Monday November 15 Analysis of Variance Monday November

Monday, November 15 Analysis of Variance

Monday, November 15 The Analysis of Variance

Monday, November 15 The Analysis of Variance ANOVA

F= Between-group variance estimate within-group variance estimate _ SST = (X - XG)2 _ _ SSB = Ni (Xi - XG)2 _ SSW = (X 1 - X 1_)2 + (X 2 - X 2)2_+ • • (Xk - Xk)2 SST = SSB + SSW

F= Between-group variance estimate within-group variance estimate MSB = SSB / df. B MSW = SSW / df. W where df. B = k-1 (k = number of groups) df. W = N - k

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

Est ω = df. B (F - 1) df. B F + df. W 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 oneway 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 2 x 2 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 B. C. D. E. F. Compute SST Compute SSB Subtract SSB from SST to obtain SSW (error) Compute SS 1 Compute SS 2 Compute SS 1 x 2 by subtracting SS 1 and SS 2 from SSB 2. Convert SS to MS by dividing SS by the appropriate d. f. 3. Test MS 1, MS 2 and MS 1 x 2 using F ratio.

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