VGAM (version 1.1-14)

amlbinomial: Binomial Logistic Regression by Asymmetric Maximum Likelihood Estimation

Description

Binomial quantile regression estimated by maximizing an asymmetric likelihood function.

Usage

amlbinomial(w.aml = 1, parallel = FALSE, digw = 4, link = "logitlink")

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm

and vgam.

Details

The general methodology behind this VGAM family function is given in Efron (1992) and full details can be obtained there. This model is essentially a logistic regression model (see binomialff) but the usual deviance is replaced by an asymmetric squared error loss function; it is multiplied by \(w.aml\) for positive residuals. The solution is the set of regression coefficients that minimize the sum of these deviance-type values over the data set, weighted by the weights argument (so that it can contain frequencies). Newton-Raphson estimation is used here.

References

Efron, B. (1992). Poisson overdispersion estimates based on the method of asymmetric maximum likelihood. Journal of the American Statistical Association, 87, 98--107.

See Also

amlpoisson, amlexponential, amlnormal, extlogF1, alaplace1, denorm.

Examples

# Example: binomial data with lots of trials per observation
set.seed(1234)
sizevec <- rep(100, length = (nn <- 200))
mydat <- data.frame(x = sort(runif(nn)))
mydat <- transform(mydat,
                   prob = logitlink(-0 + 2.5*x + x^2, inverse = TRUE))
mydat <- transform(mydat, y = rbinom(nn, size = sizevec, prob = prob))
(fit <- vgam(cbind(y, sizevec - y) ~ s(x, df = 3),
             amlbinomial(w = c(0.01, 0.2, 1, 5, 60)),
             mydat, trace = TRUE))
fit@extra

if (FALSE) {
par(mfrow = c(1,2))
# Quantile plot
with(mydat, plot(x, jitter(y), col = "blue", las = 1, main =
     paste(paste(round(fit@extra$percentile, digits = 1), collapse = ", "),
           "percentile-expectile curves")))
with(mydat, matlines(x, 100 * fitted(fit), lwd = 2, col = "blue", lty=1))

# Compare the fitted expectiles with the quantiles
with(mydat, plot(x, jitter(y), col = "blue", las = 1, main =
     paste(paste(round(fit@extra$percentile, digits = 1), collapse = ", "),
           "percentile curves are red")))
with(mydat, matlines(x, 100 * fitted(fit), lwd = 2, col = "blue", lty = 1))

for (ii in fit@extra$percentile)
    with(mydat, matlines(x, 100 *
         qbinom(p = ii/100, size = sizevec, prob = prob) / sizevec,
                  col = "red", lwd = 2, lty = 1))
}