# Interpretation of Simple Logistic Regression with Categorical Variables

I'm currently trying to interpret simple logistic regression with a categorical variable.

Description of variables: "region" = the beneficiary’s residential area in the US; a factor with levels northeast, southeast, southwest, northwest. "charges_cat"= which takes the value 0 (low) when charges are less than 10000 dollars and the value 1 (high) in all other cases.

> logm2<-glm(charges_cat~region, family=binomial)

Coefficients:
Estimate
(Intercept)      0.01235
regionnorthwest -0.17888
regionsoutheast -0.13337
regionsouthwest -0.25351


My interpretation for b1 = regionnorthwest is:

> exp(-0.17888)
[1] 0.8362062


If the beneficiary is living in the northwest region of the US, the odds of charges being more than 10000 dollars is 16.38% lower than the odds of charges being more than 10000 dollars for a beneficiary who lives in the northeast region of the US, with southeast and southwest regions are fixed.

My question is; in a single logistic regression should I state the factor levels of the "region" are fixed, such as "with southeast and southwest regions are fixed." or there is no need to state the dummy variables of the "region" fixed?

The key here is that in a saturated factor model like what you have above, "with southeast and southwest regions are fixed" is a superfluous. If you are in the northwest region, you are by definition not in the southeast or southwest region.

I find that it helps clarify my thinking to recognize that in this setting where you are just regressing on a single categorical variable, the model you fit is equivalent to (after some transformations of the coefficients) just reporting the values of $$P(outcome | category)$$, one for each value of the categorical variable. To be a bit more concrete about what I mean by this, I can read from the model coefficients you reported above that:

$$P(charges\_cat|northeast) = \exp(0.01235) / (1 + \exp(0.01235)) = 0.5031$$ $$P(charges\_cat|northwest) = \exp(0.01235-0.17888) / (1 + \exp(0.01235-0.17888)) = 0.4585$$ $$P(charges\_cat|southeast) = \exp(0.01235-0.13337) / (1 + \exp(0.01235-0.13337)) = 0.4698$$ $$P(charges\_cat|southwest) = \exp(0.01235-0.25351) / (1 + \exp(0.01235-0.25351 )) = 0.4400$$