Signed Rank Statistic

PROC CAPABILITY and General Statements

Signed Rank Statistic

The signed rank statistic S is computed as

$S =\sum_{ i:x_i \gt 0} r_i^+ - \frac{n (n+1)}4$

where r_i⁺ is the rank of |x_i| after discarding values of x_i = 0, and n is the number of nonzero x_i values. Average ranks are used for tied values.

If $n \leq 20$ , the significance of S is computed from the exact distribution of S, where the distribution is a convolution of scaled binomial distributions. When n > 20, the significance of S is computed by treating

$S \sqrt{ \frac{n - 1}{nV -S^2} }$

as a Student t variate with n - 1 degrees of freedom. V is computed as

$V = \frac{1}{24} n(n+1)(2n+1) - \frac{1}{48} \sum t_i(t_i+1)(t_i-1)$

where the sum is over groups tied in absolute value and where t_i is the number of values in the i^th group (Iman 1974, Conover 1980). The null hypothesis tested is that the mean (or median) is zero, assuming that the distribution is symmetric. Refer to Lehmann (1975).

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