Should I be controlling for all independent variables in my logistic regression model?

Question

I'm working on a project in R where I'm looking at California's census tract-level demographic data in an explanatory logistic regression model. I have 6 demographic variables of interest: percent below 150% poverty line, percent minority, percent unemployed, percent disabled, percent with no high school diploma, and percent without health insurance. I also want to control for population density since it varies so much amongst census tracts. My binary exposure variable is if the census tract contains a large animal farming operation (1=yes, 0=no). Here's an example of the model I coded in R:

cali_logit <- glm(exposure ~ percent_unemployed + percent_minority + 
    percent_no_diploma + percent_uninsured + percent_under150 + 
    percent_disabled + pop_density, family = "binomial", 
    data = cali_cafos)

I checked all variables' variance inflation factors and all are under 5, so multicollinearity is not a problem. From my (long ago) stats classes, I know that when we add all of our data into a model we are adjusting the model to control for potential confounding effects. However, do I need to "control" for census tract level data? It's not quite clicking with me how percentages of other categories are confounders or need to be adjusted for in the model. If I want to see the odds of being in an exposed tract given a 1% increase in x,y,z demographic of interest, and only control for population density, should I just do an individual glm for each variable with only pop_density included? What would be the reason for adding all variables into a logistic regression model with census tract data?

Answer 1

Vanilla OLS linear regression can suffer from coefficient bias when predictors are omitted that are correlated with included predictors. Logistic regression is worse, because omitting a predictor that is totally independent of the included predictors can still lead to biased coefficients (okay, yes, the usual estimation is biased, but the bias is even worse). Consequently, including relevant predictors can help you get better point estimates of your coefficients of interest.

Further, better performance can help tighten up confidence intervals on your coefficients.

If you have enough data to support a model with many relevant variables, including all of them can have some serious advantages.