The answer to Interpreting coefficients of an interaction between categorical and continuous variable contains a phrase that seems to have some significant impact on how coefficients are interpreted in a multiple-regression when a factor is introduced:
If treatment contrasts for a categorical variable are present in a model, the estimation of further effects is based on the reference level of the categorical variable. [..]
note that the estimation of the coefficients is based on the references categories of the factors (if treatment contrasts are employed). In this case the effects hold for $race = white$, $sex = male$, and $educa = 1$. They do not test an overall influence of the numeric variables irrespective of the levels of the factors.
Question: How does including a factor into a multiple regression affect the interpretation of the other coefficients (whether numeric predictors or interactions)?
Consider this example from Fox 2003:
require(effects)
require(lmtest)
Arrests$year <- as.factor(Arrests$year)
arrests.mod <- glm(released ~ employed + citizen + checks
+ colour*year + colour*age,
family=binomial, data=Arrests)
Which yields:
> coeftest(arrests.mod)
z test of coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 0.3444334 0.3100749 1.1108 0.2666514
employedYes 0.7350645 0.0847701 8.6713 < 2.2e-16 ***
citizenYes 0.5859841 0.1137717 5.1505 2.598e-07 ***
checks -0.3666425 0.0260322 -14.0842 < 2.2e-16 ***
colourWhite 1.2125167 0.3497751 3.4666 0.0005272 ***
year1998 -0.4311794 0.2603589 -1.6561 0.0977023 .
year1999 -0.0944343 0.2615447 -0.3611 0.7180519
year2000 -0.0108975 0.2592073 -0.0420 0.9664655
year2001 0.2430630 0.2630151 0.9241 0.3554129
year2002 0.2129549 0.3532786 0.6028 0.5466444
age 0.0287279 0.0086191 3.3330 0.0008590 ***
colourWhite:year1998 0.6519565 0.3134898 2.0797 0.0375555 *
colourWhite:year1999 0.1559504 0.3070430 0.5079 0.6115161
colourWhite:year2000 0.2957537 0.3062034 0.9659 0.3341076
colourWhite:year2001 -0.3805413 0.3040538 -1.2516 0.2107305
colourWhite:year2002 -0.6173178 0.4192551 -1.4724 0.1409086
colourWhite:age -0.0373729 0.0102003 -3.6639 0.0002484 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Given that we have employed={no,yes} and citizen={no,yes} and the factor year={1997,..,2002} in the model...
Does this imply that the coefficient colourWhite:age = -0.0373729 is strictly limited to describing only the interaction between colour and age for people who are unemployed, non-citizen and arrested in 1997?
