# Non-Parametric Tests

Parametric tests are either based on a normal distribution or on, e.g., t,t or χ2 distributions, which are related to and can be derived from normal-theory-based procedures. That is, the parametric tests require that a sample/group analyzed is taken from a population that meets the normality assumption 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). A Textbook Example: This textbook example is from Sheskin, D.J. (2004): A physician evaluates the hypothesis that the median number of hours each of his patients visit during the year equals 5 (i.e., the null hypothesis states: Θ = 5; the alternative hypothesis states: Θ = 5; ­ 5). He obtains the values from 10 randomly selected patients to test the hypothesis. The test results indicates: With Σ- > Sigma;+, but T > critical T at both 0.05 and 0.01 significance levels, the directional alternative hypothesis (H1: Θ > 5) is not supported. >With Σ- > Sigma;+, and T > critical T at both 0.05 and 0.01 significance levels, the directional alternative hypothesis (H1: Θ < 5) is not supported. With T > critical T at both 0.05 and 0.01 significance levels, the non-directional alternative hypothesis is not supported. The null hypothesis, which states the sample of ten subjects is derived from a population with a median value Θ = 5, is supported.

The Correction for Continuity for the Normal Approximation:

• The continuity correction is used when using normal distribution to estimate a discrete distribution such as Wilcoxon distribution. This correction is based on the premise that a normal approximation of an underlying discrete distribution will inflate the Type I error rate.
• When the correction for continuity is applied to a normal approximation of an underlying discrete distribution, the computed absolute value for z will be slightly less than that of the uncorrected counterpart.

### Wilcoxon Matched Pairs Signed-Ranks Test

 This test evaluates whether or not the median of the difference scores equals zero. The corresponding parametric test is the paired samples t-Test. The test requires data from two dependent samples (e.g., measurements on the same subjects under different conditions). 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). The image below shows data from a clinical study regarding number of angina attacks are recorded for the same subjects while on placebo and on active treatment. The right-hand side image displays the corresponding Wilcoxon matched pairs signed-ranks test results. Test results above indicate that at the 0.05 significance level, there is a real tendency for patients to have fewer attacks while on active treatment.

### 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), t statistic, Z statistic and the corresponding p values (two-tailed and one-tailed).

Example:

• Published data of Bland, J.M., Mutoka, C., and Hutt, M.S.R. (1977), from a study of the geographical distribution of a tumor, Kaposi's sarcoma, in mainland Tanzania is shown in the left-hand side image below. The aim of the study is too evaluate if there is a relationship between the observed (reported) incidences of Kaposi's sarcoma and access to health centers.
• The right-hand side image below displays the corresponding output from the Spearman's correlation test. Based on interpretation of the test results, the null hypothesis (H0: &rhoS = 0 can not be rejected, i.e., there is no relationship between the observed (reported) incidences of Kaposi's sarcoma and access to health centers. That is, there is no monotonous relationship between the two variables.

### 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), t statistic, Z statistic and the corresponding p values (two-tailed and one-tailed). The right-hand side image shows the Kendall's &tau report from the same data used or the Spearman's ρ above.

### 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. Example: Two processing systems were used to clean wafers. The data represent the (coded) particle counts A and particle counts B. The null hypothesis is that the two independent samples represent populations with the same median values, i.e., there is no significant difference between the two processing systems; the alternative hypothesis is that there is a difference. The test results include the Mann-Whitney U, the critical two-tailed and one-tailed U (0.05) and U (0.01), z statistic and the corresponding p values (two-tailed and one-tailed). For the current example: With U = 40, critical U (two-tailed)0.05 = 30, and critical U (two-tailed)0.01 = 21, the non-directional alternative hypothesis is not supported. With Average Rank 1 > Average Rank 2, and U less than the one-tailed critical U for the prescribed significance level, the directional alternative hypothesis (H1: Θ1 > Θ2) is not supported. >The null hypothesis that H1: Θ1 = Θ2) can not be rejected, i.e., for the given data, there is no significance difference between the two processing system used to clean wafers, i.e., the underlying populations have the same continuous distribution, at least as far as their medians are concerned.

### 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 the 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

 In Aabel, the Kolmogorov-Smirnov (goodness-of-fit) test is used to evaluate whether or not the distribution of data in a sample conform to a normal distribution with a specific hypothesized mean and standard deviation. The Kolmogorov-Smirnov goodness-of-fit test for a single sample is a test of ordinal data, because it requires that a cumulative frequency distribution be constructed (i.e., the scores need to be arranged in order of magnitude) 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.