# [R] Linear relative rate / excess relative risk models

David Winsemius dwinsemius at comcast.net
Wed Jan 8 19:06:33 CET 2014

```I would fit a Poisson model to the dose-response data with offsets for the baseline expecteds.

Sent from my iPhone

> On Jan 8, 2014, at 10:49 AM, "Wollschlaeger, Daniel" <wollschlaeger at uni-mainz.de> wrote:
>
> My question is how I can fit linear relative rate models (= excess relative risk models, ERR) using R. In radiation epidemiology, ERR models are used to analyze dose-response relationships for event rate data and have the following form :
>
> lambda = lambda0(z, alpha) * (1 + ERR(x, beta))
>
> * lambda is the event rate
> * lambda0 is the baseline rate function for non-exposed persons and depends on covariates z with parameters alpha
> * ERR is the excess relative risk function for exposed persons and depends on covariates x (among them dose) with parameters beta
> * lambda/lambda0 = 1 + ERR is the relative rate function
>
> Often, the covariates z are a subset of the covariates x (like sex and age). lambda is assumed to be log-linear in lambda0, and ERR typically has a linear (or lin-quadratic) dose term as well as a log-linear modifying term with other covariates:
>
> lambda0 = exp(alpha0 + alpha1*z1 + alpha2*z2 + ...)
> ERR = beta0*dose * exp(beta1*x1 + beta2*x2 + ...)
>
> The data is often grouped in form of life tables with the observed event counts and person-years (pyr) for each cell that results from categorizing and cross-classifying the covariates. The counts are assumed to have a Poisson-distribution with mean mu = lambda*pyr, and the usual Poisson-likelihood is used. The interest is less in lambda0, but in inference on the dose coefficient beta0 and on the modifier coefficients beta.
>
> In the literature, the specialized Epicure program is almost exclusively used. Last year, a similar question on R-sig-Epi  did not lead to a successful solution (I contacted the author). Atkinson & Therneau in  discuss excess risk models but get lambda0 separately from external data instead of fitting lambda0 as a log-linear term. Some R packages sound promising to me (eg., gnm, timereg) but I currently don't see how to correctly specify the model.
>
> Any help on how to approach ERR models in R is highly appreciated!
> With many thanks and best regards
>
> Daniel
>
>  Preston DL. Beyond Dose Response: Describing Long-Term Health Effects of Radiation Exposure.
> http://isi.cbs.nl/iamamember/CD2/pdf/519.PDF
>
>  https://stat.ethz.ch/pipermail/r-sig-epi/2012-January/000265.html
>
>  Atkinson et al. 2008. Poisson models for person-years and expected rates.
> http://www.mayo.edu/research/documents/biostat-81pdf/DOC-10026981
>
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