library("gamlss2")
data("SpirometryUS", package = "gamlss2")
## subset for females
d <- subset(SpirometryUS, gender == "female")
## fit one conditional median model
m50 <- gamlss2(fev1 ~ s(age), data = d, family = ALD(tau = 0.5))
## estimate several conditional quantiles
qu <- c(0.025, 0.5, 0.975)
m <- fit_ALD(fev1 ~ s(age), data = d, tau = qu,
max_step = 0.1, monotone = TRUE, verbose = FALSE)
## plot fitted quantile curves
fit <- fitted(m)
i <- order(d$age)
plot(fev1 ~ age, data = d)
matlines(d$age[i], fit[i, ], lty = 1, lwd = 2,
col = c(2, 4, 3))
## estimate lower limits model using full MCMC
set.seed(1328)
m2.5 <- mcmc(m$models[["tau=0.025"]],
n.iter = 12000, burnin = 2000, thin = 10)
summary(m2.5)
## predict credible intervals
nd <- data.frame(age = seq(6, 80, length = 300))
nd$p <- predict(m2.5, newdata = nd,
FUN = function(x) quantile(x, probs = c(0.025, 0.975)))
## plot
plot(fev1 ~ age, data = d)
matlines(nd$age, nd$p, lty = 1, lwd = 2, col = 2)Asymmetric Laplace Quantile Regression
Description
Asymmetric Laplace family for quantile regression and a wrapper for estimating one or more conditional quantile models.
Usage
ALD(tau = 0.5)
fit_ALD(formula, data, tau, anchor = 0.5, max_step = 0.05,
restarts = c("always", "on_failure", "never"), start = NULL,
monotone = FALSE, verbose = interactive(), ...)
Arguments
tau
|
For ALD, one finite quantile probability strictly between zero and one. For fit_ALD, a numeric vector of such probabilities.
|
formula
|
A model formula accepted by gamlss2.
|
data
|
A data frame containing the variables in formula.
|
anchor
|
Quantile probability from which the sequence of fits starts. |
max_step
|
Maximum distance between successive quantiles along the fitting path. Intermediate bridge models are fitted when necessary. Use Inf to disable bridge models.
|
restarts
|
Character. With “always”, each warm-started fit is compared with a cold fit; “on_failure” uses a cold fit only when the warm fit fails; and “never” uses continuation only.
|
start
|
Optional starting values for the anchor model, as accepted by gamlss2.
|
monotone
|
Logical. If TRUE, returned fitted values and predictions are rearranged pointwise to be nondecreasing in tau.
|
verbose
|
Logical. Should progress information be printed? |
…
|
Further arguments passed to gamlss2.
|
Details
The asymmetric Laplace density is proportional to \(\exp\{-\rho_\tau((y-\mu)/\sigma)\}\), where \(\rho_\tau(u) = u\{\tau-I(u<0)\}\). Thus, \(\mu\) is the conditional \(\tau\)-quantile and \(\sigma>0\) is a scale parameter.
fit_ALD first estimates the anchor quantile and then proceeds toward the lower and upper requested quantiles. Coefficients and smoothing parameters are transferred between successive fits. Candidate fits are compared using their asymmetric Laplace log-likelihood.
Monotone rearrangement affects only the values returned by fitted() and predict(); the individual fitted models are retained unchanged.
Value
ALD returns an object of class “gamlss2.family”.
fit_ALD returns an object of class “ALDfit”, containing the individual models, fitted values, requested quantiles, and fitting diagnostics.
References
Yu K, Moyeed RA (2001). “Bayesian Quantile Regression.” Statistics & Probability Letters, 54(4), 437–447. doi:10.1016/S0167-7152(01)00124-9
Chernozhukov V, Fernandez-Val I, Galichon A (2010). “Quantile and Probability Curves Without Crossing.” Econometrica, 78(3), 1093–1125. doi:10.3982/ECTA7880
See Also
gamlss2, predict, mcmc.