A note on the concept of invariance in two-phase sampling designs
Section 3. Implications of the invariance propertyA note on the concept of invariance in two-phase sampling designs
Section 3. Implications of the invariance property
an arbitrary two-phase sampling design, the inclusion probability of unit
is generally unknown and is defined as
denotes a realisation of the random vector
are generally unknown because they require the
for every possible
(in many cases, we do) but also of
The latter are generally unknown because
may depend on the outcome of phase 1. However,
if the sampling design is weakly invariant, then
and (3.1) reduces to
that we are interested in estimating the population total
are generally unknown, the Horvitz-Thompson
cannot be used,
in general. Instead, it is common practice to use the double expansion
differ. However, for weakly invariant
two-phase designs, it is clear from (3.2), that both are identical.
be a finite population parameter and
be an estimator of
The total variance of
can be expressed as
(3.3) is often called the two-phase decomposition of the variance; e.g.,
Särndal et al. (1992). If the two-phase sampling design is strongly
invariant, the total variance of
can alternatively be decomposed as
decomposition (3.4) is often called the reverse decomposition of the variance
as the order of sampling is reversed, which can only be justified provided the
two-phase design is strongly invariant. The decomposition (3.4) cannot be used
in the case of weakly invariant two-phase design as the vector
cannot be generated prior to the vector
The reverse decomposition was studied in the
context of nonresponse by Fay (1991), Shao and Steel (1999) and Kim and Rao
(2009), among others. In a nonresponse context, assuming that the units respond
independently of one another, the set of respondents can be viewed as a
second-phase sample selected according to Poisson sampling with unknown
inclusion probabilities, called response probabilities. If the latter remain
the same from one realization of the sample to another, we are essentially in
the presence of a strongly invariant two-phase sampling design. Decomposition (3.4)
can be used to justify simplified variance estimators for two-phase sampling
designs; see Beaumont, Béliveau and Haziza (2015).
authors are grateful to an Associate Editor and a reviewer for their comments
and suggestions, which improved the quality of this paper. David Haziza’s
research was funded by a grant from the Natural Sciences and Engineering
Research Council of Canada.
Béliveau, A. and Haziza, D. (2015). Clarifying some aspects of variance
estimation in two-phase sampling. Journal of Survey Statistics and
Methodology, 3, 524-542.
Fay, R.E. (1991). A design-based
perspective on missing data variance. Proceedings of the 1991 Annual
Research Conference, US Bureau of the Census, 429-440.
Kim, J.K., and Rao,
J.N.K. (2009). A unified approach to linearization variance estimation from
survey data after imputation for item nonresponse. Biometrika, 96,
Särndal, C.-E., Swensson,
B. and Wretman, J. (1992). Model Assisted Survey Sampling.
Springer-Verlag, New York.
Shao, J., and Steel, P.
(1999). Variance estimation for survey data with composite imputation and nonnegligible
sampling fractions. Journal of the American Statistical Association, 94,
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