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I'm working with summarized public health data, specifically mortality rates. I have the numerators and denominators for each mortality, but that's it. I'm stuck on what is the best method to calculate 95% confidence intervals for this dataset.

The numerator are very small relative to the denominators, and the rates are expressed per 100,000 (mortality rates range from roughly 150 to 1 per 100,000).

Thanks in advance.

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    $\begingroup$ What would you be using to calculate your confidence intervals? You can't just take a rate and use it to calculate a confidence interval unless it comes from sample data. The rates you're describing sound like they're not actually counts, but values from a period that are then scaled to an understandable numerical value. $\endgroup$
    – dankernler
    Commented May 5, 2018 at 2:21

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Okay so this is tricky and from what I understand, I see two scenario's

  1. If you have multiple mortality rates and wish to calculate confidence interval on the average of these rates(which i am not sure about).

$$CI = \bar X \pm Z_{\alpha/2} \frac{\sigma}{\sqrt n}$$

for large samples, replace $Z_\alpha$ with $t_\alpha$ for smaller models. Again $\bar X$ and $\hat \sigma$ can be calculated if you have multiple data points.

$\bar X$ = Sum of the proportions / $n$ = count total

$\sigma$ = standard deviation

Again this is if you have multiple mortality rates and you want to do an infernec on its average.

  1. If you either have one point, or you want to infer on proportion of multiple points

$$CI = p \pm Z_{\alpha/2}\frac{\sqrt{{\hat p\hat q/n}+{z^2_{\alpha/2}/4n^2}}}{1+{z^2_{\alpha/2}/n}}$$

Here $\hat p$ = proportion = 1 per 100,000 or 0.00001 and $\hat q$ = 1 - $\hat p$

for large data sets you can use the formula

$$p \pm z_{\alpha/2} \sqrt {\hat p\hat q/n}$$

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  • $\begingroup$ This is great! Thanks for the details in your answer. $\endgroup$
    – Shark7
    Commented May 9, 2018 at 0:42
  • $\begingroup$ @Shark7 sorry in advance, but some parts of the formula were missing. I updated it and now it should be correct. Again I apologize, with so much latex, i forgot some values. $\endgroup$
    – hxlaclhemy
    Commented May 11, 2018 at 17:18
  • $\begingroup$ I still think there's a lack of clarity in the OP. If this is a question about finding a confidence interval for a proportion (point 2 above), using rates as your sample data is incorrect. $\endgroup$
    – dankernler
    Commented May 11, 2018 at 17:34
  • $\begingroup$ If he is trying to calculate the CI on lets say 40/150 it should be okay, as 40/150 = P, Q = 1 -40/150 , N = 150. So for each point he itterates this method. $\endgroup$
    – hxlaclhemy
    Commented May 11, 2018 at 17:40

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