Wednesday, Nov. 18 Analysis of Variance

The Analysis of Variance Wednesday, Nov. 18 The Analysis of Variance

The Analysis of Variance Wednesday, Nov. 28 The Analysis of Variance ANOVA

Between-group variance estimate F = within-group variance estimate _ SST =  (X - XG)2 SSB =  Ni (Xi - XG)2 SSW =  (X1 - X1)2 +  (X2 - X2)2 + ••••  (Xk - Xk)2 SST = SSB + SSW _ _ _ _ _

Between-group variance estimate F = within-group variance estimate MSB = SSB / dfB MSW = SSW / dfW where dfB = k-1 (k = number of groups) dfW = N - k

Fisher’s Protected t-test Xi - Xj MSW ( 1/Ni + 1/Nj) _ Where df = N - k

dfB (F - 1) Est ω = dfB F + dfW Est ω bears the same relationship to F that rpb bears to t.

The factorial design is used to study the relationship of two or more independent variables (called factors) to a dependent variable, where each factor has two or more levels. - p. 333

The factorial design is used to study the relationship of two or more independent variables (called factors) to a dependent variable, where each factor has two or more levels. In this design, you can evaluate the main effects of each factor independently (essentially equivalent to doing one-way ANOVA’s for each of the factors independently), but you are also able to evaluate how the two (or more factors) interact.

Partitioning variation in a 2x2 factorial design. TOTAL VARIATION Variation within groups (error) Variation between groups Variation from Factor 1 Variation from Factor 2 Variation from Factor 1 x 2 interaction

1.A Compute SST B. Compute SSB C. Subtract SSB from SST to obtain SSW (error) D. Compute SS1 E. Compute SS2 F. Compute SS1x2 by subtracting SS1 and SS2 from SSB 2. Convert SS to MS by dividing SS by the appropriate d.f. 3. Test MS1,MS2 and MS1x2 using F ratio.

More advanced ANOVA topics N-way ANOVA Repeated Measures designs Mixed models Contrasts