Comparing k > 2 Groups - Numeric ResponsesParallel Groups - Completely Randomized Design (CRD)1-Way ANOVA for Normal Data (CRD)Analysis of Variance - Sums of SquaresAnalysis of Variance Table and F-TestExample - Relaxation Music in Patient-Controlled Sedation in ColonoscopySlide 7Slide 8Post-hoc Comparisons of TreatmentsBonferroni’s Method (Most General)Slide 11Slide 12CRD with Non-Normal Data Kruskal-Wallis TestKruskal-Wallis TestExample - Thalidomide for Weight Gain in HIV-1+ Patients with and without TBSlide 16Weight Gain Example - SPSS Output F-Test and Post-Hoc ComparisonsSlide 18Weight Gain Example - SPSS Output Kruskal-Wallis H-TestCrossover Designs: Randomized Block Design (RBD)Crossover Designs - RBDRBD - ANOVA F-Test (Normal Data)Slide 23Example - Theophylline InteractionSlide 25Example - Theophylline Interaction Post-hoc ComparisonsExample - Theophylline Interaction Plot of Data (Marginal means are raw data)RBD -- Non-Normal Data Friedman’s TestExample - tmax for 3 formulation/fasting statesSlide 30Comparing k > 2 Groups - Numeric Responses•Extension of Methods used to Compare 2 Groups•Parallel Groups and Crossover Designs•Normal and non-normal data structures DataDesignNormal Non-normalParallelGroups(CRD)F-Test1-WayANOVAKruskal-Wallis TestCrossover(RBD)F-Test2-WayANOVAFriedman’sTestParallel 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 Yij is a subject from group i:ijiijiijY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 i1-Way ANOVA for Normal Data (CRD)• For each group obtain the mean, standard deviation, and sample size:1)(2ijiijiijijinyysnyy• Obtain the overall mean and sample sizenynynynynnni jijkkk 111Analysis of Variance - Sums of Squares• Total Variation 1)(1 12  ndfyyTotalSSTotalkinjiji• Between Group Variation    kinjkiTiiiikdfyynyySST1 1 1221)()(• Within Group VariationETTotalEkiiikinjiijdfdfdfSSESSTTotalSSkndfsnyySSEi   121 12)1()(Analysis of Variance Table and F-TestSource ofVariation Sum of SquaresDegrres ofFreedom Mean Square FTreatments SST k-1 MST=SST/(k-1) F=MST/MSEError SSE n-k MSE=SSE/(n-k)Total Total SS n-1• H0: No differences among Group Means (k=0)• HA: Group means are not all equal (Not all i are 0))(:)4.(:..:..,1,obsknkobsobsFFPvalPATableFFRRMSEMSTFST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:Trt (i) niMean Std. Dev.1 55 7.8 2.12 55 6.8 2.33 55 7.4 2.3Total 165 overall mean=7.33 ---164162275.84046.80929.31162316546.809)3.2)(155()3.2)(155()1.2)(155(21329.31)33.74.7(55)33.78.6(55)33.78.7(55222222TotalETdfTotalSSdfSSEdfSSTExample - Relaxation Music in Patient-Controlled Sedation in Colonoscopy• Analysis of Variance and F-Test for Treatment effectsSource ofVariation Sum of SquaresDegrres ofFreedom Mean Square FTreatments 31.29 2 15.65 3.13Error 809.46 162 5.00Total 840.75 164•H0: No differences among Group Means (3=0)• HA: Group means are not all equal (Not all i are 0)05.0)13.3(:)4.(055.3:..13.300.565.15:..162,2,05.FPvalPATableFFRRFSTkobsobsPost-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.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: jikncjinnMSEtyy11,2/• If entire interval is positive, conclude i > j• If entire interval is negative, conclude i < j• If interval contains 0, cannot conclude i  jExample - 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)42.0,62.1(02.1)4.78.6(:32)42.1,62.0(02.1)4.78.7(:31)02.2,02.0(02.1)8.68.7(:2102.155155100.540.2115500.540.2162),3(2/05.3210083.162),3(2/05.vsvsvsnnMSEtnnnMSEztjiNote all intervals contain 0, but first is very close to 0 at lower endCRD 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 = n1+...+nk), adjusting for ties–Compute the rank sums for each group: T1,...,Tk . Note that T1+...+Tk = n(n+1)/2Kruskal-Wallis Test• H0: The k population distributions are identical (1=...=k)• HA: Not all k distributions are