R: QL/MM Estimation of Generalized Linear Models for...

aodql {aods3}

R Documentation

QL/MM Estimation of Generalized Linear Models for Overdispersed Count Data

Description

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

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

  aodql(formula,
        data,
        family = c("qbin", "qpois"),
        link = c("logit", "cloglog", "probit"),
        method = c("chisq", "dev"),
        phi = NULL,
        tol = 1e-5, ...)
        
  ## S3 method for class 'aodql'
anova(object, ...)
  ## S3 method for class 'aodql'
coef(object, ...)
  ## S3 method for class 'aodql'
deviance(object, ...)
  ## S3 method for class 'aodql'
df.residual(object, ...)
  ## S3 method for class 'aodql'
fitted(object, ...)
  ## S3 method for class 'aodql'
logLik(object, ...)
  ## S3 method for class 'aodql'
predict(object, ...)
  ## S3 method for class 'aodql'
print(x, ...)
  ## S3 method for class 'aodql'
residuals(object, ...)
  ## S3 method for class 'aodql'
summary(object, ...)
  ## S3 method for class 'aodql'
vcov(object, ...)

Arguments

`formula`	A formula for the mean `\mu`, defining the parameter vector `b` (see details). For binomial-type models, the left-hand side of the formula must be of the form `cbind(m, n - m) ~ ...` where the fitted proportion is `m/n`. For Poisson-type models, the left-hand side of the formula must be of the form `m ~ ...` where the fitted count is `m`. To fit a rate, argument `offset` must be used in the right-hand side of the formula (see examples).
`data`	A data frame containing the response (`m` and, optionnally, `n`) and the explanatory variable(s).
`family`	Define the model which is fitted: “qbin” for binomial-type models and “qpois” for Poisson-type models.
`link`	For binomial-type models only. Define the link function `g` for the mean `\mu`: “cloglog”, “logit” (default) or “probit”. For Poisson-type models, `link` is automatically set to “log”.
`method`	For function `aodql`, define the statistics used for the moment estimation of `phi`; legal values are “chisq” (default) for the chi-squared statistics or “dev” for the deviance statistics.
`phi`	An optional value defining the over-dispersion parameter `\Phi` if it is set as constant. Default to NULL (in that case, `\Phi` is estimated).
`tol`	A positive scalar (default to 0.001). The algorithm stops at iteration `r + 1` when the condition `\chi{^2}[r+1] - \chi{^2}[r] <= tol` is met by the `\chi^2` statistics .
`...`	Further arguments to passed to the appropriate functions.
`object`	An object of class “aodql”.
`x`	An object of class “aodql”.

Details

Binomial-type models

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

m | \lambda,n \sim Binomial(n, \lambda)

where \lambda follows a random variable of mean E[\lambda] = \mu and variance Var[\lambda] = \Phi * \mu * (1 - \mu). The marginal mean and variance of m are

E[m] = n * \mu

Var[m] = n * \mu * (1 - \mu) * (1 + (n - 1) * \Phi)

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

Poisson-type models

—— Simple counts (model with no offset)

For a given cluster (m), the model is

y | \lambda \sim Poisson(\lambda)

where \lambda follows a random distribution of mean \mu and variance \Phi * \mu^2. The mean and variance of the marginal distribution of m are

E[m] = \mu

Var[m] = \mu + \Phi * \mu^2

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

—— Rates (model with offset)

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

m | \lambda,n \sim Poisson(\lambda)

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

E[m | n] = \mu

Var[m | n] = \mu + \Phi * \mu^2

The response in aodql is y = m. The mean is E[y] = \mu, defined such as \mu = exp(X * b + log(n)) = exp(\nu + log(n)) = exp(\eta), where log(n) is the offset. The variance is Var[y] = \mu + \Phi * \mu^2.

Other details

Vector b and parameter \Phi 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), \Phi being set to a constant value. Then, \Phi 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 \Phi 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.

Examples


#------ Binomial-type models

data(orob2)
fm <- aodql(cbind(m, n - m) ~ seed, data = orob2, family = "qbin")
coef(fm)
vcov(fm)
summary(fm)
# chi2 tests of the seed factor in fm
wald.test(b = coef(fm), varb = vcov(fm), Terms = 2)

# chi-2 vs. deviance statistic to estimate phi
fm1 <- aodql(cbind(m, n - m) ~ seed + root, data = orob2, family = "qbin")
fm2 <- aodql(cbind(m, n - m) ~ seed + root, data = orob2, family = "qbin", method = "dev")
coef(fm1)
coef(fm2)
fm1$phi
fm2$phi
vcov(fm1)
vcov(fm2)
gof(fm1)
gof(fm2)

# estimate with fixed phi
fm <- aodql(cbind(m, n - m) ~ seed, data = orob2, family = "qbin", phi = 0.05)
coef(fm)
vcov(fm)
summary(fm)

#------ Poisson-type models

data(salmonella)
fm <- aodql(m ~ log(dose + 10) + dose, data = salmonella, family = "qpois")
coef(fm)
vcov(fm)
summary(fm)
# chi2 tests of the "log(dose + 10) + dose" factors
wald.test(b = coef(fm), varb = vcov(fm), Terms = 2:3)

# chi-2 vs. deviance statistic to estimate phi
fm1 <- aodql(m ~ log(dose + 10) + dose, data = salmonella, family = "qpois")
fm2 <- aodql(m ~ log(dose + 10) + dose, data = salmonella, family = "qpois", method = "dev")
coef(fm1)
coef(fm2)
fm1$phi
fm2$phi
vcov(fm1)
vcov(fm2)
gof(fm1)
gof(fm2)

# estimate with fixed phi
fm <- aodql(m ~ log(dose + 10) + dose, data = salmonella, family = "qpois", phi = 0.05)
coef(fm)
vcov(fm)
summary(fm)

# modelling a rate
data(dja)
# rate "m / trisk"
fm <- aodql(formula = m ~ group + offset(log(trisk)), data = dja, family = "qpois")
summary(fm)

[Package aods3 version 0.4-1.2 Index]