partial r2 vs partial correlation vs stnd coefficients

seth

New Member
#1
Hi all,

I have run a few linear regression models predicting water quality for watersheds using explanatory variables such as mean impervious surface within watersheds and others suggested by theory and the research of others. I would like to be able to compare explanatory variables measured on different scales to determine their relative strengths of effect. The standard approach I see is to multiply the slope coefficients by standard deviation of the corresponding explanatory variable. I've also seen multiplying by some range based upon percentiles (such as the difference between the 75th and 25th percentile value for the variable) to deal with outliers, etc that call using standard deviation into question. I understand what partial r-squared and partial correlation are from text descriptions.

First question are partial r2 and partial correlation equivalent to each other? They are described in separate places and with different language so I have not gotten the mathematical relationship yet. Is one just the square of the other?

Second, I know that partial r2 is related to the increase in the amount of variance in the response explained by adding a variable of interest, given that all other variables are in the model. Is this a measure of effect size similar to a standardized coefficient? Will they necessarily rank variables by "importance" the same? I am trying to decide how to compare variables and partial r2 has intuitive appeal to people used to looking at multiple r2.

Thanks,
Seth
 

Mean Joe

TS Contributor
#2
Second, I know that partial r2 is related to the increase in the amount of variance in the response explained by adding a variable of interest, given that all other variables are in the model. Is this a measure of effect size similar to a standardized coefficient? Will they necessarily rank variables by "importance" the same? I am trying to decide how to compare variables and partial r2 has intuitive appeal to people used to looking at multiple r2.
What are you trying to do? I'm not used to saying one variable has greater importance to predicting the result than another; if they're important, then they're all equally important I like to think.

Are you interested in comparing variables to determine if they should be left out? I think that you should also look if the extra predictors are significant (measured by their p value, or something like gender/age which should almost always be included in a model).

I was wondering if you were trying to compare a situation with 3 predictors (r2=0.76) vs a situation with 4 predictors (r2=0.80), and you would then take the model with 3 predictors because it has a "high enough r2", like trying to get a high enough r2 with as few predictors as possible?
 
#3
I am not clear on this point either, is a partial r-squared the same as a partial correlation? It seems that they both take the additional sum of squares and divide it by the error sum of squares of a model lacking the additional predictor.
 

Jake

Cookie Scientist
#4
Yes. If you take a partial r and square it, it is then a partial r-squared. Probably sounds kind of obvious when stated this way.
 
#6
I've never heard of a semi-partial r or a partial r, but to clarify, a partial r-squared is equal to a partial correlation? Just 2 different terms for the same thing?
 

Dragan

Super Moderator
#9
How do you figure? A semi-partial (squared) correlation seems to be something quite different entirely. How would it be that this is what you get from squaring a partial correlation.
Well, no it is not, if you consider how the OP defined the partial R^2. To wit: "Second, I know that partial r2 is related to the increase in the amount of variance in the response explained by adding a variable of interest, given that all other variables are in the model."

That description is (essentially) the definition of a squared Semi-partial correlation coefficient.