VGAM (version 0.8-1)

betabinomial: 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, shrinkage.init=0.95, nsimEIM=NULL, zero=2)

Arguments

lmu, lrho

Link functions applied to the two parameters. See Links for more choices. The defaults ensure the parameters remain in $(0,1)$, however, see the warning below.

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 obta

method.init

An integer with value 1 or 2 or ..., which specifies the initialization method for $\mu$. If failure to converge occurs try the another 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 par

shrinkage.init, nsimEIM

See CommonVGAMffArguments for more information. The argument shrinkage.init is used only if method.init=2. Using the argument nsimEIM may offer large

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

If the estimated rho parameter is close to zero then it pays to try lrho="rhobit". One day this may become the default link function.

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, nsimEIM=100, say, or else use etastart argument of vglm, etc.

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 $T|P=p \sim Binomial(N,p)$ 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 $$P(T=t) = {N \choose t} \frac{B(\alpha+t, \beta+N-t)} {B(\alpha, \beta)}$$ where $t=0,1,\ldots,N$, 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 $\eta_1 = logit(\mu)$ and $\eta_2 = logit(\rho)$ 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.

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
betabindat = data.frame(N = 10, mu = 0.5, rho = 0.8)
betabindat = transform(betabindat,
                       y = rbetabin(n=100, size=N, prob=mu, rho=rho))
fit = vglm(cbind(y,N-y) ~ 1, betabinomial, betabindat, trace=TRUE)
coef(fit, matrix=TRUE)
Coef(fit)
head(cbind(fit@y, weights(fit, type="prior")))


# Example 2
fit = vglm(cbind(R,N-R)~1, betabinomial, 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)
with(lirat, plot(hb[N>1], fit2@misc$rho,
                 xlab="Hemoglobin", ylab="Estimated rho",
                 pch=as.character(grp[N>1]), col=grp[N>1]))
# cf. Figure 3 of Moore and Tsiatis (1991)
with(lirat, plot(hb, R/N, pch=as.character(grp), col=grp, las=1,
                 xlab="Hemoglobin level", ylab="Proportion Dead",
                 main="Fitted values (lines)"))
smalldf = with(lirat, lirat[N>1,])
for(gp in 1:4) {
    xx = with(smalldf, hb[grp==gp])
    yy = with(smalldf, fitted(fit2)[grp==gp])
    ooo = order(xx)
    lines(xx[ooo], yy[ooo], col=gp)
}