# betabin

##### Beta-Binomial Model for Proportions

Fits a beta-binomial generalized linear model accounting for overdispersion in clustered binomial data \((n, y)\).

- Keywords
- regression

##### Usage

```
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), …)
```

##### Arguments

- formula
A formula for the fixed effects

`b`

. The left-hand side of the formula must be of the form`cbind(y, n - y)`

where the modelled probability is`y/n`

.- random
A right-hand formula for the overdispersion parameter(s) \(\phi\).

- link
The link function for the mean \(p\): “logit” or “cloglog”.

- data
A data frame containing the response (

`n`

and`y`

) and explanatory variable(s).- phi.ini
Initial values for the overdispersion parameter(s) \(\phi\). Default to 0.1.

- warnings
Logical to control the printing of warnings occurring during log-likelihood maximization. Default to

`FALSE`

(no printing).- 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 \((b, \phi)\) and the corresponding fixed values. For example,

`fixpar = list(c(4, 5), c(0, 0))`

means that 4th and 5th parameters of the model are set to 0.- hessian
A logical. When set to

`FALSE`

, the hessian and the variances-covariances matrices of the parameters are not computed.- control
A list to control the optimization parameters. See

`optim`

. By default, set the maximum number of iterations to 2000.- …
Further arguments passed to

`optim`

.

##### Details

For a given cluster \((n, y)\), the model is:
$$y~|~\lambda \sim Binomial(n,~\lambda)$$
with \(\lambda\) following a Beta distribution \(Beta(a1,~a2)\).
If \(B\) denotes the beta function, then:
$$P(\lambda) = \frac{\lambda^{a1~-~1} * (1~-~\lambda)^{a2 - 1}}{B(a1,~a2)}$$
$$E[\lambda] = \frac{a1}{a1 + a2}$$
$$Var[\lambda] = \frac{a1 * a2}{(a1 + a2 + 1) * (a1 + a2)^2}$$
The marginal beta-binomial distribution is:
$$P(y) = \frac{C(n,~y) * B(a1 + y, a2 + n - y)}{B(a1,~a2)}$$
The function uses the parameterization \(p = \frac{a1}{a1 + a2} = h(X b) = h(\eta)\) and \(\phi = \frac{1}{a1 + a2 + 1}\),
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, \(\eta = X b\) is the linear predictor and \(\phi\) is the overdispersion
parameter (i.e., the intracluster correlation coefficient, which is here restricted to be positive).
The marginal mean and variance are:
$$E[y] = n * p$$
$$Var[y] = n * p * (1 - p) * [1 + (n - 1) * \phi]$$
The parameters \(b\) and \(\phi\) 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 \(\phi\).

##### Value

An object of formal class “glimML”: see `glimML-class`

for details.

##### References

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.

##### See Also

`glimML-class`

, `glm`

and `optim`

##### Examples

```
# NOT RUN {
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)))
# }
```

*Documentation reproduced from package aod, version 1.3, License: GPL (>= 2)*