adjustment of covariates in linear model

Question

I am trying to understand the adjustment of covariates in the linear model such as multiple logistic regression. How does adding a covariate adjusts the coefficients for that covariate (any intuitive mathematical explanation)? Thanks in advance.

Answer 1

Imagine that you want to compare the mortality rates for 2013 between two cities: say, Cross-Validopolis and Alexiville. You could just compare the crude mortality rates (# dead in 2013/# at risk of dying in 2013) between the two cities, BUT Cross-Validopolis has relatively more aged folks that Alexisville.

Being a savvy researcher, you know that being aged puts one at a higher risk of dying. So how would you be able to say that any difference in the the mortality rates between these two cities is due to an actual difference in the force of mortality irrespective of age, or is simply a function of there being old folks in Cross-Validopolis?

You can adjust for the different distributions of age by standardizing the distribution of age (i.e. create a hypothetical new population by adding together all the people in each age bracket from both cities), and then calculating how many people would have died in the standardized population, if they died at the same rates by age as they died in Cross-Validopolis to get an age-adjusted mortality rate, and do the same thing to get an age-adjusted mortality rate for Alexisville. These age adjusted mortality rates can be compared directly to understand differences in the force of mortality with age held constant.

Example:

       C-Vopolis  C-Vopolis  Alexisville   Alexisville  Standardized
        #deaths    #people     #Deaths       #people       #people
young     5000      100000      3100          60000        160000
 aged    20000      200000      5700          60000        260000

Crude Mortality Rates:
C-Vopolis: 25000/300000 = 0.083
Alexisville: 1100/120000 = 0.073

Age adjusted rates:
Standardized population:
number of young people = 100000 + 60000 = 160000
number of old people = 200000 + 60000 = 260000 number of people total = 420000

C-Vopolis' rates: young mortality rate: 5000/100000 = 0.05
aged mortality rate: 20000/200000 = 0.10
age-adjusted mortality rate: (0.05*160000 + 0.10*260000)/420000 = 0.081

Alexisville's rates: young mortality rate: 3100/60000 = 0.052
aged mortality rate: 5700/60000 = 0.95
age-adjusted mortality rate: (0.052*160000 + 0.095*260000)/420000 = 0.079

So the force of mortality irrespective of age is very close in these cities (0.081 versus 0.079), with most of the difference in crude morality rates (0.083 versus 0.073) being due to the larger proportion of aged in C-Vopolis.

This is a simple example, simpler than a multiple regression. However, the same principle applies: what is the effect of some $x$ on $y$ while holding the values of the other variables constant?