amlbinomial {VGAM} | 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="logit", earg=list())

### Arguments

In this documentation the word quantile can often be interchangeably replaced by expectile (things are informal here).

`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, earg` | 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.

### Author(s)

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.

### See Also

`amlpoisson`, `amlexponential`, `amlnormal`, `alaplace1`.

### Examples

    # Example: binomial data with lots of trials per observation
    set.seed(1234)
    sizevec = rep(100, length=(n <- 200))
    mydat = data.frame(x = sort(runif(n)))
    mydat = transform(mydat,
                      prob=logit(-0+2.5*x+x^2, inverse=TRUE))
    mydat = transform(mydat, y = rbinom(n, size=sizevec, prob=prob))
    mydat = transform(mydat, y = y / sizevec)  # Convert to proportions
    (fit = vgam(y ~ s(x, df=3), fam=amlbinomial(w=c(0.01,0.2,1,5,60)),
               data=mydat, weight=sizevec, 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, dig=1), collapse=", "),
               "percentile-expectile curves")))
    with(mydat, matlines(x, 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, dig=1), collapse=", "),
               "percentile curves are red")))
    with(mydat, matlines(x, fitted(fit), lwd=2, col="blue", lty=1))

    for(ii in fit@extra$percentile)
        with(mydat, matlines(x, qbinom(p=ii/100, size=sizevec, prob=prob)/sizevec,
                      col="red", lwd=2, lty=1))
    ## End(Not run)

[Package VGAM version 0.7-9 Index]