PROC CAPABILITY: Signed Rank Statistic

Signed Rank Statistic

The signed rank statistic $\text{[math]}$ is computed as

$\text{[math]}$

where $\text{[math]}$ is the rank of $\text{[math]}$ after discarding values of $\text{[math]}$ , and $\text{[math]}$ is the number of $\text{[math]}$ values not equal to $\text{[math]}$ . Average ranks are used for tied values.

If $\text{[math]}$ , the significance of $\text{[math]}$ is computed from the exact distribution of $\text{[math]}$ , where the distribution is a convolution of scaled binomial distributions. When $\text{[math]}$ , the significance of $\text{[math]}$ is computed by treating

$\text{[math]}$

as a Student $\text{[math]}$ variate with $\text{[math]}$ degrees of freedom. V is computed as

$\text{[math]}$

where the sum is over groups tied in absolute value and where $\text{[math]}$ is the number of values in the $\text{[math]}$ th group (Iman 1974, Conover 1980). The null hypothesis tested is that the mean (or median) is $\text{[math]}$ , assuming that the distribution is symmetric. Refer to Lehmann (1998).

PROC CAPABILITY and General Statements

Signed Rank Statistic