# Poisson Distribution CI - are the limits scalable?

I have a rate of injury = 3 per 196 hours of exercise. Based on poisson distribution, 95% confidence interval for the 3 is 0.62 to 8.77

If I re-scale my data to be rate of injury per 1000 hours of exercise, what happens the confidence interval?

Is it just 3/196 = 15.3/1000 hours and so new CI is based just on the number 15.3 ie 8.4 to 24.74

or is it based on the original CI limits themselves and becomes 3.14 to 44.50?

thank you

• Could you explain how you obtained the interval $[8.4, 24.74]$? I'm expecting you to say it's a Poisson confidence interval for the rate based on observing $15.3$ injuries in $1000$ hours. If you do, please anticipate my response: show us your $1000$ hours of observational data (and explain what a fractional injury means)! – whuber Jan 5 '17 at 22:07
• Hi, my question is about re-scaling. The actual observed data is 3 injuries in 196 hours. – user144436 Jan 5 '17 at 22:28
• Right: so you don't actually have the data to compute a CI as if you had $15.3$ injuries in $1000$ hours, do you? That rules out the first of your choices, leaving you only the second. – whuber Jan 5 '17 at 22:31
• Is the second correct? – user144436 Jan 5 '17 at 22:31
• Well, let me ask you this. Suppose somebody reported that measurements indicated the height of a skyscraper is 100 yards, with a confidence interval from 90 to 110 yards. Being more comfortable expressing heights in feet, you intend to report the height as 3*100 = 300 feet. What confidence interval would you give for the height in feet? Converting a result from injuries per 196 hours to injuries per 1000 hours is no different. – whuber Jan 5 '17 at 22:37

The confidence interval has two endpoints, the Lower Confidence Limit (LCL) and the Upper Confidence Limit (UCL). They correspond to two possible Poisson distributions that are just barely consistent with the data:

The left plot shows probabilities for the Poisson(LCL) distribution (with rate $0.62$). The values equal to or greater than the observation $x=3$ are highlighted in red. The sum of these probabilities is $2.5\%$. If the rate of the Poisson distribution were any less than the LCL, the sum of these probabilities would be less than $2.5\%$. It is in this sense that the observation $x=3$ (or any larger value) would not be very plausible were the rate lower than the LCL.

To be thorough, let's repeat this description for the right hand plot. That plot shows probabilities for the Poisson(UCL) distribution (with rate $8.77$). The values equal to or less than the observation $x=3$ are highlighted in red. Again, the sum of these probabilities is $2.5\%$. If the rate of the Poisson distribution were any greater than the UCL, the sum of these probabilities would be less than $2.5\%$. It is in this sense that the observation $x=3$ (or any smaller value) would not be very plausible were the rate greater than the LCL.

Let us scale this picture up by converting the rates from injuries per $196$ people to injuries per $1000$ people. Because $x$ merely is multiplied by $1000/196$, this is simply a matter of relabeling the $x$ axis and re-expressing the rates. For instance, $0.62$ per $196$ people becomes $3.16$ per $1000$ people. The observation $x=3$ is scaled up to $x\approx 15.3$.

It is immediately apparent that these are not Poisson distributions: instead of showing probabilities for integral values $0, 1, 2, \ldots$, they show probabilities for the scaled values $0, 1000/196, 2\times 1000/196, \ldots$. Nevertheless, these are the correct distributions for the rescaled data.

Let's look at how the Poisson distributions with these (rescaled) rates actually behave.

They are narrower than the rescaled version of the Poisson distribution. If you were to compute a Poisson confidence interval based on $x=15.3$ (as indicated in the next figure), you would be assuming these kinds of distributions governed your observations--but that's obviously wrong. Just as obviously it would lead to a narrower confidence interval than the data deserve. These are not the distributions to use for computing the confidence interval.