Confidence interval on binomial effect size

Question

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

binom.test(9,12,p=1/3)

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.

Answer 1

The relative risk is $\text{RR}=p/p_0$ where $p_0=1/3$ in your example. The binomial 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"))
 mod

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

Coefficients:
(Intercept)  
     0.8109  

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

exp(confint(mod))
Waiting for profiling to be done...
   2.5 %   97.5 % 
1.416876 2.784264 

Answer 2

If $\theta$ is the true probability value and $\theta_0$ is your comparative value then $RR = \theta/\theta_0$ is the corresponding true relative risk (compared to your assumed comparative value). If you generate a confidence interval for $\theta$ you can easily get a corresponding confidence interval for $RR$ by applying this transformation. I recommend you use the Wilson score interval for estimating the binomial proportion (see O'Neill (2021) for details).

#Compute the 95% Wilson score interval
x     <- 9;
n     <- 12;
alpha <- 0.05
CONF  <- stat.extend::CONF.prop(alpha, 
             sample.prop = x/n, n = n);

#Alter the interval to refer to the relative risk
prob0   <- 1/3;
CONF_RR <- CONF/prob0;
attributes(CONF_RR) <- attributes(CONF)
attributes(CONF_RR)$parameter <- 
"relative risk parameter for infinite 
 population";

#View the confidence interval
CONF_RR;

        Confidence Interval (CI) 
 
95.00% CI for relative risk parameter for 
infinite population 
Interval uses 12 binary data points with sample 
proportion = 0.7500 

[1.4030839951993, 2.73317499481784]