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I have ten-year data for crude number of certain events (cases of disease [counts]) from one country. Would it be appropriate to use Poisson regression with dependent variable as number of cases and only one independent variable 'Year' (coded as 2001, 2002 . . . 2010) to test for linear trend over time? Hence, if 'Year' is statistically significant, would it imply linear time trend?

Would this approach ignore the 'serial dependence'?

How/What would be the best way to study trend over time? R/Stata code/packages would really help me please!

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    $\begingroup$ A quick and dirty way of dealing with the serial dependence would be to employ HAC standard errors when testing for the significance of the Year variable. $\endgroup$ – Richard Hardy Sep 20 '17 at 11:18
  • $\begingroup$ Thanks, could you elaborate a little more please? Would Poisson regression be the right approach? $\endgroup$ – J. Dow Sep 20 '17 at 11:36
  • $\begingroup$ I don't know at the moment, that is why I have not commented on that part. $\endgroup$ – Richard Hardy Sep 20 '17 at 11:50
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I think you are right that your approach would ignore serial dependence. There's a nice little presentation on Time Series Models for Event Count Data, I think it could be useful for you. They name two methods, PEWMA and PAR(p), which are better than simple Poisson Regression. PAR (Poisson Autoregressive Model) seems to be implemented in the R-Package gsarima. For PEWMA is couldn't find a packages, but there's this crossvalidated-post, where you can find some R-code

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  • $\begingroup$ With yearly data, for some diseases autocorelation would be on a much shorter time-span, so maybe not seen in yearly data. Which disease? $\endgroup$ – kjetil b halvorsen Sep 20 '17 at 12:45
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The problem with Poisson models is that they are single-parameter models that do not allow a free variance parameter. For this reason, when dealing with count data it is usually far better to use a negative-binomial distribution for the response variable. There is some good literature on negative-binomial time-series data with autocorrelation (see e.g., McKensie 1986, All-Osh and Aly 1992, Christou and Fokianos 2013). These sometimes refer to the model as the Negative Binomial Autoregressive model. This kind of model would allow the inclusion of serial dependence, and it would also allow sufficient parametric freedom to allow the variance to be estimated independently of the mean.

According to the documentation of the glarma package, the glarma modelling function allows you to specify the negative binomial distribution (type = 'NegBin'). This package appears to accommodate both the PAR (Poisson Autoregressive) model and also the NBAR (Negative Binomial Autoregressive) model. I would generally prefer the latter for the same reasons that the negative-binomial is generally preferred for count data in fixed settings without autocorrelation.

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