betabinomial {VGAM}R Documentation

Beta-binomial Distribution Family Function

Description

Fits a beta-binomial distribution by maximum likelihood estimation. The two parameters here are the mean and correlation coefficient.

Usage

betabinomial(lmu="logit", lrho="logit", emu=list(), erho=list(),
             irho=NULL, method.init=1, zero=2)

Arguments

lmu, lrho Link functions applied to the two parameters. See Links for more choices. The defaults ensure the parameters remain in (0,1).
emu, erho List. Extra argument for each of the links. See earg in Links for general information.
irho Optional initial value for the correlation parameter. If given, it must be in (0,1), and is recyled to the necessary length. Assign this argument a value if a convergence failure occurs. Having irho=NULL means an initial value is obtained internally, though this can give unsatisfactory results.
method.init An integer with value 1 or 2 which specifies the initialization method for mu. If failure to converge occurs try the other value and/or else specify a value for irho.
zero An integer specifying which linear/additive predictor is to be modelled as an intercept only. If assigned, the single value should be either 1 or 2. The default is to have a single correlation parameter. To model both parameters as functions of the covariates assign zero=NULL.

Details

There are several parameterizations of the beta-binomial distribution. This family function directly models the mean and correlation parameter, i.e., the probability of success. The model can be written T|P=p ~ Binomial(N,p) where P has a beta distribution with shape parameters alpha and beta. Here, N is the number of trials (e.g., litter size), T=NY is the number of successes, and p is the probability of a success (e.g., a malformation). That is, Y is the proportion of successes. Like binomialff, the fitted values are the estimated probability of success (i.e., E[Y] and not E[T]) and the prior weights N are attached separately on the object in a slot.

The probability function is

P(T=t) = choose(N,t) B(alpha+t, beta+N-t) / B(alpha, beta)

where t=0,1,...,N, and B is the beta function with shape parameters alpha and beta. Recall Y = T/N is the real response being modelled.

The default model is eta1 =logit(mu) and eta2 = logit(rho) because both parameters lie between 0 and 1. The mean (of Y) is p = mu = alpha / (alpha + beta) and the variance (of Y) is mu(1-mu)(1+(N-1)rho)/N. Here, the correlation rho is given by 1/(1 + alpha + beta) and is the correlation between the N individuals within a litter. A litter effect is typically reflected by a positive value of rho. It is known as the over-dispersion parameter.

This family function uses Fisher scoring. Elements of the second-order expected derivatives with respect to alpha and beta are computed numerically, which may fail for large alpha, beta, N or else take a long time.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm.
Suppose fit is a fitted beta-binomial model. Then fit@y contains the sample proportions y, fitted(fit) returns estimates of E(Y), and weights(fit, type="prior") returns the number of trials N.

Warning

This family function is prone to numerical difficulties due to the expected information matrices not being positive-definite or ill-conditioned over some regions of the parameter space. If problems occur try setting irho to some numerical value, or else use etastart argument of vglm, etc.

Note

This function processes the input in the same way as binomialff. But it does not handle the case N=1 very well because there are two parameters to estimate, not one, for each row of the input. Cases where N=1 can be omitted via the subset argument of vglm.

The extended beta-binomial distribution of Prentice (1986) is currently not implemented in the VGAM package as it has range-restrictions for the correlation parameter that are currently too difficult to handle in this package.

Author(s)

T. W. Yee

References

Moore, D. F. and Tsiatis, A. (1991) Robust estimation of the variance in moment methods for extra-binomial and extra-Poisson variation. Biometrics, 47, 383–401.

Prentice, R. L. (1986) Binary regression using an extended beta-binomial distribution, with discussion of correlation induced by covariate measurement errors. Journal of the American Statistical Association, 81, 321–327.

See Also

betabin.ab, Betabin, binomialff, betaff, dirmultinomial, lirat.

Examples

# Example 1
N = 10; mu = 0.5; rho = 0.8 
y = rbetabin(n=100, size=N, prob=mu, rho=rho)
fit = vglm(cbind(y,N-y) ~ 1, betabinomial, trace=TRUE)
coef(fit, matrix=TRUE)
Coef(fit)
cbind(fit@y, weights(fit, type="prior"))[1:5,]

# Example 2
data(lirat)
fit = vglm(cbind(R,N-R) ~ 1, betabinomial, data=lirat,
           trace=TRUE, subset=N>1)
coef(fit, matrix=TRUE)
Coef(fit)
t(fitted(fit))
t(fit@y)
t(weights(fit, type="prior"))

# Example 3, which is more complicated
lirat = transform(lirat, fgrp = factor(grp))
summary(lirat)   # Only 5 litters in group 3
fit2 = vglm(cbind(R,N-R) ~ fgrp + hb, betabinomial(zero=2),
           data=lirat, trace=TRUE, subset=N>1)
coef(fit2, matrix=TRUE)
## Not run: 
plot(lirat$hb[lirat$N>1], fit2@misc$rho,
     xlab="Hemoglobin", ylab="Estimated rho",
     pch=as.character(lirat$grp[lirat$N>1]),
     col=lirat$grp[lirat$N>1])
## End(Not run)
## Not run: 
data(lirat)
attach(lirat)
# cf. Figure 3 of Moore and Tsiatis (1991)
plot(hb, R/N, pch=as.character(grp), col=grp, las=1,
     xlab="Hemoglobin level", ylab="Proportion Dead",
     main="Fitted values (lines)")
detach(lirat)

smalldf = lirat[lirat$N>1,]
for(gp in 1:4) {
    xx = smalldf$hb[smalldf$grp==gp]
    yy = fitted(fit2)[smalldf$grp==gp]
    o = order(xx)
    lines(xx[o], yy[o], col=gp)
}
## End(Not run)

[Package VGAM version 0.7-7 Index]