Question about variable selection

Dear Wensui,

What you are asking about is called in psychology a "suppressor" 
variable: a predictor variable unrelated to the criterion but 
correlated with the other predictors. (X1 in the following example) 
Although it has a zero relationship with the DV, it does "really" 
help to predict the DV by removing extraneous variance from the other 
IVs.  (I am not going to touch the Wittgenstein issue of truth here). 
Should it be included in the predictor set? Yes.  Is there any easy 
way to find all possible suppressors? No.


Consider the following:

#demonstration of "suppressor effects"
library(mvtnorm)
sigma <- matrix(c(1,.5,0,.5,1,.5,0,.5,1),ncol=3)
my.data <- data.frame(rmvnorm(1000,sigma=sigma))
names(my.data) <- c("X1", "X2", "Y")
round(cor(my.data),2)
summary(lm(Y~ X1 + X2,data= my.data))

which produces
       X1   X2     Y
X1  1.00 0.45 -0.04
X2  0.45 1.00  0.51
Y  -0.04 0.51  1.00

 summary(lm(Y~ X1 + X2,data= my.data))

Call:
lm(formula = Y ~ X1 + X2, data = my.data)

Residuals:
      Min       1Q   Median       3Q      Max
-2.09350 -0.58069  0.02280  0.53436  3.02017

Coefficients:
             Estimate Std. Error t value Pr(>|t|)   
(Intercept)  0.02807    0.02557   1.098    0.273   
X1          -0.32849    0.02813 -11.680   <2e-16 ***
X2           0.65666    0.02861  22.951   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.8081 on 997 degrees of freedom
Multiple R-Squared: 0.3465,	Adjusted R-squared: 0.3452
F-statistic: 264.4 on 2 and 997 DF,  p-value: < 2.2e-16

Dear Wensui,

I don't think that it's possible to answer these questions mechanically,
especially if you're interested in the "true" relationship between the
response and a set of explanatory variables. If, however, you have a pure
prediction problem, then variable selection is a more reasonable approach,
as long as it's done carefully (in my opinion).

I don't see how resampling and repeatedly examining the marginal
relationship between Y and an X is relevant to the question of whether there
is a partial relationship in the absence of a marginal relationship. (This
is close to what Wittgenstein once called buying two copies of the same
newspaper to see whether what was said in the first one is true.) After all,
as I said (and as you understand), the partial and marginal relationship can
differ -- so evidence about the marginal relationship is not necessarily
relevant to inference about the partial relationship. (As well,
bootstrapping a linear least-squares regression likely isn't going to give
you much additional information anyway.)

Regards,
 John

--------------------------------
John Fox
Department of Sociology

.... (discussion of interaction from Andy Liaw)

 > > > From: Wensui Liu

 > > > >
 > > > > Dear Lister,
 > > > >
 > > > > I have a question about variable selection for regression.
 > > > >
 > > > > if the IV is not significantly related to DV in the bivariate
 > > > > analysis, does it make sense to include this IV into the

 > > full model

 > > > > with multiple IVs?
 > > > >
 > > > > Thank you so much!

William Revelle		http://pmc.psych.northwestern.edu/revelle.html   
Professor			http://personality-project.org/personality.html
Department of Psychology       http://www.wcas.northwestern.edu/psych/
Northwestern University	http://www.northwestern.edu/