Hans Kiesl Regensburg University of Applied Sciences, Germany … · Thank you very much for...
Transcript of Hans Kiesl Regensburg University of Applied Sciences, Germany … · Thank you very much for...
Multi-stage indirect sampling
Hans Kiesl
Regensburg University of Applied Sciences, Germany
NTTS Conference, 22-24 February 2011, Brussels
23 February 2011
Background
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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.
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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
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Inclusion probabilities for the kindergartens are difficult to come up with.
S h l d ki d tSchools and kindergartens
0ji 0ji
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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
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Example
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32
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84
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Example
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94
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Example
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32
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36353431~
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66
365
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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.
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Example
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45355
11w
6 4453534535
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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
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St t 2Strategy 2
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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!
NTTS Conference, 22-24 February 2011, Brussels
23 February 2011