17.5.1.2 Algorithms (One-Sample Wilcoxon Signed Rank Test)
The Wilcoxon Sign Rank test is used to replace the one sample t-test, when the normality is questionable. So it requires looser conditions than the one sample t-test and has a wider use than the t-test.
a) For each \(x_i\,\!\) , for\(i=1,2,\ldots ,n\) , the signed difference\(d_i=x_i-\mu _0\,\!\) is found,where \(\mu _0\,\!\)is a given test value for the median of the sample.
b) Ignore the cases where \(d_i=0\,\!\). Rank the rest of\(\left| d_i\right| \), use\(r_i\) as its rank. Pay attention to that any tied values of\(\left| d_i\right| \) are assigned the average of the tied ranks. For example, three? \(\left| d_i\right| \)ranked as 7 8 9 are ties, then their rank is (7+8+9) /3=8.?
c) To each rank is affixed the sign of \(d_i\,\!\) to which it corresponds. Let \(s_i=sign(d_i)r_i\,\!\)
d) The sum of the positive-signed ranks is calculated as
\[W_1=\sum_{s_i>0}s_i\]
Our null hypothesis is that the population median has a specific value \(\mu _0\,\!\). We test the null hypothesis against the two-sided alternative hypothesis that the population does not have a median value \(\mu _0\,\!\). The confidence interval is converted to hypothesis-test form. The test is a one-sample Wilcoxon Sign Rank test, and it is defined as:
| \[H_0\] | \[\mu =\mu _0\,\!\] |
|---|---|
| \[H_1\] | \[\mu \neq \mu _0\] |
| Test Statistic | \(z=\frac{(W-\frac{n_1(n_1+1)}4)-\frac 12\cdot sign(W-\frac{n_1(n_1+1)}4)}{\sqrt{\frac 14\cdot\sum_{i=1}^n S_i^2}}\)
Where \(W\,\!\),\(s_i\,\!\) is said above \(n_1\,\!\), and is the number of non-zero \(d_i\,\!\), . |
| Significance Level \(\alpha \,\!\): | The most commonly used value for \(\alpha \,\!\) is 0.05. |
| Critical Region: | Reject the null hypothesis that the median is a specified value, \(\mu _0\,\!\), if
\(\left| z\right| >Z_{\alpha /2}\),where Z~N(0,1) Because for large sample, for example the size of the population is more than 50, sized the distribution of is approximately standard normal. |
For more details of the algorithm, please refer to nag_wilcoxon_test (g08agc)