How to account for tank to tank variability: Whole-plot factor, covariate, blocking?

Question

I want to run a screening DOE to ID factors important for yield in a batch process. I have 4 tanks (call them A, B, C, D) to do it in, and want to investigate 5 continuous factors impact on my 1 response. However, I discovered that there is significant variability between the 4 tanks - with 1 tank consistently having significantly lower yield than 1 other tank (highest producing tank), while the other 3 tanks don't show a significant difference between each other.

I cannot vary the order the tanks are run in - the run order will always start with tank A 1st, followed by B, C, & D. I was thinking the most efficient way (the least amount of runs) to account for this variability would be to treat the tanks as a covariate w/ 2 levels (tank A, and all other tanks), but I'm not entirely sure if that's correct. I couldn't do an RCBD - it wouldn't be possible - I don't have enough batches scheduled for this product. Is treating them as a covariate in this context correct - basically I know there's variation w/ this factor, but I'm not really interested in testing it and I cannot randomize which tank gets which treatment?

statman · Answer

The way a covariate works is to assign/reduce the variation due to an uncontrollable measured noise variable (essentially the analysis adjusts the values of the response for all treatments as a function of the covariate (thus you use LS means instead of arithmetic means.) and thus reducing the size of the error term (which would have been potentially inflated but the covariate if it was not assigned).  By doing this you increase the likelihood of statistically significant terms in the model (i.e., p-values) which we call increasing the precision of the design.  This assumes a linear relationship between the covariate and the response variable.  Is there some measure of the tanks that could be a covariate?  For example, tank temperature or pressure or volume, etc.

Doug discusses ANCOVA in Chapter 17 (third edition, sorry I'm old)

But regardless.  The key question still remains:  Are the model effects the same for all tanks?  If so, then you will still get a mean shift in yields due to tank effect, but the model will be the same for all tanks.  If not, then the model may be different for all tanks.

BTW, I'm not a huge fan of using covariates for the following reasons:

1. Adds an additional measurement error into the mix

2. You can only have one value for each treatment.  What if the covariate is changing during the running of the treatment?  What number do you use?

3. Potential lurking variables that are confounded with the covariate.

4. Multicollinearity of the covariate with treatments.

5. Handling covariate lag effects is challenging.

mjz5448 · Answer

I didn't mean to apply the "accept as solution" to my previous post - I was trying to reply to this thread. Not sure how to undo that?

But looking at Montgomery's book "Design and Analysis of Experiments, 9th Edition" in chapter 15.3 he talks briefly about using covariates as an option instead of a randomized block design.

"In some situations, the experimenter may have a choice between either running a completely randomized design with a covariate or running a randomized block design with the covariate used in some fashion to form the blocks. If the relationship between the covariate and the response is really well‐approximated by a straight line and that is the form of the covariance model that the experimenter chooses, then either of these approaches is about equally effective. "

mjz5448 · Answer

The following JMP example for adding fixed covariates shows three covariates being added, two that are continuous, and one that is categorical with 3-levels (supplier with levels of A, B, C). So I'm not sure why I couldn't include one 2-level categorical covariate in some sort  of optimal screening DOE?

https://www.jmp.com/support/help/en/18.2/index.shtml#page/jmp/design-with-fixed-covariates.shtml#ww624097 is