In IQSS/Zelig: Everyone's Statistical Software
Built using Zelig version r packageVersion("Zelig")
knitr::opts_knit$set(
stop_on_error = 2L
)
knitr::opts_chunk$set(
fig.height = 11,
fig.width = 7
)
options(cite = FALSE)
Generalized Estimating Equation for Poisson Regression with poisson.gee.
The GEE poisson estimates the same model as the standard poisson
regression (appropriate when your dependent variable represents the
number of independent events that occur during a fixed period of time).
Unlike in poisson regression, GEE poisson allows for dependence within
clusters, such as in longitudinal data, although its use is not limited
to just panel data. The user must first specify a "working" correlation
matrix for the clusters, which models the dependence of each observation
with other observations in the same cluster. The "working" correlation
matrix is a $T \times T$ matrix of correlations, where $T$
is the size of the largest cluster and the elements of the matrix are
correlations between within-cluster observations. The appeal of GEE
models is that it gives consistent estimates of the parameters and
consistent estimates of the standard errors can be obtained using a
robust "sandwich" estimator even if the "working" correlation matrix is
incorrectly specified. If the "working" correlation matrix is correctly
specified, GEE models will give more efficient estimates of the
parameters. GEE models measure population-averaged effects as opposed to
cluster-specific effects.
Syntax
z.out <- zelig(Y ~ X1 + X2, model = "poisson.gee",
id = "X3", weights = w, data = mydata)
x.out <- setx(z.out)
s.out <- sim(z.out, x = x.out)
where id is a variable which identifies the clusters. The data
should be sorted by id and should be ordered within each cluster
when appropriate.
Additional Inputs
Use the following arguments to specify the structure of the "working"
correlations within clusters:
-
corstr: character string specifying the correlation structure:
"independence", "exchangeable", "ar1", "unstructured" and
"userdefined"
-
See geeglm in package geepack for other function arguments.
Examples
rm(list=ls(pattern="\\.out"))
suppressWarnings(suppressMessages(library(Zelig)))
set.seed(1234)
Example with Exchangeable Dependence
Attaching the sample turnout dataset:
data(sanction)
Variable identifying clusters
sanction$cluster <- c(rep(c(1:15), 5), rep(c(16), 3))
Sorting by cluster
sorted.sanction <- sanction[order(sanction$cluster),]
Estimating model and presenting summary:
z.out <- zelig(num ~ target + coop, model = "poisson.gee",
id = "cluster", data = sorted.sanction)
summary(z.out)
Set explanatory variables to their default values:
x.out <- setx(z.out)
Simulate quantities of interest
s.out <- sim(z.out, x = x.out)
summary(s.out)
Generate a plot of quantities of interest:
plot(s.out)
The Model
Suppose we have a panel dataset, with $Y_{it}$ denoting the
dependent variable of the number of independent events for a fixed
period of time for unit $i$ at time $t$. $Y_{i}$ is a
vector or cluster of correlated data where $y_{it}$ is correlated
with $y_{it^\prime}$ for some or all $t, t^\prime$. Note
that the model assumes correlations within $i$ but independence
across $i$.
- The stochastic component is given by the joint and marginal
distributions
$$
\begin{aligned}
Y_{i} &\sim& f(y_{i} \mid \lambda_{i})\
Y_{it} &\sim& g(y_{it} \mid \lambda_{it})
\end{aligned}
$$
where $f$ and $g$ are unspecified distributions with
means $\lambda_{i}$ and $\lambda_{it}$. GEE models make
no distributional assumptions and only require three specifications:
a mean function, a variance function, and a correlation structure.
- The systematic component is the mean function, given by:
$$
\lambda_{it} = \mathrm{exp}(x_{it} \beta)
$$
where $x_{it}$ is the vector of $k$ explanatory variables
for unit $i$ at time $t$ and $\beta$ is the vector
of coefficients.
- The variance function is given by:
$$
V_{it} = \lambda_{it}
$$
- The correlation structure is defined by a $T \times T$
"working" correlation matrix, where $T$ is the size of the
largest cluster. Users must specify the structure of the "working"
correlation matrix a priori. The "working" correlation matrix then
enters the variance term for each $i$, given by:
$$
V_{i} = \phi \, A_{i}^{\frac{1}{2}} R_{i}(\alpha) A_{i}^{\frac{1}{2}}
$$
where $A_{i}$ is a $T \times T$ diagonal matrix with the
variance function $V_{it} = \lambda_{it}$ as the $t$\ th
diagonal element, $R_{i}(\alpha)$ is the "working" correlation
matrix, and $\phi$ is a scale parameter. The parameters are
then estimated via a quasi-likelihood approach.
-
In GEE models, if the mean is correctly specified, but the variance
and correlation structure are incorrectly specified, then GEE models
provide consistent estimates of the parameters and thus the mean
function as well, while consistent estimates of the standard errors
can be obtained via a robust "sandwich" estimator. Similarly, if the
mean and variance are correctly specified but the correlation
structure is incorrectly specified, the parameters can be estimated
consistently and the standard errors can be estimated consistently
with the sandwich estimator. If all three are specified correctly,
then the estimates of the parameters are more efficient.
-
The robust "sandwich" estimator gives consistent estimates of the
standard errors when the correlations are specified incorrectly only
if the number of units $i$ is relatively large and the number
of repeated periods $t$ is relatively small. Otherwise, one
should use the "naïve" model-based standard errors, which assume that
the specified correlations are close approximations to the true
underlying correlations. See for more details.
Quantities of Interest
-
All quantities of interest are for marginal means rather than joint
means.
-
The method of bootstrapping generally should not be used in GEE
models. If you must bootstrap, bootstrapping should be done within
clusters, which is not currently supported in Zelig. For conditional
prediction models, data should be matched within clusters.
-
The expected values (qi$ev) for the GEE poisson model is the mean of
simulations from the stochastic component:
$$
E(Y) =
\lambda_{c} = \mathrm{exp}(x_{c} \beta),
$$
given draws of $\beta$ from its sampling distribution, where
$x_{c}$ is a vector of values, one for each independent
variable, chosen by the user.
- The first difference (qi$fd) for the GEE poisson model is defined as
$$
\textrm{FD} = \Pr(Y = 1 \mid x_1) - \Pr(Y = 1 \mid x).
$$
- In conditional prediction models, the average expected treatment
effect (att.ev) for the treatment group is
$$
\frac{1}{\sum_{i=1}^n \sum_{t=1}^T tr_{it}}\sum_{i:tr_{it}=1}^n \sum_{t:tr_{it}=1}^T \left{ Y_{it}(tr_{it}=1) -
E[Y_{it}(tr_{it}=0)] \right},
$$
where $tr_{it}$ is a binary explanatory variable defining the
treatment ($tr_{it}=1$) and control ($tr_{it}=0$) groups.
Variation in the simulations are due to uncertainty in simulating
$E[Y_{it}(tr_{it}=0)]$, the counterfactual expected value of
$Y_{it}$ for observations in the treatment group, under the
assumption that everything stays the same except that the treatment
indicator is switched to $tr_{it}=0$.
Output Values
The output of each Zelig command contains useful information which you
may view. For example, if you run
z.out <- zelig(y ~ x, model = poisson.gee, id, data), then you may
examine the available information in z.out by using
names(z.out), see the coefficients by using z.out$coefficients, and
a default summary of information through summary(z.out). Other
elements available through the $ operator are listed below.
From the zelig() output object z.out, you may extract:
-
coefficients: parameter estimates for the explanatory variables.
-
residuals: the working residuals in the final iteration of the
fit.
-
fitted.values: the vector of fitted values for the systemic
component, $\lambda_{it}$.
-
linear.predictors: the vector of $x_{it}\beta$
-
max.id: the size of the largest cluster.
From summary(z.out), you may extract:
-
coefficients: the parameter estimates with their associated
standard errors, $p$-values, and $z$-statistics.
-
working.correlation: the "working" correlation matrix
-
From the sim() output object s.out, you may extract quantities of
interest arranged as matrices indexed by simulation $\times$
x-observation (for more than one x-observation). Available quantities
are:
-
qi$ev: the simulated expected values for the specified values of
x.
-
qi$fd: the simulated first difference in the expected
probabilities for the values specified in x and x1.
-
qi$att.ev: the simulated average expected treatment effect for the
treated from conditional prediction models.
See also
The geeglm function is part of the geepack package by Søren Højsgaard,
Ulrich Halekoh and Jun Yan. Advanced users may wish to refer
to help(geepack) and help(family).
z5 <- zpoissongee$new()
z5$references()
IQSS/Zelig documentation built on Dec. 11, 2023, 1:51 a.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Built using Zelig version r packageVersion("Zelig")
knitr::opts_knit$set( stop_on_error = 2L ) knitr::opts_chunk$set( fig.height = 11, fig.width = 7 ) options(cite = FALSE)
Generalized Estimating Equation for Poisson Regression with poisson.gee.
The GEE poisson estimates the same model as the standard poisson regression (appropriate when your dependent variable represents the number of independent events that occur during a fixed period of time). Unlike in poisson regression, GEE poisson allows for dependence within clusters, such as in longitudinal data, although its use is not limited to just panel data. The user must first specify a "working" correlation matrix for the clusters, which models the dependence of each observation with other observations in the same cluster. The "working" correlation matrix is a $T \times T$ matrix of correlations, where $T$ is the size of the largest cluster and the elements of the matrix are correlations between within-cluster observations. The appeal of GEE models is that it gives consistent estimates of the parameters and consistent estimates of the standard errors can be obtained using a robust "sandwich" estimator even if the "working" correlation matrix is incorrectly specified. If the "working" correlation matrix is correctly specified, GEE models will give more efficient estimates of the parameters. GEE models measure population-averaged effects as opposed to cluster-specific effects.
Syntax
z.out <- zelig(Y ~ X1 + X2, model = "poisson.gee", id = "X3", weights = w, data = mydata) x.out <- setx(z.out) s.out <- sim(z.out, x = x.out)
where id is a variable which identifies the clusters. The data
should be sorted by id and should be ordered within each cluster
when appropriate.
Additional Inputs
Use the following arguments to specify the structure of the "working" correlations within clusters:
-
corstr: character string specifying the correlation structure: "independence", "exchangeable", "ar1", "unstructured" and "userdefined" -
See
geeglmin packagegeepackfor other function arguments.
Examples
rm(list=ls(pattern="\\.out")) suppressWarnings(suppressMessages(library(Zelig))) set.seed(1234)
Example with Exchangeable Dependence
Attaching the sample turnout dataset:
data(sanction)
Variable identifying clusters
sanction$cluster <- c(rep(c(1:15), 5), rep(c(16), 3))
Sorting by cluster
sorted.sanction <- sanction[order(sanction$cluster),]
Estimating model and presenting summary:
z.out <- zelig(num ~ target + coop, model = "poisson.gee", id = "cluster", data = sorted.sanction) summary(z.out)
Set explanatory variables to their default values:
x.out <- setx(z.out)
Simulate quantities of interest
s.out <- sim(z.out, x = x.out) summary(s.out)
Generate a plot of quantities of interest:
plot(s.out)
The Model
Suppose we have a panel dataset, with $Y_{it}$ denoting the dependent variable of the number of independent events for a fixed period of time for unit $i$ at time $t$. $Y_{i}$ is a vector or cluster of correlated data where $y_{it}$ is correlated with $y_{it^\prime}$ for some or all $t, t^\prime$. Note that the model assumes correlations within $i$ but independence across $i$.
- The stochastic component is given by the joint and marginal distributions
$$ \begin{aligned} Y_{i} &\sim& f(y_{i} \mid \lambda_{i})\ Y_{it} &\sim& g(y_{it} \mid \lambda_{it}) \end{aligned} $$
where $f$ and $g$ are unspecified distributions with means $\lambda_{i}$ and $\lambda_{it}$. GEE models make no distributional assumptions and only require three specifications: a mean function, a variance function, and a correlation structure.
- The systematic component is the mean function, given by:
$$ \lambda_{it} = \mathrm{exp}(x_{it} \beta) $$
where $x_{it}$ is the vector of $k$ explanatory variables for unit $i$ at time $t$ and $\beta$ is the vector of coefficients.
- The variance function is given by:
$$ V_{it} = \lambda_{it} $$
- The correlation structure is defined by a $T \times T$ "working" correlation matrix, where $T$ is the size of the largest cluster. Users must specify the structure of the "working" correlation matrix a priori. The "working" correlation matrix then enters the variance term for each $i$, given by:
$$ V_{i} = \phi \, A_{i}^{\frac{1}{2}} R_{i}(\alpha) A_{i}^{\frac{1}{2}} $$
where $A_{i}$ is a $T \times T$ diagonal matrix with the variance function $V_{it} = \lambda_{it}$ as the $t$\ th diagonal element, $R_{i}(\alpha)$ is the "working" correlation matrix, and $\phi$ is a scale parameter. The parameters are then estimated via a quasi-likelihood approach.
-
In GEE models, if the mean is correctly specified, but the variance and correlation structure are incorrectly specified, then GEE models provide consistent estimates of the parameters and thus the mean function as well, while consistent estimates of the standard errors can be obtained via a robust "sandwich" estimator. Similarly, if the mean and variance are correctly specified but the correlation structure is incorrectly specified, the parameters can be estimated consistently and the standard errors can be estimated consistently with the sandwich estimator. If all three are specified correctly, then the estimates of the parameters are more efficient.
-
The robust "sandwich" estimator gives consistent estimates of the standard errors when the correlations are specified incorrectly only if the number of units $i$ is relatively large and the number of repeated periods $t$ is relatively small. Otherwise, one should use the "naïve" model-based standard errors, which assume that the specified correlations are close approximations to the true underlying correlations. See for more details.
Quantities of Interest
-
All quantities of interest are for marginal means rather than joint means.
-
The method of bootstrapping generally should not be used in GEE models. If you must bootstrap, bootstrapping should be done within clusters, which is not currently supported in Zelig. For conditional prediction models, data should be matched within clusters.
-
The expected values (qi$ev) for the GEE poisson model is the mean of simulations from the stochastic component:
$$ E(Y) = \lambda_{c} = \mathrm{exp}(x_{c} \beta), $$
given draws of $\beta$ from its sampling distribution, where $x_{c}$ is a vector of values, one for each independent variable, chosen by the user.
- The first difference (qi$fd) for the GEE poisson model is defined as
$$ \textrm{FD} = \Pr(Y = 1 \mid x_1) - \Pr(Y = 1 \mid x). $$
- In conditional prediction models, the average expected treatment effect (att.ev) for the treatment group is
$$ \frac{1}{\sum_{i=1}^n \sum_{t=1}^T tr_{it}}\sum_{i:tr_{it}=1}^n \sum_{t:tr_{it}=1}^T \left{ Y_{it}(tr_{it}=1) - E[Y_{it}(tr_{it}=0)] \right}, $$
where $tr_{it}$ is a binary explanatory variable defining the treatment ($tr_{it}=1$) and control ($tr_{it}=0$) groups. Variation in the simulations are due to uncertainty in simulating $E[Y_{it}(tr_{it}=0)]$, the counterfactual expected value of $Y_{it}$ for observations in the treatment group, under the assumption that everything stays the same except that the treatment indicator is switched to $tr_{it}=0$.
Output Values
The output of each Zelig command contains useful information which you
may view. For example, if you run
z.out <- zelig(y ~ x, model = poisson.gee, id, data), then you may
examine the available information in z.out by using
names(z.out), see the coefficients by using z.out$coefficients, and
a default summary of information through summary(z.out). Other
elements available through the $ operator are listed below.
From the zelig() output object z.out, you may extract:
-
coefficients: parameter estimates for the explanatory variables.
-
residuals: the working residuals in the final iteration of the fit.
-
fitted.values: the vector of fitted values for the systemic component, $\lambda_{it}$.
-
linear.predictors: the vector of $x_{it}\beta$
-
max.id: the size of the largest cluster.
From summary(z.out), you may extract:
-
coefficients: the parameter estimates with their associated standard errors, $p$-values, and $z$-statistics.
-
working.correlation: the "working" correlation matrix
-
From the
sim()output object s.out, you may extract quantities of interest arranged as matrices indexed by simulation $\times$ x-observation (for more than one x-observation). Available quantities are: -
qi$ev: the simulated expected values for the specified values of x. -
qi$fd: the simulated first difference in the expected probabilities for the values specified inxandx1. -
qi$att.ev: the simulated average expected treatment effect for the treated from conditional prediction models.
See also
The geeglm function is part of the geepack package by Søren Højsgaard,
Ulrich Halekoh and Jun Yan. Advanced users may wish to refer
to help(geepack) and help(family).
z5 <- zpoissongee$new() z5$references()
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
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