1. Comparing k > 2 Groups - Numeric Responses Extension of Methods used to Compare 2 Groups Parallel Groups and Crossover Designs Normal and non-normal data structures

2. Parallel Groups - Completely Randomized Design (CRD) Controlled Experiments - Subjects assigned at random to one of the k treatments to be compared Observational Studies - Subjects are sampled from k existing groups Statistical model Y ij is a subject from group i: where  is the overall mean,  i is the effect of treatment i,  ij is a random error, and  i is the population mean for group i

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 Total Variation Between Group Variation Within Group Variation

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

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 (    3 =0) H A : Group means are not all equal (Not all  i are 0)

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 k-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 k(k-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 k>2 Groups Procedure: –Rank the observations across groups from smallest (1) to largest (n = n 1 +...+n k ), adjusting for ties –Compute the rank sums for each group: T 1,...,T k. Note that T 1 +...+T k = n(n+1)/2

14. Kruskal-Wallis Test H 0 : The k population distributions are identical (  1 =...=  k ) H A : Not all k distributions are identical (Not all  i are equal) 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 k=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. Crossover Designs: Randomized Block Design (RBD) k > 2 Treatments (groups) to be compared b individuals receive each treatment (preferably in random order). Subjects are called Blocks. Outcome when Treatment i is assigned to Subject j is labeled Y ij Effect of Trt i is labeled  i Effect of Subject j is labeled  j Random error term is labeled  ij

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

22. RBD - ANOVA F-Test (Normal Data) Data Structure: (k Treatments, b Subjects) Mean for Treatment i: Mean for Subject (Block) j: Overall Mean: Overall sample size: n = bk ANOVA:Treatment, Block, and Error Sums of Squares

23. RBD - ANOVA F-Test (Normal Data) ANOVA Table: H 0 :  1  k  0 (  1  k ) 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 k treatments within each block (1=smallest, k=largest) adjusting for ties –Compute rank sums for treatments (T i ) across blocks –H 0 : The k populations are identical (  1 =...=  k ) –H A : Differences exist among the k group means

29. Example - t max for 3 formulation/fasting states k=3 Treatments of Valproate: Capsule/Fasting (i=1), Capsule/nonfasting (i=2), Enteric-Coated/fasting (i=3) b=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 k populations are identical (  1 =...=  k ) H A : Differences exist among the k group means