5.1.1a: The Additive Model (No Interaction)
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In a factorial design, we first look at the interactions for significance. In the case where 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 5.1) 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 two 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 was '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.
The amount of glucose detected in each sample was recorded and is presented in the table below.
| Chemical Assay Method | ||||||
|---|---|---|---|---|---|---|
| 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 | |
Solution
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-value > 0.05) at a 5% level. We drop the interaction effect from the model and run the additive model. The resulting ANOVA table is:
| 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 |
| method | 1 | 263.733889 | 263.733889 | Var(Residual)+Q(method, method) | MS(Residual) | 14 | 80.26 | <.0001 |
| doping | 2 | 57026 | 28513 | Var(Residual) + Q(doping,doping) | MS(Residual) | 14 | 8677.63 | <.0001 |
| 1Residual | 14 | 46.001111 | 3.285794 | Var(Residual) | ||||
The Error SS is now 46.001, which is the sum of the interaction SS and the error SS of the model with the interaction. The df values were also added the same way. This example shows that any term not included in the model gets added into the error term, which may erroneously inflate the error especially if the impact of excluded term on the response is not negligible.
The Error SS is now 46.001, which is the sum of the interaction SS and the error SS of the model with the interaction. The df values were also added the same way. This example shows that any term not included in the model gets added into the error term, which may erroneously inflate the error especially if the impact of excluded term on the response is not negligible.
| 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 |
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| doping Least Squares Means | ||||||||
|---|---|---|---|---|---|---|---|---|
| Doping | Estimate | Standard Error | DF | t Value | Pr >|t| | Alpha | Lower | Upper |
| 1 | 43.67 | 0.7400 | 14 | 59.01 | <.0001 | 0.05 | 42.08 | 45.25 |
| 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 |
Here, we can see that the response variable, the amount of glucose detected in a sample, is the overall mean PLUS the effect of the method used PLUS the effect of the glucose amount added to the original sample. (Hence, the additive nature of this model!)