I'm somewhat new to using logistic regression, and a bit confused by a discrepancy between my interpretations of the following values which I thought would be the same:
- exponentiated beta values
- predicted probability of the outcome using beta values.
Here is a simplified version of the model I am using, where undernutrition and insurance are both binary, and wealth is continuous:
Under.Nutrition ~ insurance + wealth
My (actual) model returns an exponentiated beta value of .8 for insurance, which I would interpret as:
"The probability of being undernourished for an insured individual is .8 times the probability of being undernourished for an uninsured individual."
However, when I calculate the difference in probabilities for individuals by putting in values of 0 and 1 into the insurance variable and the mean value for wealth, the difference in undernutrition is only .04. That is calculated as follows:
Probability Undernourished = exp(β0 + β1*Insurance + β2*Wealth) / (1+exp(β0 + β1*Insurance + β2*wealth))
I would really appreciate it if someone could explain why these values are different, and what a better interpretation (particularly for the second value) might be.
Further Clarification Edits
As I understand it, the probability of being under-nourished for an uninsured person (where B1 corresponds to insurance) is:
Prob(Unins) = exp(β0 + β1*0 + β2*Wealth) / (1+exp(β0 + β1*0+ β2*wealth))
While the Probability of being under-nourished for an insured person is:
Prob(Ins)= exp(β0 + β1*1 + β2*Wealth) / (1+exp(β0 + β1*1+ β2*wealth))
The odds of being undernourished for an uninsured person compared to an insured person is:
Is there a way to translate between these values (mathematically)? I'm still a bit confused by this equation (where I should probably be a different value on the RHS):
Prob(Ins) - Prob(Unins) != exp(B)
In layman's terms, the question is why doesn't insuring an individual change their probability of being under-nourished as much as the odds ratio indicates it does? In my data, Prob(Ins) - Prob(Unins) = .04, where the exponentiated beta value is .8 (so why is the difference not .2?)