Spatial Survey Data Modeling - Sampling Spatial Units for Agricultural Surveys - page 312

In practical circumstances, we have other information about the population that
we can use in the inference process. We refer to this information as the auxiliary
variables, which are generally known. We denote them as X U.
For the sake of simplicity, in this description we refer to the case of complete
response. In summary, we restrict our narrative to an observed data distribution that
is dependent on the joint outcome of:

￿ The survey variable of interest (y U ).
￿ The auxiliary variables (X U ).
￿ The sampling process (i U ).
In addition to the complete response assumption, let us consider a
non-informative selection plan where the population generating process and the
sample selection method are independently conditional on the auxiliary informa-
tion. More formally, this hypothesis means

¼ f i U X U

f y U , i U X U j f y U X U j ð j Þ: ð 12 : 23 Þ

In practice, this occurs when the sample selection only depends on the values X U ,so
that

¼

. SRS is an important example of this, where the
sample selection is independent of the values in y U and X U . In other words, if the
selection is non-informative, the realization (i U ) of the selection process does not
contain any additional information for

f i U y U ; X U f i U X U ð j Þ

(after we have included X U in our data),
and it can be ignored in the inference of this parameter.
Under non-informative selection, the relevant likelihood for

ʸ

ʸ given X U is

ð f y U ; ð f

¼ d y s ¼ f y U ; d y s ¼

f y s ; i U ; X U i U ; X U i U y U ; X U X U

ð f y U ;

d y s ¼ :

¼ f i U X U ð j Þ X U f i U X U ð j Þ f y s ; X U ð 12 : 24 Þ

If f (i U |X U ) does not include , the likelihood inference can be based on f (y s , X U ).

The relevant likelihood in Eq. ( 12.24 ) is different from the face value likelihood in
Eq. ( 12.22 ), which was obtained by neglecting both the sample selection and
auxiliary information. In this case, the maximum likelihood estimator

ʸ

ʸ

is
obtained by maximizing the relevant likelihood in Eq. ( 12.24 ), and in particular, its
logarithmic transformation.
For more technical details about maximum likelihood estimations of the survey
data and the description of the non-complete response case, the reader can refer to
Chambers et al. ( 2012 , Chaps. 1 and 2) and the references therein.
However, there are two recently developed, alternative methods for the full
likelihood principle for inferences on sample data. These are the pseudolikelihood
and sample likelihood methods.

of ʸ