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

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

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

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

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)

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)

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

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)

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.

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)

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

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

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 n I ), adjusting for ties –Compute the rank sums for each group: R 1,...,R I. Note that R R I = N(N+1)/2

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

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)

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

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

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

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

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)

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

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)

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)

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)

Example - Theophylline Interaction Post-hoc Comparisons

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

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

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)

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

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: 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: 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: