Wilcoxon t-test

Examples of Wilcoxon t-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 $T$.
  • Other names may include the "$t$-test for matched pairs" or the "$t$-test for dependent samples."

• 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 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.
  • Finally, the significance test is done using a Chi Square distribution with k-1 degrees of freedom.

• 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.
  • Therefore, rank randomization tests are generally less powerful than randomization tests based on the original numbers.
  • The two most common are the Mann-Whitney U test and the Wilcoxon Rank Sum Test.
  • 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.

• Distribution-Free Tests

  • Anderson–Darling test: tests whether a sample is drawn from a given 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.
  • Median test: tests whether two 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

  • Some kinds of statistical tests employ calculations based on ranks.

• The t-Test

  • A t-test is any statistical hypothesis test in which the test statistic follows a Student's t-distribution if the null hypothesis is supported.
  • A t-test is any statistical hypothesis test in which the test statistic follows a Student's t-distribution if the null hypothesis is supported.
  • Gosset devised the t-test as a cheap way to monitor the quality of stout.
  • Gosset's work on the t-test was published in Biometrika in 1908.
  • The form of the test used when this assumption is dropped is sometimes called Welch's t-test.

• Multivariate Testing

  • Hotelling's $T$-square statistic allows for the testing of hypotheses on multiple (often correlated) measures within the same sample.
  • A generalization of Student's $t$-statistic, called Hotelling's $T$-square statistic, allows for the testing of hypotheses on multiple (often correlated) measures within the same sample.
  • Because measures of this type are usually highly correlated, it is not advisable to conduct separate univariate $t$-tests to test hypotheses, as these would neglect the covariance among measures and inflate the chance of falsely rejecting at least one hypothesis (type I error).
  • Hotelling's $T^2$ statistic follows a $T^2$ distribution.
  • In particular, the distribution arises in multivariate statistics in undertaking tests of the differences between the (multivariate) means of different populations, where tests for univariate problems would make use of a $t$-test.

• t-Test for Two Samples: Paired

  • Paired-samples $t$-tests typically consist of a sample of matched pairs of similar units, or one group of units that has been tested twice.
  • $t$-tests are carried out as paired difference tests for normally distributed differences where the population standard deviation of the differences is not known.
  • Paired samples $t$-tests typically consist of a sample of matched pairs of similar units, or one group of units that has been tested twice (a "repeated measures" $t$-test).
  • A typical example of the repeated measures t-test would be where subjects are tested prior to a treatment, say for high blood pressure, and the same subjects are tested again after treatment with a blood-pressure lowering medication .
  • Paired-samples $t$-tests are often referred to as "dependent samples $t$-tests" (as are $t$-tests on overlapping samples).