Small Area Estimation - Sampling Spatial Units for Agricultural Surveys - page 282

11.3.2 Unit Level Models

Unit-level models relate the unit values of a study variable to unit-specific auxiliary
data. More formally, suppose that y dk is the value of a study variable for area d and
unit k , for d ¼1,2,

... , D and k ¼1,2, ... , N d . D is the number of SAs and N d is the

number of population units in SA d .
Assume that unit-specific auxiliary information x dk ¼

t
is available for every unit in the population, where q is the number of auxiliary
variables. A basic unit-level model relates the y dk to the x dk using a nested error
regression model of the form

ð x dk 1 x dk 2 ... x dkq Þ

y dk ¼ x dk β þ ˅ d þ e dk , d ¼ 1, 2, ... , Dk ¼ 1, 2, ... , N d ; ð 11 : 31 Þ

are area-specific
random numbers, and e dk s are the sampling errors. Furthermore, the

iid N 0

where β is a fixed set of regression parameters, ˅ d

2
˅

; ˃

e

˅ d s are

e (Battese et al. 1988 ).
The sample data are assumed to obey the population model in Eq. ( 11.31 ). This
implies that sample selection bias is absent, which is satisfied by SRS within areas.
Model ( 11.31 ) may not be suitable under more complex designs such as stratified
multistage sampling: see Ghosh and Rao ( 1994 ) for a discussion about this topic.
In this case, we can also consider various extensions to the basic unit level model
in Eq. ( 11.31 ). Fuller and Harter ( 1987 ) defined a multivariate nested regression
model where a vector of variables of interest is related to covariates. Arora and
Lahiri ( 1997 ) used a unit level model that relaxed the equal error variances
assumption, applying it to estimate the average weekly consumer expenditures
for various goods and services. A general two-level model framework was applied
by Moura and Holt ( 1999 ) to Brazilian data.
Battese et al. ( 1988 ) first used the unit-level model to predict areas planted with
corn and soybeans for 12 counties in north-central Iowa. The area of corn and
soybeans in the segments (PSUs) of the 12 counties was determined by
interviewing farm operators. Crop areas for each segment were estimated using
satellite images by counting the number of individual pixels in the satellite photo-
graphs. The model assumes that there is a linear relationship between the survey
and satellite data, with county-specific random effects.
Finally, outliers and missing data are often present in satellite information. They
are mainly due to cloudy weather, which prevents researchers from correctly
identifying crop areas in digital images. These important problems are addressed
in Benedetti and Filipponi ( 2010 ).

iid N 0 ; ˃

independent from the residual errors e dk , with e dk

e