[R] Linear relative rate / excess relative risk models
Wollschlaeger, Daniel
wollschlaeger at uni-mainz.de
Wed Jan 8 16:49:28 CET 2014
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 [1]:
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 [2] did not lead to a successful solution (I contacted the author). Atkinson & Therneau in [3] 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
[1] Preston DL. Beyond Dose Response: Describing Long-Term Health Effects of Radiation Exposure.
http://isi.cbs.nl/iamamember/CD2/pdf/519.PDF
[2] https://stat.ethz.ch/pipermail/r-sig-epi/2012-January/000265.html
[3] 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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