Non-Parametric Tests
Statistics

Non-parametric tests are used when assumptions required by the parametric counterpart tests are not met or are questionable. All tests involving ranked data are non-parametric.

Wilcoxon Signed-Ranks Test

This test evaluates whether a sample of n observations is drawn from a population in which the median equals a specific (hypothesized) value.

  • The test requires one numeric data column.
  • The output results include the null median (for the parent population), the sum of positive and negative rank differences, the Wilcoxon T statistic, critical T values, and p values (two-tailed and one-tailed) (see the left-hand side table below).

Wilcoxon Matched Pairs Signed-Ranks Test

This test evaluates whether or not the median of the difference scores equals zero.

  • The test requires data from two dependent variables.
  • The test output provides a table report and a percentile chart of the raw data. The report includes the sum of positive and negative rank differences, the Wilcoxon T statistic, critical T values, and p values (two-tailed and one-tailed) (see the right-hand side table and graph below).

Spearman's Rank-Order Correlation Coefficient (Spearman's ρ)

Spearman's ρ is the non-parametric analog of the Pearson product-moment correlation coefficient. The results or the former and latter are closely similar, as the Spearman correlation is calculated in a very similar manner as Pearson, except that Spearman first ranks the data.

    There are two different ways of calculating Spearman's ρ:

  • Converting each variable to ranks and calculating the Pearson correlation coefficient between the two sets of ranks
  • Converting each variable to ranks and for each observation pair, correcting the ties and calculating the difference between the ranks (this is the method used in Aabel)
  • The method requires two numeric variables.
  • The test results include the estimated Spearman's rank-order correlation coefficient (rs), the critical one-tailed and two-tailed rs (0.05) and rs (0.01), the t and z statistics and p values (two-tailed and one-tailed).

Kendall's Rank Correlation Coefficient (Kendall's τ)

Like Spearman's ρ, Kendall's τ is a non-parametric method of correlation between two variables, but has an advantage over Spearman's ρ: Kendall's τ also indicates the difference between the probability that the observed data are in the same order for the two variables vs. the probability that the observed data are in different orders for the two variables.

  • The method requires two numeric variables.
  • The test results include the estimated Kendall's τ, the critical one-tailed and two-tailed τ(0.05) and τ(0.01), z and p values (two-tailed and one-tailed).

Mann-Whitney U Test (Wilcoxon Ranks Sum Test)

This is the non-parametric analog of the unpaired samples t-test. The Mann-Whitney test (also known as the Wilcoxon ranks sum test and the Man-Whitney-Wilcoxon test) will be performed on ranked data and the hypothesis evaluated is whether or not the median of the difference scores equals zero.

  • The test requires two numeric variables (e.g., the particle counts from two processing systems used to clean wafers).
  • The test results include the Mann-Whitney U, the critical two-tailed and one-tailed U (0.05) and U (0.01). Aabel also provides the p values (from z approximation). However, if the sample size is not sufficiently large for the normal distribution to provide a reasonable approximation of the sampling distribution of U, a warning note ("not recommended") is displayed after the z in the table.

Kruskal-Wallis Test

The Kruskal-Wallis test is the non-parametric analog of a one-way ANOVA. The Kruskal-Wallis test is used to compare independent samples, and tests the hypothesis that several populations have the same continuous distribution, at least as far as their medians are concerned.

  • This method requires >=2 numeric variables. The number of observations for different samples can be the same or different.
  • The test result output includes the H and tie-corrected H, the critical χ2 (0.05) and χ2 (0.01) values, and p values.

Friedman Two-Way Analysis of Variance by Ranks

This is a non-parametric test used to compare dependent samples (or observations) that are repeated on the same subjects. This test is the non-parametric version of the repeated measures ANOVA, and evaluates the hypothesis that several populations have the same continuous distribution, at least as far as their medians are concerned.

  • This method requires >=2 numeric variables. The number of observations for different samples should be the same.
  • The test result output includes the χ2 and tie-corrected χ2, the critical χ2 (0.05) and χ2 (0.01) values.

Kolmogorov-Smirnov Test for a Single Sample

The Kolmogorov-Smirnov (goodness-of-fit) test is used to evaluate whether or not the sample of data is consistent with a specified distribution function. In Aabel, it evaluates whether the distribution of n scores that comprise a sample conform to a normal distribution.

  • The test requires one numeric data column.
  • The output includes the hypothesized mean and standard deviation, the estimated mean and standard deviation, the resulting M statistic, the sign of the M statistic, and the critical two-tailed and one-tailed M(0.05) and M(0.01) values.