betabinomialff: Beta-binomial Distribution Family Function

Description

Fits a beta-binomial distribution by maximum likelihood estimation. The two parameters here are the shape parameters of the underlying beta distribution.

Usage

betabinomialff(lshape1 = "loglink", lshape2 = "loglink",
   ishape1 = 1, ishape2 = NULL, imethod = 1, ishrinkage = 0.95,
   nsimEIM = NULL, zero = NULL)

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 (better: depvar(fit)) contains the sample proportions $y$, fitted(fit) returns estimates of

$E(Y)$, and weights(fit, type = "prior") returns the number of trials $N$.

Arguments

lshape1, lshape2: Link functions for the two (positive) shape parameters of the beta distribution. See Links for more choices.
ishape1, ishape2: Initial value for the shape parameters. The first must be positive, and is recyled to the necessary length. The second is optional. If a failure to converge occurs, try assigning a different value to ishape1 and/or using ishape2.
zero: Can be 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 model both shape parameters as functions of the covariates. If a failure to converge occurs, try zero = 2. See CommonVGAMffArguments for more information.
ishrinkage, nsimEIM, imethod: See CommonVGAMffArguments for more information. The argument ishrinkage is used only if imethod = 2. Using the argument nsimEIM may offer large advantages for large values of $N$ and/or large data sets.

Author

T. W. Yee

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 ishape1 to be some other positive value, using ishape2 and/or setting zero = 2.

This family function may be renamed in the future. See the warnings in betabinomial.

Details

There are several parameterizations of the beta-binomial distribution. This family function directly models the two shape parameters of the associated beta distribution rather than the probability of success (however, see Note below). 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 $\eta_1 = \log(\alpha)$ and $\eta_2 = \log(\beta)$ because both parameters are positive. 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. The two diagonal 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.

Examples

Run this code

# Example 1
N <- 10; s1 <- exp(1); s2 <- exp(2)
y <- rbetabinom.ab(n = 100, size = N, shape1 = s1, shape2 = s2)
fit <- vglm(cbind(y, N-y) ~ 1, betabinomialff, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)
head(fit@misc$rho)  # The correlation parameter
head(cbind(depvar(fit), weights(fit, type = "prior")))


# Example 2
fit <- vglm(cbind(R, N-R) ~ 1, betabinomialff, data = lirat,
            trace = TRUE, subset = N > 1)
coef(fit, matrix = TRUE)
Coef(fit)
fit@misc$rho  # The correlation parameter
t(fitted(fit))
t(depvar(fit))
t(weights(fit, type = "prior"))
# A "loglink" link for the 2 shape params is a logistic regression:
all.equal(c(fitted(fit)),
          as.vector(logitlink(predict(fit)[, 1] -
                          predict(fit)[, 2], inverse = TRUE)))


# 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, betabinomialff(zero = 2),
           data = lirat, trace = TRUE, subset = N > 1)
coef(fit2, matrix = TRUE)
coef(fit2, matrix = TRUE)[, 1] -
coef(fit2, matrix = TRUE)[, 2]  # logitlink(p)
if (FALSE)  with(lirat, plot(hb[N > 1], fit2@misc$rho,
   xlab = "Hemoglobin", ylab = "Estimated rho",
   pch = as.character(grp[N > 1]), col = grp[N > 1])) 
if (FALSE)   # cf. Figure 3 of Moore and Tsiatis (1991)
with(lirat, plot(hb, R / N, pch = as.character(grp), col = grp,
   xlab = "Hemoglobin level", ylab = "Proportion Dead", las = 1,
   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)
}

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