# aodql

0th

Percentile

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

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 $y = m$.

Keywords
regression
##### Usage
aodql(formula,
data,
family = c("qbin", "qpois"),
method = c("chisq", "dev"),
phi = NULL,
tol = 1e-5, ...)

# S3 method for aodql
anova(object, ...)
# S3 method for aodql
coef(object, ...)
# S3 method for aodql
deviance(object, ...)
# S3 method for aodql
df.residual(object, ...)
# S3 method for aodql
fitted(object, ...)
# S3 method for aodql
logLik(object, ...)
# S3 method for aodql
predict(object, ...)
# S3 method for aodql
print(x, ...)
# S3 method for aodql
residuals(object, ...)
# S3 method for aodql
summary(object, ...)
# S3 method for 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.

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.

glm

##### Aliases
• aodql
• anova.aodql
• coef.aodql
• deviance.aodql
• df.residual.aodql
• fitted.aodql
• logLik.aodql
• predict.aodql
• print.aodql
• residuals.aodql
• summary.aodql
• vcov.aodql
##### Examples
# NOT RUN {
#------ 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)

# }

Documentation reproduced from package aods3, version 0.4-1, License: GPL (>= 2)

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