# 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 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-
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 $\chi{^2}[r+1] - \chi{^2}[r] ... Further arguments to passed to the appropriate functions. object An object of class aodql. x An object of class aodql. ##### Details Binomial-type models For a given cluster$(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. ##### encoding latin1 ##### 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 ##### 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 #------ 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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