Defining Factors in a Central Composite Design
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 This topic has 15 replies, 3 voices, and was last updated 4 years, 4 months ago by Robert Butler.

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March 19, 2012 at 9:01 am #53995
roartyParticipant@cognition Include @cognition in your post and this person will
be notified via email.My formulation is a simple mixture of ingredients and I intend to carry out a central composite design to establish optimum level and combination of ingredients. I wish to investigate the following four factors
1. Level of fluoropolymer (continuous factor, 0% to 10%)
2. Addition of ingredient X (two level categorical factor, present or not present)
3. Level of silica (continuous factor, 0% to 5%)
4. Type of silica (two level categorical factor, hydrophobic or hydrophilic)The first two factors are, I believe, pretty straight forward. My problem is with the last two factors, type of silica and level of silica.
Because my low level in Factor 3 is 0% (ie. no silica to be added) this means that the low level for both my hydrophobic formulation and my hydrophilic formulation are effectively the same formulation as both contain no level of silica at all!
My concern is that formulations containing 0% level of silica will be over represented in my design. This is probably not a problem but it just feels there is a more efficient solution?
One idea was to make Factor 4 a three level categorical factor i.e. hydrophobic, hydrophilic and none. But now my none level in Factor 4 becomes effectively the same formulation as the low level of Factor 3 which also contains no silica. So Im back to square one with the feeling that I am not designing this experiment as efficiently as I could.
I hope the above makes sense and any advice is gratefully received. Ive got a feeling I will kick myself when I find out the solution.
0March 19, 2012 at 9:54 am #192639
MBBinWIParticipant@MBBinWI Include @MBBinWI in your post and this person will
be notified via email.@cognition – You are correct that a measured value of zero causes over sampling of the specific point. You could still analyze this via general linear model or you can set the low value at a very low but nonzero level.
0March 19, 2012 at 10:24 am #192640
Robert ButlerParticipant@rbutler Include @rbutler in your post and this person will
be notified via email.Im missing something how are you going to run a central composite on type variables?
If we just look at the design space and recognize the problem with too many zero levels of silica then you could do a couple of things.
If your post is as it sounds – a desire for combinations that are either pure hydrophobic or pure hydrophilic then you could do as MBBinWI suggests and make the 0 value small but nonzero or you could make the exisiting design slightly nonorthogonal by taking the design experiments with values of 0 level for either silica and randomly replace the 0 level with a small non zero value for x percentage of these experiments. The choice of percent of conversions of 0 levels would be driven by your personal comfort level with respect to the number of experiments with 0 level of silica.
If you rebuild the design in this manner (we’re ignoring the type variable issue for the moment) then when you go to analyze it you will have to use the actual design matrix for analysis and not the idealized one with all zero settings. This will mean that in addition to coded levels of 1,0,1 and the star point levels you will also have some values with something like .9.
If it makes sense to run experiments with mixtures of the two silica types then a better design would be one with the following variables:
Level of fluropolymer
Presence or Absence of ingredient X
Level of Silica #1
Level of Silica #20March 19, 2012 at 10:59 am #192641
MBBinWIParticipant@MBBinWI Include @MBBinWI in your post and this person will
be notified via email.@rbutler – Robert: Good catch, I wasn’t paying attention to the central composite design. In that case, unless you are using DOE specific software, then you would need to use a GLM analysis (in Minitab).
0March 19, 2012 at 1:40 pm #192644
roartyParticipant@cognition Include @cognition in your post and this person will
be notified via email.How are you going to run a central composite on type variables?
Oops! mea culpa, I think I got a bit ahead of myself in that I anticipated my problem regarding 0% silica levels before Ive had a chance to sit down and think about the appropriate design.
In fact my plan was to initially do a 2level factorial design; then if the factors Ive mentioned were to prove significant I planned to make up some midpoint formulations to get a response surface and hopefully identify an optimum formulation.
Thank you both for your suggestion on choosing a very low level of silica instead of 0%. I think I will reject this option however. Suppose I was to opt for 0.1% as my lower silica level, I do not feel that my ever so slightly hydrophilic formulation would be very different from my ever so slightly hydrophobic formulation. Im sure you can appreciate that I would be pretty much back where I began with an over representation of very low level silica formulations. Of course I could increase the lower silica level but then I miss out on silica free formulations and I am not prepared to do that.
Robert, you have suggested two interesting alternatives. Formulations containing a mixture of silica types happen to be unsuitable for this work. Making the design nonorthogonal by replacing the zero value of one of the silica types is an attractive option however. This work will form part of a young lads final year university dissertation so although I am loath to deviate too much from a conventional design which can be easily explained to his professor, I am sure he is smart enough to explain the choice of a nonorthogonal design.
Thank you both.
0March 20, 2012 at 5:16 am #192655
Robert ButlerParticipant@rbutler Include @rbutler in your post and this person will
be notified via email.If you replace some of the 0 values with a nonzero value you will want to do this for a certain percentage of both of the silica types. After you have done this you should check the design using eigenvalues and condition indices (if you have that capability) and VIF’s to make sure that the changes do result in nothing more than a slight nonorthogonality.
0March 22, 2012 at 10:58 am #192691This can be accomplished through a couple of steps. First, ignore ingredient X and the silica type and come up with the design that makes the most sense to you (maybe a central composite design with the axial points at your limits). Then generate that design, but with four blocks. Have in mind a coding scheme – it doesn’t have to be random – ahead of time, such as:
Block 1: Ingredient X = no, Silica Type = Phobic
Block 2: Ingredient X = no, Silica Type = Phillic
Block 3: Ingredient X = yes, Silica Type = Phobic
Block 4: Ingredient X = yes, Silica Type = PhillicYour design should randomize the blocks, and then you assign the factors above according to the blocks. Then rerandomize everything, and you have a fullyrandomized, completelyorthogonal (assuming you chose an orthogonal base) design to work from that never has both types of silica in it.
0March 22, 2012 at 11:15 am #192693
MBBinWIParticipant@MBBinWI Include @MBBinWI in your post and this person will
be notified via email.@joelatminitab – You da’ man!
0March 23, 2012 at 3:20 am #192708
roartyParticipant@cognition Include @cognition in your post and this person will
be notified via email.Thank you Joel,
I had not considered blocking. Your approach is the one I will go with as it maintains the orthogonal nature of the design while providing a solution to my problem.
0May 10, 2017 at 9:54 pm #201349Mr. Joel Thanks for your valuable reply. Actually I have kind of the same problem. I have 4 factors which are time, temperature, solvent and size. Time and temperature are range between high and low point, while solvent and size factors are categorical. Solvent is methanol and ethanol. Size is 1 and 0.6. So how can I do the design in central composite design ? Same as what you have mentioned earlier by putting 4 blocks? Or in another way ?
0May 10, 2017 at 9:56 pm #201350@joelatminitab
0May 11, 2017 at 6:19 am #201351
Robert ButlerParticipant@rbutler Include @rbutler in your post and this person will
be notified via email.The short answer is that you can’t design a central composite design with the variables you have. Based on your description of the variables what you have is 3 continuous variables and possibly 1 categorical variable. The fact that size is 1 and .6 is interesting but in theory size could be anything you would want to make it so size should be treated as continuous. As for solvent – are we talking nothing more than the difference between alcohol types with no other caveats such as alcohol percent? If it really is a case of just different alcohol types and nothing else then you do have a categorical variable but there are only 2 levels so you can build the design and run the analysis as though they were continuous variables.
If you want to be able to check out all of the two way interactions then the simplest design would be a 2**4 augmented with two or three additional experiments to permit a check of curvilinear effects of time and temperature and a replicate. Another possibility would be a Doptimal design – you could get all of the above in 18 runs plus two additional experiments chosen from the initial 18 to give a measure of replication – total 20 runs.
Now, if you don’t need to know about all of the 2 way interactions then there is the possibility of running an augmented fractional factorial design to permit a check of all main effects, the curvilinear effects of time and temp, and a maximum of three of the two way interactions (the main effects will be clear of two way but two way effects will be confounded with other two way effects). Attached is a Doptimal design to give you all main effects, all two way interactions, the curvilinear effects of time and temperature and two replicates. This design is, of course, based on the earlier assumptions I made in this post.
0January 28, 2018 at 4:39 am #202188Hi Robert,
I am going to do a central composite design optimization with three factors: two continuous and one categorical (with 2 levels). I wonder if there is a published research study having a similar design. Minitab is giving me 26 runs. The response optimization equation is given in two subequations each one related to the 2 levels of the categorical variable. Like this:
Categ. Var.
High PDI = ……………………..Low PDI = …………………….
I am not sure how to report the data or if I am on the right way. A similar article can really help me or your and other friends’ ideas on how to report the data are appreciated.
0January 28, 2018 at 9:50 am #202191
Robert ButlerParticipant@rbutler Include @rbutler in your post and this person will
be notified via email.As was noted in the earlier posts – you can’t do a central composite with type variables. If you want curvature for the two continuous variables then the simplest design would be a 3**2 design run for each of the categorical levels. This would be a total of 18 experiments and would allow an investigation of all main effect, curvature of the two continuous variables and all two way interactions of continuous and categorical. If you toss in a couple of replicated points you could do it all in 20 experiments.
0July 25, 2018 at 3:14 am #202853Dear Robert
My experimental design is central composite design with 3 factors (water, nitrogen and phosphorus) and 5 levels of each factor. I have three responses (Yield, height and plant biomass). I want to know the optimum combination of water, nitrogen and phosphorus by using Minitab. I am a basic learner so I want to learn step by step. Can anyone help to solve this?
Rana Roy0July 25, 2018 at 9:33 am #202854
Robert ButlerParticipant@rbutler Include @rbutler in your post and this person will
be notified via email.I don’t follow the reasoning of your post. A design is just that – a design. You say you have 3 factors and you have identified the 5 levels of those factors corresponding to the five settings in a composite design. So, what is left is the matter of running the design, taking the values for each of the responses, using methods of backward elimination and stepwise regression to identify the reduced regression model and then use those models to identify the combinations of water, nitrogen, and phosphorous that will provide the best tradeoff in terms of optimum yield, height, and biomass.
From the standpoint of actually running the design there are a number of issues you should think about.
1. What drove your choice of a central composite? Do you have any kind of prior information that would indicate curvilinear behavior in all three factors and possible significance of most of the two way interactions?
2. You will have to grow more than one plant per experimental combination and, since plants grown in the same plot of land will be more alike than plants grown in separate plots of land you will need to run the same treatments on more than on plot of land.
3. Yield and biomass will (I assume) be based on a whole plot result but what are you going to do about plant height?
4. Given the labor involved in multiple plots of land I would first recommend a simple saturated design – 3 variables in 4 experimental plots with, in this case, a full replication of the design to get some sense of withinplot variation. This would come to a total of 8 plots of land, it would give you a sense of the main effects of the variables of interest and it may be all that you need. If it doesn’t then you can augment what you have to flesh out a full central composite.0 
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