# How do you calculate a Poisson distribution for matched data?

I am trying to find the rate of infections with 2 different treatments and to see if there is any difference between them. I want to work out whether the rate of 'no infections' is different between the 2 treatments.

Initially I checked the odds ratios and looked at conditional logistic regression to get the confidence intervals.

The odds ratio of treatment A no infection is 0.70, and treatment B is 0.69 The confidence intervals on conditional logistic regression is 0.61 and 2.2 so they both lie within the confidence intervals - i.e. treatments are equivalent.

I am told that I can use the Poisson distribution and see if there is a difference between them.

$\hat{\lambda}$ for number of infections per person for treatment A is 0.39, and for treatment B is 0.34

This gives the Poisson distribution for less than 1 infection as 0.71 (confidence interval 0.68-0.74 calculated by $\hat\lambda \pm 1.96\sqrt{\hat\lambda}$ for treatment A and 0.67 (0.64-0.70) for treatment B.

Is this a valid method for calculating rate ratios or do I have to take something else into account for matched pairs?

• Are the experimental units people? Can a person be infected more than once? – Andrew M May 26 '15 at 20:49
• Experimental units are people, they can have more than one infection over the outcome time period (2 years) – user1745691 May 26 '15 at 20:58

Ignoring pairing that exists in the data will generally give you conservative results (p-value to large, confidence intervals too wide). However, if there is a negative correlation within pairs (unusual, but not impossible) then the errors can be in the other direction.

To account for the pairing in the data, I would suggest using a generalized linear mixed effects model (GLMM) or a Bayesian approach. Neither is simple and need a fair amount of sophistication to carry out properly (so if you are not already familiar with these approaches, you should take another course or 2 (or 3) or consult with a statistician who is familiar with these techniques).

• In the end, what I did was use the gnm library in R: model<-gnm(infections, treatmentgroup, data, family=poisson, eliminate-factor(strata)). Where treatment group was 0 or 1, and used the exponential of the coefficient for the rate ratio and standard error to plug into calculating the probability of infection count = 1 – user1745691 May 27 '15 at 14:54

I would not use (simple) Poisson regression for this data. Given that in both groups, you have the vast majority of subjects not having any infection, describing this with a Poisson model seems a little backwards; you're not getting much more information out of your data, unless something is an extreme outlier (i.e a subject with 10 infections).

So basically, the only extra power you get is that you can now allow outliers to have heavy leverage. But the only times you'll have heavy outliers is when the model doesn't fit well and shouldn't be trusted anyways! In my head, this makes for a lose-lose scenario (or at least lose-tie scenario).

If you really wanted to allow number of counts to have influence, you can try something like a zero-inflated Poisson. But my guess is that this is more modeling than you are going to care for, and it's likely not to add any power (unless you really think that conditional on the treatments not preventing infection overall at different rates, it still reduces number of infections at different rates. To me that seems odd.)

So I'd suggest just sticking to odds ratios, unless the data's more complicated than I'm seeing.

• The data is matched patients (1:1) and probably not more complex than you think. I thought that odds ratios made more sense, but I am trying to see if its possible to massage the data to show that treatment A is statistically better than B. – user1745691 May 26 '15 at 21:02