1. Monotonic relationship of two variables, X and Y

2. 0 4 8 12 16 01234 Y X Deterministic monotonicity If X grows then Y grows too

3. 0 4 8 12 16 01234 Y X Stochastic monotonicity * * * * * * * * * * * ** * * * * If X grows then likely Y grows too

4. Ss X Y 1 1 35 2 1.5 34 3 2 36 4 3 37 5 7 38 6 10 39 An example

5. Ss X rank Y rank 1 1 1 35 2 2 1.5 2 34 1 3 2 3 36 3 4 3 4 37 4 5 7 5 38 5 6 10 6 39 6 Rank data separately for X and Y

6. Spearman-s rank correlation (r S ): Correlation between ranks In the above example: r = 0.91, r S = 0.94

7. DiscordancyConcordancy

8. +  A B C D X Y Concordancy and discordancy

9.  p   p  Kendall-s tau p + : Proportion of concordant pairs in the population p - : Proportion of discordant pairs in the population

10.  1    +1 If X and Y are independent:    = 0: no stochastic monotonicity    =  deterministic monotone decreasing (inreasing) relationship Features of Kendall’s 

11. A Kendall’s gamma For discrete X and Y variables

12.  1    +1 If X and Y are independent:  = 0  = 0: no stochastic monotonicuty If  =  1: p + = 0 If  = +1: p  = 0 Features of Kendall’s 

13. Testing the H 0 :  = 0 null hypothesis Sample tau: Kendall’s rank correlation coefficient (r  ) Testing stochastic monotonicity = testing the significancy of r 

14. +  A B C D X Y Computation of sample tau + +  C + c = n  = 4 d = n  = 2 r  = (4-2) /(4+2) = 2/6 = 0.33

15. c = # of concordancies d = # of discordancies T = # of total couples = n(n-1)/2 r  = (c - d)/T,  = (c - d)/(c+d) In which cases will r  =  ? Formulea of r  and 

16. Ss X Y 1 135 2 1.5 34 3 236 4 3 37 5 7 38 6 1039 An example r   (p < 0.02); r S   (p < 0.02); r   (p < 0.10);

17. Comparison of several independent samples

18. -60 -40 -20 0 20 40 60 80 GSR-decrease Agr 1 Agr 2 Agr 3 LightVerbal Groups

19. NormalPerson. disorder Holocaust group 0 0.5 1 1.5 2 2.5 Average Rorschach time (min)

20. Comparison of population means H 0 : E(X 1 ) = E(X 2 ) =... = E(X I ) H 0 :  1 =  2 =... =  I

21. One way independent sample ANOVA

22. SS total = SS b + SS w SS total : Total variability SS b : Between sample variability SS w : Within sample variability Basic identity

23. Var b = SS b /(I - 1) = SS b /df b - Treatment variance Var w = SS w /(N - I) = SS w /df w - Error variance One-way ANOVA Test statistic: F = Var b /Var w

24. Treatment variance

25. Error variance

26. H 0 :  1 =  2 =... =  I F = Var b /Var w ~ F-distribution Assumptions of ANOVA F  F  : reject H 0 at level  +

27. l Independent samples l Normality of the dependent variable l Variance homogeneity (identical population variances) Assumptions of ANOVA

28. l Welch test l James test l Brown-Forsythe test Robust ANOVA’s

29. l Levene test l O’Brien test Testing variance homogeneity

30. Var 1  Var 2 ...  Var I or (and) n 1  n 2 ...  n I Trust in the result of ANOVA

31.  Different sample sizes  Substantially different sample variances When to apply a robust ANOVA?

32. l Conventional test: Tukey- Kramer test (Tukey’s HSD test) l Robust test: Games-Howell test Post hoc analyses H ij :  i =  j

33. Nonlinear coefficient of determination Explained variance: eta 2 = SS b /SS total Nonlinear correlation coefficient: eta SS total = SS b + SS w

34. An example Agr 1 2 3 LightVerb. n i 54644 x i 14.506.755.20-13.45-30.08 s i 29.609.156.9613.1114.57

35. l Levene test: F(4, 7) = 0.784 (p > 0.10, n.s.) l O’Brien test: F(4, 8) = 1.318 (p > 0.10, n.s.) Testing variance homogeneity

36. Treatment var.: Var b = 1413.9 Error variance: Var w = 286.2 F(4, 18) = 1413.9/286.2 = 4.940** Nonlinear coeff. of determin.: eta 2 = SS b /SS total = 0.523 Conventional ANOVA

37. l Welch test: W(4, 8) = 5.544* l James test: U = 27.851 + l Brown-Forsythe test: BF(4, 9) = 5.103* Robust ANOVA’s

38. Tukey-Kramer test: T12= 0.97T13= 1.28 T14= 3.48T15= 5.55** T23= 0.20T24= 2.39 T25= 4.35*T34= 2.42 T35= 4.57*T45= 1.97 Pairwise comparison of means