Orthogonality

The following is an excerpt on DOE designed experiment techniques from Six Sigma Demystified (2011, McGraw-Hill) by Paul Keller.

Orthogonality refers to the property of a design that ensures that all specified parameters may be estimated independent of any other.

The degree of orthogonality is measured by the normalized value of the determinant of the information matrix. An orthogonal design matrix having one row to estimate each parameter (mean, factors, and interactions) has a measure of 1. It is easy to check for orthogonality: If the sum of the factors columns in standard format equals 0, then the design is orthogonal.

Some writers lump orthogonality with balance, which is different. Balance implies that the data are distributed properly over the design space. It implies a uniform physical distribution of the data and an equal number of levels of each factor.

Designs do not necessarily need balance to be good designs. Rather, some designs (such as central composite designs) sacrifice balance to achieve better distribution of the variance or predicted error. Balance also may be sacrificed by avoiding extreme combinations of factors, such as in the Box-Behnken design.

For example, a complete factorial design is both orthogonal and balanced if in fact the model that includes all possible interactions is correct. Such a design, although both balanced and orthogonal, would not be a recommended experimental design because it cannot provide an estimate of experimental error. When an additional point (row) is added (which for a complete factorial design must be a repeated run), the design is still orthogonal but unbalanced. If the complete interaction model were not correct, then the design is balanced, but the distribution of data is not balanced for the parameters to be estimated.

Fractional factorial and Plackett-Burman designs are normally constructed to have both orthogonality and balance; however, they may have more rows than are required for estimating parameters and error. In such cases, appropriate rows may be trimmed so that orthogonality is preserved but balance is lost. Central composite designs are orthogonal in that all the parameters for the central composite model may be estimated, but the design itself is unbalanced. A greater or lesser number of centerpoints is used to achieve an estimating criterion and an error estimate.

Learn more about the DOE tools for designed experiments in Six Sigma Demystified (2011, McGraw-Hill) by Paul Keller, in his online Intro. to DOE short course (only $99) or online Advanced Topics in DOE short course (only $139), or his online Black Belt certification training course ($875).