Wilcoxon Rank Sum test

Examples of Wilcoxon Rank Sum test in the following topics:

• Statistical Literacy

  • The Wilcoxon rank sum test was used to test for significance.
  • Why might the authors have used the Wilcoxon test rather than a t test?

• Alternatives to the t-Test

  • The $t$-test provides an exact test for the equality of the means of two normal populations with unknown, but equal, variances.
  • The Welch's $t$-test is a nearly exact test for the case where the data are normal but the variances may differ.
  • For example, for two independent samples when the data distributions are asymmetric (that is, the distributions are skewed) or the distributions have large tails, then the Wilcoxon Rank Sum test (also known as the Mann-Whitney $U$ test) can have three to four times higher power than the $t$-test.
  • The nonparametric counterpart to the paired samples $t$-test is the Wilcoxon signed-rank test for paired samples.
  • Explain how Wilcoxon Rank Sum tests are applied to data distributions

• Rank Randomization: Two Conditions (Mann-Whitney U, Wilcoxon Rank Sum)

  • The two most common are the Mann-Whitney U test and the Wilcoxon Rank Sum Test.
  • Rank sum = 24.
  • Rank sum = 26.
  • Rank sum = 25
  • Rank sum = 24.

• Distribution-Free Tests

  • Friedman two-way analysis of variance by ranks: tests whether $k$ treatments in randomized block designs have identical effects.
  • Kruskal-Wallis one-way analysis of variance by ranks: tests whether more than 2 independent samples are drawn from the same distribution.
  • Mann–Whitney $U$ or Wilcoxon rank sum test: tests whether two samples are drawn from the same distribution, as compared to a given alternative hypothesis.
  • Squared ranks test: tests equality of variances in two or more samples.
  • Wilcoxon signed-rank test: tests whether matched pair samples are drawn from populations with different mean ranks.

• When to Use These Tests

  • "Ranking" refers to the data transformation in which numerical or ordinal values are replaced by their rank when the data are sorted.
  • In statistics, "ranking" refers to the data transformation in which numerical or ordinal values are replaced by their rank when the data are sorted.
  • In these examples, the ranks are assigned to values in ascending order.
  • Some kinds of statistical tests employ calculations based on ranks.
  • Some ranks can have non-integer values for tied data values.

• Wilcoxon t-Test

  • The Wilcoxon $t$-test assesses whether population mean ranks differ for two related samples, matched samples, or repeated measurements on a single sample.
  • The Wilcoxon signed-rank t-test is a non-parametric statistical hypothesis test used when comparing two related samples, matched samples, or repeated measurements on a single sample to assess whether their population mean ranks differ (i.e., it is a paired difference test).
  • The test is named for Frank Wilcoxon who (in a single paper) proposed both the rank $t$-test and the rank-sum test for two independent samples.
  • In consequence, the test is sometimes referred to as the Wilcoxon $T$-test, and the test statistic is reported as a value of $T$.
  • Calculate the test statistic $W$, the absolute value of the sum of the signed ranks:

• Rank Randomization: Two or More Conditions (Kruskal-Wallis)

  • The Kruskal-Wallis test is a rank-randomization test that extends the Wilcoxon test to designs with more than two groups.
  • The test is based on a statistic H that is approximately distributed as Chi Square.
  • The first step is to convert the data to ranks (ignoring group membership) and then find the sum of the ranks for each group.
  • For the "Smiles and Leniency" case study, the sum of the ranks for the four conditions are:
  • Therefore, all values of 2.5 were assigned ranks of 8.5.

• Rank Correlation

  • A rank correlation is any of several statistics that measure the relationship between rankings.
  • If, for example, one variable is the identity of a college basketball program and another variable is the identity of a college football program, one could test for a relationship between the poll rankings of the two types of program: do colleges with a higher-ranked basketball program tend to have a higher-ranked football program?
  • Spearman developed a method of measuring rank correlation known as Spearman's rank correlation coefficient.
  • If the agreement between the two rankings is perfect (i.e., the two rankings are the same) the coefficient has value $1$.
  • Evaluate the relationship between rankings of different ordinal variables using rank correlation

• Mann-Whitney U-Test

  • The sum of these counts is $U$.
  • First, add up the ranks for the observations that came from sample 1.
  • The sum of ranks in sample 2 is now determinate, since the sum of all the ranks equals:
  • where $n_1$ is the sample size for sample 1, and $R_1$ is the sum of the ranks in sample 1.
  • As it compares the sums of ranks, the Mann–Whitney test is less likely than the $t$-test to spuriously indicate significance because of the presence of outliers (i.e., Mann–Whitney is more robust).

• Comparing Two Populations: Independent Samples

  • Nonparametric independent samples tests include Spearman's and the Kendall tau rank correlation coefficients, the Kruskal–Wallis ANOVA, and the runs test.
  • Nonparametric methods for testing the independence of samples include Spearman's rank correlation coefficient, the Kendall tau rank correlation coefficient, the Kruskal–Wallis one-way analysis of variance, and the Walk–Wolfowitz runs test.
  • $\displaystyle{\rho = \frac{\sum_i (x_i - \bar{x}) (y_i - \bar{y})}{\sqrt{\sum_i(x_i - \bar{x})^2 \sum_i(y_i - \bar{y})^2}}}$
  • If the agreement between the two rankings is perfect (i.e., the two rankings are the same) the coefficient has value $1$.
  • The Kruskal–Wallis one-way ANOVA by ranks is a nonparametric method for testing whether samples originate from the same distribution.