Using unequal probability sampling to limit antici-pated variances of regression estimators

Using unequal probability sampling to limit antici-pated variances of regression estimators

Anders Holmberg ICES III 07

Anders Holmberg

Department of Research & Development

Statistics Sweden

SE-701 89 Örebro

SwedenTel: +46 19 176905

Fax: +46 19 177084

E-mail: [email protected]

Outline

• Background

• The problem• Some theory• Auxiliary Information

• An application in a business survey• Comparisons and Results

• CommentsAnders Holmberg ICES III 07

Background(1)


• Prepare the sampling frame

• Derive and analyse diagnostic data

• Decide on a sampling design, sampling scheme and estimator

• Launch the survey

Background(2)


• Prerequisites– A well defined business population– Several parameters of interest – Design-based inference– An up-to-date frame from the business

register– Admin. data available as auxiliary

information– Attempt to find the most efficient/(robust)

design

Background(6)


(1) Number of employees (u1)

(2) Turnover (u2)

(3) Personnel expenses (u3)

(4) Investments (u4)

(t-2)


(2) Turnover (u2)



(t-1)

(1) Number of employees (y1)

(2) Turnover (y2)

(3) Personnel expenses (y3)

(4) Investments (y4)

(t)


(2) Turnover (u2)



(t-1)

A design that minimizes

is such that

( )( . . )k qk k q opt k qk qkUi e n

Minimum of ˆ( )qq y rANV t is

2 2min

1ˆ( )qq y r qk qkU U

ANV tn

Brewer, Hajek, Cassel et al., Rosén

Optimal design in the single variable case


21)1()ˆ( qkU kyq q

tANV

Population plot


nypers2

0

20

40

60

80

100

120

0 10 20 30 40 50 60 70 80 90 100

E.g. if : 22

)( qkqk u

qkk u

~~qkqkk u’Guesstimate’ to find

size measures

The multivariate case?



(2) Turnover (u2)



(t-2)


(2) Turnover (u2)



(t-1)

(1) Number of employees (y1)

(2) Turnover (y2)

(3) Personnel expenses (y3)

(4) Investments (y4)

(t)


(2) Turnover (u2)



(t-1)

The multivariate case


The least we should do is to analyse the various designs’ possible effects on different estimators, before we make the design choice.

Derive inclusion probabilities as a function of standardized (univariate) size measures

Maximal Brewer selection

The multivariate case


There is no evident criterion of optimality, but some are better than others.

),,1())ˆ(( QqvtANVg qyq

)))ˆ((,)),ˆ(((1 Qyy tANVgtANVgf Minimize

under the restrictions

Try to find a design that in some sence is optimal for all important parameters?

Scale effects are neutralized, the relations between the ANVq :s and the corresponding single parameter minimum values (The Brewer selection) are used .


The multivariate case some optimisation approaches

Minimizing a weighted sum of relative efficiency losses:

Q

q yq

pyq

q

q

iq

tANV

tANVHANOREL

1 min )ˆ(

)ˆ(

is minimized when

Q

qU qkkoptq

qkqk H

121

)(

2

~)1~(

~

If we want to put restrictions on certain parameters, e.g.

21

1 21 ( )

min ( ) ( 1) ( 1)

subject to

Qqk

q kUq q opt k qkU

f H

π

Optimization model:

0

21

1 2( )

0 1 1, ,

0

1 1, ,( 1)

k

kU

qkq qkU

q opt k qkU

k N

g n

g v q Q

π

π

min

ˆ( )1, ,

ˆ( )q i

q

q y r pq

y rq

ANV tv q Q

ANV t

Then a design that minimizes ANOREL can be obtained through non-linear programming


The multivariate case some optimisation approaches

An Application


The 4 variables studied for three branches (strata)

SNI25: Manufacturers of food products & beverages

N=749,

SNI28: Manufacturers of metal goods (except machines and devices)

N=2292,

SNI33: Manufacturers of optical instruments

N=323,

290)( snE

112)( snE

64)( snE

Analysis and comparisons made on admin data from previous reference times. Plots, Estimated correlations and gammacoefficients

An Application


• A common ratio model pictures the relationships reasonably well if the corresponding older variable is used as regressor variable. (Strongest pairwise correlation over branches and time, although doubts exist for the investment variable)

• Estimates of the gammacoefficient are sensitive.

• Estimates ranged between 0.2 and 0.9 and sometimes deteriorated!?

• For investments very weak or no heteroscedasticity

• For the other three variables,

“cannot be ruled out” and is simple as a guesstimate

5.0~

An Application


5.0

studyvariable Food Metal Optic

employees 0.5 0.5

turnover 0.5 0.5 0.5

P-costs 0.5 0.5 0.5

investment 0.2 0 0.2

StrataAuxiliary/

size variable

~ ~

)(~

qkkoptq u

~

1ku ~

1ku~

1ku

~

2ku~

2ku~

2ku

~

3ku ~

3ku~

3ku

~

4ku~

4ku ~

4ku

An Application


• Computations of inclusion probabilities and the anticipated variances using the Brewer selection (Maximal brewer selection)

• Computation of the optimisation based approaches, with the extra condition that

15.1)ˆ(

)ˆ(

min

q

iq

yq

pyq

tANV

tANV

1p

2p

3p

4p

5p

6p

Study variables

Considered Design Employees Turnover P-cost Invest Mean

Opt. on Empl 0 24.3 3.5 24.4 13.0

Opt. on Turn 24.5 0 19.1 74.4 29.5

Opt. on P-cost 3.3 16.4 0 43.0 13.0

Opt. on Invest 34.4 91.7 45.9 0 43.0

Minimizing Anorel 2.8 13.9 2.9 19.5 9.8

Minimizing Anorel with restrictions 5.7 15.0 6.5 15.0 10.6

Food & Beverages ]1))ˆ()ˆ([(100 min qiq yqpyq tANVtANV

1p

2p

3p

4p

5p

6p

Study variables


Opt. on Empl 0 13.0 3.4 30.6 11.8

Opt. on Turn 11.0 0 8.2 51.6 17.7

Opt. on P-cost 3.0 8.3 0 37.1 12.1

Opt. on Invest 44.7 73.3 53.4 0 42.8



Optical Instruments ]1))ˆ()ˆ([(100 min qiq yqpyq tANVtANV

1p

2p

3p

4p

5p

6p

Study variables


Opt. on Empl 0 7.3 2.0 21.0 7.6

Opt. on Turn 6.1 0 4.3 33.0 10.9

Opt. on P-cost 1.8 5.0 0 24.4 7.8

Opt. on Invest 31.6 51.2 36.1 0 29.7



Metal goods ]1))ˆ()ˆ([(100 min qiq yqpyq tANVtANV

Maximal Brewer selection satisfies the criteria but with 25% larger sample

3.364)( snE

Does it work on the estimator variances?


• In most cases we will never know

• However, for these variables we can check against admin. data (coming in 1.5 year later)

• Using

• Where is the Taylor expanded variance of the ratio estimator under poisson sampling

( )

*

ˆ( )100( 1)

ˆ( )PO q q i

q

T y r p

q y r

V t

V t

( )

1 2ˆ( ) 1PO qT y r k qkU

V t E

1p

2p

3p

4p

5p

6p

Study variables

Considered Design Employees Turnover P-cost Invest Mean loss

Opt. on Empl 0 19 9 72 25

Opt. on Turn 9 0 36 67 28

Opt. on P-cost 8 6 0 68 21

Opt. on Invest 146 146 222 24 134

Minimizing Anorel 2 2 31 0 9

Minimizing Anorel with restrictions 10 8 45 8 18

Metal goodsRatios of the Taylor expanded variances to the smallest variance of each estimator (%)

Summary

• Carefully choosing appropriate size measures to get limits anticipated variances of

regression estimators. And Brewer’s results can be extended to a multivariate situation.

• If there is a multivariate issue and you intend to use auxiliary information in the design, diagnostic computations are important.

• With an optimization approach we know what we are aiming to minimize and with the non-linear programming approach some practical trouble in designing a pps-sample are avoided.


qkk ~

Using unequal probability sampling to limit antici-pated variances of regression estimators

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