Introduction and notations

It is often of high interest to study simultaneously the times to several events; for example, in breast cancer, one can be interested in studying the time until a tumor appears for both breasts, and in this case, the times to failure for a female patient are correlated. Suppose that the sample of size n is composed of two groups, A and B, with respective sizes ha and hb , and that each subject may experience K > 1 events, possibly censored. For an easier understanding, let K = 2, but generalization to K > 2 is immediate. For k = 1, 2 and j = A, B, let A3k be the cumulative hazard function for event k in group j. The null hypothesis {AA = AB} can be tested with rank tests when there is no censoring, or with weighted logrank statistics in the censored case ([GEH65]; [MAN66]; [COX72]; [PETO72]; [PRE78]; [HF82]). But these test statistics can not be used if we are interested in testing H0 : {AA = AB, AA = AB } because of the dependence between the failure times. In 1984 Wei, L. J. and Lachin, J. M. [WL84] proposed a test statistic of H0 based on the marginal weighted logrank statistics, say LRk, for k = 1, 2; they proved that the vector (LR\,LR2)' converges in distribution under Ho to a normal law, with null expectation and with a variance covariance matrix for which they constructed a consistent estimator. But their simulations study showed a rather high first type error rate. In 1987, Pocock [PGT87] developed a family of tests for treatment comparison with multiple endpoints, including failure times, in this latter case using Wei and Lachin covariance estimator found in [WL84]. Then Wei, L. J., Lin, D. Y. and Weissfeld, L. [WLW89] extended in 1989 this marginal approach to construct proportional hazards model for multiple events, their method allowing multiple hypotheses testing on regression parameter estimates with Wald test statistics. We aimed to propose an improvement of weighted logrank test statistics for multiple events, by constructing a consistent estimator of the variance co-variance matrix of (LR\,LR2)' using martingales properties. We study the asymptotic distribution of the vector (LRi,LR2)' under Ho; then we produce the results of a simulations study performed to compare the observed first type error rate of our test statistic to those of [WL84] and [WLW89]. The last section reports the application of our method to data from a clinical trial studying efficacy of laser treatment on retinopathy depending on type of diabetes (juvenile or adult).

In group j (j = A,B), let (Tiji,T2ji) be the times to failure for subject i, independent and identically distributed for i = 1,...,nj with joint survival function Fj where Fj(xi,x2) = Pr {Tji > xi,T2ji > x2}, and with density fj. For k and j fixed, we also suppose the random variables Tkji independent and identically distributed, with marginal survival function F°k and with cumulative hazard function Ak.

Let (Ciji, C2ji) be the censoring times, independent of the failure times (Tiji,T2ji), independent and identically distributed with joint survival function G and with marginal survival function Gk equal in both groups A and B.

Then, for subject i, we only observe (Xiji,X2ji,Siji,S2ji) where for k = 1, 2 Xkji = (Tkji A Ckji) and Skji = 1{Tkji < Ckji}.

The maximum follow-up time is t < to, such that YkA(T) > 0, YkB(t) > 0.

The means n-1YkA, n—1YkB and n-1Yk converge uniformly to ykA = FAGk,

VkB = FB Gk and yk = PAVkA + PbVkB respectively, where pa and pb are the limits of nA/n and hb/n.

For k = 1, 2, consider the weighted logrank test statistic for event k

where Wk is a weight function that converges uniformly to a function wk on [0, t] (for example Wk constant and equals to 1 represents the logrank test, and Wk(x) = Yk(x) the Gehan test statistic).

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Diabetes 2

Diabetes 2

Diabetes is a disease that affects the way your body uses food. Normally, your body converts sugars, starches and other foods into a form of sugar called glucose. Your body uses glucose for fuel. The cells receive the glucose through the bloodstream. They then use insulin a hormone made by the pancreas to absorb the glucose, convert it into energy, and either use it or store it for later use. Learn more...

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