I would like to expand a little on the discussion about R2 values and
no-intercept vs intercept models. All no-intercept models are not created
equal. For example, there are many researchers who are interested in fitting
(no-intercept) Scheffe canonical models to data collected in a
mixture-experiment setting. In a mixture experiment, the sum of the component
proportions must add to 1.0. As pointed out by Marquardt and Snee
(Technometrics 16, 533-537 (1974)), the correct null hypothesis is not H0:
b(i)=0, all i, but rather H0: b(1) = b(2) = ... = b(q) (linear terms) and
b(j)=0 (other terms). The null model in this case is E(y)=b(0), and the least
squares estimate of b(0) is the mean response.
This leads to two differences from non-mixture no-intercept models. First, R2
values should be calculated by correcting both the model and total sums of
squares by the mean. Secondly, the number of degrees of freedom for the model
sum of squares should be reduced by 1.
I am aware of two software products that recognize this (undoubtedtly there are
others). These are Design-Expert and JMP. (Occasionally JMP gets confused with
no-intercept mixture models, and a ? appears for R2.)
As one who is interested in experimental design for formulations (i.e.,
mixtures), I would like to see software give the analyst a choice of methods for
calculating R2 - i.e., either corrected for the mean or not corrected for the
mean.
Wendell Smith (wfsmith@frontiernet.net)
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