# Clustered binary data

I have a binary outcome (success \ failure). I have 20 subjects, half of them gave 2 samples, the others just one, so I have 30 data points.

All data points without any exceptions were success (1). So my point estimate of the success rate is clearly 100%. Now I want to calculate a CI for the success rate, mainly to see what the lower limit is, I want to be able to say that with 95% confidence the success rate is higher than....(80%, 85%, whatever comes up).

The problem is, as you can see, is the clustering, I can't use n = 30 because I then ignore the correlation, I feel it is a waste to use n = 20. Is there a way to calculate the variance of a single sample proportion while taking into account the cluster ?

Thanks !

• As I understand, there is no variability at all in your sample. How then do you plan to compute a confidence interval? Jan 11 '14 at 9:47
• Well, you can calculate it, but it's from 100% to 100% if your data are completely invariant. $SD=0\rightarrow SE=\frac{0}{\sqrt{N}}$. Multiply that by 1.96 and add it to 100% or subtract it if you want, but it's still going to be 100%. Jan 11 '14 at 10:16
• If it weren't for the clustering, there is a solution for the CI; Agresti & Coull suggest adding two successes and two failures to the data and basing CI on that as a simple solution. Even though in this sample there were no failures, there clearly could be failures in the population. But I don't know how to deal with the sampling. On the other hand, the variance of this sample is 0, clustering or no. Jan 11 '14 at 11:51

For inference with clustered data you need some idea about the within-cluster correlation. Your data currently shows a perfect within-cluster correlation, so you might as well assume that the subjects will always give the same response. So I would go ahead with $n=20$ - that's the conservative approach. Of course, your data are also consistent with no within-cluster correlation and $p=1$, but that would be hard to argue without context.

There are many options for exact confidence intervals, for example:

> binom.test(20,20)

Exact binomial test

data:  20 and 20
number of successes = 20, number of trials = 20, p-value = 1.907e-06
alternative hypothesis: true probability of success is not equal to 0.5
95 percent confidence interval:
0.8315665 1.0000000
sample estimates:
probability of success
1