Advances in Sampling Estimation - Sampling Spatial Units for Agricultural Surveys - page 238

regarding the variable of interest y, which was not included in the sample selection
criterion.
The finite population is often not considered to be an element of primary interest
when making an inference using statistical models. p ( s ) is viewed as a nuisance
parameter that must be considered but not exploited in the data analysis. Neverthe-
less, there is a clear recognition that we must take account of the sampling design s

features when model fitting using survey data (see Chap. 12 ). However, the model
is not the target of the analysis when we are using auxiliary information for finite
population inference. It is only a tool for adding information to the sampling
weights, which would otherwise only depend on the inclusion probabilities.
These are the basic principles that underlie the model-assisted approach to survey
estimation. In this approach, a statistical model is introduced without many restric-
tions, and all the properties of the resulting estimators are evaluated within a purely
design-based inference or, more formally, with respect to p ( s ).
Within this framework, the pure design-based HT estimator has been
supplemented by alternative design-based estimators that take advantage of addi-
tional information. Examples include the ratio, difference, and post-stratified esti-
mators (see Fuller 2002 for a review).
Assume that a continuous covariate x is available in the frame for all units of the
population, and that we are interested in estimating the total of a survey variable y.
Focusing on the sample units, we can estimate both the totals to obtain t HT , y and
t HT , x according to the known HT rules ( 1.24 of Sect. 1.2 ). The known total t x will be
different from its estimated counterpart t HT , x , and it is intuitive to consider if the
observed difference in x can also be found in y.
We should correct the HT estimator depending on whether we evaluate this
difference in an additive or multiplicative way. If we assume that the proportion-
ality of the error on x is also likely to be detected on y, the correction to be applied
to the HT estimator is straightforward

'

X k2s ^ k ¼ X k2s g k , RAT d k y k ¼ X k2s w k , RAT y k ;

t x
X k2s ^ k

t RAT , y ¼

ð 10 : 1 Þ

t x

X k2s ^ k

where g k , RAT ¼ is a constant correction factor applied to the direct weights

d k ¼ =π k to obtain the final weights w k , RAT that makes explicit use of the auxiliary

information. Equation ( 10.1 ) is called the ratio estimator.
If we assume that the error on x can additively influence y, the correction to be
applied to the HT estimator is

1

X k2s ^ k ¼ X k2s d k y k

t DIF , y ¼

t x þ t x þ ð x k Þ; ð 10 : 2 Þ

where ^ k are the expanded differences between the two variables x and y. Equation
( 10.2 ) is referred to as the difference estimator.