VGAM (version 1.0-3)

binomialff: Binomial Family Function

Description

Family function for fitting generalized linear models to binomial responses, where the dispersion parameter may be known or unknown.

Usage

binomialff(link = "logit", dispersion = 1, multiple.responses = FALSE,
           onedpar = !multiple.responses, parallel = FALSE,
           zero = NULL, bred = FALSE, earg.link = FALSE)

Arguments

link

Link function; see Links and CommonVGAMffArguments for more information.

dispersion

Dispersion parameter. By default, maximum likelihood is used to estimate the model because it is known. However, the user can specify dispersion = 0 to have it estimated, or else specify a known positive value (or values if multiple.responses is TRUE).

multiple.responses

Multivariate response? If TRUE, then the response is interpreted as \(M\) independent binary responses, where \(M\) is the number of columns of the response matrix. In this case, the response matrix should have \(Q\) columns consisting of counts (successes), and the weights argument should have \(Q\) columns consisting of the number of trials (successes plus failures).

If FALSE and the response is a (2-column) matrix, then the number of successes is given in the first column, and the second column is the number of failures.

onedpar

One dispersion parameter? If multiple.responses, then a separate dispersion parameter will be computed for each response (column), by default. Setting onedpar = TRUE will pool them so that there is only one dispersion parameter to be estimated.

parallel

A logical or formula. Used only if multiple.responses is TRUE. This argument allows for the parallelism assumption whereby the regression coefficients for a variable is constrained to be equal over the \(M\) linear/additive predictors. If parallel = TRUE then the constraint is not applied to the intercepts.

zero

An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. The values must be from the set {1,2,…,\(M\)}, where \(M\) is the number of columns of the matrix response. See CommonVGAMffArguments for more information.

earg.link

Details at CommonVGAMffArguments.

bred

Details at CommonVGAMffArguments. Setting bred = TRUE should work for multiple responses (multiple.responses = TRUE) and all VGAM link functions; it has been tested for logit only (and it gives similar results to brglm but not identical), and further testing is required. One result from fitting bias reduced binary regression is that finite regression coefficients occur when the data is separable (see example below).

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, vgam, rrvglm, cqo, and cao.

Warning

With a multivariate response, assigning a known dispersion parameter for each response is not handled well yet. Currently, only a single known dispersion parameter is handled well.

See the above note regarding bred.

The maximum likelihood estimate will not exist if the data is completely separable or quasi-completely separable. See Chapter 10 of Altman et al. (2004) for more details, and safeBinaryRegression. Yet to do: add a sepcheck = TRUE, say, argument to detect this problem and give an appropriate warning.

Details

This function is largely to mimic binomial, however there are some differences.

If the dispersion parameter is unknown, then the resulting estimate is not fully a maximum likelihood estimate (see pp.124--8 of McCullagh and Nelder, 1989).

A dispersion parameter that is less/greater than unity corresponds to under-/over-dispersion relative to the binomial model. Over-dispersion is more common in practice.

Setting multiple.responses = TRUE is necessary when fitting a Quadratic RR-VGLM (see cqo) because the response is a matrix of \(M\) columns (e.g., one column per species). Then there will be \(M\) dispersion parameters (one per column of the response matrix).

When used with cqo and cao, it may be preferable to use the cloglog link.

References

McCullagh, P. and Nelder, J. A. (1989) Generalized Linear Models, 2nd ed. London: Chapman & Hall.

Altman, M. and Gill, J. and McDonald, M. P. (2004) Numerical Issues in Statistical Computing for the Social Scientist, Hoboken, NJ, USA: Wiley-Interscience.

Ridout, M. S. (1990) Non-convergence of Fisher's method of scoring---a simple example. GLIM Newsletter, 20(6).

See Also

quasibinomialff, Links, rrvglm, cqo, cao, betabinomial, posbinomial, zibinomial, double.expbinomial, seq2binomial, amlbinomial, simplex, binomial, simulate.vlm, safeBinaryRegression.

Examples

# NOT RUN {
quasibinomialff()
quasibinomialff(link = "probit")

shunua <- hunua[sort.list(with(hunua, altitude)), ]  # Sort by altitude
fit <- vglm(agaaus ~ poly(altitude, 2), binomialff(link = cloglog),
            data = shunua)
# }
# NOT RUN {
plot(agaaus ~ jitter(altitude), shunua, col = "blue", ylab = "P(Agaaus = 1)",
     main = "Presence/absence of Agathis australis", las = 1)
with(shunua, lines(altitude, fitted(fit), col = "orange", lwd = 2)) 
# }
# NOT RUN {

# Fit two species simultaneously
fit2 <- vgam(cbind(agaaus, kniexc) ~ s(altitude),
             binomialff(multiple.responses = TRUE), data = shunua)
# }
# NOT RUN {
with(shunua, matplot(altitude, fitted(fit2), type = "l",
     main = "Two species response curves", las = 1)) 
# }
# NOT RUN {

# Shows that Fisher scoring can sometime fail. See Ridout (1990).
ridout <- data.frame(v = c(1000, 100, 10), r = c(4, 3, 3), n = c(5, 5, 5))
(ridout <- transform(ridout, logv = log(v)))
# The iterations oscillates between two local solutions:
glm.fail <- glm(r / n ~ offset(logv) + 1, weight = n,
               binomial(link = 'cloglog'), ridout, trace = TRUE)
coef(glm.fail)
# vglm()'s half-stepping ensures the MLE of -5.4007 is obtained:
vglm.ok <- vglm(cbind(r, n-r) ~ offset(logv) + 1,
               binomialff(link = cloglog), ridout, trace = TRUE)
coef(vglm.ok)


# Separable data
set.seed(123)
threshold <- 0
bdata <- data.frame(x2 = sort(rnorm(nn <- 100)))
bdata <- transform(bdata, y1 = ifelse(x2 < threshold, 0, 1))
fit <- vglm(y1 ~ x2, binomialff(bred = TRUE),
            data = bdata, criter = "coef", trace = TRUE)
coef(fit, matrix = TRUE)  # Finite!!
summary(fit)
# }
# NOT RUN {
 plot(depvar(fit) ~ x2, data = bdata, col = "blue", las = 1)
lines(fitted(fit) ~ x2, data = bdata, col = "orange")
abline(v = threshold, col = "gray", lty = "dashed") 
# }