Description Usage Arguments Details Value Author(s) References See Also Examples
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 individuallevel 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).
1 2 3 4 
formula 
A symbolic description of the model to be fit, specifying the
column and row margins of 2 \times 2 ecological tables. 
data 
An optional data frame in which to interpret the variables in

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 individuallevel data such as survey
data for W_1 and W_2, respectively. If 
theta.start 
A numeric vector that specifies the starting values for
the mean, variance, and covariance. When 
fix.rho 
Logical. If 
context 
Logical. If 
sem 
Logical. If 
epsilon 
A positive number that specifies the convergence criterion
for EM algorithm. The square root of 
maxit 
A positive integer specifies the maximum number of iterations
before the convergence criterion is met. The default is 
loglik 
Logical. If 
hyptest 
Logical. If 
verbose 
Logical. If 
When SEM
is TRUE
, ecoML
computes the observeddata
information matrix for the parameters of interest based on SupplementedEM
algorithm. The inverse of the observeddata information matrix can be used
to estimate the variancecovariance matrix for the parameters estimated from
EM algorithms. In addition, it also computes the expected completedata
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.
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

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 
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 
W 
Insample estimation of W_1 and W_2. 
suff.stat 
The sufficient statistics for 
iters.em 
Number of EM iterations before convergence is achieved. 
iters.sem 
Number of SEM iterations before convergence is achieved. 
loglik 
The loglikelihood of the model when convergence is achieved. 
loglik.log.em 
A vector saving the value of the loglikelihood function at each iteration of the EM algorithm. 
mu.log.em 
A matrix saving the unweighted mean estimation of the logittransformed individuallevel 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 logittransformed individuallevel
proportions (i.e., W_1 and W_2) at each iteration of EM process.
Note, nontransformed variances are displayed on the screen (when

rho.fisher.em 
A matrix saving the fisher
transformation of the estimation of the correlations between the
logittransformed individuallevel proportions (i.e., W_1 and
W_2) at each iteration of EM process. Note, nontransformed
correlations are displayed on the screen (when 
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 observeddata information matrix 
Icom 
The (expected) complete data information
matrix estimated via SEM algorithm. When 
Iobs 
The observed information matrix. The dimension of 
Imiss 
The difference between 
Vobs 
The (symmetrized) variancecovariance matrix of the ML parameter
estimates. The dimension of 
Iobs 
The (expected) completedata variancecovariance matrix. The
dimension of 
Vobs.original 
The estimated variancecovariance matrix of the ML parameter
estimates. The dimension of 
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 
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. 
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.
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. 123. 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. 4169. available at http://imai.princeton.edu/research/eiall.html
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35  ## 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 outofsample 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)

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