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Although collinearity (of predictor variables) is a possible explanation, I would like to suggest it is not an illuminating explanation because we know collinearity is related to "common information" among the predictors, so there is nothing mysterious or counter-intuitive about the side effect of introducing a second correlated predictor into the model. ...

I think this issue has been discussed before on this site fairly thoroughly, if you just knew where to look. So I will probably add a comment later with some links to other questions, or may edit this to provide a fuller explanation if I can't find any. There are two basic possibilities: First, the other IV may absorb some of the residual variability ...

"How much" matters a great deal! The adjustment is unlikely to be zero, after all; this would only happen if z were totally uncorrelated with x or y. By common convention one would test the statistical significance of the relationship between z and y as a way of deciding whether it is necessary to use z to adjust x's coefficient. That said, significance ...

The main difference between the coefficients and the correlations (elements of structure matrix) is not that these are less stable than those. Coefficient shows partial (i.e. unique) contribution of the variable to the discriminant function score, it is like regression coefficient. Correlation shows omnibus (i.e. unique + shared with other variables) ...

I think this hinges on what you means by "not changes much". The estimated parameters could change and so could their standard errors. That's two separate effects. Let's focus on the variance part for now. Suppose the true DGP is really $y=x\beta+z\alpha+v$, but you leave out the relevant variable $z$ from your model, so you actually estimate ...

I agree with @ttnphns. There are good reasons to interpret both standardised and structure coefficient tables. In some senses they contribute two complementary perspectives on what it means to say that a variable is an important predictor: (a) prediction after controlling for other predictors (i.e., standardised coefficients); (b) prediction without ...