How to do indirect estimation of two odds ratios?

[Edward Sutanto]

Hi Statalist members,

I am planning to do a meta-analysis to find out the odds of having oral cancer with binge drinking, compared to cigarette use.
Currently, many epidemiological studies, however, do not compare them directly. Instead, these studies usually compared binge drinkers vs no substance users and cigarette smokers vs no substance users. So, the odds ratios of oral cancer currently reported in studies are:
- (A) Binge drinkers vs. (B) no substance user, and
- (C) Cigarette smokers vs. (B) no substance user
Can I estimate the odds ratio of developing oral cancer of (A) binge drinkers, compared to (C) cigarette smokers?
I am not aware of any Stata commands that can do this.
Any help would be greatly appreciated. Thank you very much!

[Maarten Buis]

So you have the following:

• The odds ratio of getting cancer for drinkers (A) versus no substance (B), lets say it is 2.
• The odds ratio of getting cancer for smokers (C) versus no subtance (B), lets say it is 3.

We want the odds ratio of getting cancer for drinkers (A) versus smokers (C).

So \(\frac{O(cancer|A)}{O(cancer | B)}=2\), which we can rewrite as \(O(cancer|A) = 2 \times O(cancer|B)\)

Similarly, \(\frac{O(cancer|C)}{O(cancer | B)}=3\), which we can rewrite as \(O(cancer|C) = 3 \times O(cancer|B)\)

This means that \(\frac{O(cancer|A)}{O(cancer|C)} = \frac{2\times O(cancer|B)}{3\times O(cancer|B)}=\frac{2}{3}\)

So you just divide the odds ratios.

[Edward Sutanto]

Hi Maarten,

Thank you for your reply. If I can have a follow-up question to this, is there a way to also estimate the 95% confidence interval of the odds ratios?

Using your example:

• The odds ratio of getting cancer for drinkers (A) versus no substance (B), lets say it is 2.00 (95% CI 1.90-2.15)
• The odds ratio of getting cancer for smokers (C) versus no subtance (B), lets say it is 3.00 (95% CI 2.85-3.08)

If we can estimate that the odds ratios of developing oral cancer for drinkers, compared to smokers is 2/3 . How about the 95% CI of that odd ratios?