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4.4 Non-parametric tests

Syllabus
9231–2028–2029
Topic
4.4
Level
A2

Learning objectives

4.4.1Non-parametric tests• understand the idea of a non-parametric test and appreciate situations in which such a test might be useful e.g. when sampling from a population which cannot be assumed to be normally distributed.4.4.2The basis of the sign test• understand the basis of the sign test, the Wilcoxon signed-rank test and the Wilcoxon rank-sum test Including knowledge that Wilcoxon tests are valid only for symmetrical distributions.4.4.3Single-sample sign tests• use a single-sample sign test and a single- sample Wilcoxon signed-rank test to test a hypothesis concerning a population median Including the use of normal approximations where appropriate. Questions will not involve tied ranks or observations equal to the population median value being tested.4.4.4Paired non-parametric tests• use a paired-sample sign test, a Wilcoxon matched-pairs signed-rank test and a Wilcoxon rank-sum test, as appropriate, to test for identity of populations. Including the use of normal approximations where appropriate. Questions will not involve tied ranks or zero-difference pairs.

Non-parametric tests use signs or ranks when a normal model is not credible

feature parametric mean test non-parametric sign/rank test
main numerical information original magnitudes signs or order ranks
typical target mean/model parameter median, paired shift or identity of distributions
distributional demand often normality and variance conditions fewer shape conditions, but still design/test-specific assumptions
useful when model assumptions are credible normality is doubtful, data are skewed/outlier-prone or only ordering is reliable

Choose from the sampling design first: one sample, matched pairs or two independent samples. Then identify whether only direction is defensible or whether ranks of magnitudes can be used. State population hypotheses before calculating a statistic.

Signs and ranks reduce sensitivity to extreme magnitudes and avoid a normal-mean model, but they discard some metric information. When a valid parametric model holds, that discarded information can make a non-parametric test less powerful.

Non-parametric does not mean assumption-free. Randomness and independence or valid pairing remain essential, and the Wilcoxon tests in this syllabus require symmetrical distributions.

Sign, signed-rank and rank-sum tests retain different information

test construct under H0H_0 statistic basis key condition
sign signs of observations-minus-median or paired differences positive count B∼Bin⁡(n,1/2)B\sim\operatorname{Bin}(n,1/2) independent signs/valid pairs
Wilcoxon signed-rank rank absolute differences, restore signs positive and negative rank sums symmetric difference distribution
Wilcoxon rank-sum pool two independent samples and rank all values rank sum for a named sample independent samples; symmetry condition in this syllabus

For $n$ signed ranks with no zero differences or ties, the positive-rank sum $W^+$ has, under $H_0$,E(W^+)=\frac{n(n+1)}4,\qquad \operatorname{Var}(W^+)=\frac{n(n+1)(2n+1)}{24}.Thesesupportanormalapproximationwhenappropriate.These support a normal approximation when appropriate.

For independent samples of sizes $n_1,n_2$ and $N=n_1+n_2$, the rank sum $W_1$ hasE(W_1)=\frac{n_1(N+1)}2,\qquad \operatorname{Var}(W_1)=\frac{n_1n_2(N+1)}{12}.

Signed-rank and rank-sum are different tests: the first ranks within-pair or one-sample absolute differences, while the second ranks two independent samples together. The syllabus excludes tied ranks and zero differences in application questions.

A single-sample median test chooses between signs and signed ranks

method process for testing median m0m_0 information used
sign test record signs of xi−m0x_i-m_0 and use B∼Bin⁡(n,1/2)B\sim\operatorname{Bin}(n,1/2) direction only
signed-rank rank ∣xi−m0∣|x_i-m_0|, restore signs and sum ranks direction plus ordered magnitude; requires symmetry

Test $H_0:m= m_0$ against $H_1:m>m_0$. If 10 of 12 observations exceed $m_0$, then under $H_0$P(B\ge10)=\frac{\binom{12}{10}+\binom{12}{11}+\binom{12}{12}}{2^{12}}=\frac{79}{4096}=0.0193.At5At 5%, reject $H_0$.

For a large sign test, $B$ may be approximated byN\left(\frac n2,\frac n4\right),using a continuity correction. For signed-rank, use the null mean and variance from the previous card and correct the discrete rank-sum boundary by $0.5$ when using a normal approximation.

The alternative determines the tail before counting. These are tests about a population median or symmetric location, not a mean. Syllabus application questions contain no observations equal to the tested median and no tied ranks.

Matched designs test differences; independent designs test pooled ranks

data/design appropriate test statistic
paired, direction only paired sign number of positive differences
paired, symmetric differences Wilcoxon matched-pairs signed-rank signed rank sum of within-pair differences
two independent samples, syllabus symmetry condition met Wilcoxon rank-sum rank sum for a named sample after pooling

For independent samples $n_1=20,n_2=25$, let $W_1=560$ and $N=45$. Under identical populations,E(W_1)=20(46)/2=460,\operatorname{Var}(W_1)=20(25)(46)/12=1916.67.Foranupper−tailnormalapproximation,continuitycorrectiongivesFor an upper-tail normal approximation, continuity correction givesz=\frac{559.5-460}{\sqrt{1916.67}}=2.27.

Since 2.27 exceeds the 5% one-tail critical value 1.645, reject the null in the direction attached to sample 1's high ranks. For paired tests, define every difference in one order before applying the sign or signed-rank procedure.

Rank-sum is not a test for paired data, and matched-pairs signed-rank is not formed by ranking the two columns separately. Questions in this syllabus avoid tied ranks and zero-difference pairs; use exact tables instead of a normal approximation when the sample sizes make that appropriate.