Linear Prior with Independent Normal Errors

Use a line or lines with independent normal errors to model a main effect or interaction. Typically used with time.

Usage

Lin(s = 1, sd = 1, along = NULL)

Arguments

s: Scale for the prior for the errors. Default is 1.
sd: Standard deviation in prior for slope of line. Default is 1.
along: Name of the variable to be used as the "along" variable. Only used with interactions.

Value

An object of class "bage_prior_lin".

Details

If Lin() is used with an interaction, then separate lines are constructed along the "along" variable, within each combination of the "by" variables.

Argument s controls the size of the errors. Smaller values tend to give smoother estimates.

Argument sd controls the size of the slopes of the lines. Larger values can give more steeply sloped lines.

Mathematical details

When Lin() is used with a main effect,

$$\beta_j = \alpha + j \eta + \epsilon_j$$ $$\alpha \sim \text{N}(0, 1)$$ $$\epsilon_j \sim \text{N}(0, \tau^2),$$

and when it is used with an interaction,

$$\beta_{u,v} \sim \alpha_u + v \eta_u + \epsilon_{u,v}$$ $$\alpha_u \sim \text{N}(0, 1)$$ $$\epsilon_{u,v} \sim \text{N}(0, \tau^2),$$

where

$\pmb{\beta}$ is the main effect or interaction;
$j$ denotes position within the main effect;
$v$ denotes position within the "along" variable of the interaction; and
$u$ denotes position within the "by" variable(s) of the interaction.

The slopes have priors $$\eta \sim \text{N}(0, \text{sd}^2)$$ and $$\eta_u \sim \text{N}(0, \text{sd}^2).$$

Parameter $\tau$ has a half-normal prior $$\tau \sim \text{N}^+(0, \text{s}^2).$$

Examples

Lin()
#>   Lin() 
#>        s: 1
#>       sd: 1
#>    along: NULL
Lin(s = 0.5, sd = 2)
#>   Lin(s=0.5,sd=2) 
#>        s: 0.5
#>       sd: 2
#>    along: NULL
Lin(along = "cohort")
#>   Lin(along="cohort") 
#>        s: 1
#>       sd: 1
#>    along: cohort