*Built using Zelig version 5.1.2.9000*

Bayesian Poisson Regression with `poisson.bayes`

.

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. The model is fit using a random walk Metropolis algorithm. For a maximum-likelihood estimation of this model see poisson.

Use the following argument to monitor the Markov chain:

`burnin`

: number of the initial MCMC iterations to be discarded (defaults to 1,000).`mcmc`

: number of the MCMC iterations after burnin (defaults to 10,000).`thin`

: thinning interval for the Markov chain. Only every`thin`

-th draw from the Markov chain is kept. The value of`mcmc`

must be divisible by this value. The default value is 1.`tune`

: Metropolis tuning parameter, either a positive scalar or a vector of length \(k\), where \(k\) is the number of coefficients. The tuning parameter should be set such that the acceptance rate of the Metropolis algorithm is satisfactory (typically between 0.20 and 0.5). The default value is 1.1.`verbose`

: default to FALSE. If`TRUE`

, the progress of the sampler (every \(10\%\)) is printed to the screen.`seed`

: seed for the random number generator. The default is`NA`

which corresponds to a random seed of 12345.`beta.start`

: starting values for the Markov chain, either a scalar or vector with length equal to the number of estimated coefficients. The default is`NA`

, such that the maximum likelihood estimates are used as the starting values.

Use the following parameters to specify the model’s priors:

`b0`

: prior mean for the coefficients, either a numeric vector or a scalar. If a scalar, that value will be the prior mean for all the coefficients. The default is 0.`B0`

: prior precision parameter for the coefficients, either a square matrix (with the dimensions equal to the number of the coefficients) or a scalar. If a scalar, that value times an identity matrix will be the prior precision parameter. The default is 0, which leads to an improper prior.

Zelig users may wish to refer to `help(MCMCpoisson)`

for more information.

Attaching the sample dataset:

`data(sanction)`

Estimating the Poisson regression using `poisson.bayes`

:

```
z.out <- zelig(num ~ target + coop, model = "poisson.bayes",
data = sanction, verbose = FALSE)
```

```
## How to cite this model in Zelig:
## Ben Goodrich, and Ying Lu. 2013.
## poisson-bayes: Bayesian Poisson Regression
## in Christine Choirat, Christopher Gandrud, James Honaker, Kosuke Imai, Gary King, and Olivia Lau,
## "Zelig: Everyone's Statistical Software," http://zeligproject.org/
```

You can check for convergence before summarizing the estimates with three diagnostic tests. See the section Diagnostics for Zelig Models for examples of the output with interpretation:

```
z.out$geweke.diag()
z.out$heidel.diag()
z.out$raftery.diag()
```

`summary(z.out)`

```
## Model:
##
## Iterations = 1001:11000
## Thinning interval = 1
## Number of chains = 1
## Sample size per chain = 10000
##
## 1. Empirical mean and standard deviation for each variable,
## plus standard error of the mean:
##
## Mean SD Naive SE Time-series SE
## (Intercept) -0.97979 0.17464 0.0017464 0.005591
## target -0.01762 0.05667 0.0005667 0.001885
## coop 1.21090 0.04702 0.0004702 0.001520
##
## 2. Quantiles for each variable:
##
## 2.5% 25% 50% 75% 97.5%
## (Intercept) -1.3310 -1.09338 -0.97766 -0.86116 -0.64556
## target -0.1299 -0.05657 -0.01953 0.02025 0.09443
## coop 1.1208 1.17889 1.21008 1.24150 1.30664
##
## Next step: Use 'setx' method
```

Setting values for the explanatory variables to their sample averages:

`x.out <- setx(z.out)`

Simulating quantities of interest from the posterior distribution given `x.out`

.

`s.out1 <- sim(z.out, x = x.out)`

`summary(s.out1)`

```
##
## sim x :
## -----
## ev
## mean sd 50% 2.5% 97.5%
## [1,] 3.235291 0.2375714 3.234844 2.777139 3.704941
## pv
## mean sd 50% 2.5% 97.5%
## [1,] 3.2301 1.795014 3 0 7
```

Estimating the first difference in the number of countries imposing sanctions when the number of targets is set to be its maximum versus its minimum :

```
x.max <- setx(z.out, target = max(sanction$target))
x.min <- setx(z.out, target = min(sanction$target))
```

```
s.out2 <- sim(z.out, x = x.max, x1 = x.min)
summary(s.out2)
```

```
##
## sim x :
## -----
## ev
## mean sd 50% 2.5% 97.5%
## [1,] 3.191614 0.2936585 3.183013 2.642371 3.803733
## pv
## mean sd 50% 2.5% 97.5%
## [1,] 3.2136 1.798638 3 0 7
##
## sim x1 :
## -----
## ev
## mean sd 50% 2.5% 97.5%
## [1,] 3.306252 0.3059862 3.300095 2.729466 3.944022
## pv
## mean sd 50% 2.5% 97.5%
## [1,] 3.3323 1.846515 3 0 7
## fd
## mean sd 50% 2.5% 97.5%
## [1,] 0.1146376 0.3671544 0.1265036 -0.6072035 0.8342282
```

Let \(Y_{i}\) be the number of independent events that occur during a fixed time period.

- The
*stochastic component*is given by

\[ \begin{aligned} Y_{i} & \sim & \textrm{Poisson}(\lambda_i) \end{aligned} \]

where \(\lambda_i\) is the mean and variance parameter.

- The
*systematic component*is given by

\[ \begin{aligned} \lambda_{i}= \exp(x_{i} \beta) \end{aligned} \]

where \(x_{i}\) is the vector of \(k\) explanatory variables for observation \(i\) and \(\beta\) is the vector of coefficients.

- The
*prior*for \(\beta\) is given by

\[ \begin{aligned} \beta \sim \textrm{Normal}_k \left( b_{0},B_{0}^{-1}\right) \end{aligned} \]

where \(b_{0}\) is the vector of means for the \(k\) explanatory variables and \(B_{0}\) is the \(k \times k\) precision matrix (the inverse of a variance-covariance matrix).

- The expected values (
`qi$ev`

) for the Poisson model are calculated as following:

\[ \begin{aligned} E(Y\mid X) = \lambda_i = \exp(x_i \beta), \end{aligned} \]

given the posterior draws of \(\beta\) based on the MCMC iterations.

The predicted values (

`qi$pr`

) are draws from the Poisson distribution with parameter \(\lambda_i\).The first difference (

`qi$fd`

) for the Poisson model is defined as

\[ \begin{aligned} \text{FD}=E(Y\mid X_{1})-E(Y\mid X). \end{aligned} \]

- In conditional prediction models, the average expected treatment effect (
`qi$att.ev`

) for the treatment group is

\[ \begin{aligned} \frac{1}{\sum_{i=1}^n t_{i}}\sum_{i:t_{i}=1}\{Y_{i}(t_{i}=1)-E[Y_{i}(t_{i}=0)]\}, \end{aligned} \]

where \(t_{i}\) is a binary explanatory variable defining the treatment (\(t_{i}=1\)) and control (\(t_{i}=0\)) groups.

- In conditional prediction models, the average predicted treatment effect (
`qi$att.pr`

) for the treatment group is

\[ \begin{aligned} \frac{1}{\sum_{i=1}^n t_{i}}\sum_{i:t_{i}=1}[Y_{i}(t_{i}=1)-\widehat{Y_{i}(t_{i}=0)}], \end{aligned} \]

where \(t_{i}\) is a binary explanatory variable defining the treatment (\(t_{i}=1\)) and control (\(t_{i}=0\)) groups.

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.bayes", data)`

you may examine the available information in `z.out`

by using `names(z.out)`

, see the draws from the posterior distribution of the `coefficients`

by using `z.out$coefficients`

, and view 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`

: draws from the posterior distributions of the estimated parameters.`zelig.data`

: the input data frame if save.data = TRUE.`seed`

: the random seed used in the model.From the

`sim()`

output object`s.out`

:`qi$ev`

: the simulated expected values for the specified values of`x`

.`qi$pr`

: the simulated predicted values for the specified values of`x`

.`qi$fd`

: the simulated first difference in the expected values for the values specified in`x`

and`x1`

.`qi$att.ev`

: the simulated average expected treatment effect for the treated from conditional prediction models.`qi$att.pr`

: the simulated average predicted treatment effect for the treated from conditional prediction models.

Bayesian poisson regression is part of the MCMCpack library by Andrew D. Martin and Kevin M. Quinn . The convergence diagnostics are part of the CODA library by Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines.

Martin AD, Quinn KM and Park JH (2011). “MCMCpack: Markov Chain Monte Carlo in R.” *Journal of Statistical Software*, **42** (9), pp. 22. <URL: http://www.jstatsoft.org/v42/i09/>.