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Linear Prior with Independent Normal Errors

Source: R/bage_prior-constructors.R
Lin.Rd

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

Usage

Lin(s = 1, mean_slope = 0, sd_slope = 1, along = NULL, con = c("none", "by"))

Arguments

s

Scale for the prior for the errors. Default is 1. Can be 0.

mean_slope

Mean in prior for slope of line. Default is 0.

sd_slope

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.

con

Constraints on parameters. Current choices are "none" and "by". Default is "none". See below for details.

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. s can be zero.

Argument sd_slope 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 $$\eta \sim \text{N}(\mathtt{mean_slope}, \mathtt{sd_slope}^2)$$ and $$\eta_u \sim \text{N}(\mathtt{mean_slope}, \mathtt{sd_slope}^2).$$

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

Constraints

With some combinations of terms and priors, the values of the intercept, main effects, and interactions are are only weakly identified. For instance, it may be possible to increase the value of the intercept and reduce the value of the remaining terms in the model with no effect on predicted rates and only a tiny effect on prior probabilities. This weak identifiability is typically harmless. However, in some applications, such as forecasting, or when trying to obtain interpretable values for main effects and interactions, it can be helpful to increase identifiability through the use of constraints.

Current options for constraints are:

  • "none" No constraints. The default.

  • "by" Only used in interaction terms that include 'along' and 'by' dimensions. Within each value of the 'along' dimension, terms across each 'by' dimension are constrained to sum to 0.

See also

  • Lin_AR() Linear with AR errors

  • Lin_AR1() Linear with AR1 errors

  • RW2() Second-order random walk

  • priors Overview of priors implemented in bage

  • set_prior() Specify prior for intercept, main effect, or interaction

Examples

Lin()
#>   Lin() 
#>          s: 1
#> mean_slope: 0
#>   sd_slope: 1
#>      along: NULL
#>        con: none
Lin(s = 0.5, sd_slope = 2)
#>   Lin(s=0.5,sd_slope=2) 
#>          s: 0.5
#> mean_slope: 0
#>   sd_slope: 2
#>      along: NULL
#>        con: none
Lin(s = 0)
#>   Lin(s=0) 
#>          s: 0
#> mean_slope: 0
#>   sd_slope: 1
#>      along: NULL
#>        con: none
Lin(along = "cohort")
#>   Lin(along="cohort") 
#>          s: 1
#> mean_slope: 0
#>   sd_slope: 1
#>      along: cohort
#>        con: none