Main Effects and Interactions for Experiments with Three-Level Factors
Both the Quick tab and the ANOVA/Effects tab of the Analysis of an Experiment with Three-Level Factors dialog box contain the Summary: Effect estimates button. Click this button to produce a spreadsheet with the ANOVA effect estimates and coefficients for the coded model. The effects that will be computed depend on the current model, as specified via the options in the Include in model group box on the Model tab, and on the recoding of the factor values (i.e., the parameterization of the model) as indicated via the Use centered & scaled polynomials check box on the Quick tab and the ANOVA/Effects tab. If an error term for the ANOVA is available, then this spreadsheet will also include the standard errors of the parameter estimates and coefficients, their confidence intervals (according to the setting of the Confidence interval box on the ANOVA/Effects tab), and their statistical significance.
All estimates in this spreadsheet pertain to the coded factor settings, that is, to the factor settings scaled to the ±1 range (see also the Use centered & scaled polynomials check box on the Quick tab and the ANOVA/Effects tab). To see the results for the original (untransformed) factor settings, use option Regression coefficients on the ANOVA/Effects tab.
- Mean/Intercept
- The row labeled Mean/ Interc. contains the estimate of the intercept for the regression model, based the coded factor settings (see also Multiple Regression).
- Block effects
- The block effects are computed from recoded added variables (to the design), one for each degree of freedom. For example, if you have 9 blocks in the design then 8 new variables are created to exploit the 8 degrees of freedom associated with the overall block effect. Specifically, each new variable bi is computed as:
- Linear main effects
- The values shown in the column labeled Effect, and in the rows labeled by the main effects, are the ANOVA effect estimates. These can be interpreted as the differences (for the dependent variable) between the low settings and the high settings for the respective factors. The values shown in the column labeled Coeff. are the regression coefficients for the coded model, and they can be interpreted as the differences between the high factor settings and the averages of the low and high factor settings for the respective factors. Thus, they are half the size of the Effect estimates.
- Centered/scaled polynomials and quadratic effects
- The coding and, thus, interpretation of the quadratic effects depends on the setting of the Use centered & scaled polynomials check box on the
Quick tab and the
ANOVA/Effects tab:
Recoded Factor Values Original Factor Setting Centered/ Scaled Not Centered/ Scaled Low -2/3 1 Medium 4/3 0 High -2/3 1 Thus, when option Use centered & scaled polynomials is selected (the default), then the original factor settings are recoded so that the effect estimates are comparable in size to the linear main effect estimates. In that case, the interpretation of the quadratic main effect estimates is analogous to that of the linear main effects. Namely, the quadratic main effect estimates are the differences (for the dependent variable) between the medium setting and the average of the low and high settings for the respective factors. The coefficients are half the difference.
When the Use centered & scaled polynomials check box is cleared, then the coding for the quadratic main effects is the result of squaring the ±1 coding for the linear main effects. In that case, the effect estimates are not comparable in size to the linear effect estimates.
- Centered/scaled polynomials and the ANOVA table
- The setting of the Use centered & scaled polynomials check box will also affect the sums-of-squares estimates in the ANOVA table. When you select the Use centered & scaled polynomials check box (i.e., leave this check box at its default, selected setting), then the resulting sums-of-squares for the effects reported in the ANOVA table will be identical to those you would obtain if you analyzed the design using a standard ANOVA approach.
- Linear by linear interaction effects
- The values in the Effect column, in the rows denoting the linear factor interaction effects, can be interpreted as follows (see Mason, Gunst, and Hess, 1989, page 127): The two-way interaction effects is half the difference between the main effects of one factor at the two levels of a second factor. Again, the values in the Coeff. column are half the Effect estimates.
- Quadratic interaction effects
- The interpretation of the interaction effects involving the quadratic components are also dependent on the setting of the Use centered & scaled polynomials check box on the Quick tab and the ANOVA/Effects tab. If this check box is selected (the default), the interpretation of the quadratic interaction estimates is analogous to that of the linear interaction estimates. Namely, the quadratic interaction of two factors is half the difference between the quadratic main effect of the first factor at the respective levels of the second factor. So, for linear by quadratic interactions, the effect estimate is half the difference between the quadratic main effect for the second factor at the low/high levels of the first factor; for the quadratic by quadratic interaction, it is half the difference between the quadratic effect for the second factor at the center level of the first factor, and the average quadratic effect at the extreme (low/high) settings of the first factor.
- Standard error of parameter estimates
- The standard errors for the parameter estimates (ANOVA estimates and coefficients) are computed from the current ANOVA error term, as selected in the ANOVA error term group box, and depending on the model as specified in the Include in model group box, both on the Model tab.