Linearization Variance Estimators

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Linearization Variance Estimatorsfor Survey
Data Some Recent Work
A. Demnati and J. N. K. RaoStatistics Canada /
Carleton University
A Presentation at the Third International
Conference onEstablishment SurveysJune 18-21,
2007
Montréal, Québec, CanadaJune 20, 2007
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Situation

• looking for a method of variance estimation that

• is simple

• is widely applicable

• has good properties

• provides unique choice

• for estimators

• of nonlinear finite population parameters

• SM, 2004

• defined explicitly or implicitly

• SM, 2004

• using calibration weights

• SM, 2004

• under missing data

• JSM, 2002 and JMS, 2002

• using repeated survey

• FCSM, 2003

• of model parameters

• Symposium, 2005

• of dual frames

• JSM, 2007

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Demnati Rao Approach

• General formulation

• Finite population parameters

• Model parameters

• Estimator for both parameters

• Variance estimators associated with and
  are different

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Demnati Rao Approach( Survey Methodology, 2004 )

• Write the estimator of a finite population
  parameter as

with
if element k is not in sample s
if element k is in sample s
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Demnati Rao Approach( Survey Methodology, 2004 )

• A linearization sampling variance estimator is
  given by

with
variance estimator of the H-T
estimator
of the total
is a (N1) vector of arbitrary number
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Demnati Rao Approach( Survey Methodology, 2004 )

• Example Ratio estimator of

For SRS and
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Demnati Rao Approach( Survey Methodology, 2004 )

• Example Ratio estimator of

• is a better choice over customary

• Royall and Cumberland (1981)

• Särndal et al. (1989)

• Valliant (1993)

• Binder (1996)

• Skinner (2004)

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Demnati Rao Approach

• Also in Survey Methodology, 2004

• Calibration Estimators

• the GREG Estimator

• the Optimal Regression Estimator

• the Generalized Raking Estimator

• Two-Phase Sampling

• New Extensions

• Wilcoxon Rank-Sum Test

• Cox Proportional Hazards Model

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Model parameters(Symposium, 2005)

• Finite-population assumed to be generated from a
  superpopulation model

• Inference on model parameter

• Total variance of

model expectation and variance
design expectation and variance

i) if f 0 then
ii) if f 1 then
where f is the sampling fraction. For
multistage sampling, the psu sampling fraction
plays the role of f.
In case i),
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• Example Ratio estimator when y is assumed to be
  random

• 
  for

• Define

• We have

where Ad is a 2N matrix of random variables
with kth column

• We get

where Ab is a 2N matrix of arbitrary real
numbers with kth column

where is an estimator of the total
variance of
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• Estimator of the total variance of

and
when

• A variance estimator of is
  given by

with
where
Note that is an
estimator of model covariance
when and
when
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• Hence

model variance
sampling variance
where
and

• Under SRS,

where
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• Under ratio model,

Note remains valid under
misspecification of

• Hence,

Note g-weight appears
automatically in
and the finite population correction 1-n/N is
absent in
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Simulation 1 Unconditional performance

• We generated R2,000 finite populations
  , each of size N393 from the ratio model

where
are independent observations generated from a
N(0,1)
are the number of beds for the Hospitals
population
studied in Valliant, Dorfman, and Royall (2000,
p.424-427)

• One simple random sample of specified size n is
  drawn from each generated population

• Parameter of interest

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Simulation 1 Unconditional performance

• Ratio estimator

• We calculated

• Simulated

• and its
  components and

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Simulation 1 Unconditional performance
Figure 1 Averages of variance estimates for
selected sample sizes compared to simulated MSE
of the ratio estimator.
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Simulation 2 Conditional performance

• We generate R20,000 finite populations
  , each of size N393 from the ratio model

using the number of beds as

• One simple random sample of size n100 is drawn
  from each generated population

• Parameter of interest

• We arranged the 20,000 samples in ascending order
  of -values and then grouped them into 20
  groups each of size 1,000

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Simulation 2 Conditional performance
Figure 2 Conditional relative bias of the
expansion and ratio estimators of
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Simulation 2 Conditional performance
Figure 3 Conditional relative bias of variance
estimators
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Simulation 2 Conditional performance
Figure 4 Conditional coverage rates of normal
theory confidence intervals based on
, and for nominal level of 95
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g-weighted estimating functions model parameter

• Generalized Linear Model

• is the solution of weighted estimating equation

• is solution

• Special case (GREG)

• Linear Regression Model

• Logistic Regression Model

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Simulation 3 Estimating equations

• We generated R10,000 finite populations
  , each of size N393 from the model

• Using the number of beds as

• leads to an average of about 60 for z

• One simple random sample of size n30 is drawn
  from each generated population

• Parameter of interest

• Population units are grouped into two classes
  with 271 units k having xlt350 in class 1 and 122
  units k with xgt350 in class 2

• Post-stratification X(271,122)T

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Simulation 3 Estimating equations
Table 1 Monte Carlo Variances Table 1 Monte Carlo Variances Table 1 Monte Carlo Variances
Parameter No Calibration Post-stratification
0.0133 0.0139
0.0161 0.0167
Table 2 DR variance estimator Table 2 DR variance estimator Table 2 DR variance estimator
Parameter No Calibration Post-stratification
0.0122 0.0123
0.0148 0.0150
Table 3 DR naïve variance estimator Table 3 DR naïve variance estimator Table 3 DR naïve variance estimator
Parameter No Calibration Post-stratification
0.0120
0.0145
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Multiple Weight Adjustments

• Weight Adjustments for

• Units (or complete) nonresponse

• Calibration

• Due to lack of time, not presented in the talk,

but it is included in the proceeding paper
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Concluding Remarks

• We provided a method of variance estimation for
  estimators

• of nonlinear model parameters

• using survey data

• defined explicitly or implicitly

• using multiple weight adjustments

• under missing data

• The method

• is simple

• is widely applicable

• has good properties

• provides unique choice

Thank you Very Much