R: Beta-binomial Distribution Family Function

betabinomialff {VGAM} | R Documentation

## 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 = "loge", lshape2 = "loge", ishape1 = 1,
                   ishape2 = NULL, imethod = 1, ishrinkage = 0.95,
                   nsimEIM = NULL, zero = NULL)

### 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.

### 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 ~ 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.

### 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.

### 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`.

### 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`.

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.

### 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

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

### Examples

    # 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 "loge" link for the 2 shape parameters is a logistic regression:
    all.equal(c(fitted(fit)),
              as.vector(logit(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]  # logit(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)

[Package VGAM version 1.0-3 Index]