mod_pois: Specify a Poisson Model

In bage: Bayesian Estimation and Forecasting of Age-Specific Rates

Specify a Poisson Model

Description

Specify a model where the outcome is drawn from a Poisson distribution.

Usage

mod_pois(formula, data, exposure)

Arguments

formula	An R formula, specifying the outcome and predictors.
data	A data frame containing outcome, predictor, and, optionally, exposure variables.
exposure	Name of the exposure variable, or a 1, or a formula. See below for details.

Details

The model is hierarchical. The rates in the Poisson distribution are described by a prior model formed from dimensions such as age, sex, and time. The terms for these dimension themselves have models, as described in priors. These priors all have defaults, which depend on the type of term (eg an intercept, an age main effect, or an age-time interaction.)

Value

An object of class bage_mod_pois.

Specifying exposure

The exposure argument can take three forms:

• the name of a variable in data, with or without quote marks, eg "population" or population;

• the number 1, in which case a pure "counts" model with no exposure, is produced; or

• a formula, which is evaluated with data as its environment (see below for example).

Mathematical details

The likelihood is

y_i \sim \text{Poisson}(\gamma_i w_i)

where

• subscript i identifies some combination of the classifying variables, such as age, sex, and time;

• y_i is an outcome, such as deaths;

• \gamma_i is rates; and

• w_i is exposure.

In some applications, there is no obvious population at risk. In these cases, exposure w_i can be set to 1 for all i.

The rates \gamma_i are assumed to be drawn a gamma distribution

y_i \sim \text{Gamma}(\xi^{-1}, (\xi \mu_i)^{-1})

where

• \mu_i is the expected value for \gamma_i; and

• \xi governs dispersion (i.e. variation), with lower values implying less dispersion.

Expected value \mu_i equals, on the log scale, the sum of terms formed from classifying variables,

\log \mu_i = \sum_{m=0}^{M} \beta_{j_i^m}^{(m)}

where

• \beta^{0} is an intercept;

• \beta^{(m)}, m = 1, \dots, M, is a main effect or interaction; and

• j_i^m is the element of \beta^{(m)} associated with cell i.

The \beta^{(m)} are given priors, as described in priors.

\xi has an exponential prior with mean 1. Non-default values for the mean can be specified with set_disp().

The model for \mu_i can also include covariates, as described in set_covariates().

See Also

• mod_binom() Specify binomial model

• mod_norm() Specify normal model

• set_prior() Specify non-default prior for term

• set_disp() Specify non-default prior for dispersion

• fit() Fit a model

• augment() Extract values for rates, together with original data

• components() Extract values for hyper-parameters

• forecast() Forecast parameters and outcomes

• report_sim() Check model using a simulation study

• replicate_data() Check model using replicate data

• Mathematical Details Detailed description of models

Examples

## specify a model with exposure
mod <- mod_pois(injuries ~ age:sex + ethnicity + year,
                data = nzl_injuries,
                exposure = popn)

## specify a model without exposure
mod <- mod_pois(injuries ~ age:sex + ethnicity + year,
                data = nzl_injuries,
                exposure = 1)

## use a formula to specify exposure
mod <- mod_pois(injuries ~ age:sex + ethnicity + year,
                data = nzl_injuries,
                exposure = ~ pmax(popn, 1))

bage documentation built on Aug. 8, 2025, 6:09 p.m.