How to interpret a predictor with a positive structure coefficient and a negative standardised coefficient in discriminant function analysis?

Question

I am doing a discriminant function analysis and I have four continous independent variables and one categorical dependent variable (that has 3 groups). I have chosen to do this analysis to see how these groups can be predicted by the four independent variables.

• Firstly, when we are interpreting the results, do we look at standardised coefficients or the structure matrix coefficients? From my reading of the literature, I have found contradictory opinions. Some say that structure coefficients are better since they are more stable while others say that both standarized and structure coefficients should be looked at.

• Secondly, when I do look at both structure and standardized coefficients, two of my dependent variables have the opposite signs for their standardized and structure coefficients (for both of them the structure coefficient is positive and more than .30 whereas the standardized coefficient is negative and has a very small value). I have read up some stuff on it and I think these are suppressor variables (they are related to the other IV's but do not directly predict the DV). How are we supposed to handle these suppressor variables?

Answer 1

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) contribution of the variable to the score, it is in fact usual zero-order correlation. You may look at both coefficients and correlations, keeping in mind the above differences. If you plan to interpret discriminant functions like you interpret factors in factor analysis, I think you better look at coefficients, which are formally similar to loadings of factor pattern matrix, with one important distinction though, that in factor analysis factor "loads" variable, while in discriminant analysis variable "loads" discriminant function.

I would not hurry to call the two variables with sign disagree between coefficients and correlations "suppressor variables". Yes, suppressors often show such a pattern, yet there are suppressors apart from it. Also, other things than suppresorness can cause it, for example collinearity. The definition of suppressor variable within multiple regression is that it rises R-square by greater amount than its own squared Pearson r with the DV. In the context of discriminant analysis, mean squared canonical correlation will stand for R-square, I believe. But what can stand for squared Pearson r? - at this point I'm not sure [Pillai's trace, I suspect?]; so, I'm not clear about how to define suppressor in discriminant analysis.

Answer 2

• 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 controlling for other predictors (i.e., structure coefficients). While there are various indices that can be used to assess variable importance while controlling or not controlling for other predictors, I think having at least one index from each category is useful.
• In particular, if you have two predictors that are correlated, whether it be multiple regression or discriminant function analysis, it is common for both predictors to predict the outcome on their own, but when combined, one of the variables appears to provide most of the prediction. This is possibly what is happening to you here. I wouldn't focus too much on the change in sign if, for example, the standardised coefficient is close to zero (e.g., perhaps $-.05$, or $-.10$).
• By reporting both types of indices of variable importance, you have a rather natural interpretation of the variable with the small standardised coefficient. I.e., on its own it predicts, but when included with other predictors, its effect is captured by other predictors. Of course, you might want to double check this interpretation by running the DFA with just that predictor.