95% CI for rare incidence rate increasing in time.

#1
I have a plot of 10 incidence rates across time (see picture). I want to calculate the 95% confidence interval for each rate. It would be easiest to just do a normal distribution calculation, but I'm concerned this is not applicable here. For one, they are rare events. I thought to do a poisson distribution calculation for the confidence intervals - that is what is shown with the error bars in the attached picture. This was criticized with the comment - "Poisson assumes a constant rate over time". Assuming that is correct, what distribution/formula should I use to calculate my confidence intervals?

Thank you!
 

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hlsmith

Omega Contributor
#2
Not sure if this is correct, but incidents are the same as risks. So I wonder if there is a formula for rare event CI. I am sure that will be easy enough to search for.
 
#3
You could try to bootstrap each interval at different time points to see how much the constant rate effects your estimates. Also, couldn't you just include a measure of time in the model and then use a time by x-variable interaction to account for the changed relationship over time (assuming you were using a Poisson regression).


Edit: Sorry misread your question.
 
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#4
Can I simply calculate the Poisson distribution interval for each year as I've done (thereby assuming a 'relatively' constant rate over the course of a single year).

The fact that the rate increases over the entire study period doesn't necessarily violate the 'constant rate' assumption of a Poisson distribution as the Poisson calculations are done on a yearly basis and not over the 'entire decade'.

Does my justification have any major flaws?
 

hlsmith

Omega Contributor
#5
I follow you and was thinking the same thing in regards to examining years. I was also thinking there is likely an exact Poisson calculation.

Not this but something like it. It is typical to use exact methods when sample sizes are small.
 
#6
Can I simply calculate the Poisson distribution interval for each year as I've done (thereby assuming a 'relatively' constant rate over the course of a single year).

The fact that the rate increases over the entire study period doesn't necessarily violate the 'constant rate' assumption of a Poisson distribution as the Poisson calculations are done on a yearly basis and not over the 'entire decade'.

Does my justification have any major flaws?
If you bootstrap and get really similar results, you could maybe use this to convince the reviewer that the constant rate assumption isn't violated in a material way, or isn't violated to begin with (I do agree with you that within each year, the constant rate assumption isn't necessarily violated).

Sorry I was a bit confused before. Did you use a Poisson regression or just unadjusted Poisson distribution for a CI?
 
#8
Yeah, the reviewers comment doesn't make sense to me because you're just plotting these separate intervals against time to show a trend. Their comment might be more valid if you were creating one CI for the entire time period (since your chart shows the rate may not be constant over the entire period), assuming I understand this correctly. So, I think if it's reasonable to assume that your intervals are short enough that the rate is constant in an interval, then you should be okay. Maybe another poster has a good reference for this. If they still disagree, you could show the bootstrapped intervals to see the extent of an impact this has even if their comment is valid (playing devils advocate).

Just note that I'm not an expert at all on this, I just took a shot at it because I'd like to learn from this too.
 
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#10
I have not understood the above discussion.

But from the graph it looks like the dependent variable is increasing as an exponential function. Let mu be the expected value of the dependent Poission variable. mu = exp(a + b*(time-timebar), thus

log(mu) = a + b*(time-timebar), thus a standard Poisson regression model. From the estimates you will get standard error for the parameter estimates. The standars deviation is the square rot of mu
thus: CI: exp(log(mu) +/- z*sqrt(mu*(stderr(a)^2 + stderr(b)^2*(time-timebar)^2))

Is this OK?
 
#11
I'm very confused by the above equations. Definitely not a stat expert, and I don't know what a, b, time, or timebar mean. I tried to search and I'm not getting very far.

With regards to the comment above from hlsmith, yes the point of the graph is to draw conclusions regarding an increasing trend. I was simply going to reference non-overlapping confidence intervals to provide evidence of that trend. When you say 'formally model', what exactly do you mean. My only experience with models has been with regard to multivariate regression analyses. For this experiment, I simply have ten incidence rates - provided below. What/how can I create model with just those points?

Thanks everyone for all your comments. It is really appreciated!

2005:2
2006:3
2007:1
2008:3
2009:4
2010:6
2011:2
2012:7
2013:10
2014:15
2015:23