color map on correlations and estimation efficency

[#1]

I am working on a design with 2 continuous factors and one discrete numeric factor and I can't go above 12 runs for this design. When I look at the color map of correlations of this design, it shows correlation among the factors. I have tried different number of runs to see it gets better but it doesn't. If I go below 12 run, power of design decreses and I can't go more than 12 runs. Is there anything I can do about the orthogonality of the design? Also, does looking at estimation efficiency help in this case?

[#2 · Accepted solution]

There is nothing wrong with using a Discrete Numeric factor. It is essemtially a continuous factor but you specify the levels and JMP specifies the terms in the model. You could also use a Continuous factor and specifiy the terms, then JMP specifies the levels. I got a better design in 12 runs this way. I saved the table here for you.

Your example appeared to use a model with all main effects and two-factor interactions and a single quadriatic term for exhaust temp. That model is the one I used.

[#3]

@Mark_Bailey Thank you for your reponse. Using continuous factor instead of discrete numeric factor does help but I think the quadriatic term for exhaust temp was added when I used discrete numeric factor. Now if I only include main effects and two-factor interactions I see correlations again. Is this expected or am I doing something different?

[#4]

Correct, as I said, if you define the levels using a Discrete Numeric factor then JMP defines the terms. You must have a quadratic term in the model to keep all three factor levels.

The reason for the non-zero correlations is the addition of a center point and the imbalance in the design. You will still have correlations if you omit the center point. Each main effect is correlated (r = +/- 1/3) with the interaction effect involving the other two factors in this case using 12 runs. You need to go to 8 runs or 16 runs to eliminate the correlations.

Why are you concerned about this much correlation in the estimates? Do you have high variance response to begin with? Do you have small effects of changing factor levels?

[#5]

@Mark_Bailey I am concerned about correlation because of small changes with factor levels. But correlation is better when using continuous factor instead of discrete numeric so I guess this will work for me in this case. Thank you for explaining it. Appreciated

[#6]

Do you mean that the inflation caused by the correlations is severe because you expect small effects due to limited factor ranges? Is there a reason that you cannot widen the factor ranges and produce a larger effect?

[#7]

That was my thought but with the design you suggested correlation between factors declined so that should be the case.

[#8]

If your expected effects are at least 3x the standard deviation of the response then the power is quite high with the 12 runs:

power.JPG

This result can be expected even with the correlations (+/- 1/3) seen here:

Correlations.JPG

The correlations result in NO bias of the estimates. The correlations inflate the variance of the estimates, which causes a lengthening of the confidence intervals:

estimation efficiencu.JPG

The CI are about 6% longer, a very modest inflation. I don't think these correlations compromise the design performance much.

[#9]

I forgot to address your specific request for orthogonality. As you discovered, it can be very expensive to achieve this characteristic. It requires a fully balanced design and all the estimation columns must be balanced, too. We can often produce economical and effective designs without orthogonality. The 12 runs that I found seem to have a few low correlations and, therefore, a few inflated confidence interval lengths. See if the correlation is acceptable for you case.