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Mixture Design with Response in %

bwanders

I have attached an example mixture design with response data in percentages. When you use standard least squares, it is possible to have predictions outside of 0 and 100%. Isn't there way to constrain responses to 0 and 1 or 0 and 100?. The generalized linear model can't take mixture effects.

Custom Mixture Design.jmp

Accepted answers

michael_bresnic

I believe this is because your first response will be undefined in the logit transform, log(y/[1-y]). You will end up with a denominator = 0.

Did you see my note about adjusting for this? (Change the result to something close to 1; e.g. 0.995; something supported by your sense of no practical difference and measurement precision.)

All comments

michael_bresnic

The logit transformation of the response may be suitable: y' = ln ( [y - lower limit] / [upper limit - y] ); for results between 0 and 1 y' - ln ( y / [1-y] ). This will not accommodate responses = 0 or 1, but does ensure y' stays within the boundaries.

michael_bresnic

Oops, for results between 0 and 1 y' = ln ( y / [1-y] ).

bwanders

Michael,

Could you send the sample file back with the corrected transformation? I'm having trouble with it.

michael_bresnic

Here you go.

Notice that Y=0 and Y=100 will be undefined (log(0) and denominator=0). I adjust these by ±0.01. You could use a smaller or larger adjustment, possibly relating it to the sensitivity and variability of your measurements. The smaller the adjustment, the more extreme the transformed value.

Also, as was pointed out in another reply, ln (log base e) is log in JMP.

Michael

Custom Mixture Design with Logit transform.jmp

michael_bresnic

Also, to convert the predicted transformed responses (Y’) back to native units (Y):

Y = 100*Exp(Y’) / (1 + Exp(Y’))

I notice when I ran your model, the two largest responses (runs 1 & 12) had very large residuals. This is likely due to the logarithmic nature of the transformation. You can reduce this somewhat with the adjustment to the 100% response. If your reporting as integers is indicative of precision, increase the adjustment to 0.5%, suggesting the precision makes the difference between 99.5 and 99.99 of no practical significance.