aodql: QL/MM Estimation of Generalized Linear Models for...
In aods3: Analysis of Overdispersed Data using S3 Methods
aodql 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.
See Also
glm
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)
aods3 documentation built on April 12, 2025, 1:48 a.m.
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| aodql | 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 |
data |
A data frame containing the response ( |
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 |
method |
For function |
phi |
An optional value defining the over-dispersion parameter |
tol |
A positive scalar (default to 0.001). The algorithm stops at iteration |
... |
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.
See Also
glm
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)
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