Analyze Factorial Design

Two-Level Factorial Designs
Analysis of Variance Table - P-Value

  

Use the p-values (P) in the analysis of variance table to determine which of the effects in the model are statistically significant. Typically, you look at the interaction effects in the model first, because a significant interaction will influence how you interpret the main effects. To use the p-value, you need to do the following:

·    Identify the p-value for the effect you want to evaluate.

·    Compare this p-value to your a-level. A commonly used a-level is 0.05.

-    If the p-value is less than or equal to a, conclude that the effect is significant.

-    If the p-value is greater than a, conclude that the effect is not significant.

In theory, any VIF value greater than 1 can inflate the variance of the coefficients so much that statistical significance is a less useful way to identify candidate models. In practice, values greater than 5–10 can produce unstable coefficients that are difficult to interpret and, thereby, prompt corrective measures

Example Output

Analysis of Variance

 

Source                 DF   Adj SS   Adj MS  F-Value  P-Value

Model                  11  451.357   41.032    17.99    0.007

  Covariates            1    3.591    3.591     1.58    0.278

    MeasTemp            1    3.591    3.591     1.58    0.278

  Linear                4  304.587   76.147    33.39    0.002

    Material            1   35.053   35.053    15.37    0.017

    InjPress            1  113.068  113.068    49.59    0.002

    InjTemp             1   75.533   75.533    33.12    0.005

    CoolTemp            1   38.666   38.666    16.96    0.015

  2-Way Interactions    6   20.309    3.385     1.48    0.366

    Material*InjPress   1    1.732    1.732     0.76    0.433

    Material*InjTemp    1    3.045    3.045     1.34    0.312

    Material*CoolTemp   1    0.095    0.095     0.04    0.848

    InjPress*InjTemp    1    1.538    1.538     0.67    0.458

    InjPress*CoolTemp   1    0.012    0.012     0.01    0.947

    InjTemp*CoolTemp    1   14.694   14.694     6.44    0.064

Error                   4    9.121    2.280

Total                  15  460.478

Interpretation

For the insulation data, the analysis of variance table shows the following results:

·    Interaction effects: The model contains six two-way interaction effects that you typically evaluate first.

The p-value of 0.366 for the set of two-way interactions is not less than 0.05. Therefore, the group of two-way interactions is not significant. The p-value for the interaction between the injection temperature and the cooling temperature is 0.064, which is not less than 0.05. However, this interaction is statistically significant in some alternative models. For example, the model that contains the main effects, the covariate, and only this two-way interaction.

·    Main effects: The model contains four main effects that you typically evaluate after you know whether interactions affect them.

The p-value of 0.002 for the set of main effects is less than 0.05. Therefore, there is significant evidence that at least one factor has an effect on insulation strength. All the p-values for the individual main effects are less than 0.05. Material type (p-value = 0.017), injection pressure (p-value = 0.002), injection temperature (p-value = 0.005), and cooling temperature (p-value = 0.015) all have a significant effect on insulation strength.

·    Covariates: The researchers measured the temperature at the time of measurement that was included as a covariate in the analysis. The p-value of 0.278 for the covariate is not less than 0.05. However, the VIF for the covariate that is in the coefficients table is 5.87. The temperature is moderately correlated with the material and more weakly correlated with the other factors and interactions. The temperature could have an effect that the multicollinearity hides. The coefficient could also change dramatically as the model changes.