Propagation of error (POE) is important when attributing the variation of the response in terms of variation in the factor values when the factor values are not very controllable.
In JMP’s implementation, the Profiler first looks at the factor and response variables to see whether there is a Sigma column property (a specification for the standard deviation of the column, accessed through the Cols > Column Info dialog box). If the property exists, then the Prop of Error Bars command becomes accessible in the Prediction Profiler drop-down menu. This displays the 3σ interval that is implied on the response due to the variation in the factor.
The POE is represented in the graph by a green bracket. The bracket indicates the prediction plus or minus three times the square root of the POE variance. The POE variance can be expressed as:
where σy is the user-specified sigma for the response column, and σx is the user-specified sigma for the factor column.
centered, with δ=xrange/10000
POE limits increase dramatically in response surface models when you are over a more sloped part of the response surface. One of the goals of robust processes is to operate in flat areas of the response surface so that variations in the factors do not amplify in their effect on the response.
It is possible to use a customized desirability function. For example, suppose you want to maximize using the following function.
First, create a column called MyDesire that contains the above formula. Then, select Graph > Profiler to launch the platform. Select all the Pred Formula columns and the MyDesire column and select Y, Prediction Formula. Turn on the desirability functions by selecting Desirability Functions from the red triangle menu. All the desirability functions for the individual effects must be turned off. To do this, first double-click in a desirability plot window, then select None in the window that appears (Selecting No Desirability Goal). Set the desirability for MyDesire to be maximized.
At this point, selecting Maximize Desirability uses only the custom MyDesire function.
When analyzing a mixture design, JMP constrains the ranges of the factors so that settings outside the mixture constraints are not possible. This is why, in some mixture designs, the profile traces appear to turn abruptly.
When there are mixture components that have constraints, other than the usual zero-to-one constraint, a new submenu, called Profile at Boundary, appears on the Prediction Profiler red triangle menu. It has the following two options:
The Profiler launch window has an Expand Intermediate Formulas check box. When this is checked, when the formula is examined for profiling, if it references another column that has a formula containing references to other columns, then it substitutes that formula and profiles with respect to the end references—not the intermediate column references.
For example, when Fit Model fits a logistic regression for two levels (say A and B), the end formulas (Prob[A] and Prob[B]) are functions of the Lin[x] column, which itself is a function of another column x. If Expand Intermediate Formulas is selected, then when Prob[A] is profiled, it is with reference to x, not Lin[x].
In addition, using the Expand Intermediate Formulas check box enables the Save Expanded Formulas command in the platform red triangle menu. This creates a new column with a formula, which is the formula being profiled as a function of the end columns, not the intermediate columns.
The Prediction Profiler, Custom Profiler, and Mixture Profiler can incorporate linear constraints into their operations. Linear constraints can be entered in two ways, described in the following sections.
To enter linear constraints via the red triangle menu, select Alter Linear Constraints from either the Prediction Profiler or Custom Profiler red triangle menu.
Choose Add Constraint from the resulting window, and enter the coefficients into the appropriate boxes. For example, to enter the constraint p1 + 2*p2 ≥ 0.9, enter the coefficients as shown in Enter Coefficients. As shown, if you are profiling factors from a mixture design, the mixture constraint is present by default and cannot be modified.
After you click OK, the Profiler updates the profile traces, and the constraint is incorporated into subsequent analyses and optimizations.
If you attempt to add a constraint for which there is no feasible solution, a message is written to the log and the constraint is not added. To delete a constraint, enter zeros for all the coefficients.
Constraints added in one profiler are not accessible by other profilers until saved. For example, if constraints are added under the Prediction Profiler, they are not accessible to the Custom Profiler. To use the constraint, you can either add it under the Custom Profiler red triangle menu, or use the Save Linear Constraints command described in the next section.
If you add constraints in one profiler and want to make them accessible by other profilers, use the Save Linear Constraints command, accessible through the platform red triangle menu. For example, if you created constraints in the Prediction Profiler, choose Save Linear Constraints under the Prediction Profiler red triangle menu. The Save Linear Constraints command creates or alters a Table Script called Constraint. An example of the Table Property is shown in Constraint Table Script.
The Constraint Table Property is a list of the constraints, and is editable. It is accessible to other profilers, and negates the need to enter the constraints in other profilers. To view or edit Constraint, right-click the red triangle menu and select Edit. The content of the constraint from Enter Coefficients is shown below in Example Constraint.
The Constraint Table Script can be created manually by choosing New Script from the red triangle menu beside a table name.
Note: When creating the Constraint Table Script manually, the spelling must be exactly “Constraint”. Also, the constraint variables are case sensitive and must match the column name. For example, in Example Constraint, the constraint variables are p1 and p2, not P1 and P2.
The Constraint Table Script is also created when specifying linear constraints when designing an experiment.