1. 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

2. 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, and JMS, 2002
using repeated survey
FCSM, 2003
of model parameters
Symposium, 2005
of dual frames
JSM, 2007

3. Demnati –Rao Approach General formulation Finite population parameters
Model parameters
Estimator for both parameters
Variance estimators associated with and are different

4. 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;

5. 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

6. Demnati –Rao Approach ( Survey Methodology, 2004 )
Example – Ratio estimator of
For SRS and

7. 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)

8. 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

9. 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),

10. 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

11. 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

12. Hence
= model variance sampling variance
where
and
Under SRS,
where

13. 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

14. 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 )
One simple random sample of specified size n is drawn from each generated population
Parameter of interest:

15. Simulation 1: Unconditional performance
Ratio estimator:
We calculated:
Simulated
and its components and

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

17. 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

18. Simulation 2: Conditional performance
Figure 2: Conditional relative bias of the expansion and ratio estimators of

19. Simulation 2: Conditional performance
Figure 3: Conditional relative bias of variance estimators

20. Simulation 2: Conditional performance
Figure 4: Conditional coverage rates of normal theory confidence intervals based on
, and for nominal level of 95%

21. 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

22. 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

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

24. 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

25. 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