Design of Experiments Guide • The Screening Platform • Technical Details
Previous
 • 
Next
Technical Details
Order of Effect Entry
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.
•
If no effect in the model is active after the intercept, the contrasts are just an orthogonal rotation of random independent variates into different random independent variates. These newly orthogonally rotated variates have the same variance as the original random independent variates. To the extent that some effects are active, the inactive effects still represent the same variation as the error in the model. The hope is that the effects and the design are strong enough to separate the active effects from the random error effects.
Lenth’s Pseudo-Standard Error
At this point, Lenth’s method (Lenth, 1989) identifies inactive effects from which it constructs an estimate of the residual standard error, known as the Lenth Pseudo Standard Error (PSE).
The value for Lenth’s PSE is shown at the bottom of the Screening report. From the PSE, t-ratios are obtained. To generate p-values, a Monte Carlo simulation of 10,000 runs of n – 1 purely random values is created and Lenth t-ratios are produced from each set. The p-value is the interpolated fractional position among these values in descending order. The simultaneous p-value is the interpolation along the max(|t|) of the n – 1 values across the runs. This technique is similar to that in Ye and Hamada (2000).
If you want to run more or less than the 10,000 default runs, you must assign a value to a global JSL variable named LenthSimN. As an example, using the sample data Half Reactor.jmp:
1.
Open the sample data Half Reactor.jmp.
2.
Select Analyze > Modeling > Screening.
3.
Select Percent Reacted as the response variable, Y.
4.
Select all the other continuous variables as effects, X.
5.
Click OK.
6.
Select Script > Save Script to Script Window from the red-triangle menu of the report.
7.
Add LenthSimN=50000; to the top of the Script Window (above the code).
8.
Highlight LenthSimN=50000; and the remaining code.
9.
Run the script from the Script Window.
Note that if LenthSimN=0, the standard t-distribution is used (not recommended).