# How does the presence of factors affect the interpretation of the other coefficients in a regression?

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?

Yes, that is exactly what it means. If you want to investigate this interaction for the other years, you would fit the 3-way interaction colour:age:year.