Advances in Sampling Estimation - Sampling Spatial Units for Agricultural Surveys - page 247

XX k , l2s Δ

¼

^ kl

ʳ k , CAL ^ k ʳ l , CAL ^ l

V t CAL , y

: ð 10 : 22 Þ

The instrumental vector method introduced by Estevao and S¨rndal ( 2000 , 2006 ,
2009 ) is a feasible alternative to using distance minimization to obtain calibration
weights.
The calibration approach is essentially a method to compute weights that
reproduce the specified auxiliary totals without using an assisting model. The
calibration weights are justified primarily by their consistency with the auxiliary
variables. However, some researchers prefer to view these corrections in terms of
the models. This approach suggests a statistical procedure that presupposes some
relationship between y and X. This is the idea that underlies the model calibration
approach proposed by Wu and Sitter ( 2001 ), Wu ( 2003 ), and Montanari and Ranalli
( 2005 ), among others.
A great number of software packages are available for computing calibrated
weights. For example, the SAS macro CALMAR (Deville et al. 1993 ), the SPSS
program G-CALIB, and the calibrate function included in the survey
R package (there are other packages in R performing calibration estimator such as
sampling, laeken, reweight, and EVER). These packages use different
methods to solve computational issues. These methods include: excluding negative
weights that satisfy the given calibration equations, keeping the computed weights
within desirable bounds, dropping some x variables to remove near linear depen-
dencies, reducing the weights of outliers in the auxiliary variables (a possible cause
of extreme weights).
In particular, the calibrate function is quite flexible; the option calfun can
accept user defined distances using the function make.calfun. The output of
calibrate function is another design with modified weights, which can be
used in the usual way to produce estimates and their standard errors.
The auxiliary variables are specified using a model formula (similarly to
post-stratification) and the population totals are specified as the column sums of
the population regression design matrix (i.e., the predictor matrix) corresponding
to the model formula (Lumley 2010 ). In this example, we have attempted to modify
the weights of an SRS so that they respect the first and second moments of the
coordinates of the population.
The second call to calibrate is used to obtain the same results for a ps

sample. In this case, the GREG estimator produces negative weights. A popular
solution to this problem is to use the logit distance in Eq. ( 10.21 ). This distance
function requires the user to specify the option bounds that restrict the range of the
correction factor. If possible, calibrate will return calibration weights that are
within the bounds. However, it may be impossible to satisfy both the bounds and the
calibration constraints. The force¼TRUE option constrains calibrate to
return a survey design in which the weights satisfy the bounds, even if the
calibration constraints are not met (Lumley 2010 ). The option maxit can be
used to set the maximum number of iterations and epsilon sets the convergence
criteria (the maximum acceptable difference between two successive iterations).

π