Specify a model where the outcome is drawn from a normal distribution.
mod_norm(formula, data, weights)An object of class bage_mod_norm.
An R formula, specifying the outcome and predictors.
A data frame containing outcome, predictor, and, optionally, weights variables.
Name of the weights variable,
a 1, or a formula. See below for details.
Internally, mod_norm() scales the outcome variable
to have mean 0 and standard deviation 1, and
scales the weights to have mean 1.
This scaling allows mod_norm() to use the
same menu of priors as mod_pois() and mod_binom().
augment() always returns values on the
original scale, rather than the transformed scale.
components() by default returns values on
the transformed scale. But if original_scale is
TRUE, it returns some types of values on the
original scale. See components() for details.
The weights argument can take three forms:
the name of a variable in data, with or without
quote marks, eg "wt" or wt;
the number 1, in which no weights are used; or
a formula, which is evaluated with data as its
environment (see below for example).
The likelihood is
$$y_i \sim \text{N}(\gamma_i, w_i^{-1} \sigma^2)$$ where
subscript \(i\) identifies some combination of the classifying variables, such as age, sex, and time,
\(y_i\) is the value of the outcome variable,
\(w_i\) is a weight.
In some applications, \(w_i\) is set to 1 for all \(i\).
Internally, bage works with standardized versions of \(\gamma_i\) and \(\sigma^2\):
$$\mu_i = (\gamma_i - \bar{y}) / s$$ $$\xi^2 = \sigma^2 / (\bar{w} s^2)$$ where $$\bar{y} = \sum_{i=1}^n y_i / n$$ $$s = \sqrt{\sum_{i=1}^n (y_i - \bar{y})^2 / (n-1)}$$ $$\bar{w} = \sum_{i=1}^n w_i / n$$
Mean parameter \(\mu_i\) is modelled as the sum of terms formed from classifying variables and covariates,
$$\mu_i = \sum_{m=0}^{M} \beta_{j_i^m}^{(m)}$$
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().
The model is hierarchical. The means in the normal 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.)
mod_pois() Specify Poisson model
mod_binom() Specify binomial model
set_prior() Specify non-default prior for term
set_disp() Specify non-default prior for standard deviation
fit() Fit a model
augment() Extract values for means,
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
data for a model
Mathematical Details Detailed description of models
mod <- mod_norm(value ~ diag:age + year,
data = nld_expenditure,
weights = 1)
## use formula to specify weights
mod <- mod_norm(value ~ diag:age + year,
data = nld_expenditure,
weights = ~sqrt(value))
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