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Thanks Hadley and Statman for your responses. We will try out the suggested approach but would like to give some extra context nonetheless.
I agree with Statman on the design being underpowered. This is definitively the start of a journey and the next DOE we design is very unlikely to give us the final model but we want to get closer while capturing as much data as possible besides the optimisation variables. Perhaps the DOE will turn out to be just a way to build up a larger dataset that we might analyse with more tailored models.
- You suggest you want a "medium optimization experiment", what is this?
We are optimizing the recipe for a bacterial growth substrate
- 4-5 variables is a fairly large number of variables for optimization. Do you already have a first order model?
We are still screening for the most relevant factors, hence the large number of variables. We have performed a definitive screening design with 11 factors which allowed us to select for the 5 variables with the largest effect and exclude some that are unlikely to be relevant. We saw quite a few factors where the optimum was closer to the ‘center point’ of the DSD design. We are building on these observations to make the next set of experiments with adjusted factor levels. For illustration I have included the main residuals plot below of the definitive screening design results.
- What do you mean by "more reliable results"?
Despite our best efforts to control variation in production, we see significant noise batch to batch in our process. We have fixed all non recipe sources of variation that we can control and still don't get stability (biological systems are fun). Therefore we decided to change the components of the recipe to see if we can get to more stable running conditions with hopefully some optimisation along the way as an happy coincidence.
- You suggest you want to account for the noise between incubator and between days and you KNOW these have an impact? What about noise-by-factor interactions? How useful is an optimized model if it changes incubator-to-incubator and Day-to-day?
I agree that there might be interactions but the situation is the same at the production level, so our scale down model appears to be useful. For the statistical model, there is definitely a possibility that some of these factors will interact with the variables we can control.
- What is changing incubator-to-incubator?
We know that temperature control is not perfect so I was thinking of taking this into account as a covariate. For the day to day variation we are clueless.
- What is your predicted rank order of model effects? Include main effects, 2nd order linear and 2nd order quadratic. This will help determine the appropriate resolution, and polynomial for your design.
We don’t know a priori which factors will have a quadratic effect. We have a strong suspicion for one based on biological knowledge, but we are hoping to get closer to the answer with this experiment. I am quite sure that we will not get all we need after this run to make a proper statistical model. We aim to get this done in an iterative manner.
- Blocking may be useful
Would you be able to show how this two level blocking is exactly done in JMP? Including the incubators and days as separate categorical factors feels wrong somehow. I tried setting up two blocks but I don't understand how JMP calculates the amounts of runs in a block and I can't seem to set a constraint that every incubator needs exactly 4 runs per day.
- Split-plots partition noise by using a design factor, so this is not what you are suggesting.
Thanks for clearing this up!
Hi,
Considering your description of the problem and the solution proposed by @statman in point#5, I created this design (table script below). I created a 32-run response surface with 5 continuous factors using Custom Design, and then grouped runs into random blocks of size 4 (corresponding to the number of flasks per incubator per day). In the table, I manually added a "Day" variable and adjusted "incubator" so that 1-4 repeat on both days. It would then be possible to account for the random variation by setting up the model as shown in the figure.
I'd be very interested to hear if @statman or anyone else has an opinion on this approach.
New Table( "day and incubator block design",
Add Rows( 32 ),
New Table Variable( "Design", "Custom Design" ),
New Table Variable( "Criterion", "I Optimal" ),
New Script(
"Model",
Fit Model(
Effects(
:incubator & Random, :X1 & RS, :X2 & RS, :X3 & RS, :X4 & RS,
:X5 & RS, :X1 * :X1, :X1 * :X2, :X2 * :X2, :X1 * :X3, :X2 * :X3,
:X3 * :X3, :X1 * :X4, :X2 * :X4, :X3 * :X4, :X4 * :X4, :X1 * :X5,
:X2 * :X5, :X3 * :X5, :X4 * :X5, :X5 * :X5
),
Y( :Y )
)
),
New Script(
"Evaluate Design",
DOE( Evaluate Design, X( Random Block, :X1, :X2, :X3, :X4, :X5 ) )
),
New Script(
"Fit Mixed",
Fit Model(
Y( :Y ),
Random Effects( :incubator ),
Effects(
:X1 & RS, :X2 & RS, :X3 & RS, :X4 & RS, :X5 & RS, :X1 * :X1,
:X1 * :X2, :X2 * :X2, :X1 * :X3, :X2 * :X3, :X3 * :X3, :X1 * :X4,
:X2 * :X4, :X3 * :X4, :X4 * :X4, :X1 * :X5, :X2 * :X5, :X3 * :X5,
:X4 * :X5, :X5 * :X5
),
NoBounds( 1 ),
Personality( "Mixed Model" ),
Run
)
),
New Script(
"DOE Dialog",
DOE(
Custom Design,
{Add Response( Maximize, "Y", ., ., . ),
Add Factor( Continuous, -1, 1, "X1", 0 ),
Add Factor( Continuous, -1, 1, "X2", 0 ),
Add Factor( Continuous, -1, 1, "X3", 0 ),
Add Factor( Continuous, -1, 1, "X4", 0 ),
Add Factor( Continuous, -1, 1, "X5", 0 ), Set Random Seed( 1910031543 ),
Number of Starts( 105 ), Add Term( {1, 0} ), Add Term( {1, 1} ),
Add Term( {2, 1} ), Add Term( {3, 1} ), Add Term( {4, 1} ),
Add Term( {5, 1} ), Add Term( {1, 2} ), Add Term( {1, 1}, {2, 1} ),
Add Term( {2, 2} ), Add Term( {1, 1}, {3, 1} ),
Add Term( {2, 1}, {3, 1} ), Add Term( {3, 2} ),
Add Term( {1, 1}, {4, 1} ), Add Term( {2, 1}, {4, 1} ),
Add Term( {3, 1}, {4, 1} ), Add Term( {4, 2} ),
Add Term( {1, 1}, {5, 1} ), Add Term( {2, 1}, {5, 1} ),
Add Term( {3, 1}, {5, 1} ), Add Term( {4, 1}, {5, 1} ),
Add Term( {5, 2} ), Set Runs Per Random Block( 4 ),
Set Sample Size( 32 ), Optimality Criterion( 2 ),
Simulate Responses( 0 ), Save X Matrix( 0 ), Make Design}
)
),
New Column( "Day",
Character,
"Nominal",
Set Property( "Design Role", Design Role( "Random Block" ) ),
Set Values(
{"1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1",
"1", "1", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2",
"2", "2", "2", "2"}
)
),
New Column( "incubator",
Character,
"Nominal",
Set Property( "Design Role", DesignRole( Random Block ) ),
Set Property(
"Value Order",
{Custom Order( {"1", "2", "3", "4", "5", "6", "7", "8"} )}
),
Set Values(
{"1", "1", "1", "1", "2", "2", "2", "2", "3", "3", "3", "3", "4", "4",
"4", "4", "1", "1", "1", "1", "2", "2", "2", "2", "3", "3", "3", "3",
"4", "4", "4", "4"}
)
),
New Column( "X1",
Numeric,
"Continuous",
Format( "Best", 12 ),
Set Property( "Coding", {-1, 1} ),
Set Property( "Design Role", DesignRole( Continuous ) ),
Set Property( "Factor Changes", Easy ),
Set Values(
[-1, -1, 1, 0, 1, 0, -1, 0, 1, 0, -1, 1, -1, 0, 0, 1, 1, 0, -1, 1, -1, 1,
-1, 1, -1, 1, 0, -1, 0, 1, 0, -1]
)
),
New Column( "X2",
Numeric,
"Continuous",
Format( "Best", 12 ),
Set Property( "Coding", {-1, 1} ),
Set Property( "Design Role", DesignRole( Continuous ) ),
Set Property( "Factor Changes", Easy ),
Set Values(
[-1, 0, 1, 0, -1, 1, 1, 0, -1, 0, -1, 1, 1, -1, 0, 1, -1, 1, 0, -1, 0, 1,
1, -1, -1, 0, 1, -1, 0, 1, -1, 1]
)
),
New Column( "X3",
Numeric,
"Continuous",
Format( "Best", 12 ),
Set Property( "Coding", {-1, 1} ),
Set Property( "Design Role", DesignRole( Continuous ) ),
Set Property( "Factor Changes", Easy ),
Set Values(
[-1, 0, 1, 0, -1, -1, 1, 0, 1, -1, 1, 0, -1, 0, 1, -1, 1, 1, -1, 0, 0, 1,
-1, -1, 1, 1, 0, -1, 0, -1, 1, 1]
)
),
New Column( "X4",
Numeric,
"Continuous",
Format( "Best", 12 ),
Set Property( "Coding", {-1, 1} ),
Set Property( "Design Role", DesignRole( Continuous ) ),
Set Property( "Factor Changes", Easy ),
Set Values(
[1, -1, 1, 0, 1, 1, 0, 0, 1, -1, -1, 0, 1, 0, 1, 1, -1, -1, 0, -1, 1,
-1, -1, 0, 1, 0, 0, -1, 1, -1, 0, 1]
)
),
New Column( "X5",
Numeric,
"Continuous",
Format( "Best", 12 ),
Set Property( "Coding", {-1, 1} ),
Set Property( "Design Role", DesignRole( Continuous ) ),
Set Property( "Factor Changes", Easy ),
Set Values(
[1, 0, -1, -1, -1, 0, -1, -1, 1, 1, 1, 0, -1, 0, 0, 1, -1, -1, 0, 1, 1,
1, 1, 0, -1, 0, 1, -1, 0, -1, 1, 1]
)
),
New Column( "Y",
Numeric,
"Continuous",
Format( "Best", 12 ),
Set Property(
"Response Limits",
{Goal( Maximize ), Lower( . ), Upper( . ), Importance( . )}
),
Set Values(
[., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., .,
., ., ., ., ., ., ., .]
)
)
)
This is really interesting. You effectively have "nested" blocking factors.
Having said that, I think you can just keep it simple. Here is something that you can try...
Use Custom Design to create a 32-run RSM design for 5 continuous factors and an 8-level blocking factor (4 runs per block) for incubator. This will ensure that the blocks are minimally correlated with the effects that you really care about.
Then just run blocks 1-4 on day 1 in incubators 1-4. Then run blocks 5-8 on day 2 in incubators 1-4. You can manually add Day and Incubator columns to your data table in case you need to use it later.
Once you have collected that data you can fit the RSM model for the 5 factors plus a random effect for the blocks. The advantage of using a random effect is that it does not consume all the degrees of freedom that a fixed factor effect does.
Now you could get fancy and fit the RSM model for 5 factors plus a random effect for Day and a random effect for Incubator nested within Day. Strictly speaking that is probably a more appropriate model.
However, I think this is probably overkill. I doubt that it will make any important difference to your conclusions. The simple model with a random effect for block should be able to adequately capture all the variance due to differences between incubators and days.
I have attached an example design with the 2 models saved. The Y response is just random.
I hope this helps.
Phil