Specify a Normal Model

Usage

mod_norm(formula, data, weights)

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.

Value

An object of class bage_mod_norm.

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.)

Internally, the outcome variable scaled to have mean 0 and sd 1.

Mathematical details

The likelihood is

$$y_i \sim \text{N}(\mu_i, \xi^2 / w_i)$$

where

$y_i$ is a scaled value for an, such of the log of income, for some combination $i$ of classifying variables, such as age, sex, and region;
$\mu_i$ is a mean;
$\xi$ is a standard deviation parameter; and
$w_i$ is a weight.

The scaling of the outcome variable is done internally. If $y_i^*$ is the original, then $y_i = (y_i^* - m)/s$ where $m$ and $s$ are the sample mean and standard deviation of $y_i^*$.

In some applications, $w_i$ is set to 1 for all $i$.

The means $\mu_i$ equal the sum of terms formed from classifying variables,

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

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.

The prior for $\xi$ is described in set_disp().

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).

Examples

mod <- mod_norm(value ~ diag:age + year,
                data = expenditure,
                weights = 1)

## use formula to specify weights
mod <- mod_norm(value ~ diag:age + year,
                data = expenditure,
                weights = ~sqrt(value))