By J. McDonald, N. Weiss
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Additional info for A Course in Real Analysis
Ni , to be the independent random variables of the survival times to occur with outcomes t11 , . . ,tknk . Further let Ci j , i = 1, . . , k, j = 1, . . , ni , be independent random variables with outcomes c11 , . . , cknk , where Ci j reflects the censoring time corresponding to the survival time Ti j . For each of the k groups it is assumed that the survival times Ti1 , . . , Tink of the group populations and the corresponding censoring times Ci1 , . . ,Cink follow the same distribution respectively.
28) Cox and Stuart  show that N2 leads to the best weighted sign test with respect to the efficiency of a sign test of HR . The linear rank test statistics R1 and R2 and the procedure S of Kendall compare all pairs of observations, while in (28) each observation is taken only for one comparison. Using (25) we get under H0 ⎧ 2 ⎨ nj k , n j even E(N2 ) = with E(N2, j ) = n82 −1 ∑ E(N2, j ) ⎩ j j=1 , n odd 8 k and Var(N2 ) = ∑ j=1 1 2 24 n j (n j − 1) . j (29) On Nonparametric Tests for Trend Detection 27 Cox and Stuart  also introduce a best unweighted sign test, which can be formulated for HR as follows N3 = k νj j=1 i=1 ∑ N3, j with N3, j = ∑ Ii,n j −ν j +i, j .
Here the deviation of observed and expected events of the groups are set into contrast. An analogous multiple contrast test to (21) with step contrasts for test problem (1) can be constructed on the basis of the statistic LT M(b ) = max(LT(b ) , . . , LT(b 1 k−1 ) ), (23) where bh = −b˜ h , h = 1, . . , k − 1. 1). The determination of the distribution of such a maximum statistic is very elaborate and therefore not accomplished here. Actually, the distribution of a statistic does not have to be known when a test is to carry out.