One-Way ANOVA Source Table J - 1 SS B / J - 1 MS B /MS W. Pairwise Post-Hoc Comparisons of Means

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Gerald Benson
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1 One-Way ANOVA Source Table ANOVA MODEL: ij = µ* + α j + ε ij H 0 : µ 1 = µ =... = µ j or H 0 : Σα j = 0 Source Sum of Squares df Mean Squares F Between Groups n j ( j - * ) J - 1 SS B / J - 1 MS B /MS W (Explained Variance) Within Groups (Error Variance) Total Variance ( i - j ) N - J SS W /df W ( i - * ) N - 1 s = SS T /N-1 where, N = total number of cases, J = number of groups, * = the grand mean of across all groups. i = each individual score on, and j = the mean for group j. n j = the number of cases in group j. R = η = SS B /SS T. Pairwise Post-Hoc Comparisons of Means However, a statistically significant F ratio only indicates that assuming the Null Hypothesis, H 0 : µ 1 = µ =... = µ j, the Results were not likely to have occurred by chance. Or, we interpret this as, at least one mean is different. We don t know which one(s)! A common approach is to use a post-hoc test for group comparisons. I personally like Tukey s Honestly Significant Difference (HSD) for pairwise comparisons of means. Tukey s HSD controls for inflation of the Type I error rate when J(J - 1)/ pairwise comparsions are made. Because of this adjustment of the significance level, it is possible to obtain a statistically significant F-ratio when no pairwise comparisons are significant. The q statistic for Tukey s HSD can be computed as follows: q = L - S MS W ( 1 n L + 1 n S ), Then q is compared to a critical value obtained from the Studentized Range Table (q-distribution). Thus, when two means are compared in a pairwise fashion, if the calculated q statistic is larger than the q-critical value then this pairwise difference in means is statistically significant.

2 T. Mark Beasley ANOVA handout ANOVA MODEL: ij = µ* + α j + ε ij Total Within Group Between Group * ( - * ) ( - * ) j ( - j ) ( - j ) j * ( j - * ) ( j - * ) Occup Counselor Therapy Education (j = 1) SS W1 = n 1 = S 1 = = S 1 = Maternal Educational Ch H Psychology (j = ) SS W = n = S = = S = Clinical Nutri Science Psychology (j = 3) SS W3 = n 3 = S 3 = = S 3 = Cognitive Epidemiology Psychology (j = 4) SS W4 = n 4 = S 4 = = S 4 = * = 6 SS T = 110 SS W = 0SS W = 0 SS B = 90 (Σα j ) estimated by SS Between = Σ n j ( j - * ) = 5(3-6) + 5(6-6) + 5(9-6) + 5(6-6) = 90. (Σε ij ) estimated by SS Within = Σ( S j )( n j-1) = (1x4) + (.5x4) + (.5x4) + (1x4) = 0 ANOVA Source Table H 0 : µ 1 = µ = µ 3 = µ 4 Source SS df MS F Between 90 J-1 =3 90/3 = 30 30/1.5 = 4.00 (Explained) Within 0 N-J = 0-4 =16 0/16 = 1.5 (Error) Total 110 N - 1 = /19 = 5.79 η = 90/110 =.8 F(3,16) = 4.00, p <.05, η =.8. Reject H 0 : µ 1 = µ = µ 3 = µ 4. Pairwise Comparisons using Tukey s HSD. q j k = j - k (MSw/)(1/n j + 1/n k ) q 1 = q 1 = (1.5/)(1/n 1 + 1/n ) (1.5/)(1/5 + 1/5) = 3/(.5) = 6. q 1 = 6 > 4.05 from Studentized Range Table; thus, p <.05. q 13 = 1, p <.05. q 14 = 6, p <.05. q 3 = 6, p <.05. q 4 = 0, ns, p >.05. q 34 = 6, p <.05. Conclusion: H A : µ 1 < (µ = µ 4 ) < µ 3

3 Complex Contrasts of Means. H 0 : ψ = 0, where ψ = a a a J J and a1 + a aj = 0. In this case, ψ = a a + a a 4 4 and a1 + a + a3. + a4 = 0. For example, comparing Nutrition Science (j = 3) to a combination of MCH (j = ) and Epidemiology (j = 4) yields; ψ = (0)(3) ; + (1/)(6) + (-1)(9) + (1/)(6) or a simpler method ψ = (0)(3) + (1)(6) + (-)(9) + (1)(6) = -6. The error term for any contrast of this form is: SE J a ψ = (MSW) j For this contrast SE ψ j = 1 = (1.5)(0/5 + 1/5 + 4/5 + 1/5) = (1.5)(1.) = 1.5 An F test with 1 and df w degrees-of-freedom is used to test the statistical significance of this n j contrast. F(1, df w ) = ψ / SE ψ = (-6) /1.5 = 36/1.5 = 4.00, which is statistically significant. Thus, F(1,16) = 4.00, p <.05, Reject H 0 : ψ = 0 To compare Occupational Therapy (j = 1) to a combination of the other three groups yields; ψ = (-3)(3) + (1)(6) + (1)(9) + (1)(6) = 1. SE ψ = (1.5)(9/5 + 1/5 + 1/5 + 1/5) = (1.5)(.4) = 3; and F = (1) / 3 = Thus, F(1,16) = 48.00, p <.05, Reject H 0 : ψ = 0 Pairwise Effect Sizes. ES jk = j - k (SS T /(N - 1) For example, ES 1 = (3-6)/.41 = F tests can also be converted to Effect Sizes by the following: ES = df n F df n F + df d or r = df n F df d

4 Oneway Descriptives Relationship of One-Way and Two-Way ANOVA 95% Confidence Interval for Mean N Mean Std. Deviation Std. Error Lower Bound Upper Bound Minimum Maximum OCT MCH NUT EPI Total ANOVA Sum of Mean Squares df Square F Sig. Between Groups Within Groups Total Post Hoc Tests Multiple Comparisons Dependent Variable: Tukey HSD Mean 95% Confidence Interval (I) D (J) D Difference (I-J) Std. Error Sig. Lower Bound Upper Bound OCT MCH NUT EPI MCH OCT NUT EPI NUT OCT MCH EPI EPI OCT MCH NUT * The mean difference is significant at the.05 level. Homogeneous Subsets Tukey HSD Subset for alpha =.05 D N 1 3 OCT MCH EPI NUT Sig Means for groups in homogeneous subsets are displayed. a Uses Harmonic Mean Sample Size =

5 Relationship of One-Way and Two-Way ANOVA Contrast Coefficients D Contrast Maternal Chi Health Occupation Therapy Epidemi ology Nutrition Science Contrast Value of Contrast Std. Error t df Sig. (-tailed) Assume equal Variances Does not Assume Equal Variances Univariate Analysis of Variance Descriptive Statistics Dependent Variable: Std. SCHOOL Orient Mean Deviation N SRHP Applied Exper Total SOPH Applied Exper Total Total Applied Exper Total Tests of Between-Subjects Effects Dependent Variable: Source Type III Sum of Squares df Mean Square F Sig. Corrected Model (a) Intercept SCHOOL Orien SCHOOL * Orien Error Total Corrected Total a R Squared =.818 (Adjusted R Squared =.784)

6 Relationship of One-Way and Two-Way ANOVA 10 9 Nutrition Science M = 9.00 Estimated Marginal Means Matern Health M = 6.00 SD = 1.58 Epidemiology M = 6.00 SCHOOL Orientation 3 Education Applied Occup Ther M = 3.00 SD = 0.71 ORIENTATION Department Membership SHRP applied SOPH Experimental Experimental Psychology 10 Nutrition Science M = Estimated Marginal Means Occup Therapy M = 3.00 SD = 0.71 Epidemiology M = 6 Matern Health M = 6 SD = 1.58 ORIENTATION Department 3 Education SHRP Applied SHRP SCHOOL Orientation SOPH Psychology Experimental SOPH 3

N J SS W /df W N - 1

N J SS W /df W N - 1 One-Way ANOVA Source Table ANOVA MODEL: ij = µ* + α j + ε ij H 0 : µ = µ =... = µ j or H 0 : Σα j = 0 Source Sum of Squares df Mean Squares F J Between Groups nj( j * ) J - SS B /(J ) MS B /MS W = ( N