Learning objectives. Non-parametric tests. Inhomogeneity of variance. Why non-parametric test? Non-parametric tests. NOT normally distributed

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1 Learning objectives Criteria for choosing a non-parametric test Part Why non-parametric test? Inhomogeneity of variance Violation of the assumption of normally distributed data Group Group Group Inhomogeneity of variance However: Parametric tests tend to be robust NOT normally distributed Group Group Group Parametric test t-test Related t-test One-way ANOVA Repeated measure ANOVA Non-parametric test Wilcoxon s Rank-Sum Test (Mann Whitney U test) Wilcoxon s Matched-Pairs Signed-Ranks Test Kruskal-Wallis ANOVA Friedman s Rank test

2 Wilcoxon s Rank-Sum Test. Example = Mann-Whitney U test Sample :,,,,, Sample :,,,,, Two independent samples Score Analogue of the t-test for two independent samples Rank.... Basic idea: Comparison of ranking of data Sum of Ranks: Sample : Sample :. Example Inferences from test Sample :,,,,, Sample :,,,,, If sums of ranks are similar, H is accepted Score Rank.... If sums of ranks are very different, H is rejected Sum of Ranks: Sample : Sample : is given by the Wilcoxon W statistics for the given sums of ranks SPSS output:. Example SPSS output:. Example Ranks Ranks SAMPLE.. Total N Mean Rank Sum of Ranks.... SAMPLE.. Total N Mean Rank Sum of Ranks.... Test Statistics b Mann-Whitney U. Wilcoxon W.. Asymp. Sig. (-tailed). Exact Sig. [*(-tailed. a Sig.)] a. Not corrected for ties. b. Grouping Variable: SAMPLE Test Statistics b Mann-Whitney U. Wilcoxon W. -. Asymp. Sig. (-tailed). Exact Sig. [*(-tailed. a Sig.)] a. Not corrected for ties. b. Grouping Variable: SAMPLE

3 Properties of non-parametric tests Checklist for non-parametric test No estimation of parameters, therefore works with very small sample sizes characterization of data by median instead of mean little effected by outliers Unclear which difference between data exactly causes the rejection of null hypothesis Inhomogeneity of variance between conditions Not normally distributed data Very small sample size Unequal sample size between conditions Many outliers Lower power than parametric test, if conditions for parametric test are fulfilled Data better characterized by median than by mean Experiment Descriptive statistics Influence of mood on attractiveness of women participants in a between-participants design Rate women in bad or good mood (maximal rating: ) Good mood Mean:. Standard deviation:. Median:. Bad mood Mean:. Standard deviation:. Median:. Good mood:,,,, Bad mood:,,,,,, Box plot Histogram Good mood Bad mood N = Good Bad MOOD

4 Why parametric test? SPSS output Unequal sample size Small sample size for good mood Different variability Not normally distributed data Test Statisticsb Mann-Whitney U Wilcoxon W Asymp. Sig. (-tailed) Exact Sig. [*(-tailed Sig.)] a. Not corrected for ties a b. Grouping Variable: CONDITIO Report Histograms for the two conditions were inspected separately. As data were not normally distributed and the participant numbers were small, the most appropriate statistical test was Wilcoxon s ranksum test. Descriptive statistics showed that participants who rated the attractiveness of an actress in a good mood gave higher ratings (median = ) than participants who rated her in a bad mood (median = ). However, the Wilcoxon W was found to be. with an associated probability of. which shows no significant difference between the two conditions. Wilcoxon s Matched-Pairs Two related samples Signed-Ranks Test Analogue of the t-test for related samples Basic idea: Comparison of ranking of difference between related scores Experiment Data Program of long range running reduce blood pressure participants were engaged in a -week running program Measure of blood pressure before and after Sample size: Mean: Standard deviation:. Median:. Sample size: Mean:. Standard deviation:. Median:.

5 N = N = Box plot Histogram BEFORE AFTER Why parametric test? Wilcoxon s Matched-Pairs Signed-Ranks Test sample size ok Variability ok (just) Difference - - Not normally distributed data, especially Rank Signed rank - - Sum of Ranks: Positive Ranks: Negative Ranks: - Inference SPSS output Ranks If similar amount of pos. and neg. ranks, H is accepted If there is a large difference between pos. and neg. ranks, H is rejected AFTER - BEFORE Negative Ranks Positive Ranks Ties Total a. AFTER < BEFORE b. AFTER > BEFORE c. BEFORE = AFTER N Mean Rank Sum of Ranks.. a b.. c The is given by the t-score (the smaller of the two ranks) Test Statistics b AFTER - BEFORE -. a Asymp. Sig. (-tailed). a. Based on positive ranks. b. Wilcoxon Signed Ranks Test

6 Report One-tailed Hypothesis An inspection of the descriptive statistics revealed a skewed distribution of the data and a slightly intolerable difference in standard deviations ( vs. ). Therefore it was concluded that the appropriate test was Wilcoxon s Matched-Pairs Signed-Ranks test. The test indicated no significant difference between the two conditions (t=, p =.). Directional (one-sided) hypothesis Divide by Same as in t-test

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