Modeling and Multivariate Methods - SAS

238 Analyzing Screening Designs Chapter 8
Statistical Details
Note that the three active factors have been highlighted. One other factor, X18, has also been highlighted. It
shows in the Half Normal plot close to the blue line, indicating that it is close to the 0.1 cutoff significance
value. The 0.1 critical value is generous in its selection of factors so you don’t miss those that are possibly
active.
The contrasts of 5.1, –3, and 1.8 are close to their simulated values (5, –3, 2). However, the similarity of
these values can be increased by using a regression model, without the effect of orthogonalization.
The p-values, while useful, are not entirely valid statistically, since they are based on a simulation that
assumes orthogonal designs, which is not the case for supersaturated designs.
Statistical Details
Operation
The Screening platform has a carefully defined order of operations.
• First, the main effect terms enter according to the absolute size of their contrast. All effects are
orthogonalized to the effects preceding them in the model. The method assures that their order is the
same as it would be in a forward stepwise regression. Ordering by main effects also helps in selecting
preferred aliased terms later in the process.
• After main effects, all second-order interactions are brought in, followed by third-order interactions, and
so on. The second-order interactions cross with all earlier terms before bringing in a new term. For
example, with size-ordered main effects A, B, C, and D, B*C enters before A*D. If a factor has more than
two levels, square and higher-order polynomial terms are also considered.
• An effect that is an exact alias for an effect already in the model shows in the alias column. Effects that
are a linear combination of several previous effects are not displayed. If there is partial aliasing ( a lack of
orthogonality) the effects involved are marked with an asterisk.
• The process continues until n effects are obtained, where n is the number of rows in the data table, thus
fully saturating the model. If complete saturation is not possible with the factors, JMP generates random
orthogonalized effects to absorb the rest of the variation. They are labeled Null n where n is a number.
For example, this situation occurs if there are exact replicate rows in the design.
Screening as an Orthogonal Rotation
Mathematically, the Screening platform takes the n values in the response vector and rotates them into n
new values. The rotated values are then mapped by the space of the factors and their interactions.
Contrasts = T’ × Responses
where T is an orthonormalized set of values starting with the intercept, main effects of factors, two-way
interactions, three-way interactions, and so on, until n values have been obtained. Since the first column of
T is an intercept, and all the other columns are orthogonal to it, these other columns are all contrasts, that
is, they sum to zero. Since T is orthogonal, it can serve as X in a linear model. It does not need inversion,
since T’ is also T -1 and (T’T)T’. The contrasts are the parameters estimated in a linear model.