# amlbinomial: Binomial Logistic Regression by Asymmetric Maximum Likelihood... In VGAM: Vector Generalized Linear and Additive Models

 amlbinomial R Documentation

## 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")
```

### Arguments

 `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 `w.aml` has more than one value then this argument allows the quantile curves to differ by the same amount as a function of the covariates. Setting this to be `TRUE` should force the quantile curves to not cross (although they may not cross anyway). See `CommonVGAMffArguments` for more information. `digw ` Passed into `Round` as the `digits` argument for the `w.aml` values; used cosmetically for labelling. `link` See `binomialff`.

### 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.

### Value

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

### Warning

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).

### Note

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

### References

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`.

### 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

## 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)
```

VGAM documentation built on July 6, 2022, 5:05 p.m.