Wilcoxon signed-rank test

Examples of Wilcoxon signed-rank test in the following topics:

• 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:

• 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

• Distribution-Free Tests

  • Friedman two-way analysis of variance by ranks: tests whether $k$ treatments in randomized block designs have identical effects.
  • 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.
  • Sign test: tests whether matched pair samples are drawn from distributions with equal medians.
  • 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.

• Statistical Literacy

  • The levels of troponin in subjects with and without signs of right ventricular strain in the electrocardiogram were compared in the experiment described here: http://www.bmj.com/content/326/7384/312.
  • The Wilcoxon rank sum test was used to test for significance.
  • The troponin concentration in patients with signs of right ventricular strain was higher (median = 0.03 ng/ml) than in patients without right ventricular strain (median < 0.01 ng/ml), p<0.001.
  • Why might the authors have used the Wilcoxon test rather than a t test?

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

  • State the difference between a randomization test and a rank randomization test
  • Rank randomization tests are performed by first converting the scores to ranks and then computing a randomization test.
  • The two most common are the Mann-Whitney U test and the Wilcoxon Rank Sum Test.
  • A rank randomization test on these data begins by converting the numbers to ranks.
  • The beginning of this section stated that rank randomization tests were easier to compute than randomization tests because tables are available for rank randomization tests.

• 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.
  • It tests for differences in central tendency in designs with one between-subjects variable.
  • 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.
  • Therefore, all values of 2.5 were assigned ranks of 8.5.

• 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.
  • The sign of the Spearman correlation indicates the direction of association between $X$ (the independent variable) and $Y$ (the dependent variable).
  • 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.

• Z-Scores and Location in a Distribution

  • A $z$-score is the signed number of standard deviations an observation is above the mean of a distribution.
  • A $z$-score is the signed number of standard deviations an observation is above the mean of a distribution.
  • This may include, for example, the original result obtained by a student on a test (i.e., the number of correctly answered items) as opposed to that score after transformation to a standard score or percentile rank.