Design of Experiments Guide • Examples of Custom Designs • Experiments with Covariates
Previous
 • 
Next
Experiments with Covariates
Sometimes measurements on the experimental units that are intended for an experiment are available. These measurements might affect the experimental response. It is useful to include these variables, called covariates, as design factors. Although you cannot directly control these values, you can ensure that the levels of the other design factors are chosen to yield the most precise estimates of all the effects.
The Custom Design platform constructs a design that selects covariate values in an optimal fashion. Covariate values are selected from an existing data table that provides covariate information about the potential experimental units. You can specify a number of runs that is smaller than the number of experimental units listed in your data table. You can also specify covariates that are hard-to-change.
Note: The number of rows in the covariate data table where covariate factors have nonmissing values must be greater than or equal to the specified Number of Runs.
Design with Fixed Covariates
In this example, you are interested in modeling the Shrinkage of parts produced by an injection molding process. The Thermoplastic.jmp sample data table in the Design Experiment folder lists 25 batches of raw (thermoplastic) material for potential use in your study. For each batch, material was removed to obtain measurements of Specific Gravity and Tensile Strength. A third covariate, Supplier, is also available.
You want to study the effects of three controllable factors, Temperature (mold temperature), Speed (screw speed), and Time (hold time), on Shrinkage. But you also want to study the effects of the covariates: Specific Gravity, Tensile Strength, and Supplier. Your resources allow for 12 runs.
Create your design as follows:
1.
Select Help > Sample Data Library and open Design Experiment/Thermoplastic.jmp.
2.
Select DOE > Custom Design.
3.
Double-click Y under Response Name and type Shrinkage.
4.
Click Maximize under Goal and change it to Minimize.
5.
Click Add Factor and select Covariate.
6.
Select Specific Gravity, Tensile Strength, and Supplier from the list and click OK.
These are covariates and cannot be controlled.
7.
Type 3 next to Add N Factors.
8.
Click Add Factor > Continuous.
9.
Rename the three continuous factors Temperature, Speed, and Time.
These are factors that can be controlled.
Responses and Factors Outlines
10.
Click Continue.
The Number of Runs shows the number of rows with covariate values available. You have 25 batches with measured covariates.
11.
Type 12 next to Number of Runs.
Note: Setting the Random Seed in step 12 and Number of Starts in step 13 reproduces the exact results shown in this example. In constructing a design on your own, these steps are not necessary.
12.
(Optional) From the Custom Design red triangle menu, select Set Random Seed, type 84951, and click OK.
13.
(Optional) From the Custom Design red triangle menu, select Number of Starts, type 40, and click OK.
14.
Click Make Design.
Twelve-Run Optimal Design
This design is D-optimal, given the potential covariate values. It selects the best sets of covariate values and the best settings for the three controllable factors.
15.
Open the Design Evaluation > Color Map On Correlations outline.
Color Map on Correlations
The seven terms corresponding to main effects appear in the upper left corner of the color map. Notice that these seven terms are close to orthogonal. The largest absolute correlation is between Tensile Strength and Supplier 2. This absolute correlation of about 0.43 is a consequence of the available covariate values.
Design with Hard-to-Change Covariates
In this example, you construct a design for developing a running shoe for serious runners that has good wear (Wear) properties. Your experimental factors include the following:
•
sole thickness (Thickness)
•
amount of gel cushioning (Gel)
•
outsole material (Outsole)
•
midsole material (Midsole)
To obtain generalizable conclusions, you need to test your shoes on a broad base of serious runners. To accommodate your experimental budget, each runner must test several experimental combinations.
Your company has collected data on 100 suitable runners willing to participate in your study. The concomitant variables (covariates) measured on these runners are average daily miles run (Miles), weight (Weight), and the foot’s strike point (Strike Point).
Create your design as follows:
1.
Select Help > Sample Data Library and open Design Experiment/Runners Covariates.jmp.
2.
Select DOE > Custom Design.
3.
Double-click Y under Response Name and type Wear.
4.
Click Maximize under Goal and change it to Minimize.
5.
Click Add Factor and select Covariate.
6.
Select Miles, Weight, and Strike Point from the list and click OK.
These are the hard-to-change covariates associated with the runners.
7.
For one of the factors Miles, Weight, and Strike Point, under Changes, click Easy and change it to Hard.
Note that Changes for all three covariates turn to Hard.
8.
To add the remaining factors manually, follow step 9 through step 17. Or, to load factors from a saved table, select Load Factors from the red triangle menu next to Custom Design. Open the Runners Factors.jmp sample data table, located in the Design Experiment folder. If you select Load Factors, skip step 9 through step 17.
9.
Type 2 next to Add N Factors.
10.
Click Add Factor > Continuous.
11.
Rename the two factors Thickness and Gel.
12.
Change the Values for Thickness to 5 and 20.
13.
Change the Values for Gel to 1 and 10.
14.
Type 2 next to Add N Factors.
15.
Click Add Factor > Categorical > 3 Level.
16.
Rename the two factors Outsole and Midsole.
Keep the default Values for these factors.
Responses and Factors Outlines
17.
Click Continue.
18.
Select Interactions > 2nd.
The specified model fits all two-factor interactions, including covariate by experimental factor interactions.
19.
Set the Number of Whole Plots, or runners, to 32 (if it is not already set to that number).
20.
Type 64 next to User Specified under Number of Runs (if it is not already set to that number).
Note: Setting the Random Seed in step 21 and Number of Starts in step 22 reproduces the exact results shown in this example. In constructing a design on your own, these steps are not necessary.
21.
(Optional) From the Custom Design red triangle menu, select Set Random Seed, type 12345, and click OK.
22.
(Optional) From the Custom Design red triangle menu, select Number of Starts and set it to 1(if it is not already set to that number). Click OK.
23.
Click Make Design.
First 40 Runs of Design for Hard-to-Change Covariates
Of the 100 runners, 32 are selected based on their covariate values. The rows corresponding to the selected runners are selected in the RunnersCovariates.jmp sample data table. Settings of the experimental factors Thickness, Gel, Insole, and Outsole are determined so that the design is optimal for the model described in the Model Outline.
24.
With the RunnersCovariates.jmp sample data table as the active table, select Analyze > Distribution.
25.
Select all three columns as Y, Columns.
26.
Check Histograms Only.
27.
Click OK.
Histograms for 100 Runners with Selected Runner Data Shaded
The histograms indicate that all of the runners with Miles of 14.0 or higher were selected. Runners at the extremes of the Weight distribution were selected. Almost all of the runners with a Strike Point of Forefoot were selected. Notice that the design is somewhat balanced in terms of Strike Point.
Design with a Linear Time Trend
Often, experiments conducted in a time sequence experience a linear drift in the response. If you randomize the order of the runs, then the drift’s effect does not generally bias the estimated factor effects. However, by accounting for the drift, you can reduce the variance of those effects.
Suppose that there is reason to suspect a strong linear trend in the response over time independent of the factor changes. Then you can construct a design that includes a linear covariate to account for the trend. The resulting design is optimal, given this trend covariate.
In this example, you design an experiment for 7 factors. You construct a 16-run design that is robust to linear trend.
1.
Select File > New > Data Table.
2.
Right-click Column1 and select Column Info.
3.
Change the column name to Run Order.
4.
From the list of Initialize Data options, select Sequence Data.
5.
Type 16 next to To.
6.
Click OK.
Consecutive integers from 1 to 16 have been entered in the data table.
7.
Select DOE > Custom Design.
8.
Click Add Factor > Covariate.
9.
Select Run Order and click OK.
10.
Type 7 next to Add N Factors.
11.
Click Add Factor > Continuous.
Responses and Factors Outlines
12.
Click Continue.
13.
Open the Alias Terms outline.
14.
Select all of the effects in the list and click Remove Term.
This omits the interaction effects from the correlation color map, leaving only the main effects.
Note: Setting the Random Seed in step 15 and Number of Starts in step 16 reproduces the exact results shown in this example. In constructing a design on your own, these steps are not necessary.
15.
(Optional) From the Custom Design red triangle menu, select Set Random Seed, type 1084680980, and click OK.
16.
(Optional) From the Custom Design red triangle menu, select Number of Starts, type 30000, and click OK.
17.
Click Make Design.
A progress bar displays elapsed time and D Efficiency.
Design Outline
18.
Open the Design Evaluation > Color Map On Correlations outline.
Color Map Showing Absolute Correlations with Run Order
The Color Map shows the following:
‒
The seven continuous factors, X2 through X8, are orthogonal to each other.
‒
Run Order, the linear time trend variable, has extremely low absolute correlation with X2 through X8.
19.
Open the Design Evaluation > Estimation Efficiency outline.
Estimation Efficiency Outline
The small absolute correlations of Run Order with X2 through X8 result in very small increases in confidence interval lengths, relative to an ideal orthogonal design. The increases in the lengths of confidence intervals for X2 through X8 are all less than 0.1%.
In this example, the run order factor is nearly orthogonal to the factor effects. In some cases, your design might have more substantial correlations between the run order factor and other factors. Even in such a situation, including the run order as a factor accounts for any linear trend effect. Including the run order also allows for more precise estimation of the other factor effects.