mod_binom {bage} | R Documentation |
Specify a Binomial Model
Description
Specify a model where the outcome is drawn from a binomial distribution.
Usage
mod_binom(formula, data, size)
Arguments
formula |
An R formula, specifying the outcome and predictors. |
data |
A data frame containing the outcome and predictor variables, and the number of trials. |
size |
Name of the variable giving the number of trials, or a formula. |
Details
The model is hierarchical. The probabilities in the binomial 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
.
Specifying size
The size
argument can take two forms:
the name of a variable in
data
, with or without quote marks, eg"population"
orpopulation
; ora formula, which is evaluated with
data
as its environment (see below for example).
Mathematical details
The likelihood is
y_i \sim \text{binomial}(\gamma_i; w_i)
where
subscript
i
identifies some combination of the the classifying variables, such as age, sex, and time;-
y_i
is a count, such of number of births, such as age, sex, and region; -
\gamma_i
is a probability of 'success'; and -
w_i
is the number of trials.
The probabilities \gamma_i
are assumed to be drawn
a beta distribution
y_i \sim \text{Beta}(\xi^{-1} \mu_i, \xi^{-1} (1 - \mu_i))
where
-
\mu_i
is the expected value for\gamma_i
; and -
\xi
governs dispersion (ie variance.)
Expected value \mu_i
equals, on a logit scale,
the sum of terms formed from classifying variables,
\text{logit} \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_pois()
Specify Poisson 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 probabilities, together with original data -
components()
Extract values for hyper-parameters -
forecast()
Forecast parameters and outcomes -
report_sim()
Check model using simulation study -
replicate_data()
Check model using replicate data -
Mathematical Details Detailed descriptions of models
Examples
mod <- mod_binom(oneperson ~ age:region + age:year,
data = nzl_households,
size = total)
## use formula to specify size
mod <- mod_binom(ncases ~ agegp + tobgp + alcgp,
data = esoph,
size = ~ ncases + ncontrols)