# Chapter 4 Analysis of Two-Factor Experiments Based on Cell Means Models

• treatment: a combination of one level from each factor forms a treatment
• full-factorial treatment design: each possible combination of one level from each possible combination of one level from each factor is applied to at least one experimental unit
• completely randomized design (CRD): all possible balanced assignments to the treatment groups are equally likely
• cell mean model: for $$i = 1, 2$$, $$j = 1,2,3$$ and $$k = 1,2$$, suppose $$y_{ijk} = \mu_{ij} + \epsilon_{ijk}$$
• $$Y = X_{n\times p}\beta + \epsilon$$
• each cell mean is estimable
• $$\beta$$ is estimable when $$rank(X) = p$$
• $$\bar \mu_{.j} = \frac{1}{2}\sum_{i=1}^2 \mu_{ij}$$, $$\bar \mu_{i.} = \frac{1}{3}\sum_{j=1}^3 \mu_{ij}$$ $$\bar\mu_{..} = \frac{1}{6}\sum_{i=1}^2\sum_{j=1}^3 \mu_{ij}$$
• Least square means: for example, the LSMEAN for $$i = 1$$ is $$c'\hat\beta$$ with $$c' = [1/3, 1/3, 1/3, 0, 0, 0]$$ and $$\hat\beta = [\bar y_{11.}, \bar y_{12.}, \bar y_{13.}, \bar y_{21.}, \bar y_{22.}, \bar y_{23.}]'$$. The standard error for an LSMEAN is given by $$\sqrt{\widehat{Var}(c'\hat\beta)} = \sqrt{\hat\sigma^2c'(X'X)^-c}$$.
• simple effect: the difference between cell means that differ in level for only one factor
• for example, $$\mu_{11} - \mu_{21}$$
• main effect: the difference between marginal means associated with two levels of a factor
• for example: $$\bar \mu_{1.} - \bar \mu_{2.}$$
• no main effect for $$j$$: $$\bar \mu_{.1} = \bar \mu_{.2} = \bar \mu_{.3}$$
• Interaction effect: the linear combination $$\mu_{ij} - \mu_{ij^*} - \mu_{i^*j}+\mu_{i^*j^*}$$ for $$i \neq i^*$$ and $$j \neq j^*$$
• when there are no interactions between factors, the simple effects of either factor are the same across all levels of the other factor.
• Test for non-zero effects: we can test whether simple effects, main effects or interaction effects are zero vs. non-zero using tests of the form $$H_0: C\beta = 0$$ vs. $$H_A: C\beta \neq = 0$$
• An alternative parameterization of the cell mean model is $$y_{ijk} = \mu + \alpha_i + \beta_j + \gamma_{ij} + \epsilon_{ijk}$$.
• for example, $$\mu_{11} = \mu + \alpha_1 + \beta_{1} + \gamma_{11}$$, $$\bar \mu_{1.} = \mu + \alpha_1 + \bar \beta_. + \bar \gamma_{1.}$$
• simple effect $$\mu_{11} - \mu_{12} = \alpha_1 - \alpha_2 + \gamma_{11} - \gamma_{21}$$.
• main effect $$\bar \mu_{1.} - \bar \mu_{2.} = \alpha_1 - \alpha_2 + \bar\gamma_{1.} - \bar\gamma_{2.}$$
• interaction effect $$\mu_{11}-\mu_{13}-\mu_{21}+\mu_{23} = \gamma_{11}-\gamma_{13}-\gamma_{21} + \gamma_{23}$$
• The lm function in R uses a full-rank model matrix with $$\beta = [\mu, \alpha_2, \beta_2, \beta_3, \gamma_{22}, \gamma_{23} ]'$$
• $$\mu = \mu_{11}$$ , $$\alpha = \mu_{21} - \mu_{11}$$, $$\beta_2 = \mu_{12} - \mu_{11}$$, $$\beta_3 = \mu_{13} - \mu_{11}$$
• $$\gamma_{22} = \mu_{11}+\mu_{22}-\mu_{21}-\mu_{12}$$, $$\gamma_{23} = \mu_{11}+\mu_{23} - \mu_{13} - \mu_{21}$$