I am going to design some experiments with one response and 4 factors to find response surface. I can not do three level full factorial design due to some shortages. Just around 27 experiments are possible to do. Do you suggest to do three level fractional factorial design or Two level full factorial design along with center points (and axial points)?
Well lets do the math.
Suppose you are going to use Least squares with linear polynomial model. Also suppose that your factors are independent (line speed, temperature etc) and they are not mixture components, which are connected with sum xi=1.
If the relationship between your response and the factors is linear, you have 5 coefficients for the linear polynomial. If you consider the interactions you have six more coefficients, totally 11. In full factorial experiment you have 2 power 4 runs to perform, which is 16. So you have 11 coefficients and 16 runs to do. So you have enough runs to fit the model and you have another 11 (up to 27) for checking the lack of fit, which is even too much. So, if you suspect linear relationship you have enough runs to play with.
If you suspect quadratic relationship. The quadratic linear polynomial model Y=b0+sumBiXi+BijXiXj+BiiX^2 for 4 factors has 15 coefficients. So you must perform at least 15 plus several (let say 5 runs) to have good enough model. This makes 20 runs. You will need another 3-5 or 7 runs (let say in the middle of the space - all factors at middle value) to check whether your measurements are reproducible (Fisher test for lack of fit, if I am not wrong). Again you have enough resources.
So - a lot of good news. You have enough runs for the both types of models. BUT (!!!) better decide firstly, which type of model you suspect (based on you technological, scientific, literature etc knowledge. Then do the planning of the experiment according to a.m.
There is another way though - to use sequential design. Which is to start with model Y=b0+sumBiXi+BijXiXj + several meausrements in the middle. Than if the model is not good enough, to add more experiments, generated to complete the design up to D-optimal for higher model (quadratic). I know it sounds weird if you don't know about sequentially generated optimal designs, but if you are interested we can discuss further how to plan the experiment.
If it is 4 component mixture - than it is a bit different story.
Dear Stoyanov
Thanks for your answer but I couldn't find the comparison between three level fractional factorial design and two level full factorial design for the same number of runs in order to get a quadratic model. Could you please give me your comment about selecting between these two type of design?
Another question, three of my factors can be coded to -1, 0, 1but one of them has the levels of 1, 10, 100. Haw can it be coded? would it cause any type of error?
For quadratic model you definitely need central points, therefore if you are going to use Full Factorial, you have to take the three level one. As for the coding you can use Log10(10)=1; Log10(100)=2; Log10(1000)=3 and then you can code the levels one more time to -1 0 1. When you have the optimal values, then you have to decode back them twice, firstly form -1 0 1 back to 1 2 3 then from 1 2 3 back to 10 100 1000. I am not sure whether it is clear enough. I suggest you not to use Full Factorial, which is the very "old fashioned", but non-symmetrical D-optimal design of 20 runs instead. Also perform 5 runs in the center to have data for the lack of fit test. 4 factor quadratic model has 15 coefficients, so 20 runs should be enough. I am ready to help you with all of this.
Kalin
Dear Kalin, thanks again for your answer.
Actually I am a beginner in DOE and helps from an expert like you will be really appreciated in advance.
I would like to use new fashioned method but I am confused at my first step. I don't know how can I select the most compatible method (design) for my problem and how can I define the details required for that method. At first, I tried to use "Custom Design" platform of JMP software (which automatically select the most compatible design according to your problem definition) but I had difficulties in defining my problem in the software (e.g. distinction between discrete numeric and categorical factors) and understanding whether the platform selected the best design or not. That's why I tended to the full factorial design which seemed simpler and more understandable to me.
I would be extremely grateful if you help me to start again with a new advance method. You think which method can be more adequate for my problem (explained here: https://www.researchgate.net/post/Design_of_biochemical_experiments_by_using_JMP_statistical_software2 )? and which software do you suggest?
This link ( http://www.jmp.com/support/help/Creating_a_Custom_Design.shtml ) is also for custom design in jmp whcih uses I-Optimality criterion rather than D-Optimality criterion for two-factor interactions and quadratic modeling.
As I said I will be happy to help according to my knowledge. I created a RG project and invited you. I suggest on this task, form now to communicate trough the project
If you have 4 factors and they are all continuous, you can use a Box-Behnken Design. This will give you all main effects, 2-way interactions and quadratic effects for all 4 factors. You will be able to do all of this at 3 levels and in 27 runs.
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