amlbinomial | R Documentation |

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.

`amlpoisson`

,
`amlexponential`

,
`amlnormal`

,
`extlogF1`

,
`alaplace1`

,
`denorm`

.

```
# 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)
```

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