Special Topics - Profiling Predicted Responses and Response Desirability

A typical problem in product development is to find a set of conditions, or levels of the input variables, that produces the most desirable product in terms of its characteristics, or responses on the output variables. The procedures used to solve this problem generally involve two steps: 1) predicting responses on the dependent, or Y variables, by fitting the observed responses using an equation based on the levels of the independent, or X variables, and 2) finding the levels of the X variables that simultaneously produce the most desirable predicted responses on the Y variables. Derringer and Suich (1980) give, as an example of these procedures, the problem of finding the most desirable tire tread compound. There are a number of Y variables, such as PICO Abrasion Index, 200 percent modulus, elongation at break, and hardness. The characteristics of the product in terms of the response variables depend on the ingredients, the X variables, such as hydrated silica level, silane coupling agent level, and sulfur. The problem is to select the levels for the X's that will maximize the desirability of the responses on the Y's. The solution must take into account the fact that the levels for the X's that maximize one response may not maximize a different response.

Statistica includes Response/desirability profiling as an option when analyzing 2(k-p) (two-level factorial) designs, 2-level screening designs, 2(k-p) maximally unconfounded and minimum aberration designs, 3(k-p) and Box Behnken designs, mixed 2 and 3 level designs, central composite designs, and mixture designs. Response/desirability profiling allows you to inspect the response surface produced by fitting the observed responses using an equation based on levels of the independent variables. Using the Profiler you can inspect the predicted values for the dependent variables at different combinations of levels of the independent variables, specify desirability functions for the dependent variables, and search for the levels of the independent variables that produce the most desirable responses on the dependent variables.

Prediction Profiles.  When you analyze the results of any of the designs listed above, a separate prediction equation for each dependent variable (containing different coefficients but the same terms) is fitted to the observed responses on the respective dependent variable. Once these equations are constructed, predicted values for the dependent variables can be computed at any combination of levels of the predictor variables. A prediction profile for a dependent variable consists of a series of graphs, one for each independent variable, of the predicted values for the dependent variable at different levels of one independent variable, holding the levels of the other independent variables constant at specified values, called current values. If appropriate current values for the independent variables have been selected, inspecting the prediction profile can show which levels of the predictor variables produce the most desirable predicted response on the dependent variable.

The Profiler can be used to produce not only a prediction profile for a single dependent variable, but also a compound prediction profile graph that shows the prediction profiles for multiple dependent variables. This can allow one to see whether the levels of the independent variables that maximize responses for one dependent variable also maximize responses on other dependent variables. Confidence intervals or prediction intervals for the predicted values can also be shown, to aid in the assessment of the reliability of prediction.

One might be interested in inspecting the predicted values for the dependent variables only at the actual levels at which the independent variables were set during the experiment. Alternatively, one also might be interested in inspecting the predicted values for the dependent variables at levels other than the actual levels of the independent variables used during the experiment, to see if there might be intermediate levels of the independent variables that could produce even more desirable responses. Also, returning to the Derringer and Suich (1980) example, for some response variables, the most desirable values may not necessarily be the most extreme values, for example, the most desirable value of elongation may fall within a narrow range of the possible values. The Factor grid option allows you to easily specify the range and number of levels at which to compute predicted values for each independent variable.

Several options are available in the Profiler dialog box for specifying the current levels of the predictor variables. You can set the current levels of the factors at the factor means, or at any user-specified setting. If the analysis contains a blocking factor, the current level of the blocking factor can also be set to any desired block. You can also set the current levels of the factors at their optimum values as determined by a search for the combination of factor settings that produces the most desirable response on the dependent variables. This brings us to the topic of profiling the desirability of responses on the dependent variables.

Response Desirability.  Different dependent variables might have different kinds of relationships between scores on the variable and the desirability of the scores. Less filling beer may be more desirable, but better tasting beer can also be more desirable - lower "fillingness" scores and higher "taste" scores are both more desirable. The relationship between predicted responses on a dependent variable and the desirability of responses is called the desirability function.

Derringer and Suich (1980) developed a procedure for specifying the relationship between predicted responses on a dependent variable and the desirability of the responses, a procedure that provides for up to three "inflection" points in the function. Returning to the tire tread compound example described above, their procedure involved transforming scores on each of the four tire tread compound outcome variables into desirability scores that could range from 0.0 for undesirable to 1.0 for very desirable. For example, their desirability function for hardness of the tire tread compound was defined by assigning a desirability value of 0.0 to hardness scores below 60 or above 75, a desirability value of 1.0 to mid-point hardness scores of 67.5, a desirability value that increased linearly from 0.0 up to 1.0 for hardness scores between 60 and 67.5 and a desirability value that decreased linearly from 1.0 down to 0.0 for hardness scores between 67.5 and 75.0. More generally, they suggested that procedures for defining desirability functions should accommodate curvature in the "falloff" of desirability between inflection points in the functions.

After transforming the predicted values of the dependent variables at different combinations of levels of the predictor variables into individual desirability scores, the overall desirability of the outcomes at different combinations of levels of the predictor variables can be computed. Derringer and Suich (1980) suggested that overall desirability be computed as the geometric mean of the individual desirabilities (which makes intuitive sense, because if the individual desirability of any outcome is 0.0, or unacceptable, the overall desirability will be 0.0, or unacceptable, no matter how desirable the other individual outcomes are, the geometric mean takes the product of all of the values, and raises the product to the power of the reciprocal of the number of values). Derringer and Suich's procedure provides a straightforward way for transforming predicted values for multiple dependent variables into a single overall desirability score. The problem of simultaneous optimization of several response variables then boils down to selecting the levels of the predictor variables that maximize the overall desirability of the responses on the dependent variables.

The Profiler provides several features for easily specifying the desirability function for each dependent variable in the analysis. The default "higher is better" desirability function can be used without further specifications whenever a higher predicted value on a dependent variable is more desirable, but you can specify up to three "inflection" points in the desirability function with corresponding desirability values ranging anywhere between 0.0 and 1.0, inclusive. The degree of curvature, if any, in the "falloff" of desirability between inflection points in the function can be easily specified. Desirability settings can be saved, and retrieved later to minimize data entry effort. Settings for a single dependent variable or for all dependent variables in the analysis can be reset to their respective default specifications.

The Profiler provides two methods for searching for the levels of the predictor variables that produce outcomes with the highest overall desirability. One method uses a general function optimization procedure (the simplex method of function optimization) to find the optimal settings of the independent variables (within the specified experimental range) for overall response desirability. The second method searches every specified combination of levels of the predictor variables for the combination producing the optimal overall response desirability. Both methods can be used to help find the set of conditions that produces the most desirable product, and thus provide a powerful tool for product development research.

Additional Features

If the Show desirability function check box in the Profiler dialog box is selected, a graph of the desirability function for each profiled dependent variable is included on the compound prediction profile graph. A series of graphs, one for each independent variable, profiling overall response desirability of the dependent variable at different levels of each independent variable, holding the levels of the other independent variables constant at their current values, is also included in the compound prediction profile graph. This allows you to see at a glance how changes in the levels of each predictor variable influence not only responses on each dependent variable, but also the overall desirability of the responses. This feature is especially useful in determining for each independent variable how quickly overall response desirability changes as the level of the predictor variable changes, in essence, allowing you to distinguish between "inert" and "active" ingredients.

Surface plots and contour plots for overall response desirability are available as options in the Profiler dialog box. These plots are useful for interpreting the effects on overall response desirability of different combinations of levels of all pairs of independent variables. The surface plots show these effects graphically in 3-D plots where pairs of independent variables are represented on two of the axes and overall response desirability is represented on the third axis. The contour plots show the levels of overall response desirability produced in different regions of the plane defined by pairs of independent variables, where each region of the plane represents a different combination of the levels of the two variables. Again, these graphic features can aid in distinguishing between ingredients that are "inert" and "active" with respect to other ingredients.

In some situations with multiple dependent variables, the prediction model for some dependent variables (for example, measures of the strength of a metal) may be different from the prediction model for other dependent variables (for example, measures of the corrosion-resistance of a metal). Although the response/desirability profiles are computed using a separate prediction equation for each dependent variable, each prediction equation contains the same terms for the factors, but with different coefficients. When it is not unreasonable to assume that different factors influence responses on different types of dependent variables, there is still a way to simultaneously maximize the desirability of responses on different types of dependent variables. Specifically, separately compute and save the predicted values using the desired model for each dependent variable. Then take these predicted values and use them as "data" in a subsequent analysis. Fit a model containing all of the terms used in modeling the original dependent variables (effects for terms used in computing the predicted values for the original dependent variables will equal the original effects; effects for terms not used in computing the predicted values for the original dependent variables will be zero). Then specify the desirability function for the "predicted values of the predicted values" and the desired search. The desirability maximum will be at the levels of the independent variables that simultaneously produce the maximum desirability, given the original model used in predicting each original dependent variable.

Summary.  When you are developing a product whose characteristics are known to depend on the "ingredients" of which it is constituted, producing the best product possible requires determining the effects of the ingredients on each characteristic of the product, and then finding the balance of ingredients that optimizes the overall desirability of the product. In data analytic terms, the procedure that is followed to maximize product desirability is to (1) find adequate models (i.e., prediction equations) to predict characteristics of the product as a function of the levels of the independent variables, and (2) determine the optimum levels of the independent variables for overall product quality. These two steps, if followed faithfully, will likely lead to greater success in product improvement than the fabled, but Statistically dubious technique of hoping for accidental breakthroughs and discoveries that radically improve product quality. The Experimental Design module's Response/desirability profiler is a useful tool for empirically-based attempts to produce the most desirable product possible.