For household surveys like the RIS,
estimates are required for person characteristics as well as household
denote the total of a target variable
With linear weighting, an estimator for a person based target variable
is defined as
value of the target variable for person
belonging to household
estimator for a household based target variable is given by
value of the target variable for household
for the corresponding household.
Weights are obtained by means of the
GREG estimator to use auxiliary variables which are observed in the sample and
for which the population totals are known from other sources (Särndal et al.
1992). Consequently, the weights reflect the (unequal) inclusion expectations
of the sampling units and an adjustment such that for auxiliary variables the
weighted observations sum to the known population totals. Often categorical
variables like gender, age, marital status or region are used as auxiliary
variables. Due to the fact that the values of auxiliary variables differ from
person to person within the same household, different weights can be derived
for people from the same household. To ensure that relationships between
household variables and person variables are reflected in estimated totals, it
is relevant to apply a weighting method which yields one unique household
weight for all its household members. If the weights for persons within a
household are the same, then household and person based estimates of the same
target variables are consistent with each other (for example the total income
estimated from households and that from persons). This can be achieved with
so-called integrated weighting methods.
Lemaître and Dufour (1987) apply an
integrated weighting method at the persons level and replace the original
auxiliary variables defined at the person level by the corresponding household
mean. In this way, members of the same household have the same inclusion
expectation and share the same auxiliary information, and therefore the
resulting regression weights are forced to be the same. Nieuwenbroek (1993)
proposes a slightly more general approach by applying the linear weighting
method at the household level, where the auxiliary information of person based
characteristics is aggregated at the household level. Nieuwenbroek (1993)
mentions that the linear weighting method at the household level is equal to
the linear weighting method of Lemaître and Dufour (1987) at the person level,
if the residual variance of the regression model at the household level is
chosen proportional to the number of persons within the household. Steel and
Clark (2007) and Estevao and Särndal (2006) further generalize the integrated
weighting of person and household surveys. Steel and Clark (2007) address the
issue of whether the cosmetic benefits of integrated weighting result in an
increased design variance of the GREG estimates. They show that large-sample
design variances obtained by linear weighting at the household level is less
than or equal to the design variance obtained with linear weighting at the
person level. For small samples there can be a small increase in the design
variance due to integrated weighting. As a result there is little or no loss in
efficiency by applying an integrated weighting method.
In this paper the integrated
weighting approach at the household level is applied. Let
auxiliary variables for household
Person based characteristics are aggregated to
household totals. The GREG estimator is derived from a linear regression model
that specifies the relation between the target variable and the available
auxiliary variables for which population totals are known, and is defined as:
denotes a vector containing the
regression coefficients of the regression of
the residuals and
denote the expectation and variance with
respect to the regression model. In this application, the variance structure is
taken proportional to the household size, i.e.,
Nieuwenbroek (1993) shows that in this case
the weighting applied at the household level is equal to the method of Lemaître
and Dufour (1987).
Regression weights for the
households are finally obtained by
vector containing the known population totals
of the auxiliary variables
The weights calculated at the household level
can be used for weighting person based characteristics of the corresponding
household members, using formula (5.1) since
persons belonging to the same household
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