mod_pois {bage} | R Documentation |
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 |
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"
orpopulation
;the number
1
, in which case a pure "counts" model with no exposure, is produced; ora 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 celli
.
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))