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

[#1]

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? 

[#2]

It is very challenging to provide specific advice when there is little description of the situation or process.  Are you planning on running a fractional factorial for the 5 factors?  What is the model you intend to investigate?  Must you use all 4 tanks for the experiment?  Why?  Are you planning any replication?  Is your measurement system variation known?

First question;  Why don't you understand why the 1 tank has lower yields before experimentation?  

No, I would not treat tank as a covariate.  A covariate is a measurable noise factor.  Tank is a categorical factor. When doing covariate analysis, you will have a mixed model.  That is the 5 factors (the model you are investigating) will be fixed effects and the covariate a random effect due to the levels changing randomly treatment-to-treatment.  You will need at least 1 DF for the random effect.

Options you have are to use blocks (RCBD or BIB) or split-plots.

The main question you have to answer would be; Will the results of the experiment be the same for all tanks?  You might be very interested in factor by tank interactions.

[#3]

Thank you. 

As far as the design & model planned - we don't have one yet since we don't know how to deal w/ the tank to tank variability yet - other than main effects model with no interactions for the 5 factors for screening purposes. 

Background: I'm helping another engineer w/ a DOE - they are aware of the tank to tank differences & have a good idea why - but we don't have the budget/down-time time to fix it, so we are left now trying to determine the best way to incorporate that variability. 

I like the idea of not using that tank in the DOE though given our budget - we might have to reduce the # of factors we look at as well, but not a huge issue. 

I'm a bit confused why I can't use tank as a covariate if I code the 2 tank differences as 0/1. Otherwise - we don't have the budget to run an RCBD - or at least our timeline is drawn out in years rather than months, and/or we'll have to reduce the variables we investigate (also an option I suppose). 

What does BIB mean?  

Could you elaborate on why I can't use tank as a covariate if I were to code it as (1 = for tank A, 0 = for tanks B, C, D, E)? I'm also not very familiar w/ mixed models, fixed vs. random effects - how does that complicate things? Wouldn't tank be a fixed effect rather than random since I'm only interested in those 2 levels and not a wider population? 

[#4]

Covariates are measured random variables, not categorical factors (e.g., ambient temperature).  

https://www.jmp.com/support/help/en/18.2/?os=mac&source=application#page/jmp/experiments-with-covari...

BIB is a balanced incomplete block, essentially a fractional block created by aliasing block with a DF in the experiment, typically a higher order effect.

You should know how to analyze the experiment before you run it.  For covariates, you will need to use both sequential (type 1) and partial SS (type 3) for analysis.

While it can be debated, IMHO, you can treat the tank as a fixed effect.  The interpretation is different than a typical factor.  That is, you don't want to complete the analysis and conclude tank is significant and tank 2 is better (as you need to use all tanks).  You would be most interested in tank by factor interactions.

[#5]

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#ww6... is

[#6]

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. "

[#7]

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.