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)
Poisson Regression for Event Count Dependent Variables with poisson
.
Use the Poisson regression model if the observations of your dependent variable represents the number of independent events that occur during a fixed period of time (see the negative binomial model, , for over-dispersed event counts.) For a Bayesian implementation of this model, see .
z.out <- zelig(Y ~ X1 + X2, model = "poisson", weights = w, data = mydata) x.out <- setx(z.out) s.out <- sim(z.out, x = x.out)
rm(list=ls(pattern="\\.out")) suppressWarnings(suppressMessages(library(Zelig))) set.seed(1234)
Load sample data:
data(sanction)
Estimate Poisson model:
z.out <- zelig(num ~ target + coop, model = "poisson", data = sanction)
summary(z.out)
Set values for the explanatory variables to their default mean values, while changing the $coop$ variable across its range:
x.low <- setx(z.out, coop = 1) x.high <- setx1(z.out, coop = 4)
Simulate fitted values:
s.out <- sim(z.out, x = x.low, x1=x.high) summary(s.out)
plot(s.out)
Let $Y_i$ be the number of independent events that occur during a fixed time period. This variable can take any non-negative integer.
$$ Y_i \; \sim \; \textrm{Poisson}(\lambda_i), $$
where $\lambda_i$ is the mean and variance parameter.
$$ \lambda_i \; = \; \exp(x_i \beta), $$
where $x_i$ is the vector of explanatory variables, and $\beta$ is the vector of coefficients.
$$ E(Y) = \lambda_i = \exp(x_i \beta), $$
given draws of $\beta$ from its sampling distribution.
The predicted value (qi$pr) is a random draw from the poisson distribution defined by mean $\lambda_i$.
The first difference in the expected values (qi$fd) is given by:
$$ \textrm{FD} \; = \; E(Y | x_1) - E(Y \mid x) $$
$$ \frac{1}{\sum_{i=1}^n t_i}\sum_{i:t_i=1}^n \left{ Y_i(t_i=1) - E[Y_i(t_i=0)] \right}, $$
where $t_i$ is a binary explanatory variable defining the treatment ($t_i=1$) and control ($t_i=0$) groups. Variation in the simulations are due to uncertainty in simulating $E[Y_i(t_i=0)]$, the counterfactual expected value of $Y_i$ for observations in the treatment group, under the assumption that everything stays the same except that the treatment indicator is switched to $t_i=0$.
$$ \frac{1}{\sum_{i=1}^n t_i}\sum_{i:t_i=1}^n \left{ Y_i(t_i=1) - \widehat{Y_i(t_i=0)} \right}, $$
where $t_i$ is a binary explanatory variable defining the treatment ($t_i=1$) and control ($t_i=0$) groups. Variation in the simulations are due to uncertainty in simulating $\widehat{Y_i(t_i=0)}$, the counterfactual predicted value of $Y_i$ for observations in the treatment group, under the assumption that everything stays the same except that the treatment indicator is switched to $t_i=0$.
The Zelig object stores fields containing everything needed to rerun the Zelig output, and all the results and simulations as they are generated. In addition to the summary commands demonstrated above, some simply utility functions (known as getters) provide easy access to the raw fields most commonly of use for further investigation.
In the example above z.out$get_coef()
returns the estimated coefficients, z.out$get_vcov()
returns the estimated covariance matrix, and z.out$get_predict()
provides predicted values for all observations in the dataset from the analysis.
The poisson model is part of the stats package by the R Core Team. Advanced users may
wish to refer to help(glm)
and help(family)
.
z5 <- zpoisson$new() z5$references()
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