Variance Estimation in Complex Surveys (1).ppt

7/30/2019 Variance Estimation in Complex Surveys (1).ppt

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Variance Estimation in

Complex SurveysDrew Hardin

Kinfemichael Gedif


2/27

So far..

Variance for estimated mean and totalunder

SRS, Stratified, Cluster (single, multi-stage), etc.

Variance for estimating a ratio of twomeans under

SRS (we used linearization method)


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What about other cases?

Variance for estimators that are not linearcombinations of means and totals

Ratios

Variance for estimating other statistic fromcomplex surveys

Median, quantiles, functions of EMF, etc.

Other approaches are necessary


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Outline

Variance Estimation Methods Linearization

Random Group Methods

Balanced Repeated Replication (BRR) Resampling techniques

Jackknife, Bootstrap

Adapting to complex surveysHot research areas

Reference


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Linearization (Taylor SeriesMethods)

We have seen this before (ratio estimatorand other courses).

Suppose our statistic is non-linear. It canoften be approximated using TaylorsTheorem.

We know how to calculate variances oflinear functions of means and totals.


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Linearization (Taylor SeriesMethods)

Linearize

Calculate Variance

),(

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),...,(

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t

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),...,,(),...,,,( 21321


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Linearization (Taylor Series)Methods

Pro:

Can be applied in general sampling designs

Theory is well developed

Software is available

Con:

Finding partial derivatives may be difficult

Different method is needed for each statistic

The function of interest may not be expressed asmooth function of population totals or means

Accuracy of the linearization approximation


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Based on the concept of replicating the surveydesign

Not usually possible to merely go and replicatethe survey

However, often the survey can be divided into Rgroups so that each group forms a miniatureversions of the survey


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1 2 3 4 5 6 7 8Stratum 1

1 2 3 4 5 6 7 8Stratum 2

1 2 3 4 5 6 7 8Stratum 3

1 2 3 4 5 6 7 8Stratum 4

1 2 3 4 5 6 7 8Stratum 5

Treat as miniature sample


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Unbiased Estimator (Average of Samples)

Slightly Biased Estimator (All Data)1

)~

(1

)~

( 1

2

1

RR

V

R

rr

1

)(1

1

2

2

RRV

R

r

r


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Pro: Easy to calculate General method (can also be used for non smooth

functions)

Con:Assumption of independent groups (problem when N

is small) Small number of groups (particularly if one strata is

sampled only a few times)

Survey design must be replicated in each randomgroup (presence of strata and clusters remain thesame)


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Resampling and Replication Methods

Balanced Repeated Replication (BRR)

Special case when nh=2

Jackknife (Quenouille (1949) Tukey (1958))

Bootstrap (Efron (1979) Shao and Tu (1995))

These methods Extend the idea of random group method

Allows replicate groups to overlap

Are all purpose methods

Asymptotic properties ??


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Balanced Repeated Replication

Suppose we had sampled 2 per stratum

There are 2H ways to pick 1 from eachstratum.

Each combination could treated as asample.

Pick R samples.


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Which samples should we include?Assign each value either 1 or1 within the stratum

Select samples that are orthogonal to one another to

create balanceYou can use the design matrix for a fraction factorial

Specify a vector ar of 1,-1 values for each stratum

Estimator

2

1

)(1

)(

R

r

rBRRR

V a


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Pro Relatively few computations

Asymptotically equivalent to linearization methods for

smooth functions of population totals and quantiles Can be extended to use weights

Con 2 psu per sample

Can be extended with more complex schemes


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The JackknifeSRS-with replacement

Quenoule (1949); Tukey (1958); Shao and Tu (1995)

Let be the estimator of after omitting the ithobservation

Jackknife estimate

Jackknife estimator of the

For Stratified SRS without replacement Jones (1974)

l ii

n

i

i

J nnn )1(

~where/

~~

1

n

i

J

i

n

i

in

i

i

J

nn

nn

nV

1

2

11

2

)~~

()1(

1

/where)(1

)(

i

)(V


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The Jackknifestratified multistage design

In stratum h, delete one PSU at a time

Let be the estimator of the same form aswhen PSU iof stratum his omitted

Jackknife estimate:

Or using pseudovalues

)()1/()(' ''

hihi

hihhhhh hh

hiygwherenhyynWyWy

)( hi

L

h

n

i

L

h

n

i

hi

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hiI

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hi

hh

hi

h h

nL

n

nn

1 1 1 1

)()()()(

)()(

~11~;/

~~

)1(~


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The Jackknifestratified multistage design

Different formulae for

Where

Using the pseudovalues

)(V

hn

i

methodhiL

h h

hL

n

nV

1

2)(

1

)()1

)(

LnL

h

hL

h

hihmethod/or,/,,becan

1

)(

1

)()(

IIIjn

nV

hn

i

j

J

hiL

h h

hL ,)

~~(

)1)(

1

2)()(

1


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The BootstrapNave bootstrap

Efron (1979); Rao and Wu (1988); Shao and Tu (1995)

Resample with replacement in stratum h

Estimate:

Variance:

Or approximate by

The estimator is not a consistent estimator of thevariance of a general nonlinear statistics

hnihi

y1

*

Bb

ygandyyynyb

h

b

h

b

i

b

hih

b

h

,...,2,1

)(,,*)*()*()*()*(1)*(

2*

*

*

*

*))(()( EEVNBS

B

b

b

BVNB S 1

.*)*(**

)

(1

1)

(


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The BootstrapNave bootstrap

For

Comparing with

The ratio does not converge to 1for abounded nh

*** yyW hh

2

2

* 1)( h

h

h

h

sn

n

n

WyVar h

2

2

)( hh

sn

WyVar h

)(

)(*

yVaryVar


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The BootstrapModified bootstrap

Resample with replacement in stratum h

Calculate:

Variance:

Can be approximated with Monte Carlo

For the linear case, it reduces to the customaryunbiased variance estimator

mh< nh

1,1

* h

m

ihimy

h

)~(~

,~~,/~~

)()1(

~

1

*

2/1

2/1

ygyWymyy

yyn

myy

h

m

i

L

h

hhhih

hi

h

hhhi

h

2*

*

*

*

**))

~(

~()

~( EEV

MBS


23/27

More on bootstrap

The method can be extended to stratified srswithout replacement by simply changing

For mh=nh-1, this method reduces to the nave BS

For nh=2, mh=1, the method reduces to the

random half-sample replication method For nh>3, choice of mh see Rao and Wu (1988)

))(1()1(

~to~*

2/1

2/1

hhihh

h

hhihi

yyfn

myyy


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SimulationRao and Wu (1988)

Jackknife and Linearization intervals gavesubstantial bias for nonlinear statistics in one sidedintervals

The bootstrap performs best for one-sided intervals(especially when mh=nh-1)

For two-sided intervals, the three methods havesimilar performances in coverage probabilities

The Jackknife and linearization methods are morestable than the bootstrap

B=200 is sufficient


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Hot topics

Jackknife with non-smooth functions (Raoand Sitter 1996)

Two-phase variance estimation (Graubardand Korn 2002; Rubin-Bleuer and Schiopu-Kratina 2005)

Estimating Function (EF) bootstrap method(Rao and Tausi 2004)


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Software

OSIRIS BRR, Jackknife

SAS Linearization

Stata Linearization

SUDAAN Linearization, Bootstrap, Jackknife

WesVar BRR, JackKnife, Bootstrap


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References: Effron, B. (1979). Bootstrap methods: Another look at the jackknife. Annals of

statistics 7, 1-26. Graubard, B., J., Korn, E., L. (2002). Inference for supper population parametersusing sample surveys. Statistical Science, 17, 73-96.

Krewski, D., and Rao, J., N., K. (1981). Inference from stratified samples: Propertiesof linearization, jackknife, and balanced replication methods. The annals of statistics.9, 1010-1019.

Quenouille, M., H.(1949). Problems in plane sampling. Annals of MathematicalStatistics 20, 355-375.

Rao, J.,N.,K., and Wu, C., F., J., (1988). Resampling inferences with complex surveydata. JASA, 83, 231-241.

Rao, J.,N.,K., and Tausi, M. (2004). Estimating function variance estimation understratified multistage sampling. Communications in statistics. 33:, 2087-2095.

Rao, J. N. K., and Sitter, R. R. (1996). Discussion of Shaos paper.Statistics, 27, pp.246247.

Rubin-Bleuer, S., and Schiopu-Kratina, I. (2005). On the two-phase framework for

joint model and design based framework. Annals of Statistics (to appear) Shao, J., and Tu, (1995). The jackknife and bootstrap. New York: Springer-Verlag. Tukey, J.W. (1958). Bias and confidence in not-quite large samples. Annals of

Mathematical Statistics. 29:614.Not referred in the presentation Wolter, K. M. (1985) Introduction to variance estimation. New York: Springer-Verlag. Shao, J. (1996). Resampling Methods in Sample Surveys. Invited paper, Statistics,

27, pp. 203237, with discussion, 237254.

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Variance Estimation in Complex Surveys (1).ppt