mod_norm: Specify a Normal Model

Description

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

Usage

mod_norm(formula, data, weights)

Value

An object of class bage_mod_norm.

Arguments

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

Scaling of outcome and weights

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.

Specifying weights

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). This option has been deprecated, because it makes forecasting and measurement error models more complicated.

Mathematical details

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().

Details

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.)

Examples

Run this code

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))
## but formulas are now deprecrated, and the
## recommended approach is to transform
## the input data outside the model:
nld_expenditure$wt <- sqrt(nld_expenditure$value)
mod <- mod_norm(value ~ diag:age + year,
                data = nld_expenditure,
                weights = wt)

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