By Azriel Rosenfeld

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An1 , j . We are interested in particular in three different kinds of monotone trends, with the same trend structure in each season. This means that for each j we have the same slopes. With the slope of ai, j = iθ we achieve a linear trend, where the parameter θ controls √ the straight line. We also consider a concave case with ai, j = θ n1 i, and a convex case with ai, j = θ i2 /n1 , so that all trends increase to θ n1 . We consider sample sizes n ∈ {12, 24, 32, 48, 64, 96, 120} and splittings into k ∈ {1, 4, 8, 12, 16, 24, 32} groups whenever n1 = n/k is an integer.

Hirsch et al. [8] state that already for k = 12 and n j = 3 the normal approximation of S j works well. They also claim that their test is robust against seasonality and departures from normality, but not robust against dependence. Hirsch and Slack [9] develop a test for HR , which performs better than S if the data are autocorrelated. This test uses estimates of the covariances between two seasons based on Spearman’s rank correlation coefficient. The estimated covariances are used to correct the variance of S in the normal approximation.

In order to reflect the situation of many clinical studies in which the control group contains more observations than the others, the designs (50, 20, 20) for three groups and (50, 20, 20, 20) for four groups were chosen. 4. Shifts of the survival functions For generating the differences between the survival functions according to the alternative in (1) the following location and scale shifts were used: k (t) = S(t − θlik ) and Slo,li k Ssc,li (t) = S t exp(−θlik ) ψk with θlik = a √li . 5, 1), Exp(1), Ga(3, 1), N(3, 1) or LogN(0, 1) given above.

### An Introduction to Algebraic Structures by Azriel Rosenfeld

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