1. Comparing I > 2 Groups - Numeric Responses Extension of Methods used to Compare 2 Groups Independent and Dependent Samples Normal and non-normal data structures

2. Independent Samples - Completely Randomized Design (CRD) Controlled Experiments - Subjects assigned at random to one of the I treatments to be compared Observational Studies - Subjects are sampled from I existing groups Statistical model x ij is a subject from group i: where    is the population mean of group/treatment i,  ij is a random error

3. 1-Way ANOVA for Normal Data (CRD) For each group obtain the mean, standard deviation, and sample size: Obtain the overall mean and sample size

4. Analysis of Variance - Sums of Squares/Degrees of Freedom Total Variation Among Group Variation Within Group Variation

5. Analysis of Variance Table and F-Test H 0 : No differences among Group Means (    I ) H A : Group means are not all equal (Not all  i are equal)

6. Example - Relaxation Music in Patient- Controlled Sedation in Colonoscopy Three Conditions (Treatments): –Music and Self-sedation (i = 1) –Self-Sedation Only (i = 2) –Music alone (i = 3) Outcomes –Patient satisfaction score (all 3 conditions) –Amount of self-controlled dose (conditions 1 and 2) Source: Lee, et al (2002)

7. Example - Relaxation Music in Patient- Controlled Sedation in Colonoscopy Summary Statistics and Sums of Squares Calculations:

8. Example - Relaxation Music in Patient- Controlled Sedation in Colonoscopy Analysis of Variance and F-Test for Treatment effects H 0 : No differences among Group Means (    2  3 ) H A : Group means are not all equal (Not all  i are equal)

9. Post-hoc Comparisons of Treatments If differences in group means are determined from the F-test, researchers want to compare pairs of groups. Three popular methods include: –Dunnett’s Method - Compare active treatments with a control group. Consists of I-1 comparisons, and utilizes a special table. –Bonferroni’s Method - Adjusts individual comparison error rates so that all conclusions will be correct at desired confidence/significance level. Any number of comparisons can be made. –Tukey’s Method - Specifically compares all I(I-1)/2 pairs of groups. Utilizes a special table.

10. Bonferroni’s Method (Most General) Wish to make C comparisons of pairs of groups with simultaneous confidence intervals or 2-sided tests Want the overall confidence level for all intervals to be “correct” to be 95% or the overall type I error rate for all tests to be 0.05 For confidence intervals, construct (1-(0.05/C))100% CIs for the difference in each pair of group means (wider than 95% CIs) Conduct each test at  =0.05/C significance level (rejection region cut-offs more extreme than when  =0.05)

11. Bonferroni’s Method (Most General) Simultaneous CI’s for pairs of group means: If entire interval is positive, conclude  i >  j If entire interval is negative, conclude  i <  j If interval contains 0, cannot conclude  i   j

12. Example - Relaxation Music in Patient- Controlled Sedation in Colonoscopy C=3 comparisons: 1 vs 2, 1 vs 3, 2 vs 3. Want all intervals to contain true difference with 95% confidence Will construct (1-(0.05/3))100% = 98.33% CIs for differences among pairs of group means Note all intervals contain 0, but first is very close to 0 at lower end

13. CRD with Non-Normal Data Kruskal-Wallis Test Extension of Wilcoxon Rank-Sum Test to I>2 Groups Procedure: –Rank the observations across groups from smallest (1) to largest (N = n 1 +...+n I ), adjusting for ties –Compute the rank sums for each group: R 1,...,R I. Note that R 1 +...+R I = N(N+1)/2

14. Kruskal-Wallis Test H 0 : The I population distributions have same distribution H A : Not all I distributions are identical Post-hoc comparisons of pairs of groups can be made by pairwise application of rank-sum test with Bonferroni adjustment

15. Example - Thalidomide for Weight Gain in HIV-1 + Patients with and without TB I=4 Groups, n 1 =n 2 =n 3 =n 4 =8 patients per group (N=32) Group 1: TB + patients assigned Thalidomide Group 2: TB - patients assigned Thalidomide Group 3: TB + patients assigned Placebo Group 4: TB - patients assigned Placebo Response - 21 day weight gains (kg) -- Negative values are weight losses Source: Klausner, et al (1996)

16. Example - Thalidomide for Weight Gain in HIV-1 + Patients with and without TB

17. Weight Gain Example - SPSS Output F-Test and Post-Hoc Comparisons

19. Weight Gain Example - SPSS Output Kruskal-Wallis H-Test

20. Dependent Samples: Randomized Block Design (RBD) I > 2 Treatments (groups) to be compared J individuals receive each treatment (preferably in random order). Subjects are called Blocks. Outcome when Treatment i is assigned to Subject j is labeled x ij Effect of Trt i is labeled  i Effect of Subject j is labeled  j Random error term is labeled  ij

21. Dependent Samples - RBD Model: Test for differences among treatment effects: H 0 :  1  I  0 (  1  I ) H A : Not all  i = 0 (Not all  i are equal)

22. RBD - ANOVA F-Test (Normal Data) Data Structure: (I Treatments, J Subjects or Blocks) Mean for Treatment i: Mean for Subject (Block) j: Overall Mean: Overall sample size: N = IJ ANOVA:Treatment, Block, and Error Sums of Squares

23. RBD - ANOVA F-Test (Normal Data) ANOVA Table: H 0 :  1  I  0 (  1  I ) H A : Not all  i = 0 (Not all  i are equal)

24. Example - Theophylline Interaction Goal: Determine whether Cimetidine or Famotidine interact with Theophylline 3 Treatments: Theo/Cim, Theo/Fam, Theo/Placebo 14 Blocks: Each subject received each treatment Response: Theophylline clearance (liters/hour) Source: Bachmann, et al (1995)

25. Example - Theophylline Interaction The test for differences in mean theophylline clearance is given in the third line of the table T.S.: F obs =10.59 R.R.: F obs  F.05,2,26 = 3.37 (From F-table) P-value:.000 (Sig. Level)

26. Example - Theophylline Interaction Post-hoc Comparisons

27. Example - Theophylline Interaction Plot of Data (Marginal means are raw data)

28. RBD -- Non-Normal Data Friedman’s Test When data are non-normal, test is based on ranks Procedure to obtain test statistic: –Rank the I treatments within each block (1=smallest, I=largest) adjusting for ties –Compute rank sums for treatments (R i ) across blocks –H 0 : The I populations are identical (  1 =...=  I ) –H A : Differences exist among the I group means

29. Example - t max for 3 formulation/fasting states I=3 Treatments of Valproate: Capsule/Fasting (i=1), Capsule/nonfasting (i=2), Enteric-Coated/fasting (i=3) J=11 subjects Response - Time to maximum concentration (t max ) Source: Carrigan, et al (1990)

30. Example - t max for 3 formulation/fasting states H 0 : The I populations are identical (  1 =...=  I ) H A : Differences exist among the I group means

31. Data Sources Lee,D.W., K.W. Chan, C.M. Poon, et al (2002). “Relaxation Music Decreases the Dose of Patient-Controlled Sedation During Colonoscopy: A Prospective Randomized Controlled Trial,” Gastrointestinal Endoscopy, 55:33-36. Klausner,J.D., S. Makonkawkeyoon, P. Akarasewi, et al (1996). “The Effect of Thalidomide on the Pathogenesis of HIV-1 and M. tuberculosis Infection,” Journal of Acquired Immune Deficiency Syndromes and Human Retrovirology, 11:247-257 Bachmann, K., T.J. Sullivan, J.H. Reese, et al (1995). “Controlled Study of the Putative Interaction Between Famotidine and Theophylline in Patients with Chronic Obstructive Pulmonary Disorder,” Journal of Clinical Pharmacology, 35:529-535.