Advances in Sampling Estimation - Sampling Spatial Units for Agricultural Surveys - page 252

The efficiency of Eq. ( 10.23 ) depends on how well the partition defined by the
codes of the covariates explains the variation of the outcome y or the probabilities
r k . This approach can be easily extended to a regression model using a set of
covariates, so that the residual variation and the bias of y and/or r k will be small.
Note that this is the basic assumption of the calibration approach. As a conse-
quence, it can play the role of a framework in which the weights are adjusted to
account for nonresponses (S ¨ rndal and Lundstr ¨ m 2005 , 2008 ). In this approach,
the weights originally attached to each observation k

s that belongs to the respon-
dent set r are modified, so that the estimate of the total of a set of auxiliary variables
has no error when the auxiliaries are available for all of the population, or for the
selected sample. The rationale behind this approach is quite obvious: if the cali-
brated weights predict the total of the auxiliary variables (or their complete sample
estimates) without errors, then they should also be suitable for estimating the total
of the variable of interest, providing they have a close relationship.
This approach seems promising, because it should perform well for a suitable
choice of auxiliary variables, simultaneously reducing the nonresponse bias and
increasing the accuracy of the estimates. Moreover, this approach is model-assisted
because it does not explicitly refer to any model, and its properties can be analyzed
within a design-based framework.
The calibration approach for nonresponses consists of a reweighting scheme that
makes a distinction between two different kinds of auxiliary variables.
If dealing with covariates (x k ) that are only known for the units in the sample s ,

2

the constraints on their totals X k2U x k are not required because they are estimated

using X k2s d k x k .
This set of covariates can only be used to reduce the nonresponse bias. They
are used along with any set of usual calibration variables x k , whose benchmark

totals ( X k2U x k ) for all the population must be known from other sources. The
calibration uses the combined auxiliary vectors and total information

X k2U x k

!
: ;

x k
x k

X k2s d k x k

x k ¼ t x ¼ ð 10 : 24 Þ

S ¨ rndal and Lundstr ¨ m( 2010 ) developed a bias indicator that can be used to select
auxiliary variables that will effectively reduce the nonresponse bias. The main
advantage of using calibration to deal with unit nonresponse is that auxiliary
variables do not need to be available for the population. Additionally, because we
do not require explicit response modeling, the calibration approach is simple and
flexible.
The variance estimates should account for the nonresponse adjustment, because
any additional variation in the weights will cause confidence intervals to be wider
than necessary and could lead to conservative intervals. However, this is a better
solution than using a variance estimator that does not consider missing data, which
would result in an underestimate. The effects of the weight adjustments may be