A note on the concept of invariance in two-phase sampling designs
Section 2. The concept of invarianceA note on the concept of invariance in two-phase sampling designs
Section 2. The concept of invariance
distinguish the concept of strong invariance that may also be called
distribution invariance from that of weak invariance that may also be called
Definition 1. A two-phase sampling design is said to be strongly (or
distribution) invariant provided that
consequence of Definition 1 is that
and therefore, with a strongly invariant
two-phase sampling design, the vector
can be generated prior to the vector
In practice, the concept of strong invariance
is satisfied for only few two-phase sampling designs. A first example is
Poisson sampling at the second phase. This covers the case of nonresponse,
which is often viewed as a Poisson sampling design at the second phase. An
other example is two-stage sampling. Both are described in greater detail
Example 1. At the first phase, a sample is selected according
to an arbitrary sampling design followed by Poisson sampling at the second
phase, where the units selection probability are set prior to
sampling, which means that for Since Poisson sampling
is completely characterized by its first-order selection probabilities, we have As a result, this
sampling design is strongly invariant. It can be implemented as follows: first,
generate the vector according to the
Poisson sampling design and, independently,
generate the vector according to the
Example 2. Two-stage cluster sampling can be described as follows: at
the first stage, a sample of clusters is selected randomly from the population
of clusters. Then, at the second stage, within each cluster selected at the
first stage, a sample of elements is randomly selected. Note that, even in this
case, the vector is still defined at
the element level, with its size corresponding to the
number of elements in the population. Under this set-up, the selection
indicator for an element within cluster is equal to 1 for all
elements within a selected
cluster Therefore, two-stage
sampling is a special case of two-phase sampling as described in Section 1. If
the selection within clusters is independent of which clusters have been
selected in the first phase, then we are in the presence of a strongly
invariant two-stage cluster sampling design. This is satisfied if the selection
of elements within clusters is independent of the selection of elements in any
other cluster. A strongly invariant two-stage cluster sampling designs can be
implemented by reversing the actual act of sampling: instead of sampling the clusters
first, we begin by selecting the elements in each of the population clusters,
and then sampling the clusters.
that our definition of strong invariance for two-stage designs is slightly
different from the one given in Särndal, Swensson and Wretman (1992, Chapter 4)
because the latter restrict to clusters selected at the first stage. However,
for practical purposes, both definitions are essentially equivalent. We used
Definition 1 rather the standard definition of Särndal et al. (1992)
because the latter does not extend easily to the case of two-phase sampling.
Definition 2. A two-phase sampling design is said to be weakly (or
first-two-moment) invariant if
a strongly invariant two-phase sampling design is weakly invariant but the
opposite is not true. The next example describes a sampling design that is
weakly invariant but not strongly invariant.
Example 3. At the first phase, we select a sample, of size according to an
arbitrary fixed-size sampling design. From we select a simple
random sample without replacement, of size where is fixed prior to sampling.
This two-phase sampling design is weakly invariant since and which remain the same
from one realization of to another. However,
it is not strongly invariant since it is not possible to generate prior to and meet the
fixed-size sample size constraint for In fact, this would
also be true for any fixed-size sampling design at the second phase satisfying and
we describe a non-invariant two-phase sampling design.
Example 4.At the first
phase, we select a simple random sample without replacement, of size according to an
arbitrary fixed-size sampling design. For every we record an auxiliary
variable From a second-phase sample, of fixed size is selected using an
inclusion probability proportional-to-size procedure. In this case, we have
Clearly, the inclusion probability of unit in vary from one
realization of to another. Since is a function of it is known only after
the first-phase sample is actually realized.
Survey Methodology publishes articles dealing with various aspects of statistical development relevant to a statistical agency, such as design issues in the context of practical constraints, use of different data sources and collection techniques, total survey error, survey evaluation, research in survey methodology, time series analysis, seasonal adjustment, demographic studies, data integration, estimation and data analysis methods, and general survey systems development. The emphasis is placed on the development and evaluation of specific methodologies as applied to data collection or the data themselves. All papers will be refereed. However, the authors retain full responsibility for the contents of their papers and opinions expressed are not necessarily those of the Editorial Board or of Statistics Canada.
Submission of Manuscripts
Survey Methodology is published twice a year in electronic format. Authors are invited to submit their articles in English or French in electronic form, preferably in Word to the Editor, (email@example.com, Statistics Canada, 150 Tunney’s Pasture Driveway, Ottawa, Ontario, Canada, K1A 0T6). For formatting instructions, please see the guidelines provided in the journal and on the web site (www.statcan.gc.ca/SurveyMethodology).
Note of appreciation
Canada owes the success of its statistical system to a long-standing partnership between Statistics Canada, the citizens of Canada, its businesses, governments and other institutions. Accurate and timely statistical information could not be produced without their continued co-operation and goodwill.
Standards of service to the public
Statistics Canada is committed to serving its clients in a prompt, reliable and courteous manner. To this end, the Agency has developed standards of service which its employees observe in serving its clients.
Published by authority of the Minister responsible for Statistics Canada.