Covariates and Additional Explanatory Variables

A covariate is a variable that cannot be controlled during the experiment. Instead, it freely varies and the level is measured run to run.

The key to building a design including covariates is to design for the polynomial model that best links the responses to the covariates. If the experiment only included the covariates and there was some way to control the covariates as factors, what polynomial model would be best to link the covariates to the responses? Combine the answer with the objective for the factors of interest to choose the correct design.

A response surface optimal design provides the best results for fitting a polynomial model. If the objective is screening for the important factors, including the covariate, then a factorial design may be sufficient.

Set up the design with the covariate as both a factor (to build in enough runs for modeling) and as a response (for data entry and evaluation). Leave the factor ranges on the covariate factors at the default -1 = low and +1 = high settings. Ignore the covariate factor columns by right-clicking on the factor’s column header and selecting Ignore Factor.

Record the covariate data gathered during the experiment in the response.

Once the data set is complete, there are three steps to handling the covariate. First, right-click on the column header and select Edit Info. Set the low value to match the minimum observation on the covariate and the high value to match the maximum value. Second, copy the covariate response column and paste it over the covariate factor column. Finally, save the design.

Before restoring the covariate factor, try to fit a model for the covariate response(s) in terms of the controlled factors. If the factors strongly model the covariate and there is science supporting this relationship, the covariate can be used as a surrogate for those factors. Either the covariates or the factors, but not both, can be used to model the other responses. Using surrogate measurements in the place of controlled factors requires knowledge that the covariate is the true cause of the effect. Ignore the controlled factors that appear as strong effects in the surrogate model before restoring the surrogate measurement factor. The controlled factors are being swapped for the surrogate.

To restore a factor, right-click on the factor column header and choose Restore Factor.

If a covariate is strongly linked to the factors, and it cannot be directly controlled, consider leaving it as a response. Model it in terms of the controllable factors. This opens up the option to include it during optimization.

If the covariate is not strongly linked to the factors (there wasn’t a strong model) then both the controlled factors and the covariates can be used in the model. The covariate becomes another factor.

Analyze the responses of interest using the standard methods. Covariates will usually reduce the full replicates to 0, which in turn will eliminate the lack-of-fit test. Use the “Jump to Run” feature on the model graphs Factors Tool to visually check the model for lack-of-fit. If a factorial design is used, right-click on the response columns and change the analysis type from factorial to polynomial to better represent the model for the covariate.