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)
Probit Regression for Dichotomous Dependent Variables with probit
.
Use probit regression to model binary dependent variables specified as a function of a set of explanatory variables.
z.out <- zelig(Y ~ X1 + X2, model = "probit", weights = w, data = mydata) x.out <- setx(z.out) s.out <- sim(z.out, x = x.out, x1 = NULL)
rm(list=ls(pattern="\\.out")) suppressWarnings(suppressMessages(library(Zelig))) set.seed(1234)
Attach the sample turnout dataset:
data(turnout)
Estimate parameter values for the probit regression:
z.out <- zelig(vote ~ race + educate, model = "probit", data = turnout)
summary(z.out)
Set values for the explanatory variables to their default values.
x.out <- setx(z.out)
Simulate quantities of interest from the posterior distribution.
s.out <- sim(z.out, x = x.out)
summary(s.out)
plot(s.out)
Let $Y_i$ be the observed binary dependent variable for observation $i$ which takes the value of either 0 or 1.
$$ Y_i \; \sim \; \textrm{Bernoulli}(\pi_i), $$
where $\pi_i=\Pr(Y_i=1)$.
$$ \pi_i \; = \; \Phi (x_i \beta) $$
where $\Phi(\mu)$ is the cumulative distribution function of the Normal distribution with mean 0 and unit variance.
$$ E(Y) = \pi_i = \Phi(x_i \beta), $$
given a draw of $\beta$ from its sampling distribution.
The predicted value (qi$pr) is a draw from a Bernoulli distribution with mean $\pi_i$.
The first difference (qi$fd) in expected values is defined as
$$ \textrm{FD} = \Pr(Y = 1 \mid x_1) - \Pr(Y = 1 \mid x). $$
$$ \textrm{RR} = \Pr(Y = 1 \mid x_1) / \Pr(Y = 1 \mid x). $$
$$ \frac{1}{\sum_{i=1}^n t_i}\sum_{i:t_i=1}^n \left{ Y_i(t_i=1) - E[Y_i(t_i=0)] \right}, $$
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$get_predict()
provides predicted values for all observations in the dataset from the analysis.
The probit model is part of the stats package by the R Core Team. Advanced users may
wish to refer to help(glm)
and help(family)
.
z5 <- zprobit$new() z5$references()
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