My main goal is to calculate the sampling variance. But I will start with the standard deviation. I

Deshawn Cabrera

Deshawn Cabrera

Answered question

2022-05-08

My main goal is to calculate the sampling variance. But I will start with the standard deviation. I am doing a meta-analysis and need to calculate the variance for EACH effect size (prevalence in this case) in each study.
For example, I have 10 positive cases out of 1000 people that were tested. This gives a prevalence of 0.01 (or 1%). How do I find the standard deviation from this information?

Answer & Explanation

jcholewa39v8f

jcholewa39v8f

Beginner2022-05-09Added 13 answers

This is an exercise in Bayesian parameter estimation. Your prevalence estimate is of the parameter p in a Bernoulli distribution, with a random person scoring 1 if they're a case and 0 otherwise. If k people in a sample of n are cases, p's PDF on [0,1] is adjusted by Bayes's theorem from the prior 1 to the posterior p k ( 1 p ) n k B ( k + 1 , n k + 1 ) . So the mean of p is
0 1 p k + ( 1 p ) n k d p B ( k + 1 , n k + 1 ) = B ( k + + 1 , n k + 1 ) B ( k + 1 , n k + 1 ) = ( k + ) ! ( n + 1 ) ! k ! ( n + + 1 ) ! .In particular,
E p = k + 1 n + 2 , E p 2 = ( k + 1 ) ( k + 2 ) ( n + 2 ) ( n + 3 )
and
σ p 2 = ( k + 1 ) ( k + 2 ) ( n + 2 ) ( n + 3 ) ( k + 1 ) 2 ( n + 2 ) 2 = ( k + 1 ) ( n k + 1 ) ( n + 2 ) 2 ( n + 3 )
(you'll want to double-check those calculations).

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