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