R: Beta-binomial Distribution Family Function

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", irho = NULL, imethod = 1,
                 ishrinkage = 0.95, nsimEIM = NULL, zero = "rho")

### 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.
`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.
`imethod` | 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` | Specifyies which linear/additive predictor is to be modelled as an intercept only. If assigned, the single value can 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`. See `CommonVGAMffArguments` for more information.
`ishrinkage, nsimEIM` | 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.

### 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

If the estimated rho parameter is close to 0 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.

### 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. However, try `lrho = "rhobit"`.

### 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

`betabinomialff`, `Betabinom`, `binomialff`, `betaff`, `dirmultinomial`, `lirat`, `simulate.vlm`.

### Examples

    # Example 1
    bdata <- data.frame(N = 10, mu = 0.5, rho = 0.8)
    bdata <- transform(bdata,
                       y = rbetabinom(n = 100, size = N, prob = mu, rho = rho))
    fit <- vglm(cbind(y, N-y) ~ 1, betabinomial, data = bdata, trace = TRUE)
    coef(fit, matrix = TRUE)
    Coef(fit)
    head(cbind(depvar(fit), 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(depvar(fit))
    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:  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)

[Package VGAM version 1.0-3 Index]