gnmgee: Fit Nonlinear Generalized Estimating Equations
In glmtoolbox: Set of Tools to Data Analysis using Generalized Linear Models
gnmgee R Documentation
Fit Nonlinear Generalized Estimating Equations
Description
Produces an object of the class glmgee in which the main results of a Nonlinear Generalized Estimating Equation (GEE) fitted to the data are stored.
Usage
gnmgee(
formula,
family = gaussian(),
weights = NULL,
id,
waves,
data,
subset = NULL,
corstr,
corr,
start = NULL,
scale.fix = FALSE,
scale.value = 1,
toler = 1e-05,
maxit = 50,
trace = FALSE,
...
)
Arguments
formula
a nonlinear model formula including variables and parameters, which is a symbolic description of the nonlinear predictor of the model to be fitted to the data.
family
an (optional) family object, that is, a list of functions and expressions for defining link and variance functions. Families (and links) supported are the same supported by glm using its family argument, that is,
gaussian, binomial, poisson, Gamma, inverse.gaussian, and quasi. The family negative.binomial in the library MASS are also available. By default, the argument family is set to be gaussian(identity).
weights
an (optional) vector of positive "prior weights" to be used in the fitting process. The length of weights should be the same as the total number of observations.
id
a vector which identifies the subjects or clusters. The length of id should be the same as the number of observations.
waves
an (optional) positive integer-valued variable that is used to identify the order and spacing of observations within clusters. This argument is crucial when there are missing values and gaps in the data. By default, waves is equal to the integers from 1 to the size of each cluster.
data
an (optional) data frame in which to look for variables involved in the formula expression, as well as for variables specified in the arguments id and weights. The data are assumed to be sorted by id and time.
subset
an (optional) vector specifying a subset of observations to be used in the fitting process.
corstr
an (optional) character string which allows to specify the working-correlation structure. The available options are: "Independence", "Unstructured", "Stationary-M-dependent(m)", "Non-Stationary-M-dependent(m)", "AR-M-dependent(m)", "Exchangeable" and "User-defined", where m represents the lag of the dependence. By default, corstr is set to be "Independence".
corr
an (optional) square matrix of the same dimension of the maximum cluster size containing the user specified correlation. This is only appropriate if corstr is specified to be "User-defined".
start
an (optional) vector of starting values for the parameters in the nonlinear predictor. When start is
missing (and formula is not a self-starting model, see nls and selfStart), a very cheap guess for start is tried.
scale.fix
an (optional) logical variable. If TRUE, the scale parameter is fixed at the value of scale.value. By default, scale.fix is set to be FALSE.
scale.value
an (optional) numeric value at which the scale parameter should be fixed. This is only appropriate if scale.fix=TRUE. By default, scale.value is set to be 1.
toler
an (optional) positive value which represents the convergence tolerance. The convergence is reached when the maximum of the absolute relative differences between the values of the parameters in the nonlinear predictor in consecutive iterations of the fitting algorithm is lower than toler. By default, toler is set to be 0.00001.
maxit
an (optional) integer value which represents the maximum number of iterations allowed for the fitting algorithm. By default, maxit is set to be 50.
trace
an (optional) logical variable. If TRUE, output is produced for each iteration of the estimating algorithm.
...
further arguments passed to or from other methods.
Details
The values of the multivariate response variable measured on n subjects or clusters,
denoted by y_{i}=(y_{i1},\ldots,y_{in_i})^{\top} for i=1,\ldots,n, are assumed to be
realizations of independent random vectors denoted by Y_{i}=(Y_{i1},\ldots,Y_{in_i})^{\top}
for i=1,\ldots,n. The random variables associated to the i-th subject or
cluster, Y_{ij} for j=1,\ldots,n_i, are assumed to satisfy
\mu_{ij}= E(Y_{ij}),Var(Y_{ij})=\frac{\phi}{\omega_{ij}}V(\mu_{ij})
and Corr(Y_{ij},Y_{ik})=r_{jk}(\rho),
where \phi>0 is the dispersion parameter,
V(\mu_{ij}) is the variance function, \omega_{ij}>0 is a known weight, and
\rho=(\rho_1,\ldots,\rho_q)^{\top} is a parameter vector.
In addition, \mu_{ij} is assumed to be dependent on the regressors vector x_{ij}
by g(\mu_{ij})=z_{ij} + m(x_{ij},\beta), where g(\cdot) is the link function,
z_{ij} is a known offset, \beta=(\beta_1,\ldots,\beta_p)^{\top} is
a vector of regression parameters and m(x_{ij},\beta) is a known nonlinear function of \beta.
The parameter estimates are obtained by iteratively
solving the estimating equations described by Liang and Zeger (1986).
If the maximum cluster size is 6 and for a cluster of size 4 the value
of waves is set to be 2, 4, 5, 6, then it means that the data at
times 1 and 3 are missing, which should be taken into account by
gnmgee when the structure of the correlation matrix is assumed
to be "Unstructured", "Stationary-M-dependent", "Non-Stationary-M-dependent"
or "AR-M-dependent". If in this scenario waves is not specified
then gnmgee assumes that the available data for this cluster
were taken at times 1, 2, 3 and 4.
A set of standard extractor functions for fitted model objects is
available for objects of class glmgee, including methods to generic functions such as print, summary, model.matrix, estequa,
coef, vcov, logLik, fitted, confint and predict.
In addition, the model may be assessed using functions such as anova.glmgee,
residuals.glmgee, dfbeta.glmgee and cooks.distance.glmgee.
Value
an object of class glmgee in which the main results of the GEE model fitted to the data are stored, i.e., a
list with components including
coefficients a vector with the estimates of \beta_1,\ldots,\beta_p,
fitted.values a vector with the estimates of \mu_{ij} for i=1,\ldots,n and j=1,\ldots,n_i,
start a vector with the starting values used,
iter a numeric constant with the number of iterations,
prior.weights a vector with the values of \omega_{ij} for i=1,\ldots,n and j=1,\ldots,n_i,
offset a vector with the values of z_{ij} for i=1,\ldots,n and j=1,\ldots,n_i,
terms an object containing the terms objects,
loglik the value of the quasi-log-likelihood function evaluated at the parameter
estimates and the observed data,
estfun a vector with the estimating equations evaluated at the parameter
estimates and the observed data,
formula the formula,
levels the levels of the categorical regressors,
contrasts an object containing the contrasts corresponding to levels,
converged a logical indicating successful convergence,
model the full model frame,
y a vector with the values of y_{ij} for i=1,\ldots,n and j=1,\ldots,n_i,
family an object containing the family object used,
linear.predictors a vector with the estimates of g(\mu_{ij}) for i=1,\ldots,n and j=1,\ldots,n_i,
R a matrix with the (robust) estimate of the variance-covariance,
corr a matrix with the estimate of the working-correlation,
corstr a character string specifying the working-correlation structure,
id a vector which identifies the subjects or clusters,
sizes a vector with the values of n_i for i=1,\ldots,n,
call the original function call,
References
Liang K.Y., Zeger S.L. (1986) Longitudinal data analysis using generalized linear models.
Biometrika 73:13-22.
Zeger S.L., Liang K.Y. (1986) Longitudinal data analysis for discrete and continuous outcomes.
Biometrics 42:121-130.
Hardin J.W., Hilbe J.M. (2013) Generalized Estimating Equations. Chapman & Hall, London.
See Also
glmgee, wglmgee
Examples
###### Example : Orange trees grown at Riverside, California
data(Oranges)
mod <- Trunk ~ b1/(1 + exp((b2-Days)/b3))
start <- c(b1=200,b2=760,b3=375)
fit1 <- gnmgee(mod, start=start, id=Tree, family=Gamma(identity), corstr="Exchangeable",
data=Oranges)
summary(fit1, corr.digits=2)
mod <- Trunk ~ SSlogis(Days,b1,b2,b3)
fit2 <- gnmgee(mod, id=Tree, family=Gamma(identity), corstr="Exchangeable", data=Oranges)
summary(fit2, corr.digits=2)
glmtoolbox documentation built on Oct. 10, 2023, 9:06 a.m.
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| gnmgee | R Documentation |
Fit Nonlinear Generalized Estimating Equations
Description
Produces an object of the class glmgee in which the main results of a Nonlinear Generalized Estimating Equation (GEE) fitted to the data are stored.
Usage
gnmgee(
formula,
family = gaussian(),
weights = NULL,
id,
waves,
data,
subset = NULL,
corstr,
corr,
start = NULL,
scale.fix = FALSE,
scale.value = 1,
toler = 1e-05,
maxit = 50,
trace = FALSE,
...
)
Arguments
formula |
a nonlinear model |
family |
an (optional) |
weights |
an (optional) vector of positive "prior weights" to be used in the fitting process. The length of |
id |
a vector which identifies the subjects or clusters. The length of |
waves |
an (optional) positive integer-valued variable that is used to identify the order and spacing of observations within clusters. This argument is crucial when there are missing values and gaps in the data. By default, |
data |
an (optional) |
subset |
an (optional) vector specifying a subset of observations to be used in the fitting process. |
corstr |
an (optional) character string which allows to specify the working-correlation structure. The available options are: "Independence", "Unstructured", "Stationary-M-dependent(m)", "Non-Stationary-M-dependent(m)", "AR-M-dependent(m)", "Exchangeable" and "User-defined", where m represents the lag of the dependence. By default, |
corr |
an (optional) square matrix of the same dimension of the maximum cluster size containing the user specified correlation. This is only appropriate if |
start |
an (optional) vector of starting values for the parameters in the nonlinear predictor. When |
scale.fix |
an (optional) logical variable. If TRUE, the scale parameter is fixed at the value of |
scale.value |
an (optional) numeric value at which the scale parameter should be fixed. This is only appropriate if |
toler |
an (optional) positive value which represents the convergence tolerance. The convergence is reached when the maximum of the absolute relative differences between the values of the parameters in the nonlinear predictor in consecutive iterations of the fitting algorithm is lower than |
maxit |
an (optional) integer value which represents the maximum number of iterations allowed for the fitting algorithm. By default, |
trace |
an (optional) logical variable. If TRUE, output is produced for each iteration of the estimating algorithm. |
... |
further arguments passed to or from other methods. |
Details
The values of the multivariate response variable measured on n subjects or clusters,
denoted by y_{i}=(y_{i1},\ldots,y_{in_i})^{\top} for i=1,\ldots,n, are assumed to be
realizations of independent random vectors denoted by Y_{i}=(Y_{i1},\ldots,Y_{in_i})^{\top}
for i=1,\ldots,n. The random variables associated to the i-th subject or
cluster, Y_{ij} for j=1,\ldots,n_i, are assumed to satisfy
\mu_{ij}= E(Y_{ij}),Var(Y_{ij})=\frac{\phi}{\omega_{ij}}V(\mu_{ij})
and Corr(Y_{ij},Y_{ik})=r_{jk}(\rho),
where \phi>0 is the dispersion parameter,
V(\mu_{ij}) is the variance function, \omega_{ij}>0 is a known weight, and
\rho=(\rho_1,\ldots,\rho_q)^{\top} is a parameter vector.
In addition, \mu_{ij} is assumed to be dependent on the regressors vector x_{ij}
by g(\mu_{ij})=z_{ij} + m(x_{ij},\beta), where g(\cdot) is the link function,
z_{ij} is a known offset, \beta=(\beta_1,\ldots,\beta_p)^{\top} is
a vector of regression parameters and m(x_{ij},\beta) is a known nonlinear function of \beta.
The parameter estimates are obtained by iteratively
solving the estimating equations described by Liang and Zeger (1986).
If the maximum cluster size is 6 and for a cluster of size 4 the value
of waves is set to be 2, 4, 5, 6, then it means that the data at
times 1 and 3 are missing, which should be taken into account by
gnmgee when the structure of the correlation matrix is assumed
to be "Unstructured", "Stationary-M-dependent", "Non-Stationary-M-dependent"
or "AR-M-dependent". If in this scenario waves is not specified
then gnmgee assumes that the available data for this cluster
were taken at times 1, 2, 3 and 4.
A set of standard extractor functions for fitted model objects is
available for objects of class glmgee, including methods to generic functions such as print, summary, model.matrix, estequa,
coef, vcov, logLik, fitted, confint and predict.
In addition, the model may be assessed using functions such as anova.glmgee,
residuals.glmgee, dfbeta.glmgee and cooks.distance.glmgee.
Value
an object of class glmgee in which the main results of the GEE model fitted to the data are stored, i.e., a list with components including
coefficients | a vector with the estimates of \beta_1,\ldots,\beta_p, |
fitted.values | a vector with the estimates of \mu_{ij} for i=1,\ldots,n and j=1,\ldots,n_i, |
start | a vector with the starting values used, |
iter | a numeric constant with the number of iterations, |
prior.weights | a vector with the values of \omega_{ij} for i=1,\ldots,n and j=1,\ldots,n_i, |
offset | a vector with the values of z_{ij} for i=1,\ldots,n and j=1,\ldots,n_i, |
terms | an object containing the terms objects, |
loglik | the value of the quasi-log-likelihood function evaluated at the parameter |
| estimates and the observed data, | |
estfun | a vector with the estimating equations evaluated at the parameter |
| estimates and the observed data, | |
formula | the formula, |
levels | the levels of the categorical regressors, |
contrasts | an object containing the contrasts corresponding to levels, |
converged | a logical indicating successful convergence, |
model | the full model frame, |
y | a vector with the values of y_{ij} for i=1,\ldots,n and j=1,\ldots,n_i, |
family | an object containing the family object used, |
linear.predictors | a vector with the estimates of g(\mu_{ij}) for i=1,\ldots,n and j=1,\ldots,n_i, |
R | a matrix with the (robust) estimate of the variance-covariance, |
corr | a matrix with the estimate of the working-correlation, |
corstr | a character string specifying the working-correlation structure, |
id | a vector which identifies the subjects or clusters, |
sizes | a vector with the values of n_i for i=1,\ldots,n, |
call | the original function call, |
References
Liang K.Y., Zeger S.L. (1986) Longitudinal data analysis using generalized linear models. Biometrika 73:13-22.
Zeger S.L., Liang K.Y. (1986) Longitudinal data analysis for discrete and continuous outcomes. Biometrics 42:121-130.
Hardin J.W., Hilbe J.M. (2013) Generalized Estimating Equations. Chapman & Hall, London.
See Also
glmgee, wglmgee
Examples
###### Example : Orange trees grown at Riverside, California
data(Oranges)
mod <- Trunk ~ b1/(1 + exp((b2-Days)/b3))
start <- c(b1=200,b2=760,b3=375)
fit1 <- gnmgee(mod, start=start, id=Tree, family=Gamma(identity), corstr="Exchangeable",
data=Oranges)
summary(fit1, corr.digits=2)
mod <- Trunk ~ SSlogis(Days,b1,b2,b3)
fit2 <- gnmgee(mod, id=Tree, family=Gamma(identity), corstr="Exchangeable", data=Oranges)
summary(fit2, corr.digits=2)
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