Describe the non-parametric tests and their uses.

Non-parametric Tests

• When they are used — where the data are ordinal (pain scores, tumour grade, social class); where the data are nominal; where the distribution is markedly skewed or clearly not normal; where the sample is very small, so normality cannot be assessed at all; where there are extreme outliers; and where the data are censored or open-ended ("more than 10 years")
• Their advantages — fewer assumptions, so they are more widely applicable; usable with ordinal data, which parametric tests are not; robust to outliers, since ranks are unaffected by how extreme a value is; and simpler to compute
• Their disadvantage is the important one — they are less powerful than the equivalent parametric test when the parametric assumptions are met, because converting values to ranks discards information
• The practical rule follows directly — use a parametric test where its assumptions hold, and a non-parametric one where they do not; do not use a non-parametric test simply to avoid checking the distribution
• They also give NO direct estimate of the size of the effect in the original units, which is a real limitation for clinical interpretation
• The tests and their parametric equivalents should be learned as pairs, which makes them far easier to recall:
• MANN-whitney U test (Wilcoxon rank-sum) — the non-parametric equivalent of the unpaired t test, comparing two independent groups
• Wilcoxon signed-rank test — the equivalent of the paired t test, for two related samples
• Kruskal-WALLIS test — the equivalent of one-way ANOVA, for more than two independent groups
• Friedman test — the equivalent of repeated-measures ANOVA, for more than two related samples
• Spearman’S rank correlation — the equivalent of pearson correlation
• The sign test — the simplest of all, using only the direction of the difference within pairs, ignoring its magnitude
• Chi-square and FISHER’S exact test — for categorical data, and are themselves non-parametric
• McNEMAR’S test — for paired categorical data
• The principle ON which they work — the observations from both groups are ranked together, and the ranks are then compared. If the groups do not differ, the high and low ranks should be evenly distributed between them; a systematic clustering of high ranks in one group is what the test detects
• Reporting — non-parametric results should be presented with the median and interquartile range, not the mean and standard deviation, since it would be inconsistent to test on ranks and describe with a mean
• A confidence interval for the median or for the difference between medians should still be given wherever possible, since the need to quantify the effect does not disappear because the test is non-parametric
• Learn each test with its parametric equivalent, which reduces the whole topic to four couplets
• Report median and interquartile range, since testing on ranks and describing with a mean is inconsistent
• Non-parametric tests are less powerful when the parametric assumptions actually hold, because ranking discards information
• Do not use them to avoid checking the distribution; they are for when assumptions fail, not for when they are unexamined
• Ranks are unaffected by how extreme a value is, which is why these tests are robust to outliers
• They give no effect size in original units, which is a real limitation for clinical interpretation
• Still quote a confidence interval, since the need to quantify an effect does not vanish with the test choice
• Very small samples cannot be assessed for normality, which is itself a reason to choose a non-parametric test
• Open-ended data suit rank methods, since a value recorded as more than ten years still has a rank
• The sign test uses direction only, ignoring magnitude, and is the simplest of all these methods
• Ranks are assigned across both groups together, and clustering of high ranks in one group is what the test detects
• Chi-square is itself non-parametric, which students often forget when listing these tests
• Kruskal-Wallis handles more than two groups, and Friedman the related samples equivalent