Non-parametric tests

Further Maths · Further Probability & Statistics. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A t-test assumes the population is normal. A non-parametric test makes no such assumption about the shape of the distribution, so it is used when the population cannot be taken as normal or when the data are only ranks or ratings. These tests are about the median, not the mean.

The sign test uses only whether each value is above or below the hypothesised median. The Wilcoxon tests also use the sizes of the differences, through their ranks, so they use more of the information. In return the signed-rank tests need the distribution to be symmetrical, and the rank-sum test needs the two distributions to have the same shape.

Match the test to the design: one sample, paired samples (work with the differences), or two independent samples (rank-sum).

Rules to remember

Sign test: under H₀ the number of values above the median is B(n, ½); find the tail probability and compare it with the significance level. For large n use N(n/2, n/4) with a continuity correction
Wilcoxon signed-rank: find the differences from the median, rank them by size ignoring sign (1 = smallest), then P = sum of ranks of positive differences and Q = sum for negative ones
Signed-rank test statistic: T = the smaller of P and Q; check with P + Q = n(n + 1)/2. Reject H₀ if T ≤ the critical value
Signed-rank, large n: T is approximately N(n(n + 1)/4, n(n + 1)(2n + 1)/24)
Wilcoxon rank-sum (sample sizes m ≤ n): rank all m + n values together; R_m = sum of the ranks in the smaller sample; W = the smaller of R_m and m(n + m + 1) − R_m. Reject H₀ if W ≤ the critical value
Rank-sum, large samples: R_m is approximately N(m(n + m + 1)/2, mn(n + m + 1)/12)
Paired data: sign test or Wilcoxon matched-pairs signed-rank test on the differences. Two independent samples: Wilcoxon rank-sum test. The signed-rank tests need a symmetrical distribution (of the values or of the differences); the rank-sum test needs two distributions of the same shape

Common mistake

In the Wilcoxon tests a small statistic is the significant one, so H₀ is rejected when T (or W) is less than or equal to the critical value. Students who are used to t and χ² tests reject when the statistic is larger, which is the wrong way round.

Worked example

A random sample of 7 values is used to test H₀: median = 20 against H₁: median > 20 with a Wilcoxon signed-rank test. The values minus 20 are 5, −2, 9, 3, −1, 7 and 4. The critical value for n = 7 (one-tail, 5%) is 3. Carry out the test.

Sizes in order: 1, 2, 3, 4, 5, 7, 9 take ranks 1 to 7. The negative differences −1 and −2 have ranks 1 and 2.
Q = 1 + 2 = 3 and P = 3 + 4 + 5 + 6 + 7 = 25. Check: 3 + 25 = 28 = 7 × 8/2.
T = smaller of P and Q = 3. Since 3 ≤ 3, T is in the critical region.

Answer: T = 3; reject H₀. There is evidence at the 5% level that the median is greater than 20.

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