Beta-Binomial Model for Proportions
Fits a beta-binomial generalized linear model accounting for overdispersion in clustered binomial data (n, y).
betabin(formula, random, data, link = c("logit", "cloglog"), phi.ini = NULL,
warnings = FALSE, na.action = na.omit, fixpar = list(),
hessian = TRUE, control = list(maxit = 2000), ...)formula |
A formula for the fixed effects |
random |
A right-hand formula for the overdispersion parameter(s) φ. |
link |
The link function for the mean p: “logit” or “cloglog”. |
data |
A data frame containing the response ( |
phi.ini |
Initial values for the overdispersion parameter(s) φ. Default to 0.1. |
warnings |
Logical to control the printing of warnings occurring during log-likelihood maximization.
Default to |
na.action |
A function name: which action should be taken in the case of missing value(s). |
fixpar |
A list with 2 components (scalars or vectors) of the same size, indicating which parameters are
fixed (i.e., not optimized) in the global parameter vector (b, φ) and the corresponding fixed values. |
hessian |
A logical. When set to |
control |
A list to control the optimization parameters. See |
... |
Further arguments passed to |
For a given cluster (n, y), the model is:
y | λ ~ Binomial(n, λ)
with λ following a Beta distribution Beta(a1, a2).
If B denotes the beta function, then:
P(λ) = λ^{a1 - 1} * (1 - λ)^{a2 - 1} / B(a1, a2)
E[λ] = a1 / (a1 + a2)
Var[λ] = a1 * a2 / [(a1 + a2 + 1) * (a1 + a2)^2]
The marginal beta-binomial distribution is:
P(y) = C(n, y) * B(a1 + y, a2 + n - y) / B(a1, a2)
The function uses the parameterization
p = a1 / (a1 + a2) = h(X b) = h(η) and φ = 1 / (a1 + a2 + 1),
where h is the inverse of the link function (logit or complementary log-log), X is a design-matrix, b
is a vector of fixed effects, η = X b is the linear predictor and φ is the overdispersion
parameter (i.e., the intracluster correlation coefficient, which is here restricted to be positive).
The marginal mean and variance are:
E[y] = n * p
Var[y] = n * p * (1 - p) * [1 + (n - 1) * φ]
The parameters b and φ are estimated by maximizing the log-likelihood of the marginal model (using the
function optim). Several explanatory variables are allowed in b, only one in φ.
An object of formal class “glimML”: see glimML-class for details.
Matthieu Lesnoff matthieu.lesnoff@cirad.fr, Renaud Lancelot renaud.lancelot@cirad.fr
Crowder, M.J., 1978. Beta-binomial anova for proportions. Appl. Statist. 27, 34-37.
Griffiths, D.A., 1973. Maximum likelihood estimation for the beta-binomial distribution and an application
to the household distribution of the total number of cases of disease. Biometrics 29, 637-648.
Prentice, R.L., 1986. Binary regression using an extended beta-binomial distribution, with discussion of
correlation induced by covariate measurement errors. J.A.S.A. 81, 321-327.
Williams, D.A., 1975. The analysis of binary responses from toxicological experiments involving
reproduction and teratogenicity. Biometrics 31, 949-952.
glimML-class, glm and optim
data(orob2)
fm1 <- betabin(cbind(y, n - y) ~ seed, ~ 1, data = orob2)
fm2 <- betabin(cbind(y, n - y) ~ seed + root, ~ 1, data = orob2)
fm3 <- betabin(cbind(y, n - y) ~ seed * root, ~ 1, data = orob2)
# show the model
fm1; fm2; fm3
# AIC
AIC(fm1, fm2, fm3)
summary(AIC(fm1, fm2, fm3), which = "AICc")
# Wald test for root effect
wald.test(b = coef(fm3), Sigma = vcov(fm3), Terms = 3:4)
# likelihood ratio test for root effect
anova(fm1, fm3)
# model predictions
New <- expand.grid(seed = levels(orob2$seed),
root = levels(orob2$root))
data.frame(New, predict(fm3, New, se = TRUE, type = "response"))
# Djallonke sheep data
data(dja)
betabin(cbind(y, n - y) ~ group, ~ 1, dja)
# heterogeneous phi
betabin(cbind(y, n - y) ~ group, ~ group, dja,
control = list(maxit = 1000))
# phi fixed to zero in group TREAT
betabin(cbind(y, n - y) ~ group, ~ group, dja,
fixpar = list(4, 0))
# glim without overdispersion
summary(glm(cbind(y, n - y) ~ group,
family = binomial, data = dja))
# phi fixed to zero in both groups
betabin(cbind(y, n - y) ~ group, ~ group, dja,
fixpar = list(c(3, 4), c(0, 0)))Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.