Two Factor ANOVA Under Balanced Designs
 Page ID
 210
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left#1\right}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The dependence of a response variable on two factors, A and B, say, is of interest.
 Factor A has a levels and Factor B has b levels.
 In total there are a x b treatments.
 Sample sizes for all treatment groups are equal (balanced design).

Goal: to study the simultaneous effects of the two factors on the response variable, including their respective main effects and their interaction effects.
1.1 Example: drugs for hypertension
A medical investigator studied the relationship between the response to three blood pressure lowering drugs for hypertensive males and females. The investigator selected 30 males and 30 females and randomly assigned 10 males and 10 females to each of the three drugs.
 This is a balanced randomized complete block design (RCBD)
 Two factors:
 A  gender (observational factor)
 B  drug (experimental factor)
 Factor A has a = 2 levels: male vs. female
 Factor B has b = 3 levels: drug 1, drug 2, and drug 3
 In total there are a x b = 2 x 3 = 6 treatments
 For each treatment, the sample size is n = 10
Treatment  Description  Sample Size 
1  drug 1, male  10 
2  drug 1, female  10 
3  drug 2, male  10 
4  drug 2, female  10 
5  drug 3, male  10 
6  drug 3, female  10 
1.2 Population means
 Treatment means: \(\mu_{ij}\) = population mean response of the treatment with factor A at level i and factor B at level j
 Factor level means: \(\mu_{i\cdot}\) = population mean response when the ith level of factor A is applied; \(\mu_{\cdot j}\) = population mean response when the jth level of factor B is applied:
 Overall mean (the basic line quantity in comparisons of factor effects):
FACTOR B  
j = 1  j = 2  . . . . .  j = b  Row Avg  
i = 1  \(\mu_{11}\)  \(\mu_{12}\)  . . . . .  \(\mu_{1b}\)  \(\mu_{1\cdot}\)  
i = 2  \(\mu_{21}\)  \(\mu_{22}\)  . . . . .  \(\mu_{2b}\)  \(\mu_{2\cdot}\)  
FACTOR A  .  . . .  . . .  . . . . .  . . .  . . . 
i = a  \(\mu_{a1}\)  \(\mu_{a2}\)  . . . . .  \(\mu_{ab}\)  \(\mu_{a\cdot}\)  
Column Avg  \(\mu_{\cdot 1}\)  \(\mu_{\cdot 2}\)  . . . . .  \(\mu_{\cdot b}\)  \(\mu_{\cdot \cdot}\) 
1.3 Main effects
 Main effects are defined as the differences between factor level means and the overall mean
 Factor A main effects: \(\alpha_{i}\) = main effect of factor A at the ith factor level
 Factor B main effects: \(\beta_{j}\) = main effect of factor B at the jth factor level
 For both factor A and factor B, the sum of main effects is zero
1.4 Interaction effects
 Interaction effects describe how the effects of one factor depend on the levels of the other factor
 \((\alpha\beta)_{ij}\) = interaction effect of the ith level of factor A and jth level of factor B
 Note: for \(1 \leq i \leq a, 1\leq j \leq b\)
 If all \((\alpha\beta)_{ij} = 0, i = 1, ..., a, j = 1, ..., b\), then the factor effects are additive
 This is equivalent to
 If at least one of the \((\alpha\beta)_{ij}\)'s is nonzero, then the factor effects are interacting
 This means that the effects of one factor are different for differing levels of the other factor.
1.5 Additive factor effects
 If the two are additive (i.e. no interaction)
 Each factor can be studied separately, based on their factor level means \(\{\mu_{i\cdot}\}\) and \(\{\mu_{\cdot j}\}\), respectively
 This is much simpler than the joint analysis based on the treatment means \(\{\mu_{ij}\}\)
Example 1
Factor A has a = 2 levels, Factor B has b = 3
 Check additivity for all pairs of (i,j)
\[\alpha_{1} = \mu_{1\cdot}  \mu_{\cdot \cdot} = 12  12 = 0\]
\[\beta_{1} = \mu_{\cdot 1}  \mu_{\cdot \cdot} = 9  12 = 3\]
\[\mu_{11} = 9\]
\[\mu_{\cdot \cdot} + (\alpha_{1} + \beta_{1}) = 12 + (0  3) = 9 = \mu_{11}\]
 Exercise: Complete the check for additivity
FACTOR B  
j = 1  j = 2  j = 3  \(\mu_{i\cdot}\)  
FACTOR A  i = 1  9  11  16  12 
i = 2  9  11  16  12  
\(\mu_{\cdot j}\)  9  11  16  12 (= \(\mu_{\cdot \cdot}\)) 
1.6 Graphical method: interaction plots
Interaction plots constitute a graphical tool to check additivity. Xaxis is for the factor A (or B) levels, and Yaxis is for the treatment means \(\mu_{ij}\)'s
 seperate curves are drawn for each of the factor B (or A) levels
 If the curves are all horizontal, then the factor on the Xaxis has no effect at all, i.e. the treatment means do not depend on the level of that factor
 If the curves are overlapping, then the other factor (the one not on the Xaxis) has no effect
 If the curves are parallel, then the two factors are additive, i.e. the effects of factor A do not depend on (or interact with) the level of factor B, and vice versa
 Note: "horizontal" and "overlapping" are special cases of "parallel"
For Example 1:
 The two factors are additive
 Moreover, factor A does not have any effect at all (main effects of factor A are all zero)
 Factor B does have some effects (not all main effects of factor B are zero)
Factor B  
j = 1  j = 2  j = 3  \(\mu_{i\cdot}\)  
Factor A  i = 1  11  13  18  14 
i = 2  7  9  14  10  
\(\mu_{\cdot j}\)  9  11  16  12 (=\(\mu_{\cdot \cdot}\)) 
 The two factors are additive, since the curves are parallel
 Both factors have some effects (main effects of both factors are not all zero), since the curves are neither horizontal to the Xaxis nor overlapping in any of the plots
 Note: Indeed, you only need to examine one of the two plots. If the curves in one plot are parallel, the curves in the other plot must also be parallel
Summary: Additive Model
 For all pairs of (i, j): \(\mu_{ij} = \mu_{\cdot \cdot} + \alpha_{i} + \beta_{j}\)
 The curves in an interaction plot are parallel
 The difference between treatments means for any two levels of factor B (respectively, A) is the same for all level of factor A (respectively, B):
\[\mu_{1j}  \mu_{1j'} = ... = \mu_{aj}  \mu_{aj'}, 1 \leq j, j' \leq b\]
1.7 Interacting factor effects
 Interpretation of \((\alpha\beta)_{ij}\): the difference between the treatment mean \(\mu_{ij}\) and the value that would be expected if the two factors were additive
 Factor A and factor B are interacting: if some \((\alpha\beta)_{ij} \neq 0\), i.e. \(\mu_{ij} \neq \mu_{\cdot \cdot} + \alpha_{i} + \beta_{j}\) for some (i, j)
 Equivalently, the curves are not parallel in an interaction plot
Example 3
Factor A: a = 2 levels; Factor B: b = 3 levels
Factor B  
j = 1  j = 2  j = 3  \(\mu_{i\cdot}\)  
Factor A  i = 1  9  12  18  13 
i = 2  9  10  14  11  
\(\mu_{\cdot j\)  9  11  16  12 (=\(\mu_{\cdot \cdot}\)) 
 The two curves in the interaction plot are not parallel, which means the two factors are interacting
 For example:
\[\alpha_{1} = \mu_{1\cdot}  \mu_{\cdot \cdot} = 13  12 = 1 and \beta_{1} = \mu_{\cdot 1}  \mu_{\cdot \cdot} = 9  12 = 3\]
\[Thus 9 = \mu_{11} \neq \mu_{\cdot \cdot} + \alpha_{1} + \beta_{1} = 12 + 1  3 = 10\]
\[Or (\alpha\beta)_{11} = 9  10 = 1 \neq 0\]
\[\mu_{11}  \mu_{12} = 9  12 = 3 \neq \mu_{21}  \mu_{22} = 9  10 = 1\]
 There is a larger difference among treatment means between the two levels of factor A when factor B is at the 3rd level (j = 3) than when B is at the first two levels (j = 1, 2)
Suppose we put Factor B on the Xaxis of the interaction plot
 The differences in heights of the curves reflect Factor A effects. On the other hand, if all curves are overlapping, then Factor A has no effect.
 The departure from horizontal by the curves reflects Factor B effects. On the other hand, if all curves are horizontal, then Factor B has no effect.
 The lack of parallelism among the curves reflects interaction effects.
 Any one factor with no effect means additivity
 Important: no main effects does not necessarily mean no effects or no interaction effects.
Example 4 (refer to figure 4)
 Figure (a): additive
 Figure (b) and (c): Factor B has no main effects, but Factor A and Factor B are interacting
 Figure (d): when Factor A is at level 1, treatment means increase with Factor B levels; when Factor A is at level 2, the trend becomes decreasing
 Figure (e): larger difference among treatment means between the two levels of Factor A when Factor B is at a smaller indexed level
 Figure (f): more dramatic change of treatment means among Factor B levels when Factor A is at level 1