Qi QI Q2 Qs

Bivariate Exponential Gumbel 1960

Noncentrality Parameter

Fig. 1. Power of Qopt versus Q3 for Different K

Noncentrality Parameter

Fig. 1. Power of Qopt versus Q3 for Different K

References

[VKVN02] Andersen,P.K., Gill, R.D.: Cox's regression model for counting processes: a large sample study. The Annals of Statistics, 10, 1100— 20 (1981)

[VKVN02] Cai, J., Prentice, R.L.: Estimating Equations for Hazard Ratio Parameters Based on Correlated Failure Time Data. Biometrika, 82, 151-64 (1995)

[VKVN02] Cook, R.J., Lawless, J.F., Nadeau, C.: Robust Tests for Treatment Comparisons Based on Recurrent Event. Biometrics, 52, 55771 (1996)

[VKVN02] Cox, D.R.: Regression Models and Life Tables. J. R. Statist. Soc. B, 34, 187-220 (1972)

[VKVN02] Dolin, R., Amato, D.A., Fischl, M.A., et al.: Zidovudine Compared with Didanosine in Patients with Advanced HIV Type 1 Infection and Little or No Previous Experience with Zidovudine. Archives of Internal Medicine, 155 (9), 961-74 (1995)

[VKVN02] Gumbel,E.J.: Bivariate Exponential Distribtions. JASA 55, 698707 (1960)

[VKVN02] Hauser, W.A., Rich, S.S., Lee, J.R-J., et al.: Risk of Recurrent Seizures after Two Unprovoked Seizures. New England Journal of Medicine, 338, 429-34 (1998) [VKVN02] Hughes, M.: Power Considerations for Clinical Trials Using Multivariate Time to Event Data. Statistical in Medicine, 16, 865-82 (1997)

[VKVN02] Kahn, J., Lagakos, S.W., Richman, D.D., et al.: A Controlled Trial Comparing Continued Zidovudine with Didanosine in Human Immunodeficiency Virus Infection. New England Journal of Medicine, 327, 581-7 (1992) [LI97] Li, Q.H.: The Relationship between Directional and Omnibus Tests for a Vector Parameter. Doctoral Thesis, Harvard University, Massachusetts (1997). [VKVN02] Li, Q.H., Lagakos, S.W.: Comparisons of Test Statistics Arising from Marginal Analyses of Multivariate Survival Data. Lifetime Data Analysis, 10, 389-405 (2004) [VKVN02] Liang, K.Y., Self, S.G., Chang, Y.C.: Modeling Marginal Hazards in Multivariate Failure Time Data. J. Roy. Statist. Soc. B, 55, 441-53 (1993)

[VKVN02] Lin, D.Y.: Cox Regression of Multivariate Failure Time Data: the

Marginal Approach. Statistical in Medicine, 13, 2233-47 (1994) [VKVN02] Lin, J.S., Wei, L.J.: Linear Regression Analysis for Multivariate

Failure Time Observations. JASA, 87, 1091-7 (1992) [VKVN02] Prentice, R.L., Williams, B.J., Peterson, A.V.: On the Regression Analysis of Multivariate Failure Time Data. Biometrika, 68, 373-9 (1981)

[VKVN02] Richman, D., Grimes, J., Lagakos, S.: Effect of Stage of Disease and Drug Dose on Zidovudine Susceptibilities of Isolates of Human Immunodeficiency Virus. J. Acquired Immune Deficiency Syndromes, 3, 743-6 (1990) [VKVN02] Wei, L.J., Lin, D.Y., Weissfeld, L.: Regression Analysis of Multi-variate Incomplete Failure Time Data by Modeling Marginal Distributions. JASA, 84, 1065-73 (1989) [VKVN02] Yang, Y., Ying, Z.: Marginal Proportional Hazards Models for MultipleEvent-time Data. Biometrika, 88, 581-6 (2001)

Appendix I Asymptotic Normality of WLW Method under General Contiguous Alternatives

Let Yki(t) = 1(Xki > t) and Nki(t) = 1(Xki < t,Aki = 1), and define the functions S{k](P,t), Vk(p,t), s^)(^,t) and vk(p,t) for r = 0,1, 2 by

Define the martingale Mki(t) as

First consider f3, the maximizing solution of equation (2). Under the regularity conditions, it can be shown that f3 converges to f3, the solution of the following equation:

Under contiguous alternatives, the above equation becomes

which has its solution at f = 0. Let /3(n) denote the estimate of f when the alternatives are the sequence of general contiguous alternatives defined in Section 3. Using the above argument, it also follows that f(n) 0 as n — to.

To determine the asymptotic distribution of ^Jn3(n under the general contiguous alternative, we first derive the asymptotic distribution of the score statistic U (f(n)), where fj(n) =

ELiT Vk (0, t)sk (0, t)Xk(t)dt Rewriting U (P{n)), we have

The first term of right hand side of the second equal sign is martingale. It can be shown that this term converges to the N(0,1'V 1) distribution and that the second term converge to 0 in probability as n — to. Thus, as n — to, n

Tests for Multivariate Survival Data 317 n-2 U (3(n)) —► N(0, 1V1) , where V is the KxK matrix with (k, k) element vk (3k ) = / Vk (3k ,t)sf\3k,t)Ak (t)dt, Jo and with (k, w) element (k = w)

vkw(3k,3w) = E[f °{z— ^l3^ }dMki][i°{zi— sWyw,t] }dMWi] (k = w). Jo sf(3k,t) Jo sW0)(3w ,t)

Letting

1 EK=i fo°° Vk(0,t)sk0)(0,t)Ak(t)dt , and noting that 3(n) — 0, it follows that

Was this article helpful?

0 0

Post a comment