R: Double Exponential Binomial Distribution Family Function

double.expbinomial {VGAM} | R Documentation

## Double Exponential Binomial Distribution Family Function

### Description

Fits a double exponential binomial distribution by maximum likelihood estimation. The two parameters here are the mean and dispersion parameter.

### Usage

    double.expbinomial(lmean = "logit", ldispersion = "logit",
                       idispersion = 0.25, zero = "dispersion")

### Arguments

`lmean, ldispersion` | Link functions applied to the two parameters, called mu and theta respectively below. See `Links` for more choices. The defaults cause the parameters to be restricted to (0,1).
`idispersion` | Initial value for the dispersion parameter. If given, it must be in range, and is recyled to the necessary length. Use this argument if convergence failure occurs.
`zero` | A vector specifying which linear/additive predictor is to be modelled as intercept-only. If assigned, the single value can be either `1` or `2`. The default is to have a single dispersion parameter value. To model both parameters as functions of the covariates assign `zero = NULL`. See `CommonVGAMffArguments` for more details.

### Details

This distribution provides a way for handling overdispersion in a binary response. The double exponential binomial distribution belongs the family of double exponential distributions proposed by Efron (1986). Below, equation numbers refer to that original article. Briefly, the idea is that an ordinary one-parameter exponential family allows the addition of a second parameter theta which varies the dispersion of the family without changing the mean. The extended family behaves like the original family with sample size changed from n to n*theta. The extended family is an exponential family in mu when n and theta are fixed, and an exponential family in theta when n and mu are fixed. Having 0 < theta < 1 corresponds to overdispersion with respect to the binomial distribution. See Efron (1986) for full details.

This VGAM family function implements an approximation (2.10) to the exact density (2.4). It replaces the normalizing constant by unity since the true value nearly equals 1. The default model fitted is eta1 =logit(mu) and eta2 = logit(theta). This restricts both parameters to lie between 0 and 1, although the dispersion parameter can be modelled over a larger parameter space by assigning the arguments `ldispersion` and `edispersion`.

Approximately, the mean (of Y) is mu. The effective sample size is the dispersion parameter multiplied by the original sample size, i.e., n*theta. This family function uses Fisher scoring, and the two estimates are asymptotically independent because the expected information matrix is diagonal.

### Value

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

### Warning

Numerical difficulties can occur; if so, try using `idispersion`.

### Note

This function processes the input in the same way as `binomialff`, however multiple responses are not allowed (`binomialff(multiple.responses = FALSE)`).

### Author(s)

T. W. Yee

### References

Efron, B. (1986) Double exponential families and their use in generalized linear regression. Journal of the American Statistical Association, 81, 709–721.

### See Also

`binomialff`, `toxop`, `CommonVGAMffArguments`.

### Examples

    # This example mimics the example in Efron (1986).
    # The results here differ slightly.

    # Scale the variables
    toxop <- transform(toxop,
                       phat = positive / ssize,
                       srainfall = scale(rainfall),  # (6.1)
                       sN = scale(ssize))            # (6.2)

    # A fit similar (should be identical) to Section 6 of Efron (1986).
    # But does not use poly(), and M = 1.25 here, as in (5.3)
    cmlist <- list("(Intercept)"    = diag(2),
                   "I(srainfall)"   = rbind(1, 0),
                   "I(srainfall^2)" = rbind(1, 0),
                   "I(srainfall^3)" = rbind(1, 0),
                   "I(sN)" = rbind(0, 1),
                   "I(sN^2)" = rbind(0, 1))
    fit <- vglm(cbind(phat, 1 - phat) * ssize ~
                I(srainfall) + I(srainfall^2) + I(srainfall^3) +
                I(sN) + I(sN^2),
                double.expbinomial(ldisp = extlogit(min = 0, max = 1.25),
                             idisp = 0.2, zero = NULL),
                toxop, trace = TRUE, constraints = cmlist)

    # Now look at the results
    coef(fit, matrix = TRUE)
    head(fitted(fit))
    summary(fit)
    vcov(fit)
    sqrt(diag(vcov(fit)))  # Standard errors

    # Effective sample size (not quite the last column of Table 1)
    head(predict(fit))
    Dispersion <- extlogit(predict(fit)[,2], min = 0, max = 1.25, inverse = TRUE)
    c(round(weights(fit, type = "prior") * Dispersion, digits = 1))

    # Ordinary logistic regression (gives same results as (6.5))
    ofit <- vglm(cbind(phat, 1 - phat) * ssize ~
                 I(srainfall) + I(srainfall^2) + I(srainfall^3),
                 binomialff, toxop, trace = TRUE)

    # Same as fit but it uses poly(), and can be plotted (cf. Figure 1)
    cmlist2 <- list("(Intercept)"                 = diag(2),
                    "poly(srainfall, degree = 3)" = rbind(1, 0),
                    "poly(sN, degree = 2)"        = rbind(0, 1))
    fit2 <- vglm(cbind(phat, 1 - phat) * ssize ~
                 poly(srainfall, degree = 3) + poly(sN, degree = 2),
                 double.expbinomial(ldisp = extlogit(min = 0, max = 1.25),
                              idisp = 0.2, zero = NULL),
                 toxop, trace = TRUE, constraints = cmlist2)
    ## Not run:  par(mfrow = c(1, 2))
    plot(as(fit2, "vgam"), se = TRUE, lcol = "blue", scol = "orange")  # Cf. Figure 1

    # Cf. Figure 1(a)
    par(mfrow = c(1,2))
    ooo <- with(toxop, sort.list(rainfall))
    with(toxop, plot(rainfall[ooo], fitted(fit2)[ooo], type = "l",
                     col = "blue", las = 1, ylim = c(0.3, 0.65)))
    with(toxop, points(rainfall[ooo], fitted(ofit)[ooo], col = "orange",
                       type = "b", pch = 19))

    # Cf. Figure 1(b)
    ooo <- with(toxop, sort.list(ssize))
    with(toxop, plot(ssize[ooo], Dispersion[ooo], type = "l", col = "blue",
                     las = 1, xlim = c(0, 100)))
    ## End(Not run)

[Package VGAM version 1.0-3 Index]