Optimal Designs¶

Optimal (Custom) designs are used when the process requires adjustments to the experiment that cannot be accommodated by a standard design.

These adjustments include:

• Difference between the high and low of all the mixture components not the same.

• Mixture and process variables in the same design.

• Two independent mixtures in the same design.

• Constraints in addition to the factor limits

• Models other than the full quadratic (all linear, two-factor interactions and squared terms) Too many runs in the standard designs

• Replicates need to spread throughout the design.

• Combinations of the above

The number of runs in optimal designs depends on the number of terms in the model, the number of blocks and the number of additional model, lack of fit, replicates and centroids requested during the build.

Optimal designs begin with a pseudo-random set of model points (runs) that are capable of fitting the designed for model. The initial selection can usually be improved by replacing a subset of the points with better selections. The Stat-Ease 360 ® software uses one of five criteria to decide which replacements are better and up to two exchange methods to decide how they are replaced.

By default, optimal designs are augmented with five lack of fit and five replicates. The lack of fit runs are picked to maximize the minimum distance to other runs, while not being overly detrimental to the optimality. The replicates are used to estimate pure error, or the variability in the results even though the factor settings didn’t change. Taken together they form a test for the lack of fit. A significant result from the lack of fit test is an indicator that a higher-order model may be necessary to approximate the true response surface.

Without a lack of fit test, there may be no indication that an inadequate model was fit to the data.

A minimum of one unique point per term in the designed for model is required to estimate the model coefficients. Additional model points improve the optimality criterion and the precision of the estimates. Lack of fit and replicate points improve the precision as well, but to a lesser degree than additional model points.