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6.4a. For more about test statistic (6.5), see Hettmansperger (1984, pp. 224-226). Perhaps the earliest rank-based nonparametric test of the slope in simple linear regression was given by Terry (1952). His test is what is sometimes called a linear rank test with normal scores. The test in Section 6.4 is a linear rank test with Wilcoxon scores. See Hajek and Sidak (1967, Section III.3.!). 6.4h. Why does the standard deviation of ~ LS have to be estimated whereas the standard deviation of U is known The randomness of the random variable ~LS comes from the randomness of the y/s. As seen in Note 3.4a, the SO of ~LS depends on the SO of the y;'s, which is equal to the SO of the errors e i , which is 0". The parameter 0" is unknown and must be estimated. The randomness of the random variable U also comes from the randomness of the y;'s but the dependence of U on Yi is only through rank(y). When the null hypothesis f3 = 0 is true, the y/s can be regarded as a random sample from a single population, and so their ranks are simply a random permutation of the integers 1 through n. Therefore the distribution of the ranks of the y/s does not involve any unknown parameters, and hence neither does the distribution of U. SO(U) is the correct standard deviation of U only if the null hypothesis is assumed to be true. But est.SO(~LS) is a valid estimate of the standard deviation of ~LS regardless of whether the null hypothesis is true. 6.4c. To be conservative, the t distribution could be used to calculate the p-value for test statistic (6.5) instead of the standard normal distribution. For example, the test statistic for the forearm length data is It I = 3.973. By using the t distribution with 31 degrees of freedom, the p-value is 0.00038; whereas by using the standard normal distribution, the p-value is 0.00007. In either
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case, the p-value is very small and we conclude that there is a significant relationship between forearm length and height, but note that the p-value using the t distribution is larger, which is in the conservative direction. 6.4d. The ranks r i = rank( y) are simply a permutation of the integers 1 through n. So the average rank is the average of the integers 1 through n; that is, r = (1 + 2 + ... +n)/n = i(n + 1). If we regard the integers 1 through n as a sample, the sample variance is
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6.6a. The estimation of regression coefficients by minimizing (6.6) was proposed by Jaeckel (1972). Jaeckel's estimates are essentially the same as those of Jureckova (1971); they coincide for infinitely large samples. One approach to minimizing function (6.6) would be to take its partial derivatives with respect to the b/s, equate these derivatives to 0, and try to solve for the b/s. In general there may not be a solution that makes all the derivatives exactly 0, and so Jureckova chose the b/s to minimize the sum of the absolute values of the derivatives. Jaeckel and Jureckova considered arbitrary scores (see Note 6.30, not just the Wilcoxon scores that we use in this chapter. An estimate of f3 can be obtained by minimizing the function g(b) = Ea(r)e i , where e i = Yi - b'Xi' ri is the rank of e i , and a(r) is the score. Function (6.6) (or, in vector notation, (6.7 is the special case in which a(r) = r - i(n + 1). 6.6b. The function g(b) is known to be well behaved with respect to minimization. It is a convex function (see Hettmansperger, 1984, p. 234), which implies that the condition Vg(bO) = 0 is sufficient to ensure that bO minimizes the function. However, there does not necessarily exist a point bO satisfying Vg(bO) = 0, that is, at which all the partial derivatives exist and are o. A necessary and sufficient criterion for bO to minimize g(b) is that at bO, for each j = 1, ... , p, the left-hand partial derivative with respect to bj is nonpositive and the right-hand partial derivative is nonnegative. 6.6c. The derivative of g(bO + td) with respect to t, according to the chain rule, is (J/Jt)g(bO + td) = (Vg(bO + td 'd. At t = 0 it is (Vg(bO 'd. To simplify notation, let h = Vg(bO). We want to choose d so that h'd < O. Since h'h = Eh; > 0 (assuming h"* 0), we can choose d = -h. Also,
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