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")

### 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).

### 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`, `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 = logit(-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)

[Package VGAM version 1.0-4 Index]