Fitting the Nonparametric Bayesian Models of Ecological Inference in 2x2 Tables

Description

ecoNP is used to fit the nonparametric Bayesian model (based on a Dirichlet process prior) for ecological inference in 2 \times 2 tables via Markov chain Monte Carlo. It gives the in-sample predictions as well as out-of-sample predictions for population inference. The models and algorithms are described in Imai, Lu and Strauss (2008, 2011).

Usage

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ecoNP(formula, data = parent.frame(), N = NULL, supplement = NULL,
      context = FALSE, mu0 = 0, tau0 = 2, nu0 = 4, S0 = 10, 
      alpha = NULL, a0 = 1, b0 = 0.1, parameter = FALSE, 
      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 (e.g., turnout) and X as the row margin (e.g., percent African-American). Details and specific examples are given below.

data

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

N

An optional variable representing the size of the unit; e.g., the total number of voters. N needs to be a vector of same length as Y and X or a scalar.

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 in the model. The default is NULL.

context

Logical. If TRUE, the contextual effect is also modeled, that is to assume the row margin X and the unknown W_1 and W_2 are correlated. See Imai, Lu and Strauss (2008, 2011) for details. The default is FALSE.

mu0

A scalar or a numeric vector that specifies the prior mean for the mean parameter μ of the base prior distribution G_0 (see Imai, Lu and Strauss (2008, 2011) for detailed descriptions of Dirichlete prior and the normal base prior distribution) . If it is a scalar, then its value will be repeated to yield a vector of the length of μ, otherwise, it needs to be a vector of same length as μ. When context=TRUE , the length of μ is 3, otherwise it is 2. The default is 0.

tau0

A positive integer representing the scale parameter of the Normal-Inverse Wishart prior for the mean and variance parameter (μ_i, Σ_i) of each observation. The default is 2.

nu0

A positive integer representing the prior degrees of freedom of the variance matrix Σ_i. the default is 4.

S0

A positive scalar or a positive definite matrix that specifies the prior scale matrix for the variance matrix Σ_i. If it is a scalar, then the prior scale matrix will be a diagonal matrix with the same dimensions as Σ_i and the diagonal elements all take value of S0, otherwise S0 needs to have same dimensions as Σ_i. When context=TRUE, Σ is a 3 \times 3 matrix, otherwise, it is 2 \times 2. The default is 10.

alpha

A positive scalar representing a user-specified fixed value of the concentration parameter, α. If NULL, α will be updated at each Gibbs draw, and its prior parameters a0 and b0 need to be specified. The default is NULL.

a0

A positive integer representing the value of shape parameter of the gamma prior distribution for α. The default is 1.

b0

A positive integer representing the value of the scale parameter of the gamma prior distribution for α. The default is 0.1.

parameter

Logical. If TRUE, the Gibbs draws of the population parameters, μ and Σ, are returned in addition to the in-sample predictions of the missing internal cells, W. The default is FALSE. This needs to be set to TRUE if one wishes to make population inferences through predict.eco. See an example below.

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 unit. Note that the grid method is significantly slower than the Metropolis algorithm.

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 ecoNP containing the following elements:

call

The matched call.

X

The row margin, X.

Y

The column margin, Y.

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.

a0

The prior shape parameter.

b0

The prior scale parameter.

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

A three dimensional array storing the posterior draws of the population mean parameter, μ. The first dimension indexes the Monte Carlo draws, the second dimension indexes the columns of the table, and the third dimension represents the observations.

Sigma

A three dimensional array storing the posterior draws of the population variance matrix, Σ. The first dimension indexes the Monte Carlo draws, the second dimension indexes the parameters, and the third dimension represents the observations.

alpha

The posterior draws of α.

nstar

The number of clusters at each Gibbs draw.

Author(s)

Kosuke Imai, Department of Politics, Princeton University, kimai@Princeton.Edu, http://imai.princeton.edu; Ying Lu, Center for Promoting Research Involving Innovative Statistical Methodology (PRIISM), New York University ying.lu@nyu.Edu

References

Imai, Kosuke, Ying Lu and Aaron Strauss. (2011). “eco: R Package for Ecological Inference in 2x2 Tables” Journal of Statistical Software, Vol. 42, No. 5, pp. 1-23. available at http://imai.princeton.edu/software/eco.html

Imai, Kosuke, Ying Lu and Aaron Strauss. (2008). “Bayesian and Likelihood Inference for 2 x 2 Ecological Tables: An Incomplete Data Approach” Political Analysis, Vol. 16, No. 1 (Winter), pp. 41-69. available at http://imai.princeton.edu/research/eiall.html

See Also

eco, ecoML, predict.eco, summary.ecoNP

Examples

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## load the registration data
data(reg)

## NOTE: We set the number of MCMC draws to be a very small number in
## the following examples; i.e., convergence has not been properly
## assessed. See Imai, Lu and Strauss (2006) for more complete examples.

## fit the nonparametric model to give in-sample predictions
## store the parameters to make population inference later
## Not run: res <- ecoNP(Y ~ X, data = reg, n.draws = 50, param = TRUE, verbose = TRUE)

##summarize the results
summary(res)

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

## summarize the results
summary(out)

## density plots of the out-of-sample predictions
par(mfrow=c(2,1))
plot(density(out[,1]), main = "W1")
plot(density(out[,2]), main = "W2")


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

## fit the parametric model with contextual effects and N 
## using the default prior specification

res1 <- ecoNP(Y ~ X, N = N, context = TRUE, param = TRUE, data = census, 
n.draws = 25, verbose = TRUE)

## summarize the results
summary(res1)

## out-of sample prediction 
pres1 <- predict(res1)
summary(pres1)
## End(Not run)