Description

From clustered data, the function fits generalized linear models containing an over-dispersion parameter Φ using quasi-likelihood estimating equations for the mean μ and a moment estimator for Φ.

For binomial-type models, data have the form {(n_1, m_1), (n_2, m_2), ..., (n_N, m_N)}, where n_i is the size of cluster i, m_i the number of “successes”, and N the number of clusters. The response is the proportion y = m/n.

For Poisson-type models, data can be of two forms. When modeling “simple counts”, data have the form {m_1, m_2, ..., m_N}, where m_i is the number of occurences of the event under study. When modeling rates (e.g. hazard rates), data have the same form as for the BB model, where n_i is the denominator of the rate for cluster i (considered as an “offset”, i.e. a constant known value) and m_i the number of occurences of the event. For both forms of data, the response is the count y = m.

Usage

Arguments

Details

Binomial-type models

For a given cluster (n, m), the model is

m | λ,n \sim Binomial(n, λ)

where λ follows a random variable of mean E[λ] = μ and variance Var[λ] = Φ * μ * (1 - μ). The marginal mean and variance of m are

E[m] = n * μ

Var[m] = n * μ * (1 - μ) * (1 + (n - 1) * Φ)

The response in aodql is y = m/n. The mean is E[y] = μ, defined such as μ = g^{-1}(X * b) = g^{-1}(ν), where g is the link function, X is a design-matrix, b a vector of fixed effects and ν = X * b is the corresponding linear predictor. The variance is Var[y] = (1 / n) * μ * (1 - μ) * (1 + (n - 1) * Φ).

Poisson-type models

—— Simple counts (model with no offset)

For a given cluster (m), the model is

y | λ \sim Poisson(λ)

where λ follows a random distribution of mean μ and variance Φ * μ^2. The mean and variance of the marginal distribution of m are

E[m] = μ

Var[m] = μ + Φ * μ^2

The response in aodql is y = m. The mean is E[y] = μ, defined such as μ = exp(X * b) = exp(ν). The variance is Var[y] = μ + Φ * μ^2.

—— Rates (model with offset)

For a given cluster (n, m), the model is

m | λ,n \sim Poisson(λ)

where λ follows the same random distribution as for the case with no offset. The marginal mean and variance are

E[m | n] = μ

Var[m | n] = μ + Φ * μ^2

The response in aodql is y = m. The mean is E[y] = μ, defined such as μ = exp(X * b + log(n)) = exp(ν + log(n)) = exp(η), where log(n) is the offset. The variance is Var[y] = μ + Φ * μ^2.

Other details

Vector b and parameter Φ are estimated iteratively, using procedures referred to as "Model I" in Williams (1982) for binomial-type models, and "Procedure II" in Breslow (1984) for Poisson-type models.

Iterations are as follows. Quasi-likelihood estimating equations (McCullagh & Nelder, 1989) are used to estimate b (using function glm and its weights argument), Φ being set to a constant value. Then, Φ is calculated by the moment estimator, obtained by equalizing the goodness-of-fit statistic (chi-squared X2 or deviance D) of the model to its degrees of freedom.
Parameter Φ can be set as constant, using argument phi. In that case, only b is estimated.

Value

An object of class aodql, printed and summarized by various functions.

References

Breslow, N.E., 1984. Extra-Poisson variation in log-linear models. Appl. Statist. 33, 38-44.
Moore, D.F., 1987, Modelling the extraneous variance in the presence of extra-binomial variation. Appl. Statist. 36, 8-14.
Moore, D.F., Tsiatis, A., 1991. Robust estimation of the variance in moment methods for extra-binomial and extra-poisson variation. Biometrics 47, 383-401. McCullagh, P., Nelder, J. A., 1989, 2nd ed. Generalized linear models. New York, USA: Chapman and Hall.
Williams, D.A., 1982, Extra-binomial variation in logistic linear models. Appl. Statist. 31, 144-148.

See Also

glm

Examples