1. Experimental Research Methods in Language Learning Chapter 15 Non-parametric Versions of T-tests and ANOVAs

2. Leading Questions What is a non-normal data distribution? What does it look like? How do we know whether a data set is normally distributed? Do you any know of a nonparametric test that can analyze non-normally distributed data? If so, what is it?

3. Non-parametric Tests This chapter presents four non-parametric tests: Wilcoxon Signed Ranks Test (the nonparametric version of the paired-samples t-test) Mann-Whitney U Test (the nonparametric version of the independent-samples t-test); Kruskal-Wallis H Test (the nonparametric version of the one-way ANOVA); Friedman Test (the nonparametric version of the repeated-measures ANOVA).

4. Wilcoxon Signed Ranks Test This test is the non-parametric version of the paired-samples t-test. The Z score is used for statistical testing. Table 15.1.1 reports the descriptive statistics of a pretest and a posttest to be compared.

5. Wilcoxon Signed Ranks Test Table 15.1.2 presents the score ranks using the posttest and pretest scores.

6. Wilcoxon Signed Ranks Test Negative ranks refer to the observation that an individual scored lower in the posttest than in the pretest. Positive ranks refer to the observation that an individual scored higher in the posttest than the pretes.

7. Wilcoxon Signed Ranks Test Table 15.1.3 reports the Wilcoxon signed ranks test statistic. Examine the Z score and the Assymp. Sig (2- tailed) value.

8. Wilcoxon Signed Ranks Test Effect size: r = Z ÷ √N (Larson-Hall (2010, p. 378) presents a formula to compute the r effect size for both the Mann-Whitney and Wilcoxon signed ranks tests. The formula is simple to calculate: It is important. We can use the following statistical website practical to compute effect sizes: http://www.ai-therapy.com/psychology- statistics/effect-size-calculator

9. Examples of Studies Gass, Svetics, & Lemelin 2003; Kim & McDonough 2008; Marsden & Chen 2011; Yilmaz 2011; Yilmaz & Yuksel 2011

10. Mann-Whitney U Test Has a similar function to that of the independent-samples t-test for comparing two groups of participants Table 15.2.1 reports the descriptive statistics of each test.

11. Mann-Whitney U Test Table 15.2.2 presents the mean ranks using the speaking pretest and posttest scores.

12. Mann-Whitney U Test Table 15.2.3 reports the Mann-Whitney U test statistic. We examine the Z score and the Assymp. Sig (2-tailed) value.

13. Examples of Studies Henry et al. (2009); Macaro & Masterman (2006); Marsden & Chen (2011); Yilmaz and Yuksel (2011)

14. Kruskal-Wallis H Test Can help us determine differences between two or more groups. Used when our data are not normally distributed. Table 15.3.1 reports the descriptive statistics of each test.

15. Kruskal-Wallis H Test Table 15.3.2 presents the mean ranks using the speaking posttest scores.

16. Kruskal-Wallis H Test Table 15.2.3 reports the Kruskal-Wallis H test statistic. Examine the chi-square (χ 2 ) statistic, df and the Assymp. Sig value.

17. Kruskal-Wallis H Test post ho c test for Kruskal-Wallis H test is typically a Mann-Whitney U test in SPSS Alternatively use the following website to compute a post hoc test: ; accessed 01/03/2014.http://www.ai- therapy.com/psychology- statistics/hypothesis-testing/two- samples?groups=0&parametric=1

18. Examples of Studies Chen & Truscott 2010; Li 2011; Marsden & Chen 2011

19. Friedman Test Can do more than two levels of repeated measures Note that the Friedman test cannot test a group difference like the repeated-measures ANOVA. Therefore, the Friedman test is not a full parametric version of the repeated-measures ANOVA.

20. Friedman Test Table 15.4.1 reports the descriptive statistics of each test.

21. Friedman Test Table 15.4.2 presents the mean ranks of the three test scores. In this table, we can see the delayed reading posttest had the highest rank (i.e., 2.87).

22. Friedman Test Table 15. 4.3 reports the Friedman test statistic. Examine the chi-square (χ 2 ) statistic, df and the Assymp. Sig value.

23. Examples of Studies Li (2011) Marsden and Chen (2011)

24. Discussion What do you think are analytical limitations when raw scores are ranked before being analyzed? Do you find it useful to know the logic of these nonparametric tests? Does it help you understand experimental studies using these statistical tests? What are benefits of knowing an alternative statistics when our data are not normally distributed?