Multi-stage indirect sampling

Hans Kiesl

Regensburg University of Applied Sciences, Germany

NTTS Conference, 22-24 February 2011, Brussels

23 February 2011

Background

2

Problem: How to sample among kindergarten kids?

In the German system, kids aged 4-6 (may) attend a kindergarten

These kindergartens are completely separated from primary schools. g p y p p y

No complete sampling frame for all kindergartens in Germany was available for our study.

However, a complete sampling frame for primary schools was available.

3

Problem: How to sample among kindergarten kids?

Idea:

draw a sample of primary schools

define the kindergarten sample by all kindergartens that send kids to the sampled primary schools

This is no cluster sample: schools are linked to different kindergartens and vice versa.

Inclusion probabilities for the kindergartens are difficult to come up with

4

Inclusion probabilities for the kindergartens are difficult to come up with.

S h l d ki d tSchools and kindergartens

0ji 0ji

5

Links

jiSeveral possibilities to define , e.g. :

if th i li k f ki d t i t h l j1• if there is a link from kindergarten i to school j

• number of children sent from kindergarten i to school j

1ji

ji

In general:

• Links have to be non-negative

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Links have to be non negative.

• For every unit in both populations, it must be possible todetermine all out-going links.

Indirect sampling (e.g. Lavallée 2007)

Idea total of Y among kindergartens ma be ie ed as a total of Y~

Idea: total of Y among kindergartens may be viewed as a total of among the primary schools:

Y

AA BB AB N

1jj

N

1j

N

1ii

i

jiN

1i

N

1j i

jii

N

1iiY y~yyyt

1j1j 1i i1i 1j i1i

1

7

1

Example

21

2

32

3

4

84

5

6

1

Example

21

2

32

3

4

94

5

6

1

Example

21

2

32

3

4

104

5

36353431~

6

66

365

5

354

4

341

1

313 yyyyy

Indirect sampling: estimation

AN

~tf

1j

jY ytHorvitz-Thompson estimator of :

A BA BA n n

ijin N

ijin

j yyy~t̂

1j 1i j

i

i

j

1j 1i j

i

i

j

1j j

jYt

n An

Bn

1iiiY ywt̂

An

1j ij

jiiwwith weights

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jiIf number of children sent from kindergarten i to school j, then

is the fraction of kindergarten i kids that are sent to school j.

Th l t li ti f ll i d d!iji /

Thus, no complete listing of all is needed! ji

S h l d ki d tSchools and kindergartensStep 1: Sampling of schools

Step 2: Identify linked kindergartens (indirect sample)What we actually need for estimation are the sums of out

Step 3: Identify all out-going links from sampled kindergartensWhat we actually need for estimation are the sums of out-going links for every (indirectly) sampled kindergarten.

12

1

Example

21

2

32

3

4

134

5

45355

11w

6 4453534535

5

Sampling strategy in our application

1 First sampling stage: selection of 200 primary schools1. First sampling stage: selection of 200 primary schools

2. Identify all kindergartens that have links to the sampled schools (source of information: sampled schools).

3. Due to cost restrictions, not all of these kindergartens may stay in the kindergarten sample: select 250 of them (final kindergarten sample)

4. Due to cost restrictions, not all kids in the final kindergarten sample t i th l l t 20 kid ithi hmay stay in the sample: select approx. 20 kids within each

kindergarten.

Lavallée (2007) examines 1,2,4 and calls it „two stage indirect sampling“.

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Thus we call 1-4 „three stage [or three phase] indirect sampling“.

How to select the final kindergarten sample

Two possible strategies:

For each sampled school j, select a sample of kindergartens that have links to school j (do this independently for all schools). The complete selection scheme might be called three-stage indirect sampling.

Select final kindergarten sample from preliminary kindergarten sample without regard to sampled schools. The complete selection scheme might be called three-phase indirect sampling.

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St t 1Strategy 1

16

St t 2Strategy 2

17

Define school sample preliminar kindergarten sample final

How to define the estimator (disregarding 3rd stage)

s sDefine school sample , preliminary kindergarten sample , final kindergarten sample .

Strategy 1: is the probability that kindergarten i is in the final sample given school j is in the school sample

ss kpskfs

j|igiven school j is in the school sample

ss kpsn n

ikfjiY

y)si(1t̂

Strategy 2: is the probability that kindergarten i is in the final sample given school sample

1j 1i j|iji

ss|is

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given school sample ss

kfs ssn niji

Y1yt̂

ss|i1i 1j ji

Why does it work?

n nikfji ))s|y)si(1

(E(E)t̂(Ess kps

Unbiased estimators („double expansion estimators“), e.g. strategy 1:

n niji

1stage

s1j 1i j|ij

i

i

j2stage1stageY

)y(E

))s|(E(E)t(E

ss kps

Y

1j 1i j|iji1stage

t

)(

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Final remarks

Given a direct sample from population A, it is not necessary to sample all linked units in population B.

V i t f li t ( h ) Variance components from every sampling stage (phase).

Concerning our application:

Schools seem to be able to provide accurate information.p

Kindergartens seem to provide some numbers with (small) measurement error (problem: might result in bias, even if measurement error has expected value of 0).

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Thank you very much for listening!

hans.kiesl@hs-regensburg.de

NTTS Conference, 22-24 February 2011, Brussels

23 February 2011