# Significance of ratios between study populations

I have 2 published studies I need to compare. Both use the same 2 treatments and measure the same 6 outcomes; however, population size is different between the studies. The hypothesis I'd like to test is: "The outcome ratio between treatments is the same across studies." i.e., I want to test relationships between treatments between studies.

For example, say we measure outcome 1 (O1) for treatment 1 (T1) and treatment 2 (T2) where the numerator is the affected population and the denominator is the total population.

O1:T1 = 11/121 and O1:T2 = 4/117

For study two, with the same outcome variable and treatments:

O1:T1 = 22/240 and O1:T2 = 9/230

Intuitively it seems like study two doubled the affected population when the total population is doubled. We might then expect a p-value to be very low because the studies reflect the same reality. On the other hand, if study two had '1' as the numerator on both treatments then there should be a very high p-value because the study results are drastically different from the first study.

I've attempted to find a stats test that can test this hypothesis, but haven't been able to find an appropriate one. Any help is greatly appreciated. Please let me know if any clarification is needed.

I am not sure 100% what you want, but I guess you want to see wheter there is any heterogeneity in the effect estimates obtained from the two studies. This can be analyzed in a straightforward fashion with the meta package in R, as follows.

install.packages("meta")

library(meta)

studlab <- c(1, 2)

event.e <- c(11, 22)

n.e <- c(121, 240)

event.c <- c(4, 9)

n.c <- c(117, 230)

mydata <- data.frame(cbind(studlab, event.e, n.e, event.c, n.c))

mydata

meta.1 <- metabin(event.e, n.e, event.c, n.c, data = mydata, sm="OR")

summary(meta.1)

forest(meta.1)

meta.2 <- metabin(event.e, n.e, event.c, n.c, data = mydata, sm="RR")

summary(meta.2)

forest(meta.2)

meta.3 <- metabin(event.e, n.e, event.c, n.c, data = mydata, sm="RD")

summary(meta.3)

forest(meta.3)


You end up, irrespective of the use of odds ratios (OR), risk ratios (RR), or risk differences (RR), with very similar results for effect as well as for heterogeneity, with both fixed and random effects.

So my conclusion would be that there is no evidence of significant heterogeneity in effect between the two studies. Note though that you are way underpowered for this aim.