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 $\beta = [\mu, \alpha_2, \beta_2, \beta_3, \gamma_{22}, \gamma_{23} ]'$
    • $\mu = \mu_{11}$ , $\alpha = \mu_{21} - \mu_{11}$, $\beta_2 = \mu_{12} - \mu_{11}$, $\beta_3 = \mu_{13} - \mu_{11}$
    • $\gamma_{22} = \mu_{11}+\mu_{22}-\mu_{21}-\mu_{12}$, $\gamma_{23} = \mu_{11}+\mu_{23} - \mu_{13} - \mu_{21}$