function, for example when (as is often the case) g is the log function then the model is equivalent to:

image

When Ni counts the number of events observed over a period of time, ti (years), for a known number of individuals, ki, then the person-years of observation, pyi, defined as tiki will be made a part of model as:

image

so that the mean of the counts is proportional to the person-years of observation multiplied by the effect of covariates.

In the setting described here Ni would correspond to a single entry in a cross-tabulation of events (death due to or incidence of a particular cancer) by each geographical unit, and by gender, race, age, calendar time, and any other relevant variable known (from the cancer registry) about the cases. For each cell in the table the number of events and person-years at risk, pyi, are required to be calculated (see discussion below) in addition the variable of interest, dose Di, and other covariates available for each geographical unit (i.e., indices of social economic status) are required for each table entry i.

A variation on model, known as the linear excess relative risk (ERR) model, is commonly used in radiation epidemiology. The linear ERR model incorporates dose in the model for mi as:

image

Here pyi exp(XiTa) is the background rate of disease (for unexposed cells), multiplied by person-years at risk, and the ERR parameter β is the excess relative risk associated with dose or dose surrogate Di. Much more complex models can be considered and software for generalized Poisson regression is available (Epicure, Hirosoft Software, Seattle, Washington). The background rate of disease is allowed to vary depending on race, gender, age, and calendar time (to allow for disease rates to differ by age and for age-specific rates to vary by calendar year, for example). Covariates in ecologic models are not individual covariates, but instead are summaries obtained for each geographical unit, although these can also vary in time; for example, we may have information about some socioeconomic variables at the level of census tract and these variables may change with time over the period of interest. Such variables are incorporated by including (categories of) calendar time as a cross-classification variable.



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