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Linearization Variance Estimators for Survey Data: Some Recent Work. A. Demnati and J. N. K. Rao Statistics Canada / Carleton University. A Presentation at the Third International Conference on Establishment Surveys June 18-21, 2007. Montréal, Québec, Canada June 20, 2007.

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slide1
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

situation
looking for a method of variance estimation that

Situation

  • 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
demnati rao approach
Demnati –Rao Approach
  • General formulation
  • Finite population parameters
  • Model parameters
  • Estimator for both parameters
  • Variance estimators associated with and are different
demnati rao approach survey methodology 2004
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;

demnati rao approach survey methodology 20041

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 (N×1) vector of arbitrary number

demnati rao approach survey methodology 20042

Demnati –Rao Approach( Survey Methodology, 2004 )

  • Example – Ratio estimator of

For SRS and

demnati rao approach survey methodology 20043

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)
demnati rao approach1
Also in Survey Methodology, 2004:

Demnati –Rao Approach

  • 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
model parameters symposium 2005

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),

slide10
Example: Ratio estimator when y is assumed to be random
  • for
  • Define
  • We have

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

  • We get

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

where is an estimator of the total variance of

slide11
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

slide12
Hence

= model variance + sampling variance

where

and

  • Under SRS,

where

slide13
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

simulation 1 unconditional performance
Simulation 1: Unconditional performance
  • We generated R=2,000 finite populations , each of size N=393 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:
simulation 1 unconditional performance1
Simulation 1: Unconditional performance
  • Ratio estimator:
  • We calculated:
  • Simulated
  • and its components and
simulation 1 unconditional performance2
Simulation 1: Unconditional performance

Figure 1: Averages of variance estimates for selected sample sizes compared to simulated MSE of the ratio estimator.

simulation 2 conditional performance
Simulation 2: Conditional performance
  • We generate R=20,000 finite populations , each of size N=393 from the ratio model

using the number of beds as

  • One simple random sample of size n=100 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
simulation 2 conditional performance1
Simulation 2: Conditional performance

Figure 2: Conditional relative bias of the expansion and ratio estimators of

simulation 2 conditional performance2
Simulation 2: Conditional performance

Figure 3: Conditional relative bias of variance estimators

simulation 2 conditional performance3
Simulation 2: Conditional performance

Figure 4: Conditional coverage rates of normal theory confidence intervals based on

, and for nominal level of 95%

g weighted estimating functions model parameter
Generalized Linear Model

g-weighted estimating functions: model parameter

  • is the solution of weighted estimating equation:
  • is solution
  • Special case: (GREG)
  • Linear Regression Model
  • Logistic Regression Model
simulation 3 estimating equations
Simulation 3: Estimating equations
  • We generated R=10,000 finite populations , each of size N=393 from the model
  • Using the number of beds as
  • leads to an average of about 60% for z
  • One simple random sample of size n=30 is drawn from each generated population
  • Parameter of interest:
  • Population units are grouped into two classes with 271 units k having x<350 in class 1 and 122 units k with x>=350 in class 2
  • Post-stratification: X=(271,122)T
multiple weight adjustments

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

concluding remarks

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

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