13 Research methods, statistics, critical review and evidence-based practice13.9 Differential and analytic statistics
13.9.3 Parametric and non parametric inference
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Written by Resident Doctor Committee · Published
Describing a distribution
The normal distribution is symmetrical and bell-shaped, with mean, median and mode equal. Parametric tests such as the t-test assume this shape.
Negative skew has a tail stretching to the left, as with a ceiling effect, and pulls the mean below the median.
Measure | Definition | Note |
|---|---|---|
Variance | Average of the squared deviations from the mean | Its units are squared |
Standard deviation | Square root of the variance | Same units as the data |
Standard error of the mean | Standard deviation divided by the square root of the sample size | Measures the precision of a sample mean and shrinks as the sample grows |
What makes a method parametric
Parametric methods specify a model through a finite set of parameters. They also handle non-continuous data: logistic regression models binary outcomes and Poisson regression models counts.
Non-parametric methods work on ranks and do not assume normally distributed data, unlike t-tests, analysis of variance, least-squares models and Pearson's correlation.
Non-parametric methods still need an independent or paired structure and an outcome that can be ordered.
Assumptions of parametric tests
Parametric tests assume independent observations, normally distributed populations and similar variances.
Test | Assumptions | Not an assumption |
|---|---|---|
Student's t-test | Outcome normally distributed in the population sampled | Unpaired groups, unequal group sizes and small samples do not bar it |
Analysis of variance | Normal groups, equal variances, independent observations, interval or ratio outcome | Equal population means, which is the null hypothesis under test |
Welch's t-test drops the equal-variance assumption of the classical pooled t-test.
Parametric tests are not specially suited to very small samples whose distribution cannot be verified.
Rank tests are not automatically better in tiny samples: with three patients per group, an exact two-sided Mann-Whitney test cannot give a p value below 0.10.
A rank test does not remove dependence from repeated measures, matching or clustering.
Checking normality
Formal normality tests check whether data follow a normal distribution before a parametric test is used, with normality as the null hypothesis. They include Lilliefors, Shapiro-Wilk, Kolmogorov-Smirnov, Jarque-Bera and Anderson-Darling.
Small samples can miss important departures and large samples flag trivial ones, so inspect plots and influential values as well.
Transforming skewed data
Logarithmic, cube root, arcsine square root and Box-Cox transformations can pull skewed data towards normality before a parametric test.
The log-rank test is a hypothesis test, not a transformation.
A log transformation linearises an exponential relationship, so exponential growth becomes a straight line when the logarithm of the variable is plotted against time.
Rank-based tests
The Mann-Whitney U test is the rank-based counterpart of the unpaired, independent-samples t-test. It compares how the two groups are ordered (probability of superiority) and is not generally a test of medians.
The Wilcoxon signed rank test uses paired differences and assumes they are symmetrical.
The Kruskal-Wallis test compares rank distributions across groups but does not say which pair differs.