# How does the interpretation of a binomial-logit GLM change when an offset term is included?

My GLM is as follows:

logit.final <- glm(Claim_Occurrence ~ Sum.Insured100kto200k + Sum.Insured200kto300k +
Sum.Insured30kto50k   + Sum.Insured50kto100k +
Sum.Insured300Kplus,
family = binomial(link = "logit"), offset = Exposure.Years.Earned)


I am trying to predict whether a claim will be reported in a vehicle or not, based on sum insured. The base level of the Sum.Insured categorical variable is Sum.Insured0to30K. Exposure years is the offset term, which is between 0 and 1. For example, a 0.5 would mean 6 months and 1 would mean a year.

If the fitted intercept is -2.64997, does this mean the odds of a claim occurring in a vehicle with sum insured 0 to 30K is 7.07% (i.e., $$\exp -2.64997)$$)? Would the offset term have any influence on this odds / interpretation?

EDIT:

I read somewhere that the coefficient of an offset is 1. So to incorporate the offset in my interpretation, would the odds be $$\exp(-2.64997 + 1) = 19\%$$?

EDIT 2:

Okay, as per advise in the answer, I have removed Exposure Years Earned from offset term, and included it as a predictor.

My revised glm model is now as follows:

logit.final <- glm(Claim_Occurrence ~ Sum.Insured100kto200k + Sum.Insured200kto300k + Sum.Insured30kto50k + Sum.Insured50kto100k + Sum.Insured300Kplus + Exposure.Years.Earned, family = binomial(link = "logit"))


My intercept is now -3.6464, and coeff estimate of Exposure years earned is 2.0046.

So if I want to find probability of claim occurrence of a vehicle with sum insured 0 to 30K, and exposure years earned worth of 1.083, would it be Exp(-3.6464) x Exp(2.0046) x 1.083 = 20.98% ?

• The answer from @Gung notes that an offset probably isn't helpful here. In your formula at the end of the question, you would have to multiply the fixed coefficient of 1 by the number of years involved for a case. An offset for exposure-years would thus mean you are assuming that the log-odds of a claim increases by a value of exactly 1 per year. That doesn't seem to make sense. Also, you might want to consider a Poisson model (with an offset) instead, as your binomial model doesn't distinguish between having 1 claim and having 10 claims. The number of claims seems to be worth modeling.
– EdM
Commented Sep 16, 2020 at 20:56
• First, it's always safest to calculate the linear predictor (Intercept + coefficient * predictorValue) before you do any exponentials. The value I get for the linear predictor is then -1.476, which exponentiated is 0.228. Your value of 0.2098 came from exponentiating the coefficient for exposure-years before multiplying by the number of exposure-years. Second, the exponentiated result is the estimated odds of the outcome, not the probability.
– EdM
Commented Sep 20, 2020 at 16:15

An offset is just a variable whose coefficient in the fitted model is forced to be exactly $$1$$. You can use an offset any time there is sufficient justification for that. You can also use it to fix a coefficient at a different level by multiplying the variable by the desired value and then forcing it to have a subsequent coefficient of $$1$$. In general, this should only be done when there is a really strong justification, though.
To answer your stated question, the intercept still means the same thing. It is the log odds of a claim when all other variables are exactly equal to 0. Thus, when the sum insured is 0 to 30K and exposure years is exactly 0, the odds of a claim is $$\exp(−2.64997) = 7.07\%$$.