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I want to get confidence intervals for the prevalence of a certain condition. I think it is correct to model the phenomenon with a Binomial distribution

Surfing CrossValidated I see a bunch of methods for extracting such intervals but no one cited exact binomial or exact Poisson as alternatives. Why is that? Those are not correct models for modeling prevalence?

Second questions: the two methods on R give slightly different results, eg:

binom.test(142, 742) # 0.1636684 0.2215588
poisson.test(142, T = 742) # 0.1611933 0.2255661

Why is that and what should I prefer for measures like prevalence (and incidence too

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Exact confidence intervals are acceptable ways of modeling prevalence or rates. In large samples, they approximate interval estimates based on normal approximations to the mean. In small samples, they provide better coverage than normal approximations. To find a discussion on the adequacy of these methods, I recommend "Modern Epidemiology" by Kenneth Rothman.

R gives different results to the binomial and the poisson confidence intervals because you are using a different probability model for the prevalence. In the Poisson case, the prevalence can theoretically take values greater than 100%. This may in fact be a good thing because it tends to estimate a longer right tail in prevalences than are less than 50%. It is perhaps more important to emphasize how similar they are, given the large sample of 142 cases out of 742 observations.

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  • $\begingroup$ Thanks for the answer! Why do you think it's a good thing that Poisson intervals have a larger upper interval? I understand the advantage in sensibility, but isn't it better to use a binomial distribution anyway because as you said it's a better model for 0 - 1 bound estimates? $\endgroup$ – Bakaburg Apr 5 '18 at 14:33
  • $\begingroup$ @Bakaburg I like the Poisson interval because of its connection with GLMs. The binomial prevalence is a logistic regression with intercept only. The Poisson prevalence is a loglinear model with intercept only. If you add an exposure as a covariate, the logistic model summarizes the association with an odds ratio. The loglinear model summarizes the association with a prevalence ratio. The latter is more interpretable. I use a robust variance estimate to get correct 95% CIs. Also see this question: stats.stackexchange.com/q/18595/8013 $\endgroup$ – AdamO Apr 5 '18 at 14:49
  • $\begingroup$ Thanks for the link, It's a problem I also investigated a lot, so good to have new infos! Actually for curiosity I also tried to estimate the proportion via logistic regression: glm(cbind(142, 742 - 142) ~ 1, binomial()) %>% confint.default() %>% exp() but it gave results shifted by few perc. points (0.1971068 0.2841663), with more variance e biased up. Any ideas why (didn't want to overload my question) $\endgroup$ – Bakaburg Apr 5 '18 at 15:08
  • $\begingroup$ NB: I used Wald confidence intervals; profile CIs are 0.1963998 0.2832286, so very similar. $\endgroup$ – Bakaburg Apr 5 '18 at 15:10
  • $\begingroup$ @Bakaburg First off, you'd transform binom CIs with plogis and not exp, exp just gives you an odds but you want the probability. There's still some differences. Looking at the code for binom.test, it seems the CI is calculated from the Bayes estimator. $F_1^{-1}(0.025)$ and $F_2^{-1}(0.975)$ where the $F_1$ and $F_2$ have a Beta distribution with parameters $\alpha=x+1, \beta=n-x$ $\alpha=x, \beta=n-x+1$. I think this is the Agresti method. Also covered in Rothman. See body(binom.test)[11:12]. $\endgroup$ – AdamO Apr 5 '18 at 15:32

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