Fitting Parametric Models and Quantifying Missing Information for Ecological Inference in 2x2 Tables

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Description

ecoML is used to fit parametric models for ecological inference in 2 \times 2 tables via Expectation Maximization (EM) algorithms. The data is specified in proportions. At it's most basic setting, the algorithm assumes that the individual-level proportions (i.e., W_1 and W_2) and distributed bivariate normally (after logit transformations). The function calculates point estimates of the parameters for models based on different assumptions. The standard errors of the point estimates are also computed via Supplemented EM algorithms. Moreover, ecoML quantifies the amount of missing information associated with each parameter and allows researcher to examine the impact of missing information on parameter estimation in ecological inference. The models and algorithms are described in Imai, Lu and Strauss (2008, 2011).

Usage

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   ecoML(formula, data = parent.frame(), N = NULL, supplement = NULL, 
         theta.start = c(0,0,1,1,0), fix.rho = FALSE,
         context = FALSE, sem = TRUE, epsilon = 10^(-6), 
     maxit = 1000, loglik = TRUE, hyptest = FALSE, 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 (e.g., percent African-American) as the row margin. 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 ecoML 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.

fix.rho

Logical. If TRUE, the correlation (when context=TRUE) or the partial correlation (when context=FALSE) between W_1 and W_2 is fixed through the estimation. For details, see Imai, Lu and Strauss(2006). The default is FALSE.

context

Logical. If TRUE, the contextual effect is also modeled. In this case, the row margin (i.e., X) and the individual-level rates (i.e., W_1 and W_2) are assumed to be distributed tri-variate normally (after logit transformations). See Imai, Lu and Strauss (2006) for details. The default is FALSE.

sem

Logical. If TRUE, the standard errors of parameter estimates are estimated via SEM algorithm, as well as the fraction of missing data. The default is TRUE.

theta.start

A numeric vector that specifies the starting values for the mean, variance, and covariance. When context = FALSE, the elements of theta.start correspond to (E(W_1), E(W_2), var(W_1), var(W_2), cor(W_1,W_2)). When context = TRUE, the elements of theta.start correspond to (E(W_1), E(W_2), var(W_1), var(W_2), corr(W_1, X), corr(W_2, X), corr(W_1,W_2)). Moreover, when fix.rho=TRUE, corr(W_1,W_2) is set to be the correlation between W_1 and W_2 when context = FALSE, and the partial correlation between W_1 and W_2 given X when context = FALSE. The default is c(0,0,1,1,0).

epsilon

A positive number that specifies the convergence criterion for EM algorithm. The square root of epsilon is the convergence criterion for SEM algorithm. The default is 10^(-6).

maxit

A positive integer specifies the maximum number of iterations before the convergence criterion is met. The default is 1000.

loglik

Logical. If TRUE, the value of the log-likelihood function at each iteration of EM is saved. The default is TRUE.

hyptest

Logical. If TRUE, model is estimated under the null hypothesis that means of W1 and W2 are the same. The default is FALSE.

verbose

Logical. If TRUE, the progress of the EM and SEM algorithms is printed to the screen. The default is FALSE.

Details

When SEM is TRUE, ecoML computes the observed-data information matrix for the parameters of interest based on Supplemented-EM algorithm. The inverse of the observed-data information matrix can be used to estimate the variance-covariance matrix for the parameters estimated from EM algorithms. In addition, it also computes the expected complete-data information matrix. Based on these two measures, one can further calculate the fraction of missing information associated with each parameter. See Imai, Lu and Strauss (2006) for more details about fraction of missing information.

Moreover, when hytest=TRUE, ecoML allows to estimate the parametric model under the null hypothesis that mu_1=mu_2. One can then construct the likelihood ratio test to assess the hypothesis of equal means. The associated fraction of missing information for the test statistic can be also calculated. For details, see Imai, Lu and Strauss (2006) for details.

Value

An object of class ecoML 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.

context

The assumption under which model is estimated. If context = FALSE, CAR assumption is adopted and no contextual effect is modeled. If context = TRUE, NCAR assumption is adopted, and contextual effect is modeled.

sem

Whether SEM algorithm is used to estimate the standard errors and observed information matrix for the parameter estimates.

fix.rho

Whether the correlation or the partial correlation between W_1 an W_2 is fixed in the estimation.

r12

If fix.rho = TRUE, the value that corr(W_1, W_2) is fixed to.

epsilon

The precision criterion for EM convergence. √{ε} is the precision criterion for SEM convergence.

theta.sem

The ML estimates of E(W_1),E(W_2), var(W_1),var(W_2), and cov(W_1,W_2). If context = TRUE, E(X),cov(W_1,X), cov(W_2,X) are also reported.

W

In-sample estimation of W_1 and W_2.

suff.stat

The sufficient statistics for theta.em.

iters.em

Number of EM iterations before convergence is achieved.

iters.sem

Number of SEM iterations before convergence is achieved.

loglik

The log-likelihood of the model when convergence is achieved.

loglik.log.em

A vector saving the value of the log-likelihood function at each iteration of the EM algorithm.

mu.log.em

A matrix saving the unweighted mean estimation of the logit-transformed individual-level proportions (i.e., W_1 and W_2) at each iteration of the EM process.

Sigma.log.em

A matrix saving the log of the variance estimation of the logit-transformed individual-level proportions (i.e., W_1 and W_2) at each iteration of EM process. Note, non-transformed variances are displayed on the screen (when verbose = TRUE).

rho.fisher.em

A matrix saving the fisher transformation of the estimation of the correlations between the logit-transformed individual-level proportions (i.e., W_1 and W_2) at each iteration of EM process. Note, non-transformed correlations are displayed on the screen (when verbose = TRUE).

Moreover, when sem=TRUE, ecoML also output the following values:

DM

The matrix characterizing the rates of convergence of the EM algorithms. Such information is also used to calculate the observed-data information matrix

Icom

The (expected) complete data information matrix estimated via SEM algorithm. When context=FALSE, fix.rho=TRUE, Icom is 4 by 4. When context=FALSE, fix.rho=FALSE, Icom is 5 by 5. When context=TRUE, Icom is 9 by 9.

Iobs

The observed information matrix. The dimension of Iobs is same as Icom.

Imiss

The difference between Icom and Iobs. The dimension of Imiss is same as miss.

Vobs

The (symmetrized) variance-covariance matrix of the ML parameter estimates. The dimension of Vobs is same as Icom.

Iobs

The (expected) complete-data variance-covariance matrix. The dimension of Iobs is same as Icom.

Vobs.original

The estimated variance-covariance matrix of the ML parameter estimates. The dimension of Vobs is same as Icom.

Fmis

The fraction of missing information associated with each parameter estimation.

VFmis

The proportion of increased variance associated with each parameter estimation due to observed data.

Ieigen

The largest eigen value of Imiss.

Icom.trans

The complete data information matrix for the fisher transformed parameters.

Iobs.trans

The observed data information matrix for the fisher transformed parameters.

Fmis.trans

The fractions of missing information associated with the fisher transformed parameters.

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; Aaron Strauss, Department of Politics, Princeton University, abstraus@Princeton.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, ecoNP, summary.ecoML

Examples

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## load the census data
data(census)

## NOTE: convergence has not been properly assessed for the following
## examples. See Imai, Lu and Strauss (2006) for more complete analyses.
## In the first example below, in the interest of time, only part of the
## data set is analyzed and the convergence requirement is less stringent
## than the default setting.

## In the second example, the program is arbitrarily halted 100 iterations
## into the simulation, before convergence.

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

## fit the parametric model with the default model specifications
## Not run: res <- ecoML(Y ~ X, data = census[1:100,], N=census[1:100,3], 
	     	  epsilon=10^(-6), verbose = TRUE)
## End(Not run)
## summarize the results
## Not run: summary(res)

## obtain out-of-sample prediction
## Not run: out <- predict(res, verbose = TRUE)
## summarize the results
## Not run: summary(out)

## fit the parametric model with some individual 
## level data using the default prior specification
surv <- 1:600
## Not run: res1 <- ecoML(Y ~ X, context = TRUE, data = census[-surv,], 
                   supplement = census[surv,c(4:5,1)], maxit=100, verbose = TRUE)
## End(Not run)
## summarize the results
## Not run: summary(res1)