1. Lecture 14 Non-parametric hypothesis testing The ranking of data The ranking of data eliminates outliers and non- linearities. In most cases it reduces within group variances. All parametric tests can be applied to ranked data! Spiders on Mazurian lake islands Disturbance

2. Effects of ranking Ranking often reduces the within group variances.

3. Paired comparisons of the mean; Wilcoxon’s matched pairs rank test z is approximately normally distributed. Past uses a different algorithm for the same test. W The Wicoson test is the non-parametric alternative to the one-way repeated measures ANOVA

4. Sign test

5. The rank test of Withney and Mann – U-test Expected mean if no difference Expected SE if no difference The U-test is the nonparametric alternative to the t-test. Spider abundances

6. As in the case of the t-test does the ranked ANOVA result in lower significance levels. Ranking levels off the within group heterogeneity (lower within group variance). The test is less conservative. Raw data Ranked data

7. Kruskal-Wallis test or Kruskal-Wallis one way ANOVA by ranks KW is approximately χ2 distributed. Values can be taken from a  2 table with r-1 degrees of freedom Raw data ANOVA

8. The ANOVA gave the more conservative result

9. Random skewers Diversity of ground beetles along an elevational gradient Altitude Number of species 10086 25098 35075 40080 45050 50061 63055 70045 95049 103038 110039 120036 130033 150027 Ranked altitude Ranked number of species 12 21 34 43 57 65 76 89 98 1011 10 12 13 14 r0.969231 Random samples Ranked altitude Ranked number of species Ranked altitude Ranked number of species Ranked altitude Ranked number of species Ranked altitude Ranked number of species 1212 212121 3434 434343 5757 65 76 898989 989898 1011 10 12 13 14 0.983270.957210.971050.98243 We take 1000 random samples and calculate each time Spearman’s rank order correlation.

10. If there is no trend in species richness we expect a Bernoulli distribution of positive and negative correlations. Of 1000 rank correlations 623 were positive. The associated probability is Altitude Number of species 10086 25098 35075 40080 45050 50061 63055 70045 95049 103038 110039 120036 130033 150027 It’s highly probable that there is a altitudinal trend in species richness.

14. Home work and literature Refresh: U-test Wilcoxon matched pairs test Sign test Kruskal Wallis test Raw and ranked data Tied ranks Literature: Łomnicki: Statystyka dla biologów http://statsoft.com/textbook/