# Scaling the odds-ratio of a binary logistic regression

Just a quick one - to put it simply, I am conducting a study regarding age and marriage. I have found that with a binary logistic regression (dependent variable yes/no to being married), the odds-ratio regarding age is 1.01 (significant at p < 0.001). Note: age is on a scale of 18 (the youngest respondent) to 82 (the oldest).

I am pretty sure I am correct in interpreting this as 'for every one year increase in age, the likelihood of being married, holding all other controls constant, increases by 1%'. My confusion is that I had originally interpreted this as an indication of basically a statistically significant result but of low substantive value (given it is only 1%).

But I have seen that some people may scale their odds ratio - ie. suggesting that a 20 year increase in age means the likelihood of being married increases by 1.01^20 = 1.22. This would suggest a 40 year old is 22% more likely to be married than a 20 year old.

Is this correct - it seems misguided to me that you can simply scale it up like this and assume the odds-ratio holds constant across all? If this was the case, then maybe the finding has a bigger 'real life' impact than I thought?

It is correct that you can scale an OR of XXX per year to YYY per 10 years or 20 years, and the transformation used is also correct. You would probably want to use more decimal places in the 1.01, or let the computer do the scaling.

Both 1.01 per year and 1.22 per 20 years assume that the rate is constant over time. That assumption seems very unlikely to be true. Indeed, at the high end of your age range, the likelihood of being married almost certainly goes down, because one spouse (usually the man) dies and the other spouse may not get married again. But ... that's a problem with both rates. Divorce will also have a big impact.

If you want to study age and marriage you should, I think, certainly use more categories than married/not. At least: Never married, married, formerly married (which would cover divorced and widowed). And you surely want to add covariates (sex, sexual orientation, probably religion -categorized somehow -- and probably others).

• Thank you very much!
– Vito
Commented Apr 18 at 15:39

The reason you are getting confused is that your interpretation "for every one year increase in age, the likelihood of being married, holding all other controls constant, increases by 1%" is not correct.

What you described ("the probably of this thing changes by X%") is what's called a "risk ratio." What an odds ratio of 1.01 really means is that, "for every one year increase in age, the ODDS of being married, holding all other controls constant, increases by 1%. This is different because in stats the "odds" of something is not just a synonym for "probability" or "likelihood," rather it is the probability of a thing happening divided the probability of the thing not happening. So the odds ratio of 1.01 doesn't, by itself, tell you anything about how the *probability" of being married changes as you age. There is in fact no way of getting this information (which is what everyone wants) from a logit model without doing some "extra work" (like calculated average marginal effects or predicted probabilities for hypothetical cases) which basically amounts to providing information about the baseline probability.

Misunderstanding odds ratios as risk ratios is a very common mistake in statistics (the two values do coincide when the baseline probability of the event in question is zero, but that's often not the case, and probably isn't the case in regard to marriage).

That being said, you could still "scale up" an odds ratio, to tell you that "20 year increase in age means the ODDS of being married increases by 22%" but this doesn't help you very much because changes in odds are not changes in probability.

Finally, whether an effect is "substantively" big or not is not a statistical question at all, it depends on what is relevant for the research question you are trying to answer.

• Thank you for your comments! I think I understand. I am using SPSS and the so-called 'odds-ratio' I described is calculated as Exp(B). I have seen this described as the 'relative risk ratio' and used interchangeably with the odds-ratio?
– Vito
Commented Apr 18 at 15:39
• Would it perhaps be better to say - people are 1% more likely to be married with every increase in age by 1 year? I've seen this kind of phrasing a lot but starting to doubt now if maybe others have been confused as you point out?
– Vito
Commented Apr 18 at 16:24