorical variables that were significant becoming non-significant.
Again, we examine the changes in the estimated hazard ratios for the covariates and factor levels that remain significant when using an IPCW Cox model. The results are very similar to those for the IPCW estimates given in Figure 3.5. There is slight decrease in the estimated hazard ratio for patients with metabolic liver disease, suggesting the standard Cox model slightly overestimates the hazard ratio for these patients. The hazard ratios for age, UKELD score, serum sodium and INR remain significant, with the exception of a couple of the estimated hazard ratios for serum sodium. Again there is very little change in the point estimates for these covariates.
Figures 3.4 to 3.7 show the effects of inverse probability of censoring weighting on the parameter estimates of the Cox model. We will now look at the effects that these changes in the parameter estimates can have on the survival functions for individuals in the data set.
Figure 3.8 compares the estimated survival function under the standard Cox model with the estimated survival function under the IPCW Cox model for the individual who had the largest observed value of ˆβTIP CW0xi − ˆβ0
0
Txi. The weights used for the IPCW estimates were unstabilised weights using Cox model 1 for time to censoring. We can see that there is a large difference between the two estimated survival functions. The estimated survival function under the standard model, shown by the solid line in Figure 3.8 does not fall below 0.9, whereas the estimated survival function under the IPCW Cox model, shown by the dashed line, has a median survival time of approximately 1200 days. The analyses carried out in this section show that using an IPCW version of the KM estimate of the survival function has little effect on the value of the estimated survival
Figure 3.8: Plot comparing the estimated survival function under the standard Cox model with the estimated survival function under the IPCW Cox model for the individual that has the largest observed value of ˆβTIP CW0xi− ˆβ0
0
Txi. The weights used are unstabilised weights using Cox model 1 for time to censoring.
function. However, if an IPCW Cox model is used, which allows for adjustment for significant covariates, then there can be a large effect on the estimated survival function for some individuals in the data set.