Adding repeated measurements to a DOE

[#1]

I am currently trying to set up an experimental design using JMP PRO to study the impact of 3 variables on an extraction process. Two of these variables are continuous (pressure and concentration). The third variable is the amount of times I run my sample through the machine (from 1 to 4 times). How can I add this to my design in the most efficient way? I tried making an I-optimal RSM design but this assumes that all runs are independent, meaning that I can't combine 2 runs with the same pressure and concentration with a different amount of sample recirculations. For me, it would be much more practical to design an experiment in which I vary pressure and concentration and for each combination of these, I sample after 1, 2, 3 and 4 recirculations. However, I am struggling to add the "recirculation" variable to my design since the 4 samples originating from recirculations are inherently correlated to eachother but I still want to study the effect of the number of recirculations in an efficient way. Thanks in advance for you help!

[#2]

You can define the two continuous factors as hard to change. This way will create a split-plot design. It will force JMP to recognize that you are not fully randomizing the design and the analysis.

[#3]

If I do this, will JMP also consider the large correlation between all circulation samples, given that they are all taken from the same bulk sample over time? Or will it still think that the different circulation numbers (1-4) were obtained from independent measurements?

[#4]

Hard to change factors are understood to be the same (set once, used more than once) across multiple runs.

[#5]

I fully agree regarding the hard to change factors! My question was rather about the easy to change factor of recirculations. Does the design take into account that the four runs originating from a set of pre-defined hard to change factors, are originating from the same bulk sample that has just been recirculated one or two additional times? 

To a colleague, I compared this with a chemical reaction inside an incubator (hard to change variables) from which you take several samples over time (easy to change variable). In my case, you sample the same bottle inside that incubator time after time, creating a dependency between the runs. However, I feel like the design you suggest, assumes that I have put in 4 different bottles and can just take one of them out when I sample, limiting the dependency between the different samples.

[#6]

For each whole plot, you will use a defined number of recirculations, 1 to 4. That variable is independent, if I understand your situation.

[#7]

For each whole plot (so a combination of pressure and biomass), I can easily recirculate 4 times but take a sample after each of the 4 recirculations. This would mean that for each whole plot, I can do 4 runs but these runs are not independent of eachother.

FYI, I have to manually reload the machine for each circulation. This means it is almost no additional work to just take a sample after each recirculation instead of only taking a sample after the 4th recirculation.

[#8]

You might get more information and flexibility if you create four responses, one for each circulation. They are correlated. You can treat them as a series. I don't know if four points are enough to define a function, but you might be able to use the Functional Data Explorer to treat each curve as a data point for a run.

You are not getting four runs from each whole plot. You have two factors and four responses now. A quadratic model would include six terms. If it is feasible, I would make a design with 12 or 18 runs.

[#9]

Let me start by asking what questions are you trying to answer?  Are you wanting to understand causality or just pick a winner? Can you measure the extraction Y's after each "run"? Are you trying to minimize the number of re-circulations? How are you handling noise (within lot & lot-to-lot variation of the raw materials, ambient conditions, temperature, etc)?  Are you planning on replication, blocking or something else?

There are a couple of ways to think about the re-circulation:

1. Create 4 Y's for each of the 4 treatments (2 factors, pressure/concentration at 2 levels).  Measure the critical characteristic(s) of the extrusion after each run.  Analyze the 4 Y's as multivariate (correlation et. al.).  You could also use this data to create a model with the 4 data points and use those to estimate response variables that describe the model (e.g., slope, max, min).  Or you could create summary statistics of the 4 data points (e.g., mean, standard deviation)

2. As Mark suggests, a split-plot where the whole plot contains the 2 continuous factors and the subplot contains the 4 re-circulations.  This design will take into account to restriction on randomization (which you will need to account for in analysis). Are you planning on any replication?

The most important thing is to capture the data that represents how you acquired it.  So let's say you have 4 treatments (4 rows) for the 2 continuous factors (2 columns) .  Then 4 columns for each of the 4 circulations (if this is univariate, you will have a table with 6 columns and 4 rows.). Once you have this data, it can be stacked or manipulated to perform analysis multiple ways.  It is possible to create this design "by hand".