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I want to fit a fairly "standard" Poisson model, but with an autoregressive term.

$N_i \sim \mathrm{Pois}( \lambda_i E_i)$

with $\log \lambda_i = X_i \beta + \delta$

$\delta \sim AR(1)$

$X_i$ is a vector of covariates. $\beta$ are my coefficients. $\delta$ is an autoregressive term. $E_i$ is the size of population at time t.

The idea is that the count at time step $t$ is partially dependent on the count at time step $t-1$.

Ideally, I'd like to find some R package to fit this.

Any suggestions?

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    $\begingroup$ It would be good to consider accepting answers to some of your previous questions, all of which have received multiple answers, thus giving you some choice. There is a check mark next to each answer that you can click on to indicate which one has been addressed your query. $\endgroup$ – cardinal Feb 5 '12 at 19:06
  • $\begingroup$ Can you give some more detail as to what kind of autoregressive structure you want to assume. It's a little ambiguous at the moment. Defining $E_i$ would also be helpful. Cheers. :) $\endgroup$ – cardinal Feb 5 '12 at 19:07
  • $\begingroup$ This is an epidemiological model. The dependent variable is the number of people with a disease at time t. I can fit it reasonably well with a "standard" poisson, but it was suggested that an autoregressive term might work well for this particular study. $\endgroup$ – Noah Feb 5 '12 at 19:18
  • $\begingroup$ I guess I'm wondering what sort of autoregressive formulation you want. Are you thinking of something like $\log \lambda_i= X_i \beta + \alpha \log \lambda_{i-1} + \varepsilon_i$ where $\varepsilon_i$ is some additional randomness driving the evolution of the rate parameter? And, if this is an epidemiological model, is $N_i$ some number of, say, infected individuals? If so, then it would seem $\lambda_i \ll 1$, otherwise there is nonnegligible probability of more people than exist in the population becoming infected at time $i$. But, maybe I'm misunderstanding what you're aiming for. $\endgroup$ – cardinal Feb 5 '12 at 19:44
  • $\begingroup$ You understand perfectly. Nice summary. $N_i$ is the number of people with the disease and $\lambda$ is definitely less than 1. $\endgroup$ – Noah Feb 5 '12 at 19:50
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I think you are looking for the model in Brandt et al. (2000) there called PEWMA, after the forecast function. R code to fit it is available here.

The paper also has some general discussion of possible conditionally Poisson AR models. Fro more of that, chapter 7 of Cameron and Trivedi (1998) is useful.

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Have you considered a Transfer Function between N and E and your other covariates which could encode changes in parameters over time , changes in error variance over time , any necessaery autoregressive structure possibly proxying unspecified seasonal drivers and other omitted structure like level shifts, seasonal pulses and local time trends. Accomplished without letting "unusual values" distort the model/parameters. I have seen approaches like this referred to in the literature as DARIMA MODELS where the D stands for discrete. Various forms of Power Transforms such as Logs might be needed to possibly decouple the expected value from the variance of the errors ( always a good idea when needed! ) These kinds of models can be useful in detecting the possible impact of changes in N as it relates to the prediction of E.

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    $\begingroup$ Without getting into too many details, I don't believe that model would work for this data set. What I have are the counts of disease along with some categorial factors for each year. (i.e. Asian, Men, 2005, 10 out of 500 had disease.) $\endgroup$ – Noah Feb 6 '12 at 19:41

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