# Can you add up different genes' odds ratios to get a general odds ratio?

Let's say you have three risk SNPs from different genes that have been associated with a particular disease in different studies, but, as far as I know, there is no study including all. ORs have been calculated for each risk SNP

For example.

Gene A OR = 0.8

Gene B OR = 1.2

Gene C OR = 5.8

Is it possible to, in a particular way, add the OR or compute an OR for a particular "haplotype"?

For example, if patient A has risk SNP A, and B, but not C, his OR for the disease is...?

You would have to assume SNPs A, B, and C are independent which is simply not advisable. Furthermore, an OR is not a measure of risk, so you would need to know the baseline risk of disease in a healthy individual not having mutations or expression or whatever your measure is on the A, B, and C SNPs.

If you are using a generalized linear model to obtain odds ratio estimates, assuming that there are no interactions between the genes, then you can simply multiply the odds ratios for the two present genes to get the OR for disease.

From your example, assuming the presence of A does not affect how much B contributes to the risk of disease and vice versa, then the OR is 0.8*1.2=0.96.

If the presence of one gene affects how another gene contributes to the disease (ie. there are interactions) then you would also need to multiply the ORs of the interactions.

Edit: Just noticed that these ORs are obtained from 3 different studies. In this case, no it would not be appropriate to combine the ORs in the way I described above. This is because there is no way for you to find out if there are interactions or not.

• Regarding your edit, wouldn't a better reason be that the three OR's were not all in the same model and thus adjusted for the presence of the others? – Hank Lin Feb 28 '19 at 0:23

As both AdamO and Sheep mention, the events would have to be independent.

If the events were independent, one cannot simply take the sum/product of odds ratios to find the odds ratio of both events combined. However, one can take the product of probabilities of independent events to calculate the probability of the combined event.

$$P(A \bigcap B) = P(A) * P(B)$$ if and only if A and B are independent.

Since odds ratio, O(), is defined as the probability of the event occurring divided by the probability of the event not occurring,

$$O(A) = \frac{P(A)}{1-P(A)}$$

and

$$P(A) = \frac {O(A)}{O(A) + 1 }$$

With these formulas we can find P(A and B) using the odds ratios of A and B.

$$P(A \bigcap B) = \frac{ O(A)}{O(A) + 1} * \frac{ O(B)}{O(B) + 1}$$

then we can translate that back to the odds ratio of both events,

$$O(A \bigcap B) = \frac{\frac{ O(A)}{O(A) + 1} * \frac{ O(B)}{O(B) + 1}}{1-[\frac{ O(A)}{O(A) + 1} * \frac{ O(B)}{O(B) + 1}]}$$

and simplify by

$$O(A \bigcap B) = \frac{O(A)*O(B)}{[O(A)+1]*[O(B)+1]-O(A)*O(B)}$$

Not as straightforward as a simple sum or product, but doable.

• Welcome to CV! Thank you for a clear explanation and attention to the assumptions needed to justify it. – whuber Jan 15 at 22:14