We close the section with a brief discussion of the choice of weight functions for the omnibus and ordered tests for proportionality with individual-level frailty. It is well known that, if hazards are proportional in the two samples, the logrank test for equality of hazards is the most optimal (see, for example, Andersen et al., 1992). However, this proportionality assumption does not hold under our alternative hypothesis, so there is no clear winner in terms of asymptotic relative e¢ciency (ARE).
Gill and Schumacher (1987) discuss the optimal choice of weight functions for their two sample tests of proportionality against convexity. In particu-lar, they show that the logrank weight function 1 = 12(1+2)¡1 in combination with Prentice’s Wilcoxon generalisation (Prentice, 1978) 2 =
12(1+2)¡1, whereb b is the Kaplan-Meier estimate of the survival function in the combined sample, is an optimal choice in terms of ARE un-der a couple of conditions. First, the hazard ratio unun-der proportionality is unity, which holds in our case (though not necessarily in theirs). Second, the proportion of at risk individuals in each sample is proportional. This is true if censoring is random (which we assume) and if frailty is independent of all other regressors, which is also true in our case. Therefore, the above combination of weight functions is optimal in our case.
It is perhaps possible that the lifetime0 can be chosen to achieve further optimality of the testing procedures suggested here. This is, however, a di¤erent problem and outside the scope of the current paper.
6 Discussion
In summary, the tests described in this paper add important tools to the armoury of a lifetime/ duration data analyst. Therefore, our work extends an important class of two sample tests for equality of hazards to a continuous covariate framework. This also shows that usual statistical treatment of lifetime/ duration data using counting processes are useful in analysing such continuous covariate situations.
The second important contribution of the paper is in extending tests for proportionality with respect to a continuous covariate against ordered al-ternatives to the case when there is individual level frailty with completely unrestricted distribution. Here, counting process arguments do not hold, but we can use empirical process theory to extend standard two sample tests to this setup. In conjunction with Bhattacharjee (2006), this paper therefore extends many of these two sample testing procedures to the continuous
co-variate setup, and thereby makes these tests more readily usable in real life econometric applications.
Though the discussion in this paper has largely focussed on a single con-tinuous covariate, the tests can be readily used in applications with multiple continuous covariates. Here, one can take either of two approaches. The …rst one is to test the absence of covariate dependence for one covariate, while modeling covariate dependence for other covariates more explicitly using ei-ther the Cox regression model or a model with age-varying covariate e¤ects.
Then, one can use the estimates of baseline cumulative hazard functions de-rived from the regression model (including the other covariates, but not the one under study) to construct the appropriate test statistics. Alternatively, one can jointly test for covariate dependence for two or more covariates.
APPENDIX
Proof of Theorem 1: It follows from standard counting process arguments (see, for example, Andersenet. al, 1992) that, under0, for= 1 ,
2(1 2) = X2
=1
Z 0
(1 2)()£
1¡( 1)f( 1) +( 2)g¡1¤
( )
where is the Kronecker delta function, and ( ) = 1 = 12 are the innovation martingales corresponding to the counting processes
( ) = 1 = 12.
Therefore,( ) = 1 = 12are independent Gaussian processes with zero means, independent increments and variance functions
[( )] = Z
0
¤ ( )
( ) and we have as¡! 1,
2(1 2) = 2(1 2) q d[2(1 2)]
¡! (01) = 1
The proof of the Theorem would follow, if it further holds that2(1 2)
= 1 are asymptotically independent.
This follows from a version of Rebolledo’s central limit theorem (see An-dersenet. al., 1992), noting that the innovation martingales corresponding
to components of a vector counting process are orthogonal, and the vector of these martingales asymptotically converge to a Gaussian martingale. A similar argument in a di¤erent context can be found in Bhattacharjee (2006).
It follows that
2 66 64
2(11 12)
2(21 22) ...
2(1 2) 3 77 75
¡! (0Ir)
whereIr is the identity matrix of order .
Proofs of (a), (b) and (c) follow.
¤ Proof of Corollary 1: Proof follows from the well known result in extreme value theory regarding the asymptotic distribution of the maximum of a sample of iid(01)variates (see, for example, Berman, 1992), and invoking the-method by noting that maxima and minima are continuous functions.
¤
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