Adding a covariate after definitive screening design is complete?

[#1]

Hello, 

I finished up a DSD recently and found that we had to use an alternative raw material on several runs due shortages from our regular supplier. This alternative raw has shown to boost yields in past runs, but is very expensive and not something we can regularly source, so was not part of the experimental plan. I'm afraid this alternative raw has skewed my results & I'm wondering if there's a way to account for that after the experiment has been completed as a covariate? 

I also noticed somewhat of a time trend in my yields, with earlier runs having slightly better yields. Is there also a way to account for date as a covariate? 

[#2]

The first approach would be to add a column to the table that you indicate the raw material ID, or alternatively, add columns that have the properties of materials used.  Then include those in the model.  You would probably need to use stepwise regression or generalized regression (in JMP Pro) to sort out the model effect.  

For the possible time trend, if you had a randomized order of the experimental runs, then, in general, any minor time trend is averaged across the experimental factors.  If you think the trend is real and important, then create another column with a time trend indicator (may be just as simple as run order) and include it in the model as a continuous factor.  

[#3]

Thank you!. I'll try playing around with stepwise regression as you say.  

[#4]

Hi @mjz5448,

For your first question, the response from @MathStatChem will help you analyze your experiments with covariates added a-posteriori. You can check the Evaluate Designs and Multivariate platforms to evaluate the correlations between your original factors and the added covariate, to check any blind spot that this addition may create (complete aliases between specific model term and main effect of the covariate factor for example). Specific analysis and models able to handle multicollinearity may be required : Partial Least Squares, Generalized Regression with penalization through Ridge/Elastic Net methods, use of robust Machine Learning algorithm like Bootstrap Forest/Predictor Screening, etc...

For your second question, I would recommend reading these topics with similar issue :

Covariates in defined order in custom design 

Incorporate Time lag in DoE 

The first topic describes how to create a time-robust DoE by taking into account time as a covariate directly in the DoE creation.

I hope this answer will complement the other and help you,

[#5]

Thanks, I'll take a look when I get a chance. 

[#6]

Just some added comments....

You can put one value for the covariate for each treatment.  Is the covariate actually constant during the treatment?

Have you assessed the measurement error for the measurement of the covariate?

Be careful of multicollinearity when adding the covariate (get VIFs and plot scatterplots of x's against covariate)

When evaluating the covariate, put it first in the model and use sequential tests (type 1 SS) to evaluate the covariate.  Use type 3 SS to evaluate the other terms in the model.