In a factorial design we first look at the interactions for significance. In the case where an interaction is not significant, then we can drop the interaction term from our model, and we end up with an additive model.

For a two-factor factorial, the model we initially consider (as we have discussed in Section 4.1) is:

\(Y_{ij}=\mu_{..}+\alpha_i+\beta_j+(\alpha\beta)_{ij} +\epsilon_{ijk}\)

Note that the interaction term (\((\alpha\beta)_{ij}\)) is a multiplicative term.

If the interaction is found to be non-significant, then the model reduces to:

\(Y_{ij}=\mu_{..}+\alpha_i+\beta_j+\epsilon_{ijk}\)

Here we can see that the response variable is simply a function of adding the effects of the factors.

As an example, (adapted from Kuehl, 2000), let's look at a study designed to evaluate two chemical methods used for assaying the amount of glucose in blood serum. A large volume of blood serum served as a starting point for the experiment. The blood serum was divided into three portions, each of which were “doped” or augmented by adding an additional amount of glucose. Three doping levels were used. Samples of the doped serum were then assayed for glucose concentration by one of two chemical methods. This type of ‘doping’ experiment is commonly used to compare the sensitivity of assay methods.

Here are the results (Glucose data):

Method 1 | Method 2 | |||||
---|---|---|---|---|---|---|

Doping Level | 1 | 2 | 3 | 1 | 2 | 3 |

46.5 | 138.4 | 180.9 | 39.8 | 132.4 | 176.8 | |

47.3 | 144.4 | 180.5 | 40.3 | 132.4 | 173.6 | |

46.9 | 142.7 | 183 | 41.2 | 130.3 | 174.9 |

The model was run as a two-factor factorial and produced the following results:

Type 3 Analysis of Variance | ||||||||
---|---|---|---|---|---|---|---|---|

Source | DF | Sum of Squares | Mean Square | Expected Mean Square | Error Term | Error DF | F Value | Pr > F |

method | 1 | 263.733889 | 263.733889 | Var(Residual) + Q(method, method*doping) | MS(Residual) | 12 | 98.35 | <.0001 |

doping | 2 | 57026 | 28513 | Var(Residual) + Q(doping, method*doping) | MS(Residual) | 12 | 10632.5 | <.0001 |

method*doping | 2 | 13.821111 | 6.910556 | Var(Residual) + Q(method*doping) | MS(Residual) | 12 | 2.58 | 0.1172 |

Residual | 12 | 32.180000 | 2.681667 | Var(Residual) |

Here we can see that the interaction of `method*doping`

was not significant (*p *> 0.05). We drop this from the model and re-run:

The Mixed Procedure | ||||||||
---|---|---|---|---|---|---|---|---|

Type 3 Analysis of Variance | ||||||||

Source | DF | Sum of Squares | Mean Square | Expected Mean Square | Error Term | Error DF | F Value | Pr > F |

Doping | 2 | 57026 | 28513 | Var(Residual) + Q(Doping) | MS(Residual) | 14 | 8677.63 | <.0001 |

Method | 1 | 263.733889 | 263.733889 | Var(Residual)+Q(Method) | MS(Residual) | 14 | 80.26 | <.0001 |

Residual | 14 | 46.001111 | 3.285794 | Var(Residual) |

The Error SS is now 46.001. It was 32.18, so the difference here without the interaction term, 13.821111, is now found in the Error SS.

method Least Squares Means |
||||||||
---|---|---|---|---|---|---|---|---|

method | Estimate | Standard Error | DF | t Value | Pr >|t| | Alpha | Lower | Upper |

1 | 123.40 | 0.6042 | 14 | 204.23 | <.0001 | 0.05 | 122.10 | 124.70 |

2 | 115.74 | 0.6042 | 14 | 191.56 | <.0001 | 0.05 | 114.45 | 117.04 |

Doping Least Squares Means | ||||||||
---|---|---|---|---|---|---|---|---|

Doping | Estimate | Standard Error | DF | t Value | Pr >|t| | Alpha | Lower | Upper |

1 | 43.6667 | 0.7400 | 14 | 59.01 | <.0001 | 0.05 | 42.0795 | 45.2539 |

2 | 136.77 | 0.7400 | 14 | 184.81 | <.0001 | 0.05 | 135.18 | 138.35 |

3 | 178.28 | 0.7400 | 14 | 240.92 | <.0001 | 0.05 | 176.70 | 179.87 |

The response variable, amount of glucose detected in a sample, is then an overall mean **PLUS **the effect of the method used **PLUS **the amount of glucose added to the original sample. (Hence, the additive nature of this model!)