SiStaN in Brief

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A balanced Sampling approach for multiway
stratification design for small area
estimation Piero Demetrio Falorsi - Paolo
Righi ISTAT
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Index

• The issue of multivariate-multidomain sampling
  strategy
• The proposed sampling strategy
• Balanced sample for multiway stratification
• Modified GREG estimator
• The algorithm for the sample size definition
• Application fields and experiments

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1. The issue of multivariate-multidomain sampling
strategy
When planning a sample strategy for a survey
aiming at producing estimates for several domains
(defined as non-nested partitions of the
population) an issue is to define the sample size
so that the sampling errors of domain estimates
of several parameters are lower than given
thresholds. A sampling strategy is proposed here
dealing with multivariate-multidomain surveys
when the overall sample size must satisfy budget
constraints. The standard solution of a
stratification given by cross-classification of
the domain variables is often not feasible
because the number of strata can be larger than
the overall sample size. Moreover, even if the
overall sample size allows covering all the
strata, the resulting allocation could lead to an
inefficient design.
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1. The issue of multivariate-multidomain sampling
strategy

• Population
• Planned and actual sample with cross-classificatio
  n stratification

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1. The issue of multivariate-multidomain sampling
strategy

• Example Business Structural Statistics
• 36.000 cross-classification strata

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Standard strategy
1. The issue of multivariate-multidomain sampling
strategy

• Standard solution to obtain planned domains
  adopts cross-stratified sampling design by
  combining the domains
• Consequences
• when the population size in many strata is small,
  the stratification scheme could be inefficient
• if different partitions in domains of interest
  are not nested, the allocation of the sample in
  the cross-classified strata may be substantially
  different from the optimal allocation for the
  domains of a given partition
• the sample size to cover all strata could be too
  large for the survey economical constrains
• dealing with surveys repeated over time,
  statistical burden may arise if there exist
  strata containing only few units in the
  population.

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1. The issue of multivariate-multidomain sampling
strategy

• One possible solution is the multi-way
  stratification
• Several sophisticated solutions have been
  proposed to keep under control the sample size in
  all the categories of the stratifying variables
  without using cross-classification design. These
  methods are generally referred to as multi-way
  stratification techniques, and have been
  developed under two main approaches
• Latin Squares or Latin Lattices schemes (Bryant
  et al., 1960 Jessen, 1970) the indipendece
  among rows and columns is supposed. these methods
  work only if all the cross-strata exist in the
  population.
• Controlled rounding problems via linear
  programming (Causey et al., 1985 Sitter and
  Skinner, 1994). Very computationally complex
  methods, not always get to a solution, inclusion
  probability (both simple and joint) cannot be
  computed immediately.
• The main weaknesses of these approaches derives
  from the computational complexity and moreover a
  solution is not always reached.

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2. The proposed sampling strategy

• Aim of this work is to define a sample strategy
  that is optimal with regard to the sample scheme
  and to the estimator utilized, by exploiting the
  available auxiliary information in both phases
• Define a probabilistic sample method
• Realize a multiway stratification based on
  balanced sampling, controlling the sample size of
  the margin domains
• Use a modified GREG estimator
• Define the sample allocation, aiming at
  controlling the sampling errors on margins, using
  a variance estimator taking into account jointly
  both the regression model under the GREG
  estimator and the balanced sampling design
• The strategy may take into account a simple (Fay
  Herriot) Small Area Estimator
• The proposed overall sampling strategy is easy to
  implement and a software has been developed for
  each phase
• It is possible to extend it to different contexts
  (considering the anticipated variance or the use
  of indirect small area estimators)
• It is possible to develop a sample strategy for
  small area estimation considering the sample and
  estimation phases jointly

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2. The proposed sampling strategy

• Notation
• Denote with
• U the population of size N
• Ub the b-th partition in Mb domains Ubd , b1,,
  B, d1,, Mb
• the value of the (r 1,,R) variable of
  interest in the k-th population unit
• the domain membership indicator
• n the overall fixed sample size
• r-th parameter of interest

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3. Balanced sampling and multi-way stratification

• Balanced sampling is a class of designs using
  auxiliary information.
• Properties have been studied in the
• model based approach (Royall and Herson, 1973
  Valliant et al., 2000)
• design based approach (Deville and Tillé, 2004,
  2005).
• In the following we consider the design based or
  model assisted approach

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3. Balanced sampling and multi-way stratification

• Let us define the sampling design p(.) with
  inclusion probabilities
  a design which assigns a
  probability p(s) to each sample s such that
• being a vector of sample indicators.
• Let be a
  vector of Q auxiliary variables known for each
  unit in the population. The sampling design p(s)
  is said to be balanced with respect to the Q
  auxiliary variables if and only if it satisfies
  the balancing equations given by
• being the sample weight

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3. Balanced sampling and multi-way stratification

• Multi-way stratification design can represent a
  special case of balanced design, when for unit k
  the auxiliary variable vector is the indicator of
  the belonging to the domains of the different
  partitions multiplied by its inclusion
  probability
• The z vector, in this case, is defined as
• the balancing equations assure that for each
  selected sample s, the size of the subsample
  is a non-random quantity and is

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3. Balanced sampling and multi-way stratification

• For multiway stratification the balancing
  equations become
• being the sample size for the d-th
  domain of the b-th partition
• and

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3. Balanced sampling and multi-way stratification

• A relevant drawback of balanced sampling has
  always been implementing a general procedure
  giving a multivariate balanced random sample.
• Deville and Tillé (2004) proposed a sample
  selection method (cube method) drawing a balanced
  samples for a large set of auxiliary variables
  and with respect to different vectors of
  inclusion probabilities.
• A free macro for the selection of balanced
  samples for large data sets may be downloaded
  (SAS or R routine)
• http//www.insee.fr/fr/nom_df_met/outils_stat/cube
  /accueil_cube.htm
• Deville and Tillé (2000) show that with our
  specification of the auxiliary vectors, the
  balancing equations can be exactly satisfied,
  while in general the balancing equation are
  approximately respected

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4. Modified GREG estimator

• In the context of multi-variate estimation, the
  r-th parameter of interest is
• The modified GREG estimator is (through a
  specific domain weight)
• The superpopulation working model is

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Variance of the Horvitz-Thompson estimator with
the balanced sampling
4. Modified GREG estimator variance

• Deville and Tillé (2005) proposed an
  approximation of the variance expression for HT
  estimator and the overall domain
•  
• with

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4. Modified GREG estimator variance

• Starting from the result by Deville (2005) it is
  possible to derive the approximate expression of
  the variance for the modified GREG estimator
  under balanced sampling
• being
• and

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5. The algorithm for the sample size definition

• In order to calculate the inclusion probabilities
  it is necessary to fix the sample size for each
  domain so that the constraints on the sampling
  errors were accomplished
• When considering separately each marginal
  partition we would have for each of them a
  different set of inclusion probabilities
• In our methodology we calculate a single
  inclusion probability through a two step
  procedure
• Optimisation (calculating of optimal
  probabilities)
• Calibration (calculating of working
  probabilities)

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5. The algorithm for the sample size definition

• Optimisation the calculus of the inclusion
  probabilities (sample size and domain allocation)
  is carried out with the aim of minimizing the
  expected sampling errors on several domains and
  estimates
• Multi domains
• Multi variable
• The problem is solved through the system

The solution can be obtained through the Chromy
algorithm (the one used in the software for
allocation MAUSS, which can be can be downloaded
from www.istat.it)
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5. The algorithm for the sample size definition

• Calibration optimal inclusion probabilities lead
  to non integer values for the domain sample size
• Rounding of the expected domain sample size to
  next integer
• Calculating working probabilities nearest to
  the optimal ones
• The problem is defined through the system

Solution obtained by means of the Newton
algorithm (with some change), the same used in
calibration software Genesees which can be can
be downloaded from www.istat.it)
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6. Application fields and experiments Artificial
data

• Population Contingency table
• Variable for the allocation and estimation model

,
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6. Application fields and experiments Artificial
data
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• Compared sampling designs and expected CV()

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6. Application fields and experiments

• Real data
• A simulation on real enterprises data (N10,392)
  has been carried out to evaluate the effects of
  planned sample size for small domain of estimate
  (Falorsi et al., 2006)
• U1 partition regions (20 domains)
• U2 partition economic activity by size class (24
  domains)
• Cross-classification strata with population
  units 360.
• Variables of interest value added and labour
  cost
• the sample sizes of U1 and U2 partitions have
  been planned separately by means of a compromise
  allocation
• the 2 allocations guarantee a CV of 34.5 for U1
  and 8.7 for U2 with regard to the variables
  number of employers (supposed known at sampling
  stage)
• the overall sample size is n360

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6. Application fields and experiments Real data

• The experiment examines a situation
  characterizing many real survey contexts in which
  the overall sample size n is fixed and the
  marginal sample sizes are determined by a quite
  simple rule being a compromise between the
  Allocation Proportional to Population size (APP)
  and the allocation uniform for each domain of a
  given partition
• The probabilities of both designs for U1 and U2
  partitions have been obtained as solution of the
  calibration problem below where the initial
  probabilities are set uniformly equal to

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6. Application fields and experiments Real data
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7. Extension to the Fay Herriot Model
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7. Extension to the Fay Herriot Model
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7. Extension to the Fay Herriot Model
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