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NonParametric Tests
Parametric tests are either based on a normal distribution or on, e.g., t,t or χ^{2} distributions, which are related to and can be derived from normaltheorybased procedures. That is, the parametric tests require that a sample/group analyzed is taken from a population that meets the normality assumption Nonparametric tests are used when assumptions required by the parametric counterpart tests are not met or are questionable. All tests involving ranked data are nonparametric.
Wilcoxon Matched Pairs SignedRanks Test
Spearman's RankOrder Correlation Coefficient (Spearman's ρ)
Kendall's Rank Correlation Coefficient (Kendall's τ)
MannWhitney U Test (Wilcoxon Ranks Sum Test)
Wilcoxon SignedRanks Test
This test evaluates whether a sample of n observations is drawn from a population in which the median equals a specific (hypothesized) value.
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:


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 SignedRanks Test
This test evaluates whether or not the median of the difference scores equals zero. The corresponding parametric test is the paired samples tTest.
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 righthand side image displays the corresponding Wilcoxon matched pairs signedranks 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 RankOrder Correlation Coefficient (Spearman's ρ)
Spearman's ρ is the nonparametric analog of the Pearson productmoment 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.
 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)
There are two different ways of calculating Spearman's ρ:
 The method requires two numeric variables.
 The test results include the estimated Spearman's rankorder correlation coefficient (r_{s}), the critical onetailed and twotailed r_{s} _{(0.05)} and r_{s}_{(0.01)}, t statistic, Z statistic and the corresponding p values (twotailed and onetailed).
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 lefthand 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 righthand side image below displays the corresponding output from the Spearman's correlation test. Based on interpretation of the test results, the null hypothesis (H_{0}: &rho_{S} = 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 nonparametric 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 righthand side image shows the Kendall's &tau report from the same data used or the Spearman's ρ above. 

MannWhitney U Test (Wilcoxon Ranks Sum Test)
This is the nonparametric analog of the unpaired samples ttest. The MannWhitney test (also known as the Wilcoxon ranks sum test and the ManWhitneyWilcoxon 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 MannWhitney U, the critical twotailed and onetailed U _{(0.05)} and U _{(0.01)}, z statistic and the corresponding p values (twotailed and onetailed). For the current example:


KruskalWallis TestThe KruskalWallis test is the nonparametric analog of a oneway ANOVA. The KruskalWallis 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.

Friedman TwoWay Analysis of Variance by RanksThis is a nonparametric test used to compare dependent samples (or observations) that are repeated on the same subjects. This test is the nonparametric 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.

KolmogorovSmirnov Test for a Single Sample
In Aabel, the KolmogorovSmirnov (goodnessoffit) 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 KolmogorovSmirnov goodnessoffit 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)

