4.1.1a - The Additive Model (No Interaction)

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):

Chemical Assay
  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!)