Fitting the dependence model to the Liver Registration data set

To be able to assess the accuracy of the sensitivity analysis developed in Chapter 4, it is necessary to fit the dependence model described in Section 5.1 to the Liver Registration data set. Firstly, this will be done assuming only scalar parameters in each interval for the model for time to death and time to censoring. This will allow the accuracy of the sensitivity analysis applied in Section 4.3.1 to be assessed. Covariates will then be included to allow the accuracy of the sensitivity analyses applied in Section 4.3.2 to be assessed. This should indicate whether a sensitivity analysis for the linear predictor w(x) or a sensitivity analysis for the parameter vector θ is more accurate.

The parameter estimates obtained by fitting the dependence model to the Liver Reg- istration data set with scalar parameters in each interval for the model for time to death and time to censoring are given in Table 5.1. The parameter estimates obtained by fitting the corresponding independence model are included for comparison.

The sensitivity analysis from Section 4.3.1 which uses scalar parameters will now be reapplied using δ = 0.2698, which is the fitted value from the dependence model. The results of this sensitivity analysis are given in Table 5.2. The estimated values of ˆθ0.2698k− ˆ

θ0kfound using the sensitivity analysis are compared to the observed values of ˆθ0.2698k− ˆθ0k found by taking the difference of the values in Table 5.1. We can see that the sensitivity analysis overestimates the change in the parameter estimates for the first interval, but underestimates the change in the parameter estimates in the second and third intervals.

The dependence model including covariates will now be fitted to the Liver Registration data set. The explanatory variables for time to death used are age at registration, recipient

Parameter Estimate from the Estimate from the independence model dependence model

δ - 0.2698 θ1 -6.7799 -6.7571 θ2 -6.9056 -6.8206 θ3 -7.6375 -7.4458 γ1 -4.9189 -4.9137 γ2 -5.2320 -5.1008 γ3 -5.5695 -5.2806

Table 5.1: The parameter estimates obtained by fitting the dependence model to the Liver registration data set assuming scalar parameters in each interval for the models for time to death and time to censoring. The parameter estimates from the independence model are also given for comparison.

k Estimated value of Observed value of ˆ

θ0.2698k− ˆθ0k θˆ0.2698k− ˆθ0k

1 0.0370 0.0228

2 0.0819 0.0850

3 0.1812 0.1917

Table 5.2: The estimated values of ˆθ0.2698k− ˆθ0k found using the sensitivity analysis from 4.3.1 and the observed values of ˆθ0.2698k− ˆθ0k found using the values in Table 5.1.

ethnicity, primary liver disease category and UKELD score at registration. These are the same covariates included in the model for time to death in Section 4.3.2 when applying the sensitivity analysis for θ. However, only a scalar parameter is used for the model for time to censoring. This is because the model is already has a fairly large number of dimensions and including covariates for time to censoring would add enough extra dimensions to make the convergence of the algorithms in Section 5.1 too slow. This means that the results from this dependence model cannot be compared directly to the sensitivity analyses in Section 4.3.2, as they use covariates in their models for time to censoring.

The estimates for the dependence model obtained are given in Table 5.3. To see how much these vary from the estimates given by fitting the independence model, ˆθ0 is also included in Table 5.3. The sensitivity analyses for w(x) and θ are carried out using δ = 0.2769, which is the fitted value from the dependence model. This allows the direct comparison of the results of the sensitivity analyses with the results of the dependence model in Table 5.3.

The sensitivity analysis for w(x) requires the same vector of covariates to be used in the model for time to death and the model for time to censoring. This means that the sensitivity analysis that is used for comparison with the results of the fitted depen- dence model includes primary liver disease category, recipient age, recipient ethnicity and UKELD score as covariates in the models for time to death and time to censoring. The plot in Figure 5.1 shows the estimated change in the linear predictor over the range of ˆ

z0j(x) observed in each of the intervals for the data for several values of δ. The solid line is the sensitivity analysis for δ = 0.2769, which is the fitted value from the dependence model. The dashed lines are the sensitivity analyses for δ = 0.1377 and δ = 0.4162 which are the limits of 95% confidence interval for δ given in Table 5.3. These are included to show how the change in estimated linear predictor is greatly affected by the value of δ used.

The maximum change in the linear predictor estimated by the sensitivity analysis using δ = 0.2769 is 0.6248, but the dashed lines suggest that this change could be anywhere between 0.3107 and 0.9391. However, when calculating the difference between ˆwδ(x) and

ˆ

w0(x) using the parameter estimates in Table 5.3, the largest difference observed was 0.3868.

This result shows that for the Liver Registration data, the sensitivity analysis tends to overestimate the change in the estimated linear predictors. However, only a small number of the patients in the data will have a discrepancy that is large. We already know that the sensitivity analysis gives the largest changes in ˆwδ(x) and ˆw0(x) for the patients with the largest values of ˆz0j(x). From Figure 4.1, we know that only a small number of patients have values of ˆz0j(x) that are that large. So, for the majority of individuals in the Liver

Parameter Estimate Estimate Standard 95% Confidence from from Error Interval independence dependence model model δ - 0.2769 0.0711 (0.1377,0.4162) η - -5.0981 0.0221 (-5.1415,-5.0548) θ Intercept -20.5499 -20.0591 1.2500 (-22.5092,-17.6090) Age 0.0302 0.0298 0.0061 (0.0180,0.0417) Ethnicity - White 0.9787 1.0762 1.0105 (-0.9044,3.0568) Ethnicity - Asian -0.0273 0.0598 1.0495 (-1.9972,2.1169) Ethnicity - Black 0.9224 0.9775 1.1223 (-1.2221,3.1771) Ethnicity - Chinese -0.7265 -0.5046 1.4260 (-3.2994,2.2903) Ethnicity - Other 0 0 PLD - PBC -0.2318 -0.2548 0.3363 (-0.9140,0.4045) PLD - PSC -0.9330 -0.9391 0.3927 (-1.7089,-0.1694) PLD - ALD -0.4680 -0.4741 0.3083 (-1.0785,0.1302) PLD - AID -0.0243 -0.0693 0.3288 (-0.7137,0.5751) PLD - HCV 0.2319 0.1965 0.3248 (-0.4401,0.8331) PLD - HBV -0.4405 -0.4647 0.5793 (-1.6001,0.6707) PLD - Cancer -1.4646 -1.5034 0.7639 (-3.0007,-0.0062) PLD - Metabolic 0.6445 0.6279 0.3518 (-0.0616,1.3174) PLD - Other 0.3608 0.2560 0.3357 (-0.4019,0.9139) PLD - Acute 0 0 UKELD 0.1914 0.1858 0.0099 (0.1664,0.2053) j - Interval 1 0.2128 0.0241 0.1755 (-0.3198,0.3679) j - Interval 2 0.4775 0.3317 0.1575 (0.0231,0.6403) j - Interval 3 0 0

Table 5.3: Parameter estimates, standard errors and 95% confidence intervals for the dependence model when fitted to the Liver Registration data set. The parameter estimates obtained when fitting the independence model to the Liver Registration data set are included for comparison.

Figure 5.1: The results of the sensitivity analysis for the linear predictor for time to failure using the value of δ estimated by the dependence model

Registration data the discrepancy between the results of the sensitivity analysis and the change in ˆwδ(x) and ˆw0(x) using the results of the dependent model is small.

The results of the sensitivity analysis for θ are given in Table 5.4. The estimated values of ˆθ0.2769 − ˆθ0 found using the sensitivity analysis are compared to the observed values of ˆθ0.2769− ˆθ0 found by taking the difference of the parameter estimates in Table 5.3. We can see from Table 5.4 that we have mixed results concerning the accuracy of the sensitivity analysis. For most parameters the sensitivity analysis does correctly identify the direction of the change in the parameter estimates. However for patients with metabolic liver disease and white, Asian or black patients this is not the case. Even if the sensitivity analysis correctly identifies the direction of the change, then it may either overestimate or underestimate the magnitude of the change.

Approximate values of ˆθ0.2769 can be found by adding the estimated values of ˆθ0.2769− ˆ

θ0 given in Table 5.4 to the values of ˆθ0 from Table 5.3. These values of ˆθ0.2769 can then be used to find the change in the estimated linear predictor for T under this sensitivity analysis. This is done for each individual in the data set using the expression

ˆ w0.2769(xij) − ˆw0(xij) = ˆθ 0 0.2769xij − ˆθ 0 0xij.

The largest value of this change that is estimated by the sensitivity analysis for θ is 0.3869. This is very close to the observed change in the estimated linear predictor which was 0.3868. These results suggest that the sensitivity analysis for θ is more accurate than the sensitivity analysis for w(x). Therefore, the sensitivity analysis for θ should be used when we wish to apply a sensitivity analysis to a piecewise exponential model with covariates.

Parameter Estimated values of Observed value of ˆ θ0.2769− ˆθ0 θˆ0.2769− ˆθ0 Intercept 0.6085 0.4908 PLD - PBC -0.0317 -0.0230 PLD - PSC -0.0161 -0.0061 PLD - ALD -0.0110 -0.0061 PLD - AID -0.0280 -0.0450 PLD - HCV -0.0300 -0.0354 PLD - HBV -0.0279 -0.0242 PLD - Cancer -0.0303 -0.0388 PLD - Metabolic 0.0096 -0.0166 PLD - Other -0.0192 -0.1048 Ethnicity - White -0.0647 0.0975 Ethnicity - Asian -0.0402 0.0871 Ethnicity - Black -0.0901 0.0551 Ethnicity - Chinese 0.0019 0.2219 UKELD -0.0036 -0.0055 Age -0.0005 -0.0004 j - Interval 1 -0.2617 -0.1887 j - Interval 2 -0.2198 -0.1458

Table 5.4: Comparison of the estimated values of ˆθ0.2769− ˆθ0 found using the sensitivity analysis from 4.3.2 with the observed values of ˆθ0.2769− ˆθ0found using the values in Table 5.3.