4.4 Non-parametric tests

A non-parametric test uses limited assumptions, often the signs or ranks of observations rather than a normal model. It is useful when outliers, skew or ordinal data make a parametric test unsuitable.

State the null and alternative, preserve pairing or independence, and use the exact sampling distribution or critical region specified for the statistic.

For paired measurements, analyse the signs of within-pair differences rather than treating the two columns as independent samples.

“Non-parametric” does not mean assumption-free: independence, symmetry or a meaningful ordering may still be required.

For paired or one-sample data, the sign test counts positive and negative differences after removing zeros. Under a null median or no directional effect, the count follows a binomial model with p=0.5.

Choose the direction before counting, discard exact ties as specified, and use the smaller tail probability for a two-sided alternative only after defining the alternative.

With 9 non-zero paired differences, 8 positive results are assessed using X~Bin(9,0.5); the observed count is evidence about direction, not magnitude.

The sign test ignores how large differences are, and a tie is not automatically positive or negative.

To test a population median m₀, compare each observation with m₀ and count signs. Under H₀: median=m₀, the positive count has a Bin(n,0.5) distribution after ties are removed.

For H₁: median>m₀ use an upper tail; for median<m₀ use a lower tail; for a two-sided alternative use both tails. State the conclusion in the original units.

If 12 non-tied measurements contain 10 above m₀, calculate P(X≥10) for a one-sided test, then compare with α.

A rejection says the data are inconsistent with the stated median, not that every observation lies on one side or that the effect is practically large.

For matched observations, calculate a difference within each pair, then apply a sign or signed-rank procedure to those differences. Pairing removes between-subject variation only when the pairing is meaningful.

Keep the order of subtraction consistent, remove zero differences as required, and do not analyse the two samples as independent if the design is paired.

Before-and-after measurements on the same 10 people produce 10 differences; the test asks whether the typical change is zero, not whether the two raw columns have equal means.

Pairing is not merely putting two samples side by side; a random pairing can add noise and invalidate the intended interpretation.