Independent variation explained (partial correlation analysis)

[John Williams]

My study will use traditional multivariate regression on a continuous variable. We need a way for stata to tell us the *independent* variation that is attributed to *each* explanatory variable, again independent of the others. Imagine a model in which student test scores are regressed on family income, parental education, parent-child communication and parent monitoring of the child.

Basic two-variable correlation does not provide what we need because two-variable correlations will pick up "noise" from "third party" variables that are omitted in basic correlation.

I would think the command-tool we need is pcorr to produce the partial correlation.

Do you agree? Is there a pcorr-like function in stata in logistic models (dummy dependent variable models)?

If pcorr is appropriate, then why was dominance [domin] invented?

Does dominance analysis have meaningful advantages over partial correlation?

http://econpapers.repec.org/software...de/s457629.htm

[Roman Mostazir]

Hi John, to reply you partially, for independent variation explained by each of the predictor variable, perhaps you should be looking for partial-eta-square. Assuming you have run your regression model. Now, type: estat esize; this will give you ''partial eta square''; for more conservative version of effect size (explained variation), type: estat esize, omega ( this will give you omega suared''). I am not quite sure why are you coming up with logistic model later on while your model is a linear one.

[John Williams]

Thanks! In looking at the below link, it looks like pcorr does not produce the *independence that I was looking for. http://www.psychstatistics.com/2011/...-correlations/ It looks like the "partial correlations" may not sum to the total r-squared. And to clarify, when I said logistic models, I raise a new question which is to ask if pcorr works with a 0/1 dependent variable, but it is not my primary question.

[John Williams]

I will look into partial-eta-square

[David Hoaglin]

John,

In traditional multi-predictor regression, it is possible to get contributions that add up to the total R-squared if the predictors are mutually uncorrelated (mutually orthogonal). Otherwise, you can get such a decomposition of R-squared if you specify the order in which the predictors enter the regression. I think you are looking for a decomposition of R-squared that does not depend on the order of entry, but in a general multi-predictor regression that is not possible. (It is not hard to find articles that report such a decomposition of R-squared, but the authors have deceived themselves.) The predictors work together, not independently, in the least-squares fit to y.

Thus, in general a predictor does not have an "independent" contribution. That, unfortunately, is a confusing use of the word "independent," which already has other standard meanings in statistics. It is meaningful to talk about a predictor's "unique contribution," meaning its contribution after the contributions of the other predictors have been accounted for. In the correlation scale, the partial correlation between y and xj is such a measure; it equals the correlation between residual y (from the regression of y on the predictors other than xj) and residual xj (from the regression of xj on the predictors other than xj).

David Hoaglin

[Joseph Luchman]

I realize this thread has been inactive for months now - but I thought I'd provide a few citations to highlight what dominance analysis can do for anyone coming upon this thread.

As David mentioned, there is no completely unambiguous way to decompose the R2 with correlated predictors - the dominance method is simply a one way to attempt to do so that is based on averaging the uncertainty in terms of attribution of variance explained, though I'd say the primary purpose of the method is to compare predictors in terms of their relative contribution in a way that is not unduly affected by order of entry (as was discussed previously). Some papers below also discuss (semi-)partial correlations as they compare to dominance statistics.

Some papers to look at re: the dominance method are:

Azen, R., & Budescu, D. V. (2003). The dominance analysis approach for comparing predictors in multiple regression. Psychological methods, 8(2), 129.

Budescu, D. V. (1993). Dominance analysis: A new approach to the problem of relative importance of predictors in multiple regression. Psychological Bulletin,114(3), 542.

Grömping, U. (2007). Estimators of relative importance in linear regression based on variance decomposition. The American Statistician, 61(2).

Johnson, J. W., & LeBreton, J. M. (2004). History and use of relative importance indices in organizational research. Organizational Research Methods, 7(3), 238-257.

- joe