http://spssx-discussion.165.s1.nabble.com/Question-about-the-linearity-assumption-for-Discriminant-Analysis-tp5121318p5123489.html
What we are creating with discriminant function (or regression, or logistic
regression) is a linear combination of variables to make a predictor equation.
Or several predictor equations, if there are several groups as criteria.
For the result to meet the assumptions for valid testing at the end, the
residuals of prediction must be homogeneous. - I don't know where you
are reading about a "linearity assumption" for DF, but I would usually
think about it more as an assumption of homogeneity of residuals.
If each variable has similar variance across groups, then their scaling
looks "right" and linear for predicting membership in those groups.
You are correct in saying that we prefer to reduce redundancy among
predictors. I can say that we should much prefer *linear* relations
between the predictors in contrast to accidental, *nonlinear* relations.
When we intentionally put in non-linear relations (X, X-squared, X-cubed),
we (should) know that we have to be careful in our interpretation, because
it is all the same variable. (Problems are smaller when those contrasts
are designed to be orthogonal or nearly so, by subtracting the mean (say)
before squaring.) But when two predictors, W and X, happen to have a
non-linear relationship, we are entering too *blindly* into the
X, X-squared as predictors,
highly correlated, highly confounded,
artifacts-expected-from-poor-scaling
sort of paradigm. Even if the residuals end up homogeneous, we have a
worse case for interpretation than if we started with X and X-squared.
So - though I do see some sense in referring to a linearity assumption
for predictors in DF, I would usually choose to talk about it differently,
or with some larger amount of detail attached.
--
Rich Ulrich
> Date: Wed, 4 Jan 2012 15:57:44 -0800
> From:
[hidden email]> Subject: Question about the linearity assumption for Discriminant Analysis
> To:
[hidden email]>
> I'd appreciate an clarification of the linearity assumption for discriminant
> analysis. I am told that the model requires a linear relationship among the
> predictor variables within each group. At the same time, it would seem that
> we would prefer to reduce redundancy among the predictors -- i.e., have them
> be unrelated. I can't seem to reconcile these two demands. Could someone
> help?
>
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