Function fh estimates indicators using the Fay-Herriot approach by
Fay and Herriot (1979). Empirical best linear unbiased predictors
(EBLUPs) and mean squared error (MSE) estimates are provided. Additionally,
different extensions of the standard Fay-Herriot model are available:
Adjusted estimation methods for the variance of the random effects (see
Li and Lahiri (2010) and Yoshimori and Lahiri (2014)) are
offered. Log and arcsin transformation for the dependent variable and two
types of backtransformation can be chosen - a crude version and the one
introduced by Slud and Maiti (2006) for log transformed variables
and a naive and bias-corrected version following Hadam et al. (2020)
for arcsin transformed variables. A spatial extension to the Fay-Herriot
model following Petrucci and Salvati (2006) is also included. In
addition, it is possible to estimate a robust version of the standard and of
the spatial model (see Warnholz (2016)). Finally, a Fay-Herriot model
can be estimated when the auxiliary information is measured with error
following Ybarra and Lohr (2008).
fh(
fixed,
vardir,
combined_data,
domains = NULL,
method = "reml",
interval = NULL,
k = 1.345,
mult_constant = 1,
transformation = "no",
backtransformation = NULL,
eff_smpsize = NULL,
correlation = "no",
corMatrix = NULL,
Ci = NULL,
tol = 1e-04,
maxit = 100,
MSE = FALSE,
mse_type = "analytical",
B = c(50, 0),
seed = 123
)An object of class "fh", "emdi" that provides estimators
for regional disaggregated indicators like means and ratios and optionally
corresponding MSE estimates. Several generic functions have methods for the
returned object. For a full list and descriptions of the components of
objects of class "emdi", see emdiObject.
a two-sided linear formula object describing the fixed-effects part of the linear mixed regression model with the dependent variable on the left of a ~ operator and the explanatory variables on the right, separated by + operators.
a character string indicating the name of the variable
containing the domain-specific sampling variances of the direct estimates
that are included in
combined_data.
a data set containing all the input variables that are needed for the estimation of the Fay-Herriot model: the direct estimates, the sampling variances, the explanatory variables and the domains. In addition, the effective sample size needs to be included, if the arcsin transformation is chosen.
a character string indicating the domain variable that is
included in combined_data. If NULL, the domains are numbered
consecutively.
a character string describing the method for the estimation of
the variance of the random effects. Methods that can be chosen
(i) restricted maximum likelihood (REML) method ("reml"),
(ii) maximum likelihood method ("ml"),
(iii) adjusted REML following Li and Lahiri (2010) ("amrl"),
(iv) adjusted ML following Li and Lahiri (2010) ("ampl"),
(v) adjusted REML following Yoshimori and Lahiri (2014)
("amrl_yl"), (vi) adjusted ML following
Yoshimori and Lahiri (2014) ("ampl_yl"), (vii) robustified
maximum likelihood with robust EBLUP prediction following
Warnholz (2017) ("reblup"), (viii) robustified maximum
likelihood with robust and bias-corrected EBLUP prediction following
Warnholz (2017) ("reblupbc"), (ix) estimation of the
measurement error model of Ybarra and Lohr (2008) ("me").
Defaults to "reml".
optional argument, if method "reml" and
"ml" in combination with correlation equals "no"
is chosen or for the adjusted variance estimation methods "amrl",
"amrl_yl", "ampl" and "ampl_yl". Is internally
set to c(0, var(direct estimates)). If a transformation is
applied, the interval is internally set to
c(0, var(transformed(direct estimates))). If desired, interval
can be specified to a numeric vector containing a lower and upper limit for
the estimation of the variance of the random effects. Defaults to
NULL.
numeric tuning constant. Required argument when the robust version
of the standard or spatial Fay-Herriot model is chosen. Defaults to
1.345. For detailed information, please refer to
Warnholz (2016).
numeric multiplier constant used in the bias corrected
version of the robust estimation methods. Required argument when the robust
version of the standard or spatial Fay-Herriot model is chosen. Default is to
make no correction for realizations of direct estimator within
mult_constant = 1 times the standard deviation of direct estimator.
For detailed information, please refer to Warnholz (2016).
a character that determines the type of transformation
of the dependent variable and of the sampling variances. Methods that can be
chosen (i) no transformation ("no"), (ii) log transformation
("log") of the dependent variable and of the sampling variances,
(iii) arcsin transformation ("arcsin") of the dependent variable and
of the sampling variances following. Defaults to "no". For more
information, how the direct estimate and its variance are transformed, please
see the package vignette "A Framework for Producing Small Area Estimates
Based on Area-Level Models in R".
a character that determines the type of
backtransformation of the EBLUPs and MSE estimates. Required argument when a
transformation is chosen. Available methods are (i) crude bias-correction
following Rao (2015) when the log transformation is chosen
("bc_crude"), (ii) bias-correction following
Slud and Maiti (2006) when the log transformations is chosen
("bc_sm"), (iii) naive back transformation when the arcsin
transformation is chosen ("naive"), (iii) bias-corrected back
transformation following Hadam et al. (2020) when the arcsin
transformation is chosen ("bc"). Defaults to NULL.
a character string indicating the name of the variable
containing the effective sample sizes that are included in
combined_data. Required argument when the arcsin transformation is
chosen. Defaults to NULL.
a character determining the correlation structure of the
random effects. Possible correlations are
(i) no correlation ("no"),
(ii) incorporation of a spatial correlation in the random effects
("spatial"). Defaults to "no".
matrix or data frame with dimensions number of areas times
number of areas containing the row-standardized proximities between the
domains. Values must lie between 0 and 1. The columns and rows
must be sorted like the domains in fixed. For an example how to
create the proximity matrix, please refer to the vignette. Required argument
when the correlation is set to "spatial". Defaults to NULL.
array with dimension number of estimated regression coefficients
times number of estimated regression coefficients times number of areas
containing the variance-covariance matrix of the explanatory variables for
each area. For an example of how to create the array, please refer to the
vignette. Required argument within the Ybarra-Lohr model
(method = me). Defaults to NULL.
a number determining the tolerance value for the estimation of the
variance of the random effects. Required argument when method "reml"
and "ml" in combination with correlation ="spatial" are
chosen or for the variance estimation methods "reblup",
"reblupbc" and "me". Defaults to 0.0001.
a number determining the maximum number of iterations for the
estimation of the variance of the random effects. Required argument when
method "reml" and "ml" in combination with correlation
equals "spatial" is chosen or for the variance estimation methods
"reblup", "reblupbc" and "me". Defaults to 100.
if TRUE, MSE estimates are calculated. Defaults
to FALSE.
a character string determining the estimation method of the
MSE. Methods that can be chosen
(i) analytical MSE depending on the estimation method of the variance of the
random effect ("analytical"),
(ii) a jackknife MSE ("jackknife"),
(iii) a weighted jackknife MSE ("weighted_jackknife"),
(iv) bootstrap ("boot"),
(v) approximation of the MSE based on a pseudo linearisation
("pseudo"),
(vi) naive parametric bootstrap for the spatial Fay-Herriot model
("spatialparboot"),
(vii) bias corrected parametric bootstrap for the spatial Fay-Herriot model
("spatialparbootbc"),
(viii) naive nonparametric bootstrap for the spatial Fay-Herriot model
("spatialnonparboot"),
(ix) bias corrected nonparametric bootstrap for the spatial Fay-Herriot model
("spatialnonparbootbc").
Options (ii)-(iv) are of interest when the arcsin transformation is selected.
Option (ii) must be chosen when an Ybarra-Lohr model is selected
(method = me). Options (iv) and (v) are the MSE options for the
robust extensions of the Fay-Herriot model. For an extensive overview of the
possible MSE options, please refer to the vignette. Required argument when
MSE = TRUE. Defaults to "analytical".
either a single number or a numeric vector with two elements. The
single number or the first element defines the number of bootstrap iterations
when a bootstrap MSE estimator is chosen. When the standard FH
model is applied and the information criteria by Marhuenda et al. (2014)
should be computed, the second element of B is needed and must be
greater than 1. Defaults to c(50,0). For practical applications, values
larger than 200 are recommended.
an integer to set the seed for the random number generator. For
the usage of random number generation see details. If seed is set to
NULL, seed is chosen randomly. Defaults to 123.
In the bootstrap approaches, random number generation is used. Thus,
a seed is set by the argument seed.
Out-of-sample EBLUPs are available for all area-level models except for the
bc_sm backtransformation and for the robust models.
Out-of-sample MSEs are available for the analytical MSE estimator of the
standard Fay-Herriot model with reml and ml variance estimation, the crude
backtransformation in case of log transformation and the bootstrap MSE
estimator for the arcsin transformation.
For a description of how to create the proximity matrix for the
spatial Fay-Herriot model, see the package vignette. If the presence
of out-of-sample domains, the proximity matrix needs to be
subsetted to the in-sample domains.
Chen S., Lahiri P. (2002), A weighted jackknife MSPE estimator in small-area
estimation, "Proceeding of the Section on Survey Research Methods", American
Statistical Association, 473 - 477.
Datta, G. S. and Lahiri, P. (2000), A unified measure of uncertainty of
Statistica Sinica 10(2), 613-627.
Fay, R. E. and Herriot, R. A. (1979), Estimates of income for small places:
An application of James-Stein procedures to census data, Journal of the
American Statistical Association 74(366), 269-277.
González-Manteiga, W., Lombardía, M. J., Molina, I., Morales, D. and
Santamaría, L. (2008) Analytic and bootstrap approximations of prediction
errors under a multivariate Fay-Herriot model. Computational Statistics &
Data Analysis, 52, 5242–5252.
Hadam, S., Wuerz, N. and Kreutzmann, A.-K. (2020), Estimating
regional unemployment with mobile network data for Functional Urban Areas in
Germany, Refubium - Freie Universitaet Berlin Repository, 1-28.
Jiang, J., Lahiri, P., Wan, S.-M. and Wu, C.-H. (2001), Jackknifing in the
Fay–Herriot model with an example. In Proc. Sem. Funding Opportunity in
Survey Research, Washington DC: Bureau of Labor Statistics, 75–97.
Jiang, J., Lahiri, P.,Wan, S.-M. (2002), A unified jackknife theory for
empirical best prediction with M-estimation, Ann. Statist., 30,
1782-810.
Li, H. and Lahiri, P. (2010), An adjusted maximum likelihood method for
solving small area estimation problems, Journal of Multivariate Analyis 101,
882-902.
Marhuenda, Y., Morales, D. and Pardo, M.C. (2014). Information criteria for
Fay-Herriot model selection. Computational Statistics and Data Analysis 70,
268-280.
Neves, A., Silva, D. and Correa, S. (2013), Small domain estimation for the
Brazilian service sector survey, ESTADISTICA 65(185), 13-37.
Prasad, N. and Rao, J. (1990), The estimation of the mean squared error of
small-area estimation, Journal of the American Statistical
Association 85(409), 163-171.
Petrucci, A., Salvati, N. (2006), Small Area Estimation for Spatial
Correlation in Watershed Erosion Assessment, Journal of Agricultural,
Biological and Environmental Statistics, 11(2), 169–182.
Rao, J. N. K. (2003), Small Area Estimation, New York: Wiley.
Rao, J. N. K. and Molina, I. (2015), Small area estimation,
New York: Wiley.
Slud, E. and Maiti, T. (2006), Mean-squared error estimation in transformed
Fay-Herriot models, Journal of the Royal Statistical Society:Series B 68(2),
239-257.
Warnholz, S. (2016), saeRobust: Robust small area estimation.
R package.
Warnholz, S. (2016b). Small area estimation using robust extensions to area
level models. Ph.D. thesis, Freie Universitaet Berlin.
Ybarra, L. and Lohr, S. (2008), Small area estimation when auxiliary
information is measured with error, Biometrika, 95(4), 919-931.
Yoshimori, M. and Lahiri, P. (2014), A new adjusted maximum likelihood method
for the Fay-Herriot small area model, Journal of Multivariate Analysis 124,
281-294.
# \donttest{
# Loading data - population and sample data
data("eusilcA_popAgg")
data("eusilcA_smpAgg")
# Combine sample and population data
combined_data <- combine_data(
pop_data = eusilcA_popAgg,
pop_domains = "Domain",
smp_data = eusilcA_smpAgg,
smp_domains = "Domain"
)
# Example 1: Standard Fay-Herriot model and analytical MSE
fh_std <- fh(
fixed = Mean ~ cash + self_empl, vardir = "Var_Mean",
combined_data = combined_data, domains = "Domain", method = "ml",
MSE = TRUE
)
# Example 2: arcsin transformation of the dependent variable
fh_arcsin <- fh(
fixed = MTMED ~ cash + age_ben + rent + house_allow,
vardir = "Var_MTMED", combined_data = combined_data, domains = "Domain",
method = "ml", transformation = "arcsin", backtransformation = "bc",
eff_smpsize = "n", MSE = TRUE, mse_type = "boot", B = c(50, 0)
)
# Example 3: Spatial Fay-Herriot model
# Load proximity matrix
data("eusilcA_prox")
fh_spatial <- fh(
fixed = Mean ~ cash + self_empl, vardir = "Var_Mean",
combined_data = combined_data, domains = "Domain", method = "reml",
correlation = "spatial", corMatrix = eusilcA_prox, MSE = TRUE,
mse_type = "analytical"
)
# Example 4: Robust Fay-Herriot model
fh_robust <- fh(
fixed = Mean ~ cash + self_empl, vardir = "Var_Mean",
combined_data = combined_data, domains = "Domain", method = "reblupbc",
k = 1.345, mult_constant = 1, MSE = TRUE, mse_type = "pseudo"
)
# Example 5: Ybarra-Lohr model
# Create MSE array
P <- 1
M <- length(eusilcA_smpAgg$Mean)
Ci_array <- array(data = 0, dim = c(P + 1, P + 1, M))
for (i in 1:M) {
Ci_array[2, 2, i] <- eusilcA_smpAgg$Var_Cash[i]
}
fh_yl <- fh(
fixed = Mean ~ Cash, vardir = "Var_Mean",
combined_data = eusilcA_smpAgg, domains = "Domain", method = "me",
Ci = Ci_array, MSE = TRUE, mse_type = "jackknife"
)
# }
Run the code above in your browser using DataLab