Analysis of an Experiment with Three-Level Factors - ANOVA/Effects Tab

Analyzing the 3(k-p) Design

Select the ANOVA/Effects tab of the Analysis of an Experiment with Three-Level Factors dialog to access options to review more detailed information about the ANOVA effects than that available on the Quick tab. Note that these results are for the currently specified model. You can specify a new model on the Model tab.

Summary: Effect estimates

Click the Summary: Effect estimates button to display a spreadsheet with the ANOVA effect estimates and coefficients for the coded model. The effects 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 option Use centered & scaled polynomials described below. 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, see below), and their Statistical significance. Statistically significant parameters will be highlighted in this spreadsheet; the criterion for Statistical significance can be set via the Alpha (highlighting) option (the default Alpha is .05, see below). If the Effects sorted by size check box (see below) is selected, the estimates in this spreadsheet will be sorted by their absolute size (except for the intercept, which is always listed first).

All estimates in this spreadsheet pertain to the coded factor settings, that is, to the factor settings scaled to the ±1 range (see also option Use centered and scaled polynomials, below). To see the results for the original (untransformed) factor settings, click the Regression coefficients. button (see below). See Main Effects and Interactions for Experiments with Three-Level Factors for a detailed description of this spreadsheet.

Use centered and scaled polynomial

When the Use centered and scaled polynomial check box 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. When this 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.

Regression coefficients

Click the Regression coefficients button to display the multiple regression estimates for the original factor values. The coefficients computed depend on the current model, as specified via the options in the Include in model group box on the Model tab. Please note that the specified model will be applied to all selected dependent variables. If an error term for the ANOVA is available, then this spreadsheet will also include the standard errors of the regression coefficients, their confidence intervals (according to the setting of the Confidence interval box, see below), and their Statistical significance. Note that the ANOVA error term not only depends on the currently specified model, but also on the choice of error term in the ANOVA error term group box on the Model tab. Statistically significant parameters will be highlighted in this spreadsheet; the criterion for Statistical significance can be set via the Alpha (highlighting) option (the default Alpha is .05). If the Effects sorted by size check box is selected, the estimates in this spreadsheet will be sorted by their absolute size (except for the intercept, which will always be listed first).

Note: Quadratic effects. To compute the quadratic effects (and interactions by quadratic effects, if selected), Statistica does not perform any recoding of factor values. For example, the quadratic main effects are computed by adding to the design (when computing the correlation matrix from which the regression coefficients are estimated) new variables that are set as equal to the squared original variable values (factor settings). (This note refers to the regression coefficients only.)

Effects sorted by size

Select the Effects sorted by size check box to review the Summary: Effect estimates as well as the Regression coefficients (see above) sorted by (absolute) size. This option is particularly useful in order to identify the important effects in a large design with many factors and interactions.

Confidence interval

The value in the Confidence interval box determines the confidence intervals that will be computed for all relevant options on this dialog. Confidence intervals for the parameter estimates are computed (if available) when you click the Summary: Effect estimates or the Regression coefficients buttons (see above). Confidence intervals for the observed means (if replicate observations are available) will be computed when you choose the Display design and observed means option on the Design tab. Confidence intervals for marginal means will be computed if you choose the Observed marginal means options (spreadsheet or plot) on the Means tab. Confidence intervals for predicted values will be computed when you choose the Predict dependent variable values option on the Prediction & profiling tab.

Alpha (highlighting)

The value in the Alpha (highlighting) box determines the criterion for Statistical significance for all relevant options on this dialog. When you click the Summary: Effect estimates or Regression coefficients buttons (see above), this criterion will be used for highlighting significant effects and coefficients (if an error term is available). When you choose the Pareto chart of effects and plot standardized effects (see below), this criterion will be used to draw a vertical line across the columns of the chart, to indicate the minimum magnitude for a significant effect.

ANOVA table

Click the ANOVA table button to display two ANOVA tables for the current model, as specified in the Include in model group box and based on the chosen error term in the ANOVA error term group box, both on the Model tab, and based on the parameterization that follows from the setting of the Use centered & scaled polynomials check box (see above). Please refer to the note on Main Effects and Interactions for a detailed discussion of the different effects, the coding of the blocking variables, and the effect of clearing the (default) Use centered & scaled polynomials check box.

Note: Pure error and lack-of-fit. Note that if you selected the estimate of Pure error for the error term on the Model tab (if it was available), then the ANOVA table will also include a Lack of fit test (see the Introductory Overview). This is a test of the residual variance, after controlling for all effects in the model, against the estimate of pure error. If significant, then there is indication of additional significant effects, or differences between means of the design that cannot be accounted for by the parameters currently in the model. For example, there may be higher-order interactions between the factors in the design.

Plots of effects

Use the Plots of effects group box to access the options described here.

Normal probability plot

Click the Normal probability plot button to display a normal probability plot of the ANOVA parameter estimates for the current model (as specified in the Include in model group box on the Model tab). In this plot, the normal probabilities of the rank-ordered parameters are plotted on the y-axis, and the actual parameter estimates (optionally standardized) are plotted on the x-axis. If all estimates come from a population with a mean parameter estimate of zero and a common variance, then the points in this plot will approximate a straight line. "Real" effects will show in this plot as outliers. To summarize, this plot helps you distinguish between random noise and real effects.

Half-normal probability plot

Click the Half-normal probability plot button to display a half-normal probability plot of the ANOVA parameter estimates for the current model (as specified in Include in model group box on the Model tab).

Pareto chart

Click the Pareto chart button to display a Pareto chart of the ANOVA effect estimates, or, optionally, the standardized effect estimates (if the Plot standardized effects check box is selected, see below). The Pareto chart shows the effect estimates sorted by their absolute size. If you plot the standardized effects, a vertical line will also be shown to indicate the minimum magnitude of Statistically significant effects, given the current model and choice of error term, and using the criterion of Statistical significance selected in the Alpha (highlighting) box (see above). The Pareto chart is very useful for reviewing a large number of factors, and for presenting the results of an experiment to an audience that is not familiar with standard Statistical terminology.

Label points in normal plots

If the Label points in normal plots check box is selected, then the points in the normal probability plot of effects will be labeled.

Exclude block effects

Select the Exclude block effects check box to omit the block effects from the Pareto chart of effects or the Normal probability plot of effects. Note that this check box is only available if the current design contains blocking.

Plot standardized effects

If the Plot standardized effects check box is selected, then the Pareto chart of effects and the probability plot of effects will be produced for the standardized effects, that is, for the effects divided by their respective standard errors. Note that this option is only available if the standard error for the parameter estimates can be computed. Also, the standard errors are dependent on the current model and choice of error term (see Main Effects and Interactions for details).