Beta-binomial Distribution Family Function
Fits a beta-binomial distribution by maximum likelihood estimation. The two parameters here are the shape parameters of the underlying beta distribution.
betabinomialff(lshape1 = "loglink", lshape2 = "loglink", ishape1 = 1,
ishape2 = NULL, imethod = 1, ishrinkage = 0.95,
nsimEIM = NULL, zero = NULL)lshape1, lshape2 |
Link functions for the two (positive) shape parameters
of the beta distribution.
See |
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 |
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
|
ishrinkage, nsimEIM, imethod |
See |
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 ~ 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 = log(alpha) and eta2 = 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.
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.
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.
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.
Although the two linear/additive predictors given above are in terms of alpha and beta, basic algebra shows that the default amounts to fitting a logit link to the probability of success; subtracting the second linear/additive predictor from the first gives that logistic regression linear/additive predictor. That is, logit(p) = eta1 - eta2. This is illustated in one of the examples below.
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.
T. W. Yee
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.
# 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 parameters 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)
## Not run: with(lirat, plot(hb[N > 1], fit2@misc$rho,
xlab = "Hemoglobin", ylab = "Estimated rho",
pch = as.character(grp[N > 1]), col = grp[N > 1]))
## End(Not run)
## Not run: # 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)
}
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.