Interpret odds ratios log variable

[Marius Hez]

Dear all,

I'm running a similar code as this one:

sysuse auto.dta,clear
gen domestic=1-foreign
gen lweight=ln(weight)
logit domestic lweight,or

I need to interpret the resulting odd ratio of 818.6201 to answer about the relationship on the weight of cars and their probability to be domestic ones.

I see that increased weights of cars largely increases their probability to be domestic

But what I don't see precisely is to how much increase in weights, I have a 818 times higher proba for the car to be domestic.

Can anyone help me with that?
Ps: I do need the Odds and not the coeffs for other reason, and I do need the logs too

[Clyde Schechter]

You are confusing odds with probabilities.

If we have two cars whose weights differ by 1%, that is, one of them weighs 1.01 times the other, then their log-weights differ by log(1.01) = 0.01 (to two decimal places). The model then says that the log-odds of being domestic will differ by 818.6201*0.01 = 8.186201. Which implies that the odds ratio is exp(8.186201) = 3591.05 (again, to two decimal places). That is, the heavier car's odds of being domestic are 3591.05 times higher than the lighter car's odds of being domestic.

Now that may sound astronomical. And, in one sense, it is. But translating from odds to probabilities, this can be entirely reasonable. The probability-odds relationship is highly nonlinear. In the auto.dta, the overall probability of a car's being domestic is close to 0.7. So if you were starting from a car with this probability of being domestic, its odds of being domestic is 2.333333333. When you multiply that by 3591.05, the odds of the 1% heavier car's being domestic is 8379.1167, which translates back to a probability of 0.99988. This, to most people, feels a lot easier to grasp.

If you must stay in the odds metric, then it takes some getting used to, and bearing in mind that astronomical odds ratios can correspond to modest, or even imperceptible changes in probabilities, particularly at the extremes of probability.

If you are presenting these results to audiences not comfortable with the odds metric, then you might have more success working in terms of probabilities. For that purpose, the -margins- command is your friend:

sysuse auto.dta,clear
gen domestic=1-foreign
gen lweight=ln(weight)
logit domestic lweight,or

summ lweight
margins, at(lweight = (`r(min)' `r(mean)' `r(max)'))

The -margins- output gives you the model's predicted probabilities of being domestic for cars at the minimum, mean, and average values of log weight in the data. (These correspond to the minimum, geometric mean, and maximum raw weights.)