STAT 312 11.1 - Two-Factor ANOVA with Kij = 1 Chapter 11 - Multifactor Analysis of Variance Introduction 11.1 - Two-Factor ANOVA with Kij = 1 11.2 - Two-Factor ANOVA with Kij > 1 11.3 - Three-Factor ANOVA 11.4 - 2n Factorial Experiments

One-Way ANOVA Example: Different levels of a treatment (quick review) k groups “Grand Mean” “residuals” Each Yij response is assumed to be a normally-distributed random variable, centered around the group mean i, with the same variance .

One-Way ANOVA Example: Different levels of a treatment (quick review) k groups “Grand Mean” “residuals” Each residual εij is therefore assumed to be a normally-distributed random variable, centered around 0, and with the same variance . Each Yij response is assumed to be a normally-distributed random variable, centered around the group mean i, with the same variance .

(Linear Regression form) One-Way ANOVA Example: Different levels of a treatment (quick review) k groups “Grand Mean” “residuals” ANOVA Model (Linear Regression form)

How are the i estimated? One-Way ANOVA Example: Different levels of a treatment (quick review) k groups “Grand Mean” “residuals” How are the i estimated?

k groups One-Way ANOVA Example: Different levels of a treatment (quick review) k groups “Grand Mean” “residuals” Many generalizations of ANOVA exist, depending on application, assumptions, design, etc. (Covered in remainder of Ch 11.) Example: Different levels of two treatments

Suppose there are two factors, A and B. Two-Way ANOVA Example: Different levels of two treatments Suppose there are two factors, A and B. Factor A has levels i = 1, 2, 3,…, I, and B has levels j = 1, 2, 3,…, J. “Grand Mean” ij = mean response of Yij when Factor A is fixed at level i, Factor B is fixed at level j. 1 2 3 … J .  I “residuals”

Suppose there are two factors, A and B. Two-Way ANOVA Example: Different levels of two treatments Suppose there are two factors, A and B. Factor A has levels i = 1, 2, 3,…, I, and B has levels j = 1, 2, 3,…, J. “Grand Mean” ij = mean response of Yij when Factor A is fixed at level i, Factor B is fixed at level j. “Additive Model” “residuals” i measures the effect of Factor A on the response Yij, while holding level j fixed. j measures the effect of Factor B on the response Yij, while holding level i fixed.

ANOVA Table SSTot = SSA + SSB + SSErr dfTot = dfA + dfB + dfErr Source MS F-ratio p-value Factor A I – 1 SSA MSA pA Factor B J – 1 SSB MSB pB Error (I – 1)(J – 1) SSErr MSErr Total IJ – 1 SSTot –

Suppose there are two factors, A and B. Two-Way ANOVA Example: Different levels of two treatments Suppose there are two factors, A and B. Factor A has levels i = 1, 2, 3,…, I, and B has levels j = 1, 2, 3,…, J. “Grand Mean” ij = mean response when Factor A is fixed at level i, Factor B is fixed at level j. “Additive Model” i measures the effect of Factor A on the response Yij, while holding level j fixed. j measures the effect of Factor B on the response Yij, while holding level i fixed. “residuals”

Suppose there are two factors, A and B. Two-Way ANOVA Example: Different levels of two treatments Suppose there are two factors, A and B. Factor A has levels i = 1, 2, 3,…, I, and B has levels j = 1, 2, 3,…, J. “Grand Mean” ij = mean response when Factor A is fixed at level i, Factor B is fixed at level j. “Interaction term” “residuals” i measures the effect of Factor A on the response Yij, while holding level j fixed. j measures the effect of Factor B on the response Yij, while holding level i fixed.