Second-Order Random Walk Prior with First Order Autoregressive Errors

Use one or more second-order random walks, combined with an AR1 error term, to model a main effect or an interaction. Typically used with time.

Usage

RW2_AR1(
  s_rw = 1,
  sd = 1,
  sd_slope = 1,
  s_ar = 1,
  shape1 = 5,
  shape2 = 5,
  min = 0.8,
  max = 0.98,
  along = NULL,
  con = c("none", "by")
)

Arguments

s_rw: Scale for the innovations in the RW2 process. Default is 1.
sd: Standard deviation for initial term in RW2 process. Default is 1. Can be 0.
sd_slope: Standard deviation in the prior for the initial slope of RW2 process. Larger values imply steeper slopes. Default is 1.
s_ar: Scale for the innovations in the AR1 process. Default is 1.
shape1, shape2: Parameters for beta-distribution prior for coefficients. Defaults are 5 and 5.
min, max: Minimum and maximum values for autocorrelation coefficient in AR1 process. Defaults are 0.8 and 0.98.
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_rw2randomar" or "bage_prior_rw2zeroar".

Details

If RW2_AR1() is used with an interaction, separate random walks are constructed along the 'along' variable, within each combination of the 'by' variables.

Parameters controlling the RW2 process:

s_rw
sd
sd_slope

Parameters controlling the AR1 process:

s_ar
shape1
shape2
min
max

Mathematical details

When RW2_AR1() is used with a main effect,

$$\beta_j = \alpha_j + \epsilon_j$$ $$\alpha_1 \sim \text{N}(0, \mathtt{sd}^2)$$ $$\alpha_2 \sim \text{N}(\alpha_1, \mathtt{sd\_slope}^2)$$ $$\alpha_j \sim \text{N}(2\alpha_{j-1} - \alpha_{j-2}, \tau^2), \quad j = 3, \cdots, J$$ $$\epsilon_j = \phi \epsilon_{j-1} + \varepsilon_j$$ $$\varepsilon_j \sim \text{N}(0, \omega^2),$$

and when it is used with an interaction,

$$\beta_{u,v} = \alpha_{u,v} + \epsilon_{u,v}$$ $$\alpha_{u,1} \sim \text{N}(0, \mathtt{sd}^2)$$ $$\alpha_{u,2} \sim \text{N}(\alpha_{u,1}, \mathtt{sd\_slope}^2)$$ $$\alpha_{u,v} \sim \text{N}(2\alpha_{u,v-1} - \alpha_{u,v-2}, \tau^2), \quad v = 3, \cdots, V$$ $$\epsilon_{u,v} = \phi \epsilon_{u,v-1} + \varepsilon_{u,v}$$ $$\varepsilon_{u,v} \sim \text{N}(0, \omega^2),$$

where

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

The $\tau$ parameter in the random walk has prior $$\tau \sim \text{N}^+(0, \mathtt{s\_rw}^2)$$

Internally, RW2_AR() derives a value for $\omega$ that gives $\epsilon_j$ or $\epsilon_{u,v}$ a marginal variance of $\nu^2$. Parameter $\nu$ has a half-normal prior $$\nu \sim \text{N}^+(0, \mathtt{s\_ar}^2).$$

Coefficient $\phi$ is constrained to lie between min and max. Its prior distribution is

$$\phi = (\mathtt{max} - \mathtt{min}) \phi' - \mathtt{min}$$

where

$$\phi' \sim \text{Beta}(\mathtt{shape1}, \mathtt{shape2}).$$

Constraints

With some combinations of terms and priors, the values of the intercept, main effects, and interactions are only weakly identified. This weak identifiability is typically harmless. However, in some applications, such as when trying to obtain interpretable values for main effects and interactions, it can be helpful to increase identifiability through the use of constraints, specified through the con argument.

Current options for con 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.

Examples

RW2_AR1()
#>   RW2_AR1() 
#>       s_rw: 1
#>         sd: 1
#>   sd_slope: 1
#>       s_ar: 1
#>     shape1: 5
#>     shape2: 5
#>        min: 0.8
#>        max: 0.98
#>      along: NULL
#>        con: none
RW2_AR1(sd_slope = 2, s_ar = 0.5)
#>   RW2_AR1(sd_slope=2,s_ar=0.5) 
#>       s_rw: 1
#>         sd: 1
#>   sd_slope: 2
#>       s_ar: 0.5
#>     shape1: 5
#>     shape2: 5
#>        min: 0.8
#>        max: 0.98
#>      along: NULL
#>        con: none