VGAM (version 1.1-9)

double.expbinomial: 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 = "logitlink", ldispersion = "logitlink",
                   idispersion = 0.25, zero = "dispersion")

Value

An object of class "vglmff" (see

vglmff-class). The object is used by modelling functions such as vglm.

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.

Author

T. W. Yee

Warning

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

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 \(\eta_1 = logit(\mu)\) and \(\eta_2 = logit(\theta)\). 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.

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 Sec.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 = extlogitlink(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 <- extlogitlink(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. Fig.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 = extlogitlink(min = 0, max = 1.25),
                          idisp = 0.2, zero = NULL),
       toxop, trace = TRUE, constraints = cmlist2)
if (FALSE)  par(mfrow = c(1, 2))  # Cf. Fig.1
plot(as(fit2, "vgam"), se = TRUE, lcol = "blue", scol = "orange")

# 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)))