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P-Spline Prior

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

Use a p-spline (penalised spline) to model main effects or interactions. Typically used with age, but can be used with any variable where outcomes are expected to vary smoothly from one element to the next.

Usage

Sp(
  n_comp = NULL,
  s = 1,
  sd = 1,
  sd_slope = 1,
  along = NULL,
  con = c("none", "by")
)

Arguments

n_comp

Number of spline basis functions (components) to use.

s

Scale for the prior for the innovations. Default is 1.

sd

Standard deviation in prior for first element of random walk.

sd_slope

Standard deviation in prior for initial slope of random walk. 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_spline".

Details

If Sp() is used with an interaction, separate splines are used for the 'along' variable within each combination of the 'by' variables.

Mathematical details

When Sp() is used with a main effect,

$$\pmb{\beta} = \pmb{X} \pmb{\alpha}$$

and when it is used with an interaction,

$$\pmb{\beta}_u = \pmb{X} \pmb{\alpha}_u$$

where

  • \(\pmb{\beta}\) is the main effect or interaction, with \(J\) elements;

  • \(\pmb{\beta}_u\) is a subvector of \(\pmb{\beta}\) holding values for the \(u\)th combination of the 'by' variables;

  • \(J\) is the number of elements of \(\pmb{\beta}\);

  • \(U\) is the number of elements of \(\pmb{\beta}_u\);

  • \(X\) is a \(J \times n\) or \(V \times n\) matrix of spline basis functions; and

  • \(n\) is n_comp.

The elements of \(\pmb{\alpha}\) or \(\pmb{\alpha}_u\) are assumed to follow a second-order random walk.

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.

References

  • Eilers, P.H.C. and Marx B. (1996). "Flexible smoothing with B-splines and penalties". Statistical Science. 11 (2): 89–121.

See also

  • RW() Smoothing via random walk

  • RW2() Smoothing via second-order random walk

  • SVD() Smoothing of age via singular value decomposition

  • priors Overview of priors implemented in bage

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

  • splines::bs() Function used by bage to construct spline basis functions

Examples

Sp()
#>   Sp() 
#>     n_comp: NULL
#>          s: 1
#>         sd: 1
#>   sd_slope: 1
#>      along: NULL
#>        con: none
Sp(n_comp = 10)
#>   Sp(n_comp=10) 
#>     n_comp: 10
#>          s: 1
#>         sd: 1
#>   sd_slope: 1
#>      along: NULL
#>        con: none