Parametric test: (b) the data show inhomogeneity of variance; or. (c) the data are not measurements on an interval or ratio scale.

Transcription

1 Non-parametric tests: Non-parametric tests make no assumptions about the characteristics or "parameters" of your data. Use them if (a) the data are not normally distributed; (b) the data show inhomogeneity of variance; or (c) the data are not measurements on an interval or ratio scale. Examples of parametric tests and their non-parametric equivalents: Parametric test: Pearson correlation (No equivalent) Independent-means t-test Dependent-means t-test One-way Independent Measures Analysis of Variance (ANOVA) One-way Repeated-Measures ANOVA [covered in nd yr.] Non-parametric counterpart: Spearman's correlation Chi-Square test Mann-Whitney test Wilcoxon test Kruskal-Wallis test Friedman's test Non-parametric tests for comparing two groups or conditions: (a) The Mann-Whitney test: Equivalent to an independent-measures t-test. Used when you have two conditions, each performed by a separate group of subjects. Each subject produces one score. Tests whether there a statistically significant difference between the two groups. (b) The Wilcoxon test: Equivalent to a repeated-measures t-test. Used when you have two conditions, both performed by the same subjects. Each subject produces two scores, one for each condition. Tests whether there a statistically significant difference between the two conditions. 1

2 Mann-Whitney test, step-by-step: Does it make any difference to students' comprehension of statistics whether the lectures are in English or in Serbo-Croat? Group 1: statistics lectures in English. Group : statistics lectures in Serbo-Croat. DV: performance on a statistics exam. E n g lis h (r a w s c o r e s ) E n g lis h (r a n k s ) S e r b o -C ro a t (r a w s c o r e s ) S e r b o -C r o a t (r a n k s ) Step 1: Rank all the scores together, regardless of group. Revision of how to Rank scores: Same method as for Spearman's correlation. (a) Lowest score gets rank of 1 ; next lowest gets ; and so on. (b) Two or more scores with the same value are tied. (i) Give each tied score the rank it would have had, had it been different from the other scores. (ii) Add the ranks for the tied scores, and divide by the number of tied scores. Each of the ties gets this average rank. (iii) The next score after the set of ties gets the rank it would have obtained, had there been no tied scores. Step : Add up the ranks for group 1, to get T1. Here, T1 = 83. Add up the ranks for group, to get T. Here, T = 70. Step 3: N1 is the number of subjects in group 1; N is the number of subjects in group. Here, N1 = 8 and N = 9. Step 4: Call the larger of these two rank totals Tx. Here, Tx = 83. Nx is the number of subjects in this group; here, Nx = 8. e.g. raw score: original rank: actual rank:

3 Step 5: Find U: Nx (Nx + 1) U = N1 * N Tx In our example, 8 * (8 + 1) U = 8 * U = = 5 If there are unequal numbers of subjects - as in the present case - calculate U for both rank totals and then use the smaller U. In the present example, for T1, U = 5, and for T, U = 47. Therefore, use 5 as U. Step 6: Look up the critical value of U, (e.g. with the table on my website), taking into account N1 and N. If our obtained U is smaller than the critical value of U, we reject the null hypothesis and conclude that our two groups do differ significantly. N N Here, the critical value of U for N1 = 8 and N = 9 is 15. Our obtained U of 5 is larger than this, and so we conclude that there is no significant difference between our two groups. Conclusion: performance in the statistics exam is unaffected by whether the lectures are given in English or in Serbo-Croat Wilcoxon test, step-by-step: Does background music affect the performance of factory workers? Eight workers: each tested twice. Condition A: background music. Condition B: silence. DV: productivity (number of flangle-grommets manufactured per hour) 3

4 W o r k e r : S i le n c e M u s ic d i f f e r e n c e r a n k ig n o r e Step 1: Find the difference between each pair of scores, keeping track of the sign of the difference. Step : Rank the differences, ignoring their sign. Lowest = 1. Tied scores dealt with as before. Ignore zero difference-scores. Step 3: Add together the positive-signed ranks. =. Add together the negative-signed ranks. = 6. Step 4: "W" is the smaller sum of ranks; W = 6. N is the number of differences, omitting zero differences. N = 8-1 = 7. Step 5: Use table (e.g. on my website) to find the critical value of W, for your N. Your obtained W has to be smaller than this critical value, for it to be statistically significant. One Tailed Significance levels: Two Tailed significance levels: N High/low sensation seeking: No. car accidents in a 5 year period. High: mean= 3, sd = 1.3 Low: mean = 6, sd =.0 Size of politician s nose (small vs large) No. lies told per week. Large: mean= 100, sd = 5 Small: mean = 10, sd = 15 The critical value of W (for an N of 7) is. Our obtained W of 6 is bigger than this. Our two conditions are not significantly different. Conclusion: worker productivity appears to be unaffected by presence or absence of background music. Interest-level of lectures (clinical vs statistics). Same students in both lectures: no. yawns per lecture. Interest-level of lectures (clinical vs statistics). Same students in both lectures: rating on sevenpoint scale. 4

5 Mann-Whitney output from SPSS (Analyse > Nonparametric tests > independent samples): Wilcoxon output from SPSS (Analyse > Nonparametric tests > related samples): SCORE LANGUAGE english serbo-croat Total Ranks N Mean Rank Sum of Ranks Test Statistics b Mann-Whitney U Wilcoxon W Z Asymp. Sig. (-tailed) Exact Sig. [*(1-tailed Sig.)] a. Not corrected for ties. SCORE a b. Grouping Variable: LANGUAGE MUSIC - SILENCE a. MUSIC < SILENCE b. MUSIC > SILENCE c. SILENCE = MUSIC Negative Ranks Positive Ranks Ties Total Ranks Test Statistics b Z Asymp. Sig. (-tailed) a. Based on positive ranks. N Mean Rank Sum of Ranks 4 a b c 8 MUSIC - SILENCE a.175 b. Wilcoxon Signed Ranks Test 5