I am familiar with meta analysis and meta regression techniques (using the R package
metafor from Viechtbauer), but I recently stumbled on a problem I can't easily solve. Say we have a disease that can go from mother to the unborn child, and it has been studied already a number of times. Mother and child were tested for the virus right after birth. As an unborn child can impossibly get the virus other than from the mother, one would expect crosstabulations like :
| neg kid | pos kid mother neg | A | C=0 -----------|---------|-------- mother pos | B | D
Obviously using odds ratios (OR) gives errors as one would be dividing by 0. Same for relative risks :
Now the researchers want to test the (senseless) hypothesis whether infection of the child is related to the infection of the mother (which seems very, very obvious). I'm trying to reformulate the hypothesis and come up with something that makes sense, but I can't really find something.
To complicate things, some kids with negative moms actually are positive, probably due to infection in the first week. So I only have a number of studies where C = 0.
Anybody an idea on how to statistically summarize the data of different studies following such a pattern. Links to scientific papers are also more than welcome.
Seems to me this is one of the rare situations where it might well be better to meta-analyse risk differences rather than risk ratios or odds ratios. The risk difference $P(Kid_+ | Mum_+) – P(Kid_+|Mum_-)$ is estimated in each study by $D/(B+D) – C/(A+C)$. That should be finite in all studies even when $C=0$, so there should be no problem meta-analysing it.
I agree it seems pretty pointless to consider testing the hypothesis that this risk difference is zero. But it's meaningful to estimate how large it is, i.e. how much more likely a kid is to have the virus when their mum has it than when their mums doesn't.
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