Regression Analysis with Right Censoring

In practice, patients may drop out from the study or do not develop the event of interest during the study period. Therefore, is often subject to right censoring. In this chapter, we discuss how the aforementioned methods adjust for the presence of censoring. In Section 4.1, we review three ways of modification for the moment-based estimators. In Section 4.2, we review the likelihood method in presence of censoring. Suppose that under model (1.4), is subject to censoring by C with the survival function

T 4.1 Moment-based Inference

The chosen response variables discussed in Chapter 3 are not completely observed. To handle this problem, two useful techniques for analyzing missing data, namely the weighting and imputation approaches, are frequently used.

Now we illustrate the technique of weighting. For the response variable , a natural proxy under censoring is , which however is biased.

(^{T t}

Since is often unknown, the Kaplan-Meier estimator is a suitable candidate to replace . Specifically,

( )^t

4.1.A The pairwise order indicator as the chosen response Notice that

( )

This implies that

( )

if the denominator is not zero. See Appendix 2 for the details.

The estimating equation in (3.1) can be modified as

( ) ( )

Despite its simplicity and convenience, the weighting approach has a series drawback.

First of all, equation (4.2) produces (asymptotically) unbiased result only if the censoring support lies within the support of T . Specifically define sup{t:G

( )

t 0}

. However, this assumption rarely holds in practice since the study period is often limited which makes

( )^{T 0}

G _i =

c

T τ

τ > . This situation is developed in Figure (4.1).

Figure 4.1 Problem for the weighting technique

To overcome this problem, Fine, Ying and Wei (1998) suggested to impose a truncation point t₀ (Figure 4.2) such that

( )

( )

( )

j j j i

X G

X ,t X I

2

0 , 1

min ≥ δ =

,

where G

( )

t₀ >0 and is a prespecified constant satisfying. t₀

Figure 4.2 Imposing a truncation point to overcome the problem The corresponding expected value for the adjusted response is given by

(

2

) _{( )} (

0

)

which is a function of and . See Appendix 3 for the details. Furthermore instead of using the first moment condition, Fine et al. (1998) proposed to use the least square principle by minimizing the objective function:

η h

( )

t₀

which leads to the estimating equation

( ) ( ) _{( )}

For estimating h

( )

t₀ , they also proposed another estimating function.

Subramanian (2004) proposed a different way of modifying equation (3.1). The idea of Subramanian is to replace the original response I(T_i ≥T_j) by an estimator of its nonparametric estimation. However, this method assumes that the covariate Z takes discrete values.

( )

The Kaplan-Meier estimator can be applied to estimate ^S( )^{t|Z such}

( )

∏

Therefore, Subramanian develop its estimating equation,

_{∑∑ ∫} ( ) ( ) ( )

(4.4) also vulnerable to the tail problem since the Kaplan-Meier estimator can not catch the tail information either if

(

t|Z_i

τ > . Therefore, Subramanian used the same technique, imposing the truncation point t₀, to develop the modified estimating equation,

( )

^{( ) h( )t}

( )

^{^}

( )

^*

( )

^0.

4.1.B The at-risk process as the chosen response

Recall 3.1.B, Cai, Wei and Wilcox (2000) suggested to use Y

( )

t as the response. Its expected value under the model is

( )

^{t Z φ}

(

^h

( )

^{t Z η}

)

S | = ^-1 + ^T .

Under right censoring data, the corresponding response variable is I

(

X_i ≥ . Thus, we can t

)

derived the expectation:

( )

[I X t ] (T t C T ) h( )t Z G( )t E _i ≥ =Pr _i ≥ , _i > _i =ϕ⁻¹{ + _i^Tη} . Cai et al. modify the equation (3.2a) as

[

( )

{

( )

t }

( )

t ] 0, ( , ) . (4.5a)

Note that (4.5) provides a set of equations for being the observed values of . If

t T_i (i =1,...,n)

η is one-dimensional, there are n+1 unknown parameters in (4.5). Therefore we need one more equation. Cai et al. (2000) suggested the following equation

( ) ( ) ( )

where )(τ_a,τ_b is a re-specified range that contains enough data information. Solve equations (4.5a) and (4.5b) iteratively. The following numerical operation is the same as 3.1.B we

mentioned.

4.1.C The counting process as the chosen response

With censoring data structure, using the estimating equation based on counting process is easily modifying. We would not change the formation we mentioned in 3.1.B. That is to say, it is very generalized method in constructing the estimating equation in linear transformation model.

4.2 Likelihood Inference

The likelihood function in (3.6) can be extended to the censoring situation as follows:

( )

Since direct maximization is impossible, how to handle the nuisance function h(.) is the key.

4.2.1 Partial Likelihood – Cox model

Here we illustrate the way Cox (1972, 1975) used to handle the nuisance baseline hazard function ^λ₀

( )

^t under the model ^λ

( )

^t|Z = ^λ0

( )

^t ×^exp

( )

^ZTη . At time , the probability that the same appears in both the numerator and denominator and hence gets cancelled out. Thus the above conditional probability is only the function of . The so-called partial likelihood can be written as

can be written as

number of observed failure events.

∑=

4.2.2 Conditional Profile Likelihood

The amazing cancellation for the Cox partial likelihood does not happen to the more general class of transformation models. Therefore if the likelihood approach is pursued, the nuisance function has to be dealt with directly.

For the general transformation model in (1.4),

( )

^Pr

{ ( ) } ( ( ) )

^,

Pr T >t|Z = z = ε>ht +z^Tη =ϕ^-1 h t +Z^Tη

Chen, H. Y. (2001) proposed a likelihood approach for the case-cohort study, the covariate has an unknown distribution

Z

( )^{z =Pr}(^Z≤z)

π , which is a more complex data structure than that considered in the thesis. Now we organize his method based on our data structure. By writing the full likelihood as

( ) ( )

Chen suggested to express the function in terms of and the marginal survival distribution of , such that

η T

^R

( )

^{t =Pr}

(

^{T ≥t}

)

⁼

∫

^ϕ-1

(

^h

( )

^{t +ZTη}

)

^dπ

( )

^z , (4.7) where π( )^{z =Pr}(^Z≤z) is the distribution function of Z . The motivation of the above transformation is that R

( )

t can be estimated by the Kaplan-Meier estimator

( )

}

The distribution function

( )

t

R η h(.) H(.)

()^.

π can also be estimated explicitly by .

Based on (4.7), one can derive the relationship between and

∑

= focus of the thesis so that we do not state the details. Finally, after the transformation, can be estimated by maximizing the following profile likelihood function:

η

This approach is very complicated and difficult to implement. The validity of the resulting estimator depends on whether the suggested transformation has to be a one-to-one mapping.

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