NOTAS Y REFERENCIAS DEL CAPÍTULO

In this section the assumption in (3.11) will be compared to the assumption used in Scharfstein and Robins (2002). The aim of this is to make the interpretation of the assumption in (3.11) easier to understand. Scharfstein and Robins (2002) assume that the censoring process follows a proportional hazards model, so that the conditional hazard function for C can be expressed as

hC(c|T, T > c) = hC0(c) exp(q(c, T )), (3.24)

that is the conditional hazard for C given T is the baseline hazard multiplied by a function of T . The function q(c, T ) quantifies the dependence between T and C just after time c, for those who are still at risk at time c. This “censoring bias function” determines the way T enters the proportional hazards model for the cause-specific hazard of censoring.

So that the two assumptions can be compared, the corresponding conditional hazard function for the conditional density function in (3.11) needs to be found. The form of this conditional hazard function is given in Siannis et al (2005) and we shall now give the derivation of this term. Firstly, we use that

SC(c|T, γ, δ, θ) = Z ∞ c fC(c|T, γ, δ, θ)dc ' SC(c, γ)[1 − δi−1/2γ B(t, θ) ∂ ∂γHC(c, γ)],

which means that the conditional hazard can be expressed as hC(c|T, γ, δ, θ) = − ∂ ∂clog SC(c|T, γ, δ, θ) ' −∂ ∂c  log SC(c, γ) + log  1 − δi−1/2_γ B(t, θ) ∂ ∂γHC(c, γ)  (3.25)

The approximation log(1 + x) ' x is used to simplify the second term in (3.25), so that the conditional hazard becomes

hC(c|T, γ, δ, θ) ' − ∂ ∂clog SC(c, γ) − ∂ ∂c  − δi−1/2_γ B(t, θ) ∂ ∂γHC(c, γ)  .

This can be rearranged to give

hC(c|T, γ, δ, θ) ' hC(c, γ)  1 + δi−1/2_γ B(t, θ) ∂ ∂γlog hC(c, γ)  . (3.26)

To be able to compare (3.11) with (3.24), the conditional hazard in (3.11) needs to be expressed as a proportional hazards model, with the baseline hazard function being mul- tiplied by some function. To do this, the approximation ex ' 1 + x is used in (3.26), so that the conditional hazard function is now

hC(c, γ) exp  δi−1/2_γ B(T, θ) ∂ ∂γlog hC(c, γ)  . (3.27)

If (3.27) is compared with (3.24), then we can see the two hazard functions have a similar form. The baseline hazard in (3.24), has been replaced with a parametric baseline hazard in (3.27). Also, we see that the specification of q(c, T ) in (3.24) is the same as choosing δB(T, θ) in (3.27). This means that δB(T, θ) also quantifies the dependence between T and C just after time c and determines the way that T enters the proportional hazards model for censoring.

3.3.2 Application to the Liver Registration data set

This sensitivity analysis is now applied to the Liver Registration data set. Firstly, the sensitivity analysis will be performed on w(x) and then the sensitivity analysis for θ will be applied. Siannis et al. (2005) assumed exponential marginal models for T and C and Siannis (2004) used Weibull marginal models for T and C. When applying this method to the Liver Registration data set, Weibull marginal models are used as these are more flexible than exponential marginal models.

When applying the sensitivity analysis to w(x), the marginal density functions are given by

fT(t, w(x), ηT) = ew(x)ηTtηT−1exp(−ew(x)tηT) and fC(t, z(x), ηC) = ez(x)ηCtηC−1exp(−ez(x)tηC).

This means that the integrated hazard functions are

HT(t, w(x), ηT) = ew(x)tηT and

HC(t, z(x), ηC) = ez(x)tηC. (3.28) Here the scale parameters w(x) and z(x) are linear predictors that incorporate the follow- ing covariates: age at registration, ethnicity, primary liver disease category and UKELD score at registration. The same covariates need to be included in the models for time to death and time to censoring for this sensitivity analysis.

The sensitivity analysis will be conducted on the scale parameter for T , w(x), as this is the parameter of interest and the shape parameters, ηT and ηC, are treated as nuisance parameters. The scale parameter for C, z(x), is also treated as a nuisance parameter. If the integrated hazards in (3.28) are substituted into (3.23) then the sensitivity analysis equation becomes ˆ wδ(x) − ˆw0(x) ' δ Pn i=1 n ez(x)_tηT+ηC i − Zi(1 − ∆i)tηiT o Pn i=1t ηT i . (3.29)

This can be thought of as δ multiplied by a sensitivity index, U . As in (3.23), we have parameter estimates on the LHS of (3.29) and parameters on the RHS. To overcome this issue when applying the sensitivity analysis, z(x), ηT and ηC are replaced by their estimates from the Weibull proportional hazards model that assumes non-informative censoring. It is found that ˆηT 0 = 1.03 and ˆηC0 = 0.9297. The estimate for ηT was not found to be significantly different from one so an exponential model could be used for T , however the estimate for ηC was significantly different from one so the use of Weibull marginal models is justified.

As there are many different combinations of the covariates in the Liver Registration data set, ˆz0(x) takes a range of values so the sensitivity index needs to be computed over this range. The easiest way of displaying the results is to plot δU over the range of ˆz0(x), which is shown in Figure 3.9 for δ = 0.2 and 0.3. The range of values for ˆz0(x) used on the horizontal axis in Figure 3.9 is the observed range of ˆz0(x) for the Liver Registration data set. The largest values of ˆwδ(x) − ˆw0(x) are observed for patients with the largest values of ˆz0(x). These are the patients which have the greatest hazard of censoring. We see that for these individuals, the change in the estimated linear predictor seems large enough that results obtained assuming non-informative censoring could be misleading. However, to be sure of this the effect on a value of interest, such as the survival function of individuals in the data set, should be examined. When we apply the sensitivity analysis derived in Chapter 4 to the Liver Registration data set, this will be investigated.

The sensitivity analysis for θ will now be applied to the Liver Registration data set. Again Weibull marginal models are assumed for T and C. For simplicity, z(x) will be used

Figure 3.9: Plot showing δ times the sensitivity index, U , over the range of observed values for ˆz0(x) for the individuals in the Liver registration data set, using δ = 0.2 and 0.3.

as the scale parameter for C, rather than the vector γ. This means the marginal density functions are now given by

fT(t, θ, x, ηT) = eθ 0_x ηTtηT−1exp(−eθ 0_x tηT_{) and} fC(t, z(x), ηC) = ez(x)ηCtηC−1exp(−ez(x)tηC).

The integrated hazard functions are now

HT(t, θ, x, ηT) = eθ

0_x

tηT _and

HC(t, z(x), ηC) = ez(x)tηC. (3.30)

It is the vector of parameters for T , θ, that is of interest. So, ηT, ηC and z(x) will again be treated as nuisance parameters. For notational simplicity, it is assumed that the same covariate vector is used in both the model for time to death and the model for time to censoring. However, it is not a requirement for this sensitivity analysis. Therefore, age, ethnicity, primary liver disease category and UKELD score are used in the model for time to death and primary liver disease category, UKELD score, height and blood group are used in the model for time to censoring.

The sensitivity analysis equation in (3.21) will be used to carry out the sensitivity analysis for θ. When substituting the integrated hazard functions in (3.30) into (3.22), the expression for the kth component of rδ(θ) − r0(θ) becomes

n X i=1 xik n eθ0xi_ez(xi)_tηT+ηC i − Zi(1 − ∆i)e θ0xi_tηT i o . (3.31)

and the (k, l)th element of the information matrix i(θ, x) in (3.21) becomes n X i=1 xikxileθ 0_x i_tηT i .

We can see that in (3.31) we have the parameter vector θ as well as z(x), ηT and ηC. These all need to be replaced with their estimates from the Weibull proportional hazards model that assumes non-informative censoring.

Table 3.1 shows the estimated values of the components of ˆθδ− ˆθ0 for δ = 0.2 and δ = 0.3. We see that for some covariates there are positive changes in the parameter estimates, while others have negative changes in the parameter estimates. Positive values in Table 3.1 mean that the element of ˆθδfor that covariate is larger than the corresponding element of ˆθ0. So, this suggests that the hazard ratio of the covariate is being underestimated by the model assuming non-informative censoring. Conversely, negative values in Table 3.1 mean that the parameter estimate for the covariate from the model assuming informative censoring is smaller than the corresponding parameter estimate from the model assuming

non-informative censoring. Therefore, the sensitivity analysis is suggesting that the hazard ratio for these covariates are overestimated by the model that assumes δ = 0.

So, the sensitivity analysis for θ suggests the hazard ratio for patients with hepatitis B virus infection is being underestimated, whereas the hazard ratios for patients with other levels of primary liver disease are being overestimated. Patients of either white or black ethnic origin are having their hazard ratios overestimated, whereas the hazard ratios for patients of asian or oriental ethnic origin are being underestimated. The sensitivity analysis also suggests that the hazard ratios for UKELD score and age are being slightly overestimated by the model that assumes non-informative censoring.

The effects that the estimated changes in Table 3.1 have on the parameter estimates are shown in Table 3.2. The p-values of the estimates are also shown. These are all calcu- lated using the standard errors of the estimates from the model assuming non-informative censoring. This can be done as Siannis et al. (2005) show that

{Var(ˆθδ)}1/2' {Var(ˆθ0)}1/2+ O(δ2).

Only linear values of δ are considered in the sensitivity analysis so the standard error of the parameter estimate from the model assuming informative censoring can be approximated by the standard error of the parameter estimate from the model assuming non-informative censoring. This approximation should only be used if the value of δ is fairly small.