Computing confidence intervals for prevalence for several types of infection

I have a dataset about a population in hospital, and what type of infections patients have.

Let say the number of patients is 100, 10 of them have pneumonia (group A), 20 of them have urinary tract infection (group B); keep in mind that group A and B can be overlapping, that is a patient suffering from pneumonia can also have urinary tract infection.

I need to estimate the prevalence of different infection type in this population (i.e., prevalence of pneumonia, prevalence of urinary tract infection). I'm not sure whether assuming binomial distribution, like the one below from here, is appropriate:

$$\text{SE}=\sqrt{\frac{p\times (1-p)}{n}}\times\sqrt{1-f}$$

Using this formula, I will compute multiple "binomial" estimates (i.e., one for each type of infection). I would feel comfortable to use this if only I need to describe prevalence of one type of infection, but in this case, I need to describe several ones from the same population. I am not sure if using the formula is appropriate or not in this case. Can anyone here enlighten me? Thanks!

• One question is, what exactly is it you are trying to estimate. An estimate usually implies that you take a uniform subsample of the population, measure the frequency in that subsample and try to guess how different the result is from the whole population. This doesn't seem to be the case here. If you want to know the prevalence among hospital patients, then you don't need to sample. Just take the entire population of the hospital and compute the exact result. If you want to know prevalence among general population, then hospital patients are certainly not a uniform sample of it. – SheldonCooper Mar 3 '11 at 19:46
• And the second question is, can you assume the infections are independent. I.e. for someone who has pneumonia, are they just as likely to get urinary infection as anyone else, or are they more likely to get it (maybe because they are already sick and it's easier for them to succumb to another infection)? – SheldonCooper Mar 3 '11 at 19:48

So you have a population each of whom can have zero or more conditions. To answer the question: How many hospital patients have A? It seems to me that the best you can do is take your favourite proportion estimator and offer it up with your favourite confidence interval. There are lots of choices, which will make a difference for very high or very low proportions. If you have such a situation, the estimator above may not be optimal.

If you are interested in the population of just your hospital then you can, as SheldonCooper points out, dispense with the statistics altogether. I suspect however that you are interested in hospital patients more generally, so your standard errors and intervals might be interpreted relative to this population. In your suggested estimator the identity of the population will determine what 1-F is. Certainly hospital patients don't look like non-hospital patients with respect to the conditions you're counting, but that need not matter.

Following Sheldon's second observation, it is probable that the conditions correlate. But as far as I can see this is only useful information if you are asking conditional questions, e.g. the prevalence of A among B sufferers. In probabilistic terms your question is about estimating marginals, and correlation information only tells you about estimating conditionals.

If you were interested in these sorts of subgroups, you'd certainly want to model this information. You'd also want it if there were differential measurement errors or sample selection issues, etc. e.g. only getting tested for A if you have a B diagnosis... That might also make certain sample marginals problematic as estimates of population marginals. Thankfully, I don't know much about hospital populations, but I'd be willing to bet that there are some of these issues around.

Finally, about reporting: If you in fact want to report confidence regions rather than condition-wise intervals, then again the correlation structure matters, and things get considerably trickier. I seem to remember that Agresti had a paper on simultaneous confidence intervals for multivariate Binomial proportions, which might be helpful for this approach.

• Nice job! I'd like to ask what may be a naive question. I've seen and used the formula for the standard error of a proportion umpteen times, but I've never encountered the sqrt{1-f} portion of it. Under what conditions is this needed? – rolando2 Mar 6 '11 at 2:36
• Roughly speaking it indicates that the inference will be based on randomisation (over a finite but large population) rather than on a model (of a theoretically infinite population). You'll see it a lot in survey research. Notice that as the population converges on the sample the SE shrinks to 0, which is SheldonCooper's first case. – conjugateprior Mar 6 '11 at 10:45

A few thoughts:

1. As people have mentioned, if you have the entire hospital population worth of data, and all the questions you have are restricted to that hospital, you can dispense with a confidence interval entirely. However, assuming that's not the case, and you either have a subsample of the hospital, or want to talk about the hospital as a sample of the general population...

• You can probably ignore the dependency between infections. They're unlikely to be perfectly uncorrelated with each other, but the harder you look at the correlation between infections and the various and sundry other violations of independent happenings (basically, the risk for you is independent of my disease state), the more simple statistics in ID begin to break down. For something like this, you're probably okay.

• I'm pretty sure you can use the formula as stated. You're not pooling the results together, and as far as you've said, you're not going to be making any sort of comparisons between groups. If we're assuming they're independent, that's no less valid than independently estimating the prevalence any two other unrelated things in the same population. This isn't true if you want to start talking about joint prevalences or the like, but you seem to just want a table of Condition Prev (95% CI).