Analysis of a Mixture Experiment - Prediction & Profiling Tab

Select the Prediction & profiling tab of the Analysis of a Mixture Experiment dialog to access options to generate plots for the fitted response surface and predicted values as well as to perform response desirability profiling. Note that these results are for the currently specified model. You can specify a new model on the Model tab.

Response desirability profiling

Click the Response desirability profiling button to display the Profiler dialog. Use this dialog to inspect the predicted values for the dependent variables at different combinations of levels of the independent variables, to specify desirability functions for the dependent variables, and to specify a search for the levels of the independent variables that produce the most desirable response on the dependent variables.

Surface plot (fitted response). Click the Surface plot (fitted response) button to plot the currently fitted model in a surface plot, along with the observed points in the experiment. If there are more than three factors in the current design, the Select factors for triangular graph dialog is displayed when you click the Surface plot (fitted response) button. From this dialog, select the three variables for the plot.

When there are more than three factors in the current experiment, the Select factor values dialog also is displayed, in which you specify the values for those additional factors, for which to compute the surface plot (i.e., for which to compute the fitted values). Note that, even though the triangular graph (optionally) shows the fitted function for the pseudo-components, Statistica expects the values to be entered in terms of the original (untransformed) components.

If the Show fitted function check box is selected (see below), the plot will also contain, as custom text, the currently fitted function (model). Note that the parameters shown in this function pertain to the estimates for the pseudo-components (for additional details, see also options Summary: Estimates, pseudo-components and Estimates, original components on the Quick tab or ANOVA/Effects tab). If there are more than three factors in the experiment, then the pseudo-components are further rescaled in the graph to reflect the residual mixture after holding other factors constant.

Contour plot (fitted response)

Click the Contour plot (fitted response) button to plot the currently fitted model in a contour plot, along with the observed points in the experiment. If there are more than three factors in the current design, the Select factors for triangular graph dialog is displayed when you click the Contour plot (fitted response) button. From this dialog, select the three variables for the plot.

When there are more than three factors in the current experiment, the Select factor values dialog also is displayed, in which you specify the values for those additional factors, for which to compute the contour plot (i.e., for which to compute the fitted values). Note that, even though the triangular graph (optionally) shows the fitted function for the pseudo-components, Statistica expects the values to be entered in terms of the original (untransformed) components.

If the Show fitted function check box is selected (see below), the plot will also contain, as custom text, the currently fitted function (model). Note that the parameters shown in this function pertain to the estimates for the pseudo-components (for additional details, see also options Summary: Estimates, pseudo-components and Estimates, original components on the Quick tab or ANOVA/Effects tab). If there are more than three factors in the experiment, then the pseudo-components are further rescaled in the graph to reflect the residual mixture after holding other factors constant.

Show fitted function

If the Show fitted function check box is selected, then the surface or contour plot will include, as custom text, the currently fitted function. The coefficient estimates shown in that function pertain to the pseudo-components, rather than the original component values (see also options Summary: Estimates, pseudo-components and Estimates, original components on the Quick tab or ANOVA/Effects tab).

When there are more than three factors in the current experiment, then you can specify the values for the additional factors, for which to compute the surface or surface contours (i.e., for which to compute the fitted values). In that case, the function that will be displayed will be further adjusted to reflect the response surface for the residual mixture, i.e., after holding the other factors in the design constant at a particular value. Specifically, when the other factors (not shown in the graph) are held constant at values greater than their respective minima, then each x, y, and z (symbol) in the displayed function will be multiplied by a constant to reflect the reduced residual mixture total.

Show projected points

Note that Statistica plots the points along with the fitted surface in the triangular surface or contour plot. If the Show projected points check box is not selected, then only those points will be shown for which the sum of the component values is equal to the mixture total. Of course, all points will be shown if there are only 3 components (factors) in the current experiment. However, if there are 4 or more factors, and those additional factors are held at their respective minima, then only those points will be shown (by default) for which the fourth, fifth, etc. component values are equal to 0 (zero, after rescaling the original values of the pseudo-components). If you select the Show projected points check box, then all points will be plotted, and those points for which the component values do not sum to the mixture total will be proportionately adjusted for the plot (so that the plotted pseudo-component values will sum to the mixture total).

The setting of this switch has no effect when there are 4 or more factors in the mixture design, and the additional factors are specified to be held constant at values greater than their respective minima. In that case, the observed values cannot be unambiguously represented, and no points will be shown in the surface or contour plot.

Area contours

If the Area contours check box is selected, then the contours in the contour plot will be area (filled) contours; otherwise the surface contours will be indicated by lines.

Trace plot of expected responses

Click the Trace plot of expected responses button to display the Values for Reference Blend dialog, in which you select the values for the reference blend, for which to compute the response traces (see the explanation of  trace plots). Click the OK button on this dialog to display a trace plot for the predicted dependent variable values, by the pseudo-component values.

Predict dependent variable values

Use this option to compute predicted values for the dependent variable based on values for the components that you have defined and based on the current model as specified on the Model tab. Click the Predict dependent variable values button to display the Select factor values dialog, in which you specify the factor values from which to compute the predicted value. Click the OK button on this dialog to display a spreadsheet containing the results. If available, the spreadsheet also displays the confidence interval for the respective predicted value. Those confidence limits are computed from the current sum-of-squares residual term and the percentile value specified in the Confidence interval box on the ANOVA/Effects tab.

Critical values (min, max, saddle)

Click the Critical values button to display three spreadsheets that provide results of the analysis of the quadratic response surface. For a description of these spreadsheets, see Critical Values for Mixture Experiments. The Critical values button is available whenever a quadratic response surface model is used to predict the dependent variable, i.e., when the Quadratic model option button is selected and no effects are being ignored on the Model tab.

Response surface

The Response surface spreadsheet displays the second-order effects of the predictor variables on the response (i.e., the linear by linear interactions), and the first-order effects of the predictor variables on the response (i.e., the main effects). Note that these effects correspond to those in the spreadsheet with the Estimates, original components.

Eigenvalues and eigenvectors

The Eigenvalues and eigenvectors spreadsheet is useful in identifying the shape and the orientation of the quadratic response surface. The determinant of the matrix of second-order effects of the predictor variables on the response is displayed in the information box at the top of the spreadsheet. If the determinant is close to zero, the response surface is nearly flat in at least one direction. The eigenvalues of the second-order effects represent the curvature of the quadratic response surface. The eigenvalues are positive if the response surface curves upward from a minimum, and are negative if the surface curves downward from a maximum. Mixed eigenvalues indicate that the surface is shaped like a saddle, curving upward in one direction and downward in another. The eigenvectors show the orientations of the principal axes of the quadratic response surface relative to the axes of the original predictor variables. A high "loading" of a predictor variable on a principal axis indicates that the axis of the response surface is oriented in the same direction as the axis of the predictor variable. Inspecting the eigenvectors and their corresponding eigenvalues provides useful information about the curvature (upward or downward), or lack thereof (flatness) of the quadratic response surface in each direction defined by the axes of the original predictor variables.

Critical values

The Critical values spreadsheet displays information that identifies the point on the quadratic response surface that defines the curvature of the surface. The critical values for the predictor variables are the coordinates (on the axes of the predictor variables) of the origin of the quadratic response surface. The information box at the top of the spreadsheet displays whether this point represents a minimum, a maximum, or a saddle point on the response surface. The predicted value of the dependent variable at the critical values for each of the predictor variables is also displayed in the information box. The three columns of the Critical values spreadsheet list the Observed minimum values of the predictor variables, the Critical values of the predictor variables, and the Observed maximum values of the predictor variables. Rows in the Critical values spreadsheet in which the critical value lies outside the observed range of the predictor variable are highlighted. This draws attention to an origin for the response surface that lies outside the experimental region, and to the predictor variable (or variables) for which the origin is beyond the observed range.