Clarification regarding odds ratios in logistic regression

Question

I am trying to understand odds ratios (OR) in logistic regression. So far what I read says that "OR is a measure of association between exposure and outcome".

I got a kid dental dataset in which the independent variable is disability status, i.e., whether a child has a disability or not (1 for disability and 2 for a normal kid), and the dependent variable is whether a kid gets dental attendance on time or not (1 for on-time dental attendance and 0 for late dental attendance).

If I am not mistaken, in the above case, there are two exposure variables: disabled and normal kid, and the outcome is on-time dental attendance or not.

If I am right, my question is: Do I need to calculate the OR for each type of exposure variable (i.e. one OR for disabled and one for normal) or is it only one OR value for both values of the exposure variable? If it is one single value, then how could we interpret OR=1.5? Could we say that there is 50% more chance that disabled kids do not (or do) get dental attendance on time? Could anyone help me in understanding the concept?

Answer 1

Suppose your data looked like

             on-time   late

Disabled       22        5

Other         125       40

Then the odds for disabled children to be treated on time is $\frac{22}{5} =4.4$ and their odds for being treated late is the reciprocal of that, about $0.2273$

While the odds for other children to be treated on time is $\frac{125}{40} =3.125$ and their odds for being treated late is the reciprocal of that, $0.32$

So the odds ratio for disabled children to be treated on time compared to other children is $\frac{4.4}{3.125} = 1.408$.

This is also the odds ratio for other children to be treated late compared to disabled children. The odds ratio for disabled children to be treated late compared to other children and the odds ratio for other children to be treated on time compared to disabled children are each the reciprocal of that, about $0.7102$.

None of these are how much "more chance that disabled kids do not (or do) get dental attendance on time", as that is a comparison of probabilities rather than odds. The proportion of times disabled children are treated on time ($0.8148$) is about $1.0756$ times the proportion of times other children are treated on time ($0.7639$); the proportion of times disabled children are treated late ($0.1852$) is about $0.7639$ times the proportion of times other children are treated on time ($0.2424$).

While it is easy to move between odds and probabilities, translating from odds ratios to probability ratios needs information about prevalence.