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Contrast parametric with non-parametric tests and recognise when each is appropriate

CFA® Level I · Quantitative Methods · Estimation and Hypothesis Testing · QM.16.a

The idea

A parametric test concerns a population parameter such as a mean or variance and relies on assumptions about the population's distribution; the z- and t-tests of means and the chi-square and F-tests of variances are examples. A non-parametric test either makes few distributional assumptions or tests something other than a parameter. Non-parametric tests are appropriate when the data are ranks rather than values, when a small sample comes from a clearly non-normal population, when outliers dominate, or when the question is itself about ranks or randomness. Examples include the Spearman rank correlation test, the chi-square test of independence for a contingency table, the Wilcoxon signed-rank test and the sign test. When parametric assumptions hold, the parametric test is preferred because it is more powerful.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which of the following is a non-parametric test?

  1. The Spearman rank correlation test — correct
  2. The t-test of a population meanWhy not: The t-test concerns a parameter and assumes a roughly normal population, so it is parametric.
  3. The F-test of two variancesWhy not: The F-test compares population variances and assumes normality, so it is parametric.

Spearman's test works on ranks and needs no assumption about the shape of the underlying distribution, which makes it non-parametric.

A harder one

An analyst has only the rank order of 12 fund managers on two criteria, not the underlying scores. To test whether the rankings are related, the most appropriate choice is:

  1. a t-test on the correlationWhy not: The t-test on Pearson correlation needs the actual values and a normality assumption, not ranks.
  2. a paired comparison t-testWhy not: A paired t-test compares means of differences in measured values, which ranks do not provide.
  3. a non-parametric test — correct

When data are ordinal (ranks only), parametric tests on means or Pearson correlation are not usable. A rank-based test such as Spearman's is the natural choice.

The hardest tier

Five funds are ranked on two measures. Their ranks on measure X are 1, 2, 3, 4, 5 and on measure Y are 2, 1, 4, 3, 5 respectively. The Spearman rank correlation coefficient is closest to:

  1. 0.97Why not: Left the factor of 6 out of the numerator of 6 x sum of d^2.
  2. 0.80 — correct
  3. 0.00Why not: Used n^2 - 1 = 24 as the denominator and forgot to multiply by n.

The rank differences are -1, 1, -1, 1, 0, so the sum of d^2 = 4. rs = 1 - 6 x 4 / [5 x (25 - 1)] = 1 - 24 / 120 = 0.80.

In the app: 6 practice questions on this outcome, a narrated walkthrough with diagrams, spaced review that brings it back before you forget it.

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