Use a line or lines with autoregressive errors to model a main effect or interaction. Typically used with time.
Arguments
- n_coef
Number of lagged terms in the model, ie the order of the model. Default is
2.- s
Scale for the innovations in the AR process. Default is
1.- sd
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.
Details
If Lin_AR() is used with an interaction,
separate lines are constructed along
the "along" variable, within each combination
of the "by" variables.
The order of the autoregressive errors is
controlled by the n_coef argument. The
default is 2.
Argument s controls the size of the innovations.
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_AR() is used with a main effect,
$$\beta_1 = \alpha + \epsilon_1$$ $$\beta_j = \alpha + (j - 1) \eta + \epsilon_j, \quad j > 1$$ $$\alpha \sim \text{N}(0, 1)$$ $$\epsilon_j = \phi_1 \epsilon_{j-1} + \cdots + \phi_n \epsilon_{j-n} + \varepsilon_j$$ $$\varepsilon_j \sim \text{N}(0, \omega^2),$$
and when it is used with an interaction,
$$\beta_{u,1} = \alpha_u + \epsilon_{u,1}$$ $$\beta_{u,v} = \eta (v - 1) + \epsilon_{u,v}, \quad v = 2, \cdots, V$$ $$\alpha_u \sim \text{N}(0, 1)$$ $$\epsilon_{u,v} = \phi_1 \epsilon_{u,v-1} + \cdots + \phi_n \epsilon_{u,v-n} + \varepsilon_{u,v},$$ $$\varepsilon_{u,v} \sim \text{N}(0, \omega^2).$$
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;
\(u\) denotes position within the "by" variable(s) of the interaction; and
\(n\) is
n_coef.
The slopes have priors $$\eta \sim \text{N}(0, \text{sd}^2)$$ and $$\eta_u \sim \text{N}(0, \text{sd}^2).$$
Internally, Lin_AR() 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
$$\tau \sim \text{N}^+(0, \text{s}^2),$$
where a value for s is provided by the user.
The \(\phi_1, \cdots, \phi_k\) are restricted to values between -1 and 1 that jointly lead to a stationary model. The quantity \(r = \sqrt{\phi_1^2 + \cdots + \phi_k^2}\) has boundary-avoiding prior
$$r \sim \text{Beta}(2, 2).$$
See also
Lin_AR1()Special case ofLin_AR()Lin()Line with independent normal errorsAR()AR process with no linepriors Overview of priors implemented in bage
set_prior()Specify prior for intercept, main effect, or interaction