Suppose I have a binomial test, with 9 successes in 12 trials, expected p=1/3. In other words, in R you might do:


Now, I'm under the impression that relative risk is an appropriate effect size measure. In this case, RR=0.75/(1/3)=2.25.

First - is this actually an appropriate effect size measure? Second - how would I put a confidence interval on this? The resources I have found for calculating RR confidence intervals (e.g., http://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_confidence_intervals/bs704_confidence_intervals8.html) assume that the p=1/3 came from a sample with a known n, rather than a theoretical probability.


The relative risk is $\text{RR}=p/p_0$ where $p_0=1/3$ in your example. The binomia count $X$ have distribution $\mathcal{Bin}(n,p)$ which we can write as $\mathcal{Bin}(n,p_0\cdot\text{RR})$. Then we can use a binomial regression with log link function, see Calculation of Relative Risk Confidence Interval. That gives an estimate of the RR, and we can calculate a CI by likelihood profiling. An example with your data in R:

n <- 12
x <- 9
p0 <- 1/3

mod <- glm(cbind(x, n-x) ~ offset(log(p0)), family=binomial(link="log"))

Call:  glm(formula = cbind(x, n - x) ~ offset(log(p0)), family = binomial(link = "log"))


Degrees of Freedom: 0 Total (i.e. Null);  0 Residual
Null Deviance:      -6.661e-16 
Residual Deviance: -6.661e-16   AIC: 4.709

Waiting for profiling to be done...
   2.5 %   97.5 % 
1.416876 2.784264 

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