eco (version 3.0-1)

eco: Fitting the Parametric Bayesian Model of Ecological Inference in 2x2 Tables

Description

eco is used to fit the parametric Bayesian model (based on a Normal/Inverse-Wishart prior) for ecological inference in $2 \times 2$ tables via Markov chain Monte Carlo. It gives the in-sample predictions as well as the estimates of the model parameters. The model and algorithm are described in Imai, Lu and Strauss (2006). The contextual effect can also be modeled by following the strategy described in Imai, Lu, and Strauss (2006).

Usage

eco(formula, data = parent.frame(), N = NULL, supplement = NULL, 
    context = FALSE, mu0 = 0, tau0 = 2, nu0 = 4, S0 = 10,
    mu.start = 0, Sigma.start = 10, parameter = TRUE,
    grid = FALSE, n.draws = 5000, burnin = 0, thin = 0, 
    verbose = FALSE)

Arguments

formula

A symbolic description of the model to be fit, specifying the column and row margins of $2 \times 2$ ecological tables. Y ~ X specifies Y as the column margin and X as the row margin. Details and specif

data

An optional data frame in which to interpret the variables in formula. The default is the environment in which eco is called.

N

An optional variable representing the size of the unit; e.g., the total number of voters.

supplement

An optional matrix of supplemental data. The matrix has two columns, which contain additional individual-level data such as survey data for $W_1$ and $W_2$, respectively. If NULL, no additional individual-level data are included

context

Logical. If TRUE, the contextual effect is also modeled. See Imai and Lu (2005) for details. The default is FALSE.

mu0

A scalar or a numeric vector that specifies the prior mean for the mean parameter $\mu$. If it is a scalar, then its value will be repeated to yield a vector of the length of $\mu$, otherwise, it needs to be a vector of same length as $\mu$.

tau0

A positive integer representing the prior scale for the mean parameter $\mu$. The default is 2.

nu0

A positive integer representing the prior degrees of freedom of the variance matrix $\Sigma$. the default is 4.

S0

A positive scalar or a positive definite matrix that specifies the prior scale matrix for the variance matrix $\Sigma$. If it is a scalar, then the prior scale matrix will be a diagonal matrix with the same dimensions as $\Sigma$ and the diagonal

mu.start

A scalar or a numeric vector that specifies the starting values of the mean parameter $\mu$. If it is a scalar, then its value will be repeated to yield a vector of the length of $\mu$, otherwise, it needs to be a vector of same length

Sigma.start

A scalar or a positive definite matrix that specified the starting value of the variance matrix $\Sigma$. If it is a scalar, then the prior scale matrix will be a diagonal matrix with the same dimensions as $\Sigma$ and the diagonal elements

parameter

Logical. If TRUE, the Gibbs draws of the population parameters, $\mu$ and $\Sigma$, are returned in addition to the in-sample predictions of the missing internal cells, $W$. The default is TRUE.

grid

Logical. If TRUE, the grid method is used to sample $W$ in the Gibbs sampler. If FALSE, the Metropolis algorithm is used where candidate draws are sampled from the uniform distribution on the tomography line for each

n.draws

A positive integer. The number of MCMC draws. The default is 5000.

burnin

A positive integer. The burnin interval for the Markov chain; i.e. the number of initial draws that should not be stored. The default is 0.

thin

A positive integer. The thinning interval for the Markov chain; i.e. the number of Gibbs draws between the recorded values that are skipped. The default is 0.

verbose

Logical. If TRUE, the progress of the Gibbs sampler is printed to the screen. The default is FALSE.

Value

• An object of class eco containing the following elements:
• call
  The matched call.
• X
  The row margin, $X$.
• Y
  The column margin, $Y$.
• N
  The size of each table, $N$.
• burnin
  The number of initial burnin draws.
• thin
  The thinning interval.
• nu0
  The prior degrees of freedom.
• tau0
  The prior scale parameter.
• mu0
  The prior mean.
• S0
  The prior scale matrix.
• W
  A three dimensional array storing the posterior in-sample predictions of $W$. The first dimension indexes the Monte Carlo draws, the second dimension indexes the columns of the table, and the third dimension represents the observations.
• Wmin
  A numeric matrix storing the lower bounds of $W$.
• Wmax
  A numeric matrix storing the upper bounds of $W$.
• The following additional elements are included in the output when parameter = TRUE.
• mu
  The posterior draws of the population mean parameter, $\mu$.
• Sigma
  The posterior draws of the population variance matrix, $\Sigma$.

Details

An example of $2 \times 2$ ecological table for racial voting is given below: lccc{ black voters white voters Voted $W_{1i}$ $W_{2i}$ $Y_i$ Not voted $1-W_{1i}$ $1-W_{2i}$ $1-Y_i$ $X_i$ $1-X_i$ } where $Y_i$ and $X_i$ represent the observed margins, and $W_1$ and $W_2$ are unknown variables. All variables are proportions and hence bounded between 0 and 1. For each $i$, the following deterministic relationship holds, $Y_i=X_i W_{1i}+(1-X_i)W_{2i}$.

References

Imai, Kosuke, Ying Lu and Aaron Strauss. (2006) Bayesian and Likelihood Inference for 2 x 2 Ecological Tables: An Incomplete Data Approach Technical Report, Princeton University, available at http://imai.princeton.edu/research/eiall.html

See Also

ecoML, ecoNP, predict.eco, summary.eco

Examples

## load the registration data
data(reg)

## NOTE: convergence has not been properly assessed for the following
## examples. See Imai, Lu and Strauss (2006) for more complete analyses.

## fit the parametric model with the default prior specification
res <- eco(Y ~ X, data = reg, verbose = TRUE)
## summarize the results
summary(res)

## obtain out-of-sample prediction
out <- predict(res, verbose = TRUE)
## summarize the results
summary(out)

## load the Robinson's census data
data(census)

## fit the parametric model with contextual effects and N 
## using the default prior specification
res1 <- eco(Y ~ X, N = N, context = TRUE, data = census, verbose = TRUE)
## summarize the results
summary(res1)

## obtain out-of-sample prediction
out1 <- predict(res1, verbose = TRUE)
## summarize the results
summary(out1)