" . £ Because of the asymptotic normality of 3, it follows that under Ho, Q2 —►
Xi and Q3 Xk as n ^to. Q2 is proposed by WLW for situations where it is felt that the components of (3 are approximately equal, whereas Q3 is intended to be an omnibus test.
Another test of H0, proposed by Lin [LIN94], arises from maximization of (1) under the constraint that fa = fa = ■ ■ ■ = /3k = fa, i.e., by maximizing
If 3 denotes the maximizing value of fa, Lin [LIN94] shows that the test statistic
a is asymptotically Xi under H0, where a2 is the estimate of a2 = 2
obtained by replacing V and VD by the same estimators of these used by
WLW to estimate S. Li [LI97] proves that Q\ is asymptotically equivalent under Ho (and under Ha defined below) to a test, Q*, defined in the same way as Q2, but with weight c* = -.Yd \ . Thus, in studying the relative properties
of directional tests of Ho, we restrict attention to those based on an arbitrary linear combination of 3, say Qc = n(C>3C . It is easy to see that under H0, Qc —— x2 as n — to.
3 Asymptotic Properties of the Test Statistics under Contiguous Alternatives
In this section we derive the asymptotic distributions of the test statistics Q1, Q2, Q3 and Qc under a sequence of arbitrary contiguous alternatives to Ho. The results indicate the alternatives for which each test is asymptotically optimal, and provide the basis for their comparison in sections 5 and 6. Since Q1 is asymptotically equivalent to the linear combination test Q*, these tests are used interchangeably in this section.
The test statistics Q2 and Q3 introduced in the previous section were derived under the proportional hazard assumption Xk(t\Z) = Xk(t)[email protected] for k = 1, 2, ••• ,K. We now consider their behavior under an arbitrary alternative to H0. Consider the following sequence of alternatives to H0:
for k = 1, 2, • • • ,K. For simplicity we assume that the functions gk (t) are bounded and, without loss of generality, take supte^o,°]\gk(t)\ = 1- We further assume that the family of alternatives Ha is contiguous to Ho by taking a/na.k ^ 5k, where §1,62, • • • ,5k are fixed constants, where we suppress the dependency of ak on n for simplicity of notation. The special case of proportional hazards alternatives is obtained by taking gk (t) = 1, in which case 5k represents the limiting treatment group hazard ratio for the kth failure time.
Consider the estimator f, which arises from model (2). The asymptotic distribution of f3 under Ha is shown in Appendix I to be ^Jn,3 —> N(f, (£K=1 vk)-2l'V 1), where
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