Random Effects in Factorial Design

In a factorial design, whether factor effects are fixed or random will affect the f-tests that determine the significance of the effect if there is more than one factor.
Does not change the way the sums of squares (SSA,SSB,etc.) or mean squares (MSA,MSB,etc.) are computed.

Plastic Fork Example

A = Thickness of Plastic (1mm…2mm) <- Random
B = Type of Plastic (PVC, ABX) <- Fixed

\(2^2\) Design

Model Equation: \(y\_{ijk} = \mu + \alpha_{i} + \beta_{j} + \alpha\beta_{ij} + \epsilon_{ijk}\)
where \(\epsilon_{ijk} \sim N(0,\sigma^2)\)

When Factors A & B are Fixed

\(E[y_{ijk}] = \mu + \alpha_{i} + \beta_{j} + \alpha\beta_{ij}\)
\(Var[y_{ijk}] = \sigma^2\)

\(H_o: \alpha_{i} = 0 , H_a: \alpha_{i} \neq 0\)
\(H_o: \beta_{j} = 0 , H_a: \beta_{j} \neq 0\)
\(H_o: \alpha\beta_{ij} = 0 , H_a: \alpha\beta_{ij} \neq 0\)

ANOVA Table

##   Source       d.f.   SS   MS        f
## 1      A        I-1  SSA  MSA  MSA/MSE
## 2      B        J-1  SSB  MSB  MSB/MSE
## 3     AB (I-1)(J-1) SSAB MSAB MSAB/MSE
## 4  error             SSE  MSE         
## 5  Total      IJK-1

Expected Mean Squared Value of Sources

\(E[MSA] = \sigma^2 + \frac{JK\sum(\alpha_{i}^2)}{(I-1)}\)
\(E[MSB] = \sigma^2 + \frac{IK\sum(\beta_{j}^2)}{(J-1)}\)
\(E[MSAB] = \sigma^2 + \frac{K\sum(\alpha\beta_{ij}^2)}{(I-1)(J-1)}\)
\(E[MSE] = \sigma^2\)

f Ratios of Factors

\(f = \frac{E[MSA]}{E[MSE]} = \frac{\sigma^2 + \frac{JK\sum(\alpha_{i}^2)}{(I-1)}}{\sigma^2} = 1\)
\(f = \frac{E[MSAB]}{E[MSE]} = \frac{\sigma^2 + \frac{K\sum(\alpha\beta_{ij}^2)}{(I-1)(J-1)}}{\sigma^2} = 1\)

When Factors A & B are Random

Model Equation: \(y\_{ijk} = \mu + \alpha_{i} + \beta_{j} + \alpha\beta_{ij} + \epsilon_{ijk}\)

\(E[y_{ijk}] = \mu\)
\(Var[y_{ijk}] = \sigma_{\alpha}^2 + \sigma_{\beta}^2 + \sigma_{\alpha\beta}^2 + \sigma^2\)

\(H_o: \sigma_{\alpha}^2 = 0 , H_a: \sigma_{\alpha}^2 \neq 0\)
\(H_o: \sigma_{\beta}^2 = 0 , H_a: \sigma_{\beta}^2 \neq 0\)
\(H_o: \sigma_{\alpha\beta}^2 = 0 , H_a: \sigma_{\alpha\beta}^2 \neq 0\)

Expected Mean Squared Value of Sources

\(E[MSE] = \sigma^2\)
\(E[MSA] = JK\sigma_{\alpha}^2 + K\sigma_{\alpha\beta}^2 + \sigma^2\)
\(E[MSB] = IK\sigma_{\beta}^2 + K\sigma_{\alpha\beta}^2 + \sigma^2\)
\(E[MSAB] = K\sigma_{\alpha\beta}^2 + \sigma^2\)

f Ratios of Factors

\(f = \frac{E[MSA]}{E[MSAB]} = \frac{JK\sigma_{\alpha}^2 + K\sigma_{\alpha\beta}^2 + \sigma^2}{K\sigma_{\alpha\beta}^2 + \sigma^2} = 1\)
\(f = \frac{E[MSAB]}{E[MSE]} = \frac{K\sigma_{\alpha\beta}^2 + \sigma^2}{\sigma^2} = 1\)

ANOVA Table

##   Source       d.f.   SS   MS        f
## 1      A        I-1  SSA  MSA MSA/MSAB
## 2      B        J-1  SSB  MSB MSB/MSAB
## 3     AB (I-1)(J-1) SSAB MSAB MSAB/MSE
## 4  error             SSE  MSE         
## 5  Total      IJK-1

When Factor A is Fixed & Factor B is Random (Mixed Effects)

Model Equation: \(y\_{ijk} = \mu + \alpha_{i} + \beta_{j} + \alpha\beta_{ij} + \epsilon_{ijk}\)

\(\alpha = Fixed, \beta = Random, \alpha\beta = Random\)

\(H_o: \alpha_{i} = 0 , H_a: \alpha_{i} \neq 0\)
\(H_o: \sigma_{\beta}^2 = 0 , H_a: \sigma_{\beta}^2 \neq 0\)
\(H_o: \sigma_{\alpha\beta}^2 = 0 , H_a: \sigma_{\alpha\beta}^2 \neq 0\)

Expected Mean Squared Value of Sources

\(E[MSA] = \sigma_{\alpha}^2 + K\sigma_{\alpha\beta}^2 + \frac{JK\sum(\alpha_{i}^2)}{(I-1)}\)
\(E[MSB] = \sigma^2 + IK\sigma_{\beta}^2\)

\(Note: K\sigma_{\alpha\beta}^2 \ is \ restricted\)

\(E[MSAB] = \sigma^2 + K\sigma_{\alpha\beta}^2\)
\(E[MSE] = \sigma^2\)

ANOVA Table

##   Source       d.f.   SS   MS        f
## 1      A        I-1  SSA  MSA MSA/MSAB
## 2      B        J-1  SSB  MSB  MSB/MSE
## 3     AB (I-1)(J-1) SSAB MSAB MSAB/MSE
## 4  error             SSE  MSE         
## 5  Total      IJK-1