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How exactly does one “control for other variables”?

In my linear model fitted.model <- lm(spending ~ sex + status + income, data=spending), my results were as follows:

                       Estimate  Std. Error t value   Pr(>|t|)    
(Intercept)    22.55565   17.19680   1.312   0.1968    
sex         **-22.11833**  8.21111  -2.694   0.0101 *  
status          0.05223    0.28111   0.186   0.8535    
income          4.96198    1.02539   4.839 1.79e-05 ***
verbal         -2.95949    2.17215  -1.362   0.1803 

Now, when I held sex and all other predictors constant in new lm model mydata<-lm(spending ~ sex, data=spending) my coefficient was

            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   29.775      5.498   5.415 2.28e-06 ***
sex        **-25.909**    8.648  -2.996  0.00444 ** 

Questions: Should the sex value of -22.118 be the same in my new model? Because I now get -25.909. Since I am holding all constant, I should get a different value, please clarify?...

  • $\begingroup$ Your two models appear to involve two dataset names, dataset and spending. If the data differ, then of course you should expect the results to differ! Please, then, explain the relationship between those two datasets. $\endgroup$ – whuber Sep 14 '12 at 16:18
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    $\begingroup$ Even if the same data (response variable values) are used in two models removing covariates could change coefficients simply because a covariate removed is correlated with one that remains. I believe that this can be avoided if the covariates can be constructed to be orthogonal such as with orthogonal polynomials. $\endgroup$ – Michael R. Chernick Sep 14 '12 at 16:49
  • $\begingroup$ Actually the data set is the same, spending. I typed the wrong name of the dataset. I needed to determine the difference in predicted spending for a male vs. a female with other variables held constant. But i was questioning if the value of the sex in my 1st model using all constants should be the same value when one variable and all other constants. $\endgroup$ – MsSnowy Sep 15 '12 at 1:07
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    $\begingroup$ Your question is related to this CV question: How exactly does one "control for other variables"?. It is explained why regression coefficients change when we include other variables (which is also referred to as "controlling for"). $\endgroup$ – Bernd Weiss Sep 15 '12 at 3:03
  • $\begingroup$ @berndWeiss Good reference to a duplicate or near duplicate. I gave an answer to the OPs specific question but the responses in the link should give the OP a deeper understanding of what is going on and what the issues are. Maybe it would be appropriate to close this question. $\endgroup$ – Michael R. Chernick Sep 15 '12 at 3:38

The answer must be that the variables are correlated. One model uses sex and income which are eboth very significant and two other variables that appear to be non-significant. The other model uses only sex as a covariate. The coefficients for the intercept and sex both change a little and the intercept becomes significant when income and the other variables are removed. This should not be a surprise. I suspect that a lot of this change is due to removing income which is an important variable in the model. For the full model you should look at the sample covariance matrix and I am sure you will see that some of these variables are correlated and probably at least one with sex. Check that out.

As I mentioned in my comments the variables need to be uncorrelated for the coefficients to be guaranteed not to change when other covariate are removed as in the case when the variables are polynomial terms that are orthogonal.

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  • $\begingroup$ Thanks for the information and I will read the sections indicated above. I'm new to R so I dont know how to do a covariance matrix but I will look in related topics. $\endgroup$ – MsSnowy Sep 16 '12 at 1:15

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