Binomial Logistic Regression by Asymmetric Maximum Likelihood Estimation
Binomial quantile regression estimated by maximizing an asymmetric likelihood function.
amlbinomial(w.aml = 1, parallel = FALSE, digw = 4, link = "logitlink")
w.aml |
Numeric, a vector of positive constants controlling the percentiles. The larger the value the larger the fitted percentile value (the proportion of points below the “w-regression plane”). The default value of unity results in the ordinary maximum likelihood (MLE) solution. |
parallel |
If |
digw |
Passed into |
link |
See |
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.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm
and vgam.
If w.aml has more than one value then the value returned by
deviance is the sum of all the (weighted) deviances taken over
all the w.aml values. See Equation (1.6) of Efron (1992).
On fitting, the extra slot has list components "w.aml"
and "percentile". The latter is the percent of observations
below the “w-regression plane”, which is the fitted values. Also,
the individual deviance values corresponding to each element of the
argument w.aml is stored in the extra slot.
For amlbinomial objects, methods functions for the generic
functions qtplot and cdf have not been written yet.
See amlpoisson about comments on the jargon, e.g.,
expectiles etc.
In this documentation the word quantile can often be interchangeably replaced by expectile (things are informal here).
Thomas W. Yee
Efron, B. (1992). Poisson overdispersion estimates based on the method of asymmetric maximum likelihood. Journal of the American Statistical Association, 87, 98–107.
# 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
## Not run:
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))
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.