Use a line or lines with AR1 errors to model a main effect or interaction. Typically used with time.
Usage
Lin_AR1(
s = 1,
shape1 = 5,
shape2 = 5,
min = 0.8,
max = 0.98,
mean_slope = 0,
sd_slope = 1,
along = NULL,
con = c("none", "by")
)
Arguments
- s
Scale for the innovations in the AR process. Default is
1
.- shape1, shape2
Parameters for beta-distribution prior for coefficients. Defaults are
5
and5
.- min, max
Minimum and maximum values for autocorrelation coefficient. Defaults are
0.8
and0.98
.- mean_slope
Mean in prior for slope of line. Default is 0.
- sd_slope
Standard deviation in the prior for the slope of the line. Larger values imply steeper slopes. 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.
Details
If Lin_AR1()
is used with an interaction,
separate lines are constructed along
the 'along' variable, within each combination
of the 'by' variables.
Arguments min
and max
can be used to specify
the permissible range for autocorrelation.
Argument s
controls the size of the innovations.
Smaller values tend to give smoother estimates.
Argument sd_slope
controls the slopes of
the lines. Larger values can give more steeply
sloped lines.
Mathematical details
When Lin_AR1()
is being used with a main effect,
$$\beta_1 = \alpha + \epsilon_1$$ $$\beta_j = \alpha + (j - 1) \eta + \epsilon_j, \quad j > 1$$ $$\alpha \sim \text{N}(0, 1)$$ $$\epsilon_j = \phi \epsilon_{j-1} + \varepsilon_j$$ $$\varepsilon \sim \text{N}(0, \omega^2),$$
and when it is used with an interaction,
$$\beta_{u,1} = \alpha_u + \epsilon_{u,1}$$ $$\beta_{u,v} = \eta (v - 1) + \epsilon_{u,v}, \quad v = 2, \cdots, V$$ $$\alpha_u \sim \text{N}(0, 1)$$ $$\epsilon_{u,v} = \phi \epsilon_{u,v-1} + \varepsilon_{u,v},$$ $$\varepsilon_{u,v} \sim \text{N}(0, \omega^2).$$
where
\(\pmb{\beta}\) is the main effect or interaction;
\(j\) denotes position within the main effect;
\(u\) denotes position within the '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 $$\eta_u \sim \text{N}(\mathtt{mean\_slope}, \mathtt{sd\_slope}^2).$$
Internally, Lin_AR1()
derives a value for \(\omega\) that
gives \(\epsilon_j\) or \(\epsilon_{u,v}\) a marginal
variance of \(\tau^2\). Parameter \(\tau\)
has a half-normal prior
$$\tau \sim \text{N}^+(0, \mathtt{s}^2),$$
where a value for s
is provided by the user.
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 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()
Generalization ofLin_AR1()
Lin()
Line with independent normal errorsAR1()
AR1 process with no linepriors Overview of priors implemented in bage
set_prior()
Specify prior for intercept, main effect, or interaction