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STK 4600: Statistical methods for social sciences. Survey sampling and statistical demography Surveys for households and individuals. Survey sampling: 4 major topics. Traditional design-based statistical inference 6 weeks Likelihood considerations 1 weeks Model-based statistical inference - PowerPoint PPT Presentation
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1
STK 4600: Statistical methods for social sciences.
Survey sampling and statistical demography
Surveys for households and individuals
2
Survey sampling: 4 major topics
1. Traditional design-based statistical inference • 6 weeks
2. Likelihood considerations• 1 weeks
3. Model-based statistical inference• 3 weeks
4. Missing data - nonresponse• 2 weeks
3
Statistical demography
• Mortality
• Life expectancy
• Population projections• 2-3 weeks
4
Course goals
• Give students knowledge about:– planning surveys in social sciences
– major sampling designs
– basic concepts and the most important estimation methods in traditional applied survey sampling
– Likelihood principle and its consequences for survey sampling
– Use of modeling in sampling
– Treatment of nonresponse
– A basic knowledge of demography
5
But first: Basic concepts in sampling
Population (Target population): The universe of all units of interest for a certain study
• Denoted, with N being the size of the population: U = {1, 2, ...., N}
All units can be identified and labeled
• Ex: Political poll – All adults eligible to vote
• Ex: Employment/Unemployment in Norway– All persons in Norway, age 15 or more
• Ex: Consumer expenditure : Unit = household
Sample: A subset of the population, to be observed. The sample should be ”representative” of the population
6
Sampling design:
• The sample is a probability sample if all units in the sample have been chosen with certain probabilities, and such that each unit in the population has a positive probability of being chosen to the sample
• We shall only be concerned with probability sampling
• Example: simple random sample (SRS). Let n denote the sample size. Every possible subset of n units has the same chance of being the sample. Then all units in the population have the same probability n/N of being chosen to the sample.
• The probability distribution for SRS on all subsets of U is an example of a sampling design: The probability plan for selecting a sample s from the population:
nssp
nsn
Nsp
|| if 0)(
|| if /1)(
7
Basic statistical problem: Estimation
• A typical survey has many variables of interest
• Aim of a sample is to obtain information regarding totals or averages of these variables for the whole population
• Examples : Unemployment in Norway– Want to estimate the total number t of individuals unemployed.
For each person i (at least 15 years old) in Norway:
otherwise 0 ,unemployed is person if 1 iyi
Ni iyt 1
:Then
8
• In general, variable of interest: y with yi equal to the value of y for unit i in the population, and the total is denoted
Ni iyt 1
• The typical problem is to estimate t or t/N
•Sometimes, of interest also to estimate ratios of totals:
Example- estimating the rate of unemployment:
otherwise 0 force,labor in the is person if 1
otherwise 0 ,unemployed is person if 1
ix
iy
i
i
xy tt , swith total
Unemployment rate: xy tt /
9
Sources of error in sample surveys
1. Target population U vs Frame population UF
Access to the population is thru a list of units – a register UF . U and UF may not be the same: Three possible errors in UF:
– Undercoverage: Some units in U are not in UF
– Overcoverage: Some units in UF are not in U
– Duplicate listings: A unit in U is listed more than once in UF
• UF is sometimes called the sampling frame
10
2. Nonresponse - missing data• Some persons cannot be contacted• Some refuse to participate in the survey• Some may be ill and incapable of responding• In postal surveys: Can be as much as 70%
nonresponse• In telephone surveys: 50% nonresponse is not
uncommon
• Possible consequences:– Bias in the sample, not representative of the
population– Estimation becomes more inaccurate
• Remedies: – imputation, weighting
11
3. Measurement error – the correct value of yi is not measured
– In interviewer surveys:• Incorrect marking
• interviewer effect: people may say what they think the interviewer wants to hear – underreporting of alcohol ute, tobacco use
• misunderstanding of the question, do not remember correctly.
12
4. Sampling error– The error caused by observing a sample
instead of the whole population– To assess this error- margin of error:
measure sample to sample variation
– Design approach deals with calculating sampling errors for different sampling designs
– One such measure: 95% confidence interval:
If we draw repeated samples, then 95% of the calculated confidence intervals for a total t will actually include t
13
• The first 3 errors: nonsampling errors– Can be much larger than the sampling error
• In this course:– Sampling error– nonresponse bias– Shall assume that the frame population is
identical to the target population– No measurement error
14
Summary of basic concepts
• Population, target population• unit• sample• sampling design• estimation
– estimator– measure of bias – measure of variance– confidence interval
15
• survey errors:– register /frame population– mesurement error– nonresponse– sampling error
16
Example – Psychiatric Morbidity Survey 1993 from Great Britain
• Aim: Provide information about prevalence of psychiatric problems among adults in GB as well as their associated social disabilities and use of services
• Target population: Adults aged 16-64 living in private households
• Sample: Thru several stages: 18,000 adresses were chosen and 1 adult in each household was chosen
• 200 interviewers, each visiting 90 households
17
Result of the sampling process• Sample of addresses 18,000
Vacant premises 927Institutions/business premises 573Demolished 499Second home/holiday flat 236
• Private household addresses 15,765Extra households found 669
• Total private households 16,434Households with no one 16-64 3,704
• Eligible households 12,730• Nonresponse 2,622• Sample 10,108
households with responding adults aged 16-64
18
Why sampling ?• reduces costs for acceptable level of accuracy
(money, manpower, processing time...)• may free up resources to reduce nonsampling error
and collect more information from each person in the sample– ex:
400 interviewers at $5 per interview: lower sampling error
200 interviewers at 10$ per interview: lower nonsampling error
• much quicker results
19
When is sample representative ?• Balance on gender and age:
– proportion of women in sample proportion in population
– proportions of age groups in sample proportions in population
• An ideal representative sample: – A miniature version of the population: – implying that every unit in the sample represents the
characteristics of a known number of units in the population
• Appropriate probability sampling ensures a representative sample ”on the average”
20
Alternative approaches for statistical inference based on survey sampling
• Design-based: – No modeling, only stochastic element is the
sample s with known distribution• Model-based: The values yi are assumed to be
values of random variables Yi: – Two stochastic elements: Y = (Y1, …,YN) and s– Assumes a parametric distribution for Y– Example : suppose we have an auxiliary
variable x. Could be: age, gender, education. A typical model is a regression of Yi on xi.
21
• Statistical principles of inference imply that the model-based approach is the most sound and valid approach
• Start with learning the design-based approach since it is the most applied approach to survey sampling used by national statistical institutes and most research institutes for social sciences. – Is the easy way out: Do not need to model. All
statisticians working with survey sampling in practice need to know this approach
22
Design-based statistical inference• Can also be viewed as a distribution-free
nonparametric approach• The only stochastic element: Sample s, distribution
p(s) for all subsets s of the population U={1, ..., N}• No explicit statistical modeling is done for the
variable y. All yi’s are considered fixed but unknown • Focus on sampling error• Sets the sample survey theory apart from usual
statistical analysis• The traditional approach, started by Neyman in 1934
23
Estimation theory-simple random sample
Estimation of the population mean of a variable y: NyN
i i /1
A natural estimator - the sample mean: nyy si is / Desirable properties:
)ˆ(
ifunbiased is ˆestimator An :ess UnbiasednI)(
Edesign SRSfor unbiased is sy
SRS of size n: Each sample s of size n has
n
Nsp /1)(
Can be performed in principle by drawing one unit at time at random without replacement
24
The uncertainty of an unbiased estimator is measured by its estimated sampling variance or standard error (SE):
)ˆ(ˆ)ˆ(
)ˆ( of estimate (unbiased)an is )ˆ(ˆ
)ˆ( if ,)ˆ()ˆ( 2
VSE
VarV
EEVar
Some results for SRS:
)( )2(
fraction sampling the, /Then
sample, in the is unit y that probabilit thebe Let )1(
s
i
i
yE
fNn
i
25
correction population finite called the is )-(1factor theHere,
)1()(
)(1
1 :variance population thebe Let )3(
2
1222
f
fn
yVar
yN
s
Ni i
• usually unimportant in social surveys:
n =10,000 and N = 5,000,000: 1- f = 0.998
n =1000 and N = 400,000: 1- f = 0.9975
n =1000 and N = 5,000,000: 1-f = 0.9998
• effect of changing n much more important than effect of changing n/N
26
si si yyn
s 22
2
)(1
1
variance sample
by thegiven is ofestimator unbiased An
The estimated variance )1()(ˆ2
fn
syV s
Usually we report the standard error of the estimate:
)(ˆ)( ss yVySE
Confidence intervals for is based on the Central Limit Theorem:
)1,0(~/)1(/)(:, large For NnfyZnNn s
)(96.1)(96.1 ),(96.1
:for CI95% eApproximat
ssssss ySEyySEyySEy
27
Example
N = 341 residential blocks in Ames, Iowa
yi = number of dwellings in block i
1000 independent SRS for different values of n
n Proportion of samples with |Z| <1.64
Proportion of samples with |Z| <1.96
30 0.88 0.93
50 0.88 0.93
70 0.88 0.94
90 0.90 0.95
28
For one SRS with n = 90:
14.53) 11.47,( 1.5313 0.781.9613 :CI95% eApproximat
78.090/75)341/901()(
75
132
s
s
ySE
s
y
29
The coefficient of variation for the estimate:
sss yySEyCV /)()(
•A measure of the relative variability of an estimate.
•It does not depend on the unit of measurement.
• More stable over repeated surveys, can be used for planning, for example determining sample size
• More meaningful when estimating proportions
Absolute value of sampling error is not informative when not related to value of the estimate
For example, SE =2 is small if estimate is 1000, but very large if estimate is 3
%606.013/78.0)( :exampleIn syCV
30
Estimation of a population proportion pwith a certain characteristic A
p = (number of units in the population with A)/N
Let yi = 1 if unit i has characteristic A, 0 otherwise
Then p is the population mean of the yi’s.
Let X be the number of units in the sample with characteristic A. Then the sample mean can be expressed as
nXyp s /ˆ
31
1
)1( equals variance population thesince
)1
11(
)1()ˆ(
and
)ˆ(
:SRSunder Then
2
N
pNpN
n
n
pppVar
ppE
)ˆ1(ˆ1
2 ppn
ns
So the unbiased estimate of the variance of the estimator:
)1(1
)ˆ1(ˆ)ˆ(ˆ
N
n
n
pppV
32
Examples
A political poll: Suppose we have a random sample of 1000 eligible voters in Norway with 280 saying they will vote for the Labor party. Then the estimated proportion of Labor votes in Norway is given by:
2801000280 ./p
01440999
7202801
1
1.
..)
N
n(
n
)p(p)p(SE
Confidence interval requires normal approximation. Can use the guideline from binomial distribution, when N-n is large: 5)1(and 5 pnnp
33
In this example : n = 1000 and N = 4,000,000
0.308) (0.252, 0.028 0.280
961 :CI 95% eApproximat
)p(SE.p
Ex: Psychiatric Morbidity Survey 1993 from Great Britain
p = proportion with psychiatric problems
n = 9792 (partial nonresponse on this question: 316)
N
47)(0.133,0.1 0.0070.14 0.00351.96 0.14 :CI %95
0035.09791/86.014.0)00024.01()ˆ(
14.0ˆ
pSE
p
34
General probability sampling• Sampling design: p(s) - known probability of selection for each subset s of the population U
• Actually: The sampling design is the probability distribution p(.) over all subsets of U
• Typically, for most s: p(s) = 0 . In SRS of size n, all s with size different from n has p(s) = 0.
• The inclusion probability:
}:{)()(
sample) in the is unit (
sis
i
spsiP
iP
35
Illustration
U = {1,2,3,4}Sample of size 2; 6 possible samplesSampling design: p({1,2}) = ½, p({2,3}) = 1/4, p({3,4}) = 1/8, p({1,4}) = 1/8
The inclusion probabilities:
}4:{4
}3:{3
}2:{2
}1:{1
8/2})4,1({})4,3({)(
8/3})4,3({})3,2({)(
8/64/3})3,2({})2,1({)(
8/5})4,1({})2,1({)(
ss
ss
ss
ss
ppsp
ppsp
ppsp
ppsp
36
Some results
n
nII
nnEI
N
N
...
:advancein be todetermined is size sample If )(
size sample theis ; )(... )(
21
21
N
i i
N
i i
N
i i
iii
i
ZEnEZn
ZEZP
iZLet
111)()(
)()1(
otherwise 0 sample, in the included is unit if 1
:Proof
37
Estimation theory probability sampling in general
Problem: Estimate a population quantity for the variable y
For the sake of illustration: The population total
N
iiyt
1
tt ˆ :sample on thebased ofestimator An
)ˆ( ifunbiased is ˆ
)ˆ( :Bias
)(]ˆ)(ˆ[]ˆˆ[)ˆ( :Variance
)()(ˆ)ˆ( :valueExpected 22
ttEt
ttE
sptEsttEtEtVar
spsttE
s
s
38
ttSEtCVt
tVtSEt
tVartV
ˆ/)ˆ()ˆ( :ˆ ofvariation oft Coefficien
)ˆ(ˆ)ˆ( :ˆ oferror standard The
)ˆ( of estimate possible) if(unbiased an be )ˆ(ˆLet
CV is a useful measure of uncertainty, especially when standard error increases as the estimate increases
Because, typically we have that
nNntSEtttSEtP , largefor 95.0))ˆ(2ˆ)ˆ(2ˆ(
)ˆ(2 :error ofMargin tSE
nNnt , largefor d distributenormally ely approximat is ˆ Since
CI 95% aely approximat is 2 )t(SEt
39
Some peculiarities in the estimation theoryExample: N=3, n=2, simple random sample
3313232
3213122
112112
2
1
321
2
1)(ˆ)
3
1
2
1(3)(ˆ
2
1)(ˆ)
3
2
2
1(3)(ˆ
)(ˆ)(2
13)(ˆ
:bygiven be ˆLet
unbiased ,3ˆLet
1,2,3 for 3/1)(
}3,2{},3,1{},2,1{
ystyyst
ystyyst
styyst
t
yt
ksp
sss
s
k
40
ttstspsttE
t
k ks 33
1)(ˆ
3
1)()(ˆ)ˆ(
:unbiased is ˆ Also
31 222
2
)33(6
1)ˆ()ˆ( 312321 yyyytVartVar
1,0when happens thisvariables,1/0 If
33and 0 if )ˆ()ˆ(
321
312321
yyyy
yyyytVartVar
i
For this set of values of the yi’s:
5.2)(ˆ ,2)(ˆ ,5.1)(ˆ
correctnever : 3)(ˆ ,5.1)(ˆ ,5.1)(ˆ
322212
312111
ststst
ststst
values- for these ˆy than variabilit lessclearly has ˆ12 ytt
41
Let y be the population vector of the y-values.
This example shows that
syNis not uniformly best ( minimum variance for all y) among linear design-unbiased estimators
Example shows that the ”usual” basic estimators do not have the same properties in design-based survey sampling as they do in ordinary statistical models
In fact, we have the following much stronger result:
Theorem: Let p(.) be any sampling design. Assume each yi can take at least two values. Then there exists no uniformly best design-unbiased estimator of the total t
42
Proof:
0
0
yy
yy
when 0)ˆ(with ˆunbiased exists Then there
. of value possible one be let and unbiased, be ˆLet
00 tVart
t
00 yyyy for total theis ,),(ˆ),(ˆ),(ˆ000 ttststst
0)ˆ( samples allfor ˆ:When )2
)(),(ˆ)ˆ( :unbiased is ˆ )1
000
000
tVarstt
ttspstttEt s
0
0
yy
y
This implies that a uniformly best unbiased estimator must have variance equal to 0 for all values of y, which is impossible
43
Determining sample size
• The sample size has a decisive effect on the cost of the survey
• How large n should be depends on the purpose for doing the survey
• In a poll for detemining voting preference, n = 1000 is typically enough
• In the quarterly labor force survey in Norway, n = 24000
Mainly three factors to consider:
1. Desired accuracy of the estimates for many variables. Focus on one or two variables of primary interest
2. Homogeneity of the population. Needs smaller samples if little variation in the population
3. Estimation for subgroups, domains, of the population
44
It is often factor 3 that puts the highest demand on the survey
• If we want to estimate totals for domains of the population we should take a stratified sample
• A sample from each domain
• A stratified random sample: From each domain a simple random sample
H
H
n...nnn
n,...,n,n
H
21
21
: size sample Total
:sizes Sample
population whole theconstitute that strata
hneach determineMust
45
Assume the problem is to estimate a population proportion p for a certain stratum, and we use the sample proportion from the stratum to estimate p
Let n be the sample size of this stratum, and assume that n/N is negligible
Desired accuracy for this stratum: 95% CI for p should be %5
n
pppp
)ˆ1(ˆ96.1ˆ:for CI95%
The accuracy requirement:
384)ˆ1(ˆ2096.1
20
105.0
)ˆ1(ˆ96.1
22
ppn
n
pp
46
The estimate is unkown in the planning fase
Use the conservative size 384 or a planning value p0 with n = 1536 p0(1- p0 )
F.ex.: With p0 = 0.2: n = 246
In general with accuracy requirement d, 95% CI dp ˆ
200 /)1(84.3 dppn
edpCVpdn
pp
pp
p
96.1/)ˆ(ˆ )ˆ1(ˆ
96.1
) -1 estimate otherwise ,5.0ˆ(when
ˆ toalproportion is CI95% ofLength
:trequiremenaccuracy eAlternativ
47
With e = 0.1, then we require approximately that
900and 02.0ˆ CI %95:1.0when
100and 10.0ˆ CI %95:5.0when
0
0
npp
npp
0
020
2
11: value Planning
ˆ
ˆ11ˆ/)ˆ(
p
p
enp
p
p
eneppSE
48
Example: Monthly unemployment rate
Important to detect changes in unemployment rates from month to month
planning value p0 = 0.05
7300005.0
600,45002.0
182,400 0.1%) error of(margin 001.0
/1824.0/)1(84.3)ˆ(1.96
:accuracyDesired 22
00
nd
nd
nd
ddppndpSE
%5051.05.0/00255.0)ˆ(005.0 :Note pCVd
49
Two basic estimators:Ratio estimator
Horvitz-Thompson estimator
• Ratio estimator in simple random samples
• H-T estimator for unequal probability sampling: The inclusion probabilities are unequal
• The goal is to estimate a population total t for a variable y
50
Ratio estimator
),...,( 21 Nxxxx
N
i ixX1
Let
Suppose we have known auxiliary information for the whole population:
Ex: age, gender, education, employment status
The ratio estimator for the y-total t:
s
s
si i
si iR x
yX
x
yXt
ˆ
51
We can express the ratio estimator on the following form:
)(ˆs
sR yN
xN
Xt
It adjusts the usual “sample mean estimator” in the cases where the x-values in the sample are too small or too large.
Reasonable if there is a positive correlation between x and y
Example: University of 4000 students, SRS of 400
Estimate the total number t of women that is planning a career in teaching, t=Np, p is the proportion
yi = 1 if student i is a woman planning to be a teacher, t is the y-total
52
Results : 84 out of 240 women in the sample plans to be a teacher
840ˆˆ
21.0400/84ˆ
pNt
p
HOWEVER: It was noticed that the university has 2700 women (67,5%) while in the sample we had 60% women. A better estimate that corrects for the underrepresentation of women is obtained by the ratio estimate using the auxiliary
x = 1 if student is a woman
945)840(6.04000
2700ˆ
Rt
53
In business surveys it is very common to use a ratio estimator.
Ex: yi = amount spent on health insurance by business i
xi = number of employees in business i
We shall now do a comparison between the ratio estimator and the sample mean based estimator. We need to derive expectation and variance for the ratio estimator
54
First: Must define the population covariance
variables and theof means population are ,
))((1
11
xy
yxN
yx
N
i yixixy
N
i xix
N
i yiy
xN
yN
1
22
1
22
)(1
1
)(1
1
The population correlation coefficient: yx
xyxy
55
),ˆ()ˆ( :Bias )(
//ˆ and
//Let 11
sR
ssss
N
i i
N
i i
xNRCovttEI
xyxNyNR
XtxyR
),ˆ()()ˆ(
)1(ˆ
Proof
sss
sR
s
ss
s
sR
xNRCovXxNxN
yNEttE
txN
XxNyNtX
xN
yNtt
56
It follows that
)(|),ˆ(|)(
)()()ˆ(
|),ˆ(|
)ˆ(
|)ˆBias(|
sss
s
s
s
R
R
xCVxNRCorrxNCV
xNVarxNVarRVarX
xNRCov
tVar
t
Hence, in SRS, the absolute bias of the ratio estimator is small relative to the true SE of the estimator if the coefficient of variation of the x-sample mean is small
Certainly true for large n
57
nt)t(E)II( R largefor ,
N
i ii
xxyyR
RxyNn
fN
RRn
fNtVarIII
1
22
2222
)(1
11
)2(1
)ˆ( )(
58
Note: The ratio estimator is very precise when the population points (yi , xi) lie close around a straight line thru the origin with slope R.
The regression model generates the ratio estimator
59
N
i iiR RxyNn
fNtVar
1
22 )(1
11 )ˆ(
N
i yi
N
i iisR yRxyyNVartVar1
2
1
2 )()()()ˆ(
The ratio estimator is more accurate if Rxi predicts yi better than y does
N
i yis yNn
fNyNVar
1
22 )(1
11)(
that recalling and
60
Estimated variance for the ratio estimator
)1/()ˆ(by
)1/()( Estimate
2
1
2
nxRy
NRxy
si ii
N
i ii
atreflect th larger to becomes estimate variancethe
anduncertain more is ˆ then small, very is If :
)ˆ(1
11)ˆ(ˆ 22
2
RxNote
xRynn
fN
xtV
s
si iis
xR
61
For large n, N-n: Approximate normality holds and an approximate 95% confidence interval is given by
si iis
R xRynn
f
x
Xt 2)ˆ(
1
1196.1ˆ
62
Unequal probability sampling
tobelongs individual
that household in the 64-16 adults ofnumber
/1
i
M
M
i
ii
Example:
Psychiatric Morbidity Survey: Selected individuals from households
Inclusion probabilities:NisiPi ,...,1 allfor 0)(
63
Horvitz-Thompson estimator- unequal probability sampling
NisiPi ,...,1 allfor 0)(
syNLet’s try and use
unbiasednot
)/()(1
)(
)( otherwise. 0 , if 1Let
11
N
i ii
N
i iis
iii
tynNZyEn
NyNE
ZEsiZ
Bias is large if inclusion probabilities tend to increase or decrease systematically with yi
64
Use weighting to correct for bias:
ii
i
N
i iii
N
i iii
isi ii
w
yt
ywZywEtE
swywt
/1 ifonly and if
valuespossible allfor unbiased is ˆ and
)ˆ(
on dependnot does ; ˆ
11
sii
iHT
yt
ˆ
sHTi yNtNn ˆ and / SRS,In
65
2
1
1 1
1
1 1
2
1
)()ˆ( )
then,|| If
21
)ˆ( )
N
i
N
ijj
j
i
iijjiHT
ji
N
i
N
ijji
jiiji
N
ii
iHT
yytVarb
ns
yyytVara
)1(),( jiij ZZPsjiP
Horvitz-Thompson estimator is widely used f.ex., in official statistics
66
Note that the variance is small if we determine the inclusion probabilities such that
ii
ii
y
y
increasing with increases i.e.
equal,ely approximat are /
Of course, we do not know the value of yi when planning the survey, use known auxiliary xi and choose
Xnxx iiii /
nN
i i 1 since
67
unequal are ' h theeven thoug
estimator,-HT usenot should one and enormous becan )ˆ(
"correlated" negativelyor relatednot are and If
s
tVar
y
i
HT
ii
Example: Population of 3 elephants, to be shipped. Needs an estimate for the total weight
•Weighing an elephant is no simple matter. Owner wants to estimate the total weight by weighing just one elephant.
• Knows from earlier: Elephant 2 has a weight y2 close to the average weight. Wants to use this elephant and use 3y2 as an estimate
• However: To get an unbiased estimator, all inclusion probabilities must be positive.
68
• Sampling design:
05.0 ,90.0 and 1|| 312 s
• The weights: 1,2, 4 tons, total = 7 tons
}3{ if 80
{2} if 22.2
}1{ if 20
}{ if /ˆ
s
s
s
isyt iiHT • H-T estimator:
Hopeless! Always far from true total of 7
ttE HT 7)ˆ(Can not be used, even though
69
Problem:
46.295
05.0.)780(90.0)722.2(05.0)720()ˆ( 222
HTtVar
!!! 2.17)ˆ()ˆ( True HTHT tVartSE
The planned estimator, even though not a SRS:
}{ if 33ˆ isyyt iseleph
Possible values: 3, 6, 12
70
49122752
atlook but unbiased,not
156
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.)t(E
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eleph
elephelepheleph
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tVarttEtMSE
topreferableclearly is HTeleph tt
71
Variance estimate for H-T estimator
2
)ˆ(ˆ
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all provided),ˆ( ofestimator unbiasedAn
j
j
i
i
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ijjiHT
ij
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yytV
tVar
Assume the size of the sample is determined in advance to be n.
)ˆ(ˆ96.1ˆ
:, largefor CI, 95% eApproximat
HTHT tVt
nNn
72
• Can always compute the variance estimate!!Since, necessarily ij > 0 for all i,j in the sample s
• But: If not all ij > 0 , should not use this estimate! It can give very incorrect estimates
• The variance estimate can be negative, but for most sampling designs it is always positive
73
A modified H-T estimator
Consider first estimating the population mean
Nty HTHT /ˆˆ
Nty /
An obvious choice:
Alternative: Estimate N as well, whether N is known or not
),1( 1ˆ iyN isi
i
NZENEN
i ii
N
i ii
11
11)ˆ(
Nn
NNNn
si
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74
si i
si iiHTw
yNty
/1
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estimator ratio a isit that note Wenot.or known is
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75
If sample size varies then the “ratio” estimator performs better than the H-T estimator, the ratio is more stable than the numerator
Example:
tlyindependen ,y probabilit
with selected is population in theunit Each
:sampling Bernoulli design Sampling
,...,1for ,
Nicyi
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76
tNcn
ncNt
tNccN
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cn
t
w
HT
HT
/
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H-T estimator varies because n varies, while the modified H-T is perfectly stable
77
Review of Advantages of Probability Sampling
• Objective basis for inference• Permits unbiased or approximately unbiased
estimation• Permits estimation of sampling errors of
estimators– Use central limit theorem for confidence interval
– Can choose n to reduce SE or CV for estimator
78
Outstanding issues in design-based inference
• Estimation for subpopulations, domains• Choice of sampling design –
– discuss several different sampling designs
– appropriate estimators
• More on use of auxiliary information to improve estimates
• More on variance estimation
79
Estimation for domains• Domain (subpopulation): a subset of the
population of interest• Ex: Population = all adults aged 16-64
Examples of domains:
– Women
– Adults aged 35-39
– Men aged 25-29
– Women of a certain ethnic group
– Adults living in a certain city
• Partition population U into D disjoint domains U1,…,Ud,..., UD of sizes N1,…,Nd,…,ND
80
Estimating domain means Simple random sample from the population
dUi did Ny / :meandomain True
• e.g., proportion of divorced women with psychiatric problems.
||
/
in sample theofpart the
: frommean sample by the Estimate
dd
dsi is
dd
dd
sn
nyy
Uss
U
dd
Note: nd is a random variable
81
The estimator is a ratio estimator:
otherwise 0
if 1
otherwise 0
if
Define
di
dii
Uix
Uiyu
Rxuxuy
Rxu
sssi si iis
N
i
N
i iid
d
ˆ//
/1 1
82
si isid
d
ds
s
xyunn
fN
nn
NN
NyV
ny
dd
d
22
2
2)(
1
11
/
/1)(ˆ
largefor unbiasedely approximat is
d dsi sid
d
d
yyn
s
s
22
2
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domain, for the variancesample thebe Let
d
ddd
ds n
sfsn
nn
f
n
nyV
d
22
2
2
)1()1()1(
1)(ˆ
NnfnsfySE ddsd/ , /)1()( 2
83
fNnf ddd / samples largeFor
• Can then treat sd as a SRS from Ud
• Whatever size of n is, conditional on nd, sd is a SRS from Ud – conditional inference
Example: Psychiatric Morbidity Survey 1993
Proportions with psychiatric problems
Domain d nd SE
women 4933 0.18
Divorced women
314 0.29
dsy )(dsy
005.04932/82.018.
026.0313/71.029.0
84
Estimating domain totals
dsd yN
ssi isdd
dsd
d
uNun
NyNt
nnNxNN
xN
d
1ˆˆ
/ˆ
:total- theis Since
• Nd is known: Use
• Nd unknown, must be estimated
nsfNtSE ud /)1()ˆ( 2
85
Stratified sampling• Basic idea: Partition the population U into H
subpopulations, called strata. • Nh = size of stratum h, known• Draw a separate sample from each stratum, sh of size nh
from stratum h, independently between the strata• In social surveys: Stratify by geographic regions, age
groups, gender• Ex –business survey. Canadian survey of employment.
Establishments stratified by o Standard Industrial Classification – 16 industry
divisionso Size – number of employees, 4 groups, 0-19, 20-49, 50-
199, 200+o Province – 12 provinces
Total number of strata: 16x4x12=768
86
Reasons for stratification
1. Strata form domains of interest for which separate estimates of given precision is required, e.g. strata = geographical regions
2. To “spread” the sample over the whole population. Easier to get a representative sample
3. To get more accurate estimates of population totals, reduce sampling variance
4. Can use different modes of data collection in different strata, e.g. telephone versus home interviews
87
Stratified simple random sampling
• The most common stratified sampling design
• SRS from each stratum
hsi hhih
hhi
hhi
H
h h
hh
nyy
siy
Niyh
nn
nsh
/ :mean Sample
):( :Sample
,...,1,: stratum from Values
:size sample Total
size of sample: stratum From
1
• Notation:
88
th = y-total for stratum h: hN
i hih yt1
H
h htt1
: totalpopulation The
Consider estimation of th: hhh yNt ˆ
Assuming no auxiliary information in addition to the “stratifying variables”
The stratified estimator of t:
h
H
h h
H
h hst yNtt
11ˆˆ
89
h
H
hh
stst yN
NNty
Nt
1/ˆ :mean Stratified
:/mean population theestimate To
A weighted average of the sample stratum means.
•Properties of the stratified estimator follows from properties of SRS estimators.
•Notation:
h
h
N
i hhih
h
h
N
i hih
yN
h
Nyh
1
22
1
)(1
1: stratumin Variance
/: stratumin Mean
90
Hh
Hh h
h
hhhst
stst
)f(n
N)t(Var)t(Var
tt)t(E
1 1
22 1
unbiased is ,
Estimated variance is obtained by estimating the stratum variance with the stratum sample variance
hsi hhih
h yyn
s 22 )(1
1
)1()ˆ(ˆ1
22
h
H
hh
hhst f
n
sNtV
Approximate 95% confidence interval if n and N-n are large:
)(ˆ96.1ˆstst tVt
91
Estimating population proportion in stratified simple random sampling
A sticcharacteri has stratumin unit if 1 where
ˆ
hiy
yp
hi
hh
NpNH
h hh /1
h
H
h hstst pNNyp ˆ)/(ˆ1
ph : proportion in stratum h with a certain characteristic A
p is the population mean: p = t/N
Stratum mean estimator:
Stratified estimator of the total t = number of units in thewith characteristic A:
H
h hhstst pNpNt1
ˆˆˆ
92
Estimated variance:
31) (slide 11
1)
N
n(
n
)p(p)p(V
h
h
h
hhh
H
hh
hh
h
hh
H
h hhst
hh
H
h
H
hh
hh
h
hhhhst
n
pp
N
nWNpWNVtV
NNW
n
pp
N
nWpWVpV
1
22
1
1 1
2
1
)ˆ1(ˆ)1()ˆ(ˆ)ˆ(ˆ
and
/ where
1
)ˆ1(ˆ)1()ˆ(ˆ)ˆ(ˆ
93
Allocation of the sample units• Important to determine the sizes of the stratum samples,
given the total sample size n and given the strata partitioning – how to allocate the sample units to the different strata
• Proportional allocation– A representative sample should mirror the population– Strata proportions: Wh=Nh/N– Strata sample proportions should be the same:
nh/n = Wh
– Proportional allocation:
hN
n
N
n
N
Nnn
h
hhh allfor
94
The stratified estimator under proportional allocation
SRS anot isit but ,population in the units allfor same the
// : iesprobabilitInclusion NnNn hhhi
s
H
h si hi
si hih
H
h hh
H
h hst
yNyn
N
yn
NyNt
h
h
1
11
1ˆ
/ˆ :mean stratified The sstst yNty
The equally weighted sample mean ( sample is self-weighting: Every unit in the sample represents the same number of units in the population , N/n)
95
Variance and estimated variance under proportional allocation
NNWNnfWn
fN
fn
NtVar
hh
H
h hh
H
h hh
hhst
/ ,/ , 1
)1()ˆ(
1
22
1
22
H
h hhst sWn
fNtV
1
22 1)ˆ(ˆ
96
• The estimator in simple random sample:
sSRS yNt ˆ
• Under proportional allocation:
SRSst tt ˆˆ
• but the variances are different:
H
h hhst
SRSSRS
Wn
fNtVar
n
fNtVar
1
22
22
1)ˆ( :allocation alproportionUnder
1)ˆ( :SRSUnder
97
H
h hh
H
h hh
h
hhh
WW
N
N
N
N
N
N
1
2
1
22 )(
:11
and 1
1 ionsapproximat theUsing
Total variance = variance within strata + variance between strata
Implications:1. No matter what the stratification scheme is: Proportional allocation gives more accurate estimates of population total than SRS2. Choose strata with little variability, smaller strata variances. Then the strata means will vary more and between variance becomes larger and precision of estimates increases compared to SRS
2
1
22 1)ˆ(
fromseen as general,in y trueessentiall also is This .3
hh
hH
h hst n
fWNtV
98
Optimal allocationIf the only concern is to estimate the population total t:
• Choose nh such that the variance of the stratified estimator is minimum
• Solution depends on the unkown stratum variances• If the stratum variances are approximately equal,
proportional allocation minimizes the variance of the stratified estimator
99
H
k kk
hhh
N
Nnn
1
:allocation Optimal
)()11
(
Minimize :method multiplier Lagrange Use
fixed is subject to
sizes sample therespect to with )ˆ( Minimize
:Proof
1
2
1
2
1
nnNn
NQ
nnn
tVar
H
h hhh
h
H
h h
H
h hh
st
01
0 222
hh
hh
Nnn
Q /hhh Nn
Result follows since the sample sizes must add up to n
100
• Called Neyman allocation (Neyman, 1934)• Should sample heavily in strata if
– The stratum accounts for a large part of the population
– The stratum variance is large
• If the stratum variances are equal, this is proportional allocation
• Problem, of course: Stratum variances are unknown– Take a small preliminary sample (pilot)
– The variance of the stratified estimator is not very sensitive to deviations from the optimal allocation. Need just crude approximations of the stratum variances
101
Optimal allocation when considering the cost of a survey
• C represents the total cost of the survey, fixed – our budget
• c0 : overhead cost, like maintaining an office• ch : cost of taking an observation in stratum h
– Home interviews: traveling cost +interview– Telephone or postal surveys: ch is the same for all
strata– In some strata: telephone, in others home interviews
h
H
h hcncC
10
• Minimize the variance of the stratified estimator for a given total cost C
102
H
h hh
hhh
H
h hst
Ccnc
NnWNtVar
10
2
1
22
:subject to
)11
()ˆ( Minimize
Solution: hhhh cWn /
H
k kkkh
hhh
cW
cC
c
Wn
1
0 )(
H
h hhh
H
h hhh
cN
cNcCn
C
1
10 /)(
:cost totalfixed afor Hence,
103
allocation alproportion
:equal are ' theand equal are ' theIf 3.
allocationNeyman :equal are ' theIf .2
strata einexpensivin samples Large 1.
ssc
sc
hh
h
We can express the optimal sample sizes in relation to n
H
k kkk
hhhh
cW
cWnn
1/
/
In particular, if ch = c for all h: ccCn /)( 0
104
Other issues with optimal allocation• Many survey variables• Each variable leads to a different optimal solution
– Choose one or two key variables– Use proportional allocation as a compromise
• If nh > Nh, let nh =Nh and use optimal allocation for the remaining strata
• If nh=1, can not estimate variance. Force nh =2 or collapse strata for variance estimation
• Number of strata: For a given n often best to increase number of strata as much as possible. Depends on available information
105
• Sometimes the main interest is in precision of the estimates for stratum totals and less interest in the precision of the estimate for the population total
• Need to decide nh to achieve desired accuracy for estimate of th, discussed earlier
– If we decide to do proportional allocation, it can mean in small strata (small Nh) the sample size nh must be increased
106
Poststratification
• Stratification reduces the uncertainty of the estimator compared to SRS
• In many cases one wants to stratify according to variables that are not known or used in sampling
• Can then stratify after the data have been collected• Hence, the term poststratification• The estimator is then the usual stratified estimator
according to the poststratification• If we take a SRS and N-n and n are large, the
estimator behaves like the stratified estimator with proportional allocation
107
Poststratification to reduce nonresponse bias
• Poststratification is mostly used to correct for nonresponse
• Choose strata with different response rates• Poststratification amounts to assuming that the
response sample in poststratum h is representative for the nonresponse group in the sample from poststratum h
108
Systematic sampling• Idea:Order the population and select every kth unit• Procedure: U = {1,…,N} and N=nk + c, c < n
1. Select a random integer r between 1 and k, with equal probability
2. Select the sample sr by the systematic rule
sr = {i: i = r + (j-1)k: j= 1, …, nr}
where the actual sample size nr takes values [N/k] or [N/k] +1 k : sampling interval = [N/n]
• Very easy to implement: Visit every 10th house or interview every 50th name in the telephone book
109
• k distinct samples each selected with probability 1/k
otherwise 0
,...,1 , if /1)(
krssk
sp r
• Unlike in SRS, many subsets of U have zero probability
Examples:
1) N =20, n=4. Then k=5 and c=0. Suppose we select r =1. Then the sample is {1,6,11,16}
5 possible distinct samples. In SRS: 4845 distinct samples
2) N= 149, n = 12. Then k = 12, c=5. Suppose r = 3. s3 = {3,15,27,39,51,63,75,87,99,111,123,135,147} and sample size is 13 3) N=20, n=8. Then k=2 and c = 4. Sample size is nr =10
4) N= 100 000, n = 1500. Then k = 66 , c=1000 and c/k =15.15 with [c/k]=15. nr = 1515 or 1516
110
Estimation of the population total
)(
)()(ˆ 2)
)(]/[)()(ˆ 1)
:) when (equal estimators Two
size sample )( ,)(
sn
stNyNst
stnNsktst
nkN
snyst
s
si i
1 ]/[or ]/[)( kNkNsn
These estimators are approximately the same:
)/(
1/
kNN
N
kNnN
111
kr r
kr r
kr rr
t)s(tk
)s(kt
)s(p)s(t)t(E
t
11
1
1
:unbiased is
t
t
ˆ than riancesmaller vaslightly usually -
estimator) ratio a s(it' unbiasedely approximatonly is ˆ
• Advantage of systematic sampling: Can be implemented even where no population frame exists
•E.g. sample every 10th person admitted to a hospital, every 100th tourist arriving at LA airport.
112
totalssample theof average theis
/)( where
))(())((1
)())(ˆ()ˆ()ˆ(
1
1
2
1
2
1
22
kstt
tstktstkk
sptstttEtVar
k
r r
k
r r
k
r r
k
r rr
• The variance is small if
shomogeneou very are ,,..}21{}1{
strata"" theif i.e., little, varies)(
etc.k, ...,k, ,..,k
st r
• Or, equivalently, if the values within the possible samples sr are very different; the samples are heterogeneous
• Problem: The variance cannot be estimated properly because we have only one observation of t(sr)
113
Systematic sampling as Implicit StratificationIn practice: Very often when using systematic sampling (common design in national statistical institutes):
The population is ordered such that the first k units constitute a homogeneous “stratum”, the second k units another “stratum”, etc.
Implicit strata Units
1 1,2….,k
2 k+1,…,2k
: :
n = N/k assumed (n-1)k+1,.., nk
Systematic sampling selects 1 unit from each stratum at random
114
Systematic sampling vs SRS
• Systematic sampling is more efficient if the study variable is homogeneous within the implicit strata– Ex: households ordered according to house numbers
within neighbourhooods and study variable related to income
• Households in the same neighbourhood are usually homogeneous with respect socio-economic variables
• If population is in random order (all N! permutations are equally likely): systematic sampling is similar to SRS
• Systematic sampling can be very bad if y has periodic variation relative to k: – Approximately: y1 = yk+1, y2 = yk+2 , etc
115
Variance estimation
•No direct estimate, impossible to obtain unbiased estimate
• If population is in random order: can use the variance estimate form SRS as an approximation
• Develop a conservative variance estimator by collapsing the “implicit strata”, overestimate the variance
• The most promising approach may be:
Under a statistical model, estimate the expected value of the design variance
• Typically, systematic sampling is used in the second stage of two-stage sampling (to be discussed later), may not be necessary to estimate this variance then.
116
Cluster sampling and multistage sampling
• Sampling designs so far: Direct sampling of the units in a single stage of sampling
• Of economial and practical reasons: may be necessary to modify these sampling designs
– There exists no population frame (register: list of all units in the population), and it is impossible or very costly to produce such a register
– The population units are scattered over a wide area, and a direct sample will also be widely scattered. In case of personal interviews, the traveling costs would be very high and it would not be possible to visit the whole sample
117
• Modified sampling can be done by 1. Selecting the sample indirectly in groups , called
clusters; cluster sampling– Population is grouped into clusters– Sample is obtained by selecting a sample of
clusters and observing all units within the clusters
– Ex: In Labor Force Surveys: Clusters = Households, units = persons
2. Selecting the sample in several stages; multistage sampling
118
3. In two-stage sampling: • Population is grouped into primary sampling
units (PSU)• Stage 1: A sample of PSUs• Stage 2: For each PSU in the sample at stage
1, we take a sample of population units, now also called secondary sampling units (SSU)
• Ex: PSUs are often geographical regions
119
Examples1. Cluster sampling. Want a sample of high school
students in a certain area, to investigate smoking and alcohol use. If a list of high school classes is available,we can then select a sample of high school classes and give the questionaire to every student in the selected classes; cluster sampling with high school class being the clusters
2. Two-stage cluster sampling. If a list of classes is not available, we can first select high schools, then classes and finally all students in the selected classes. Then we have 2-stage cluster sample.
1. PSU = high school2. SSU = classes3. Units = students
120
Psychiatric Morbidity Survey is a 4-stage sample
– Population: adults aged 16-64 living in private households in Great Britain
– PSUs = postal sectors
– SSUs = addresses
– 3SUs = households
– Units = individuals
Sampling process:
1) 200 PSUs selected
2) 90 SSUs selected within each sampled PSU (interviewer workload)
3) All households selected per SSU
4) 1 adult selected per household
121
Cluster sampling
N
i iMM1
:size Population
advancein fixednot
: sample final of Size
in units all :units of sample Final
|| clusters, of sample
Isi i
I
II
Mm
s
ss
sns
Number of clusters in the population : N
Number of units in cluster i: Mi
M/ty
tt,ilustertotal in cyt
y
N
i ii
:variable for themean Population
1
122
Simple random cluster samplingRatio-to-size estimator
I
I
si i
si i
R M
tMt
MtMty
yMNn
fNtVar
iii
N
i iiR
/ and mean,cluster the, / where
)(1
11)ˆ(
1
222
Use auxiliary information: Size of the sampled clusters
Approximately unbiased with approximate variance
123
mean sample usual theis
and where
1
11
by estimated
2222
II
I
si isi is
si siiR
M/tyN/nf
)yy(Mnn
fN
n/m
N/M)t(V
Note that this ratio estimator is in fact the usual sample mean based estimator with respect to the y- variable
sR yMt ˆ
And corresponding estimator of the population mean of y is
sy
Can be used also if M is unknown
124
• Estimator’s variance is highly influenced by how the clusters are constructed.
similar values themaking clusters, in the lies
values- in the variation theofmost such that
ous,heterogene clusters themake
small )( make toclusters Choose 22
i
ii
y
y
yM
• Note: The opposite in stratified sampling• Typically, clusters are formed by “nearby units’ like households, schools, hospitals because of economical and practical reasons, with little variation within the clusters:
Simple random cluster sampling will lead to much less precise estimates compared to SRS, but this is offset by big cost reductions
Sometimes SRS is not possible; information only known for clusters
125
Design Effects
A design effect (deff) compares efficiency of two design-estimation strategies (sampling design and estimator) for same sample size
Now: Compare
Strategy 1:simple random cluster sampling with ratio estimator
Strategy 2: SRS, of same sample size m, with usual sample mean estimator
In terms of estimating population mean:
s
sR
y
yMt
:estimator 2Strategy
/ˆ :estimator 1Strategy
126
)(/)(),( sSRSsSCSs yVaryVarySCSdeff
The design effect of simple random cluster sampling, SCS, is then
Estimated deff: )(ˆ/)(ˆsSRSsSCS yVyV
In probation example:
200387.0)1/()ˆ1(ˆ)1)](1/()ˆ1(ˆ[)ˆ(ˆ mppfmpppVSRS
9.6000387.0/0.0302 deff Estimated 22
Conclusion: Cluster sampling is much less efficient
! 99615.0/9.60
615.026/16/1ˆ/ -1 and
)/(ˆ lettingby /-1factor p.c. theestimatecan We:Note
estdeff
NnMm
nmNMMm
127
Two-stage sampling• Basic justification: With homogeneous clusters
and a given budget, it is inefficient to survey all units in the cluster- can instead select more clusters
• Populations partioned into N primary sampling units (PSU)
• Stage 1: Select a sample sI of PSUs• Stage 2: For each selected PSU i in sI: Select a
sample si of units (secondary sampling units, SSU) • The cluster totals ti must be estimated from the
sample
128
|| :size sample Total
||
PSUs of sample 1 stage of size||
smm
sm
sn
Isi i
ii
I
General two-stage sampling plan:
)| (
) (
| Iiij
IIi
sisjSSUP
siPSUP
ijIiij
ij
|
:cluster in (SSU)unit for y probabilit Inclusion
129
ijyy
yt
sit
ij
sjij
ijHTi
Ii
i
cluster in unitfor of value where
ˆ
:, totalofestimator Thompson -Horvitz
|,
Suggested estimator for population total t :
IsiIi
HTitt,
ˆˆ
HTsi sj si sjij
ij
ij
ij
Ii
tyy
tI i I i
ˆ1ˆ|
Unbiased estimator
130
N
iIi
HTi
siIi
itVart
VartVarI 1
, )ˆ()ˆ(
1. The first component expresses the sampling uncertainty on stage 1, since we are selecting a sample of PSU’s. It is the variance of the HT-estimator with ti as observations
2. The second component is stage 2 variance and tells us how well we are able to estimate each ti in the whole population
3. The second component is often negligible because of little variability within the clusters
131
A special case: Clusters of equal size and SRS
on stage 1 and stage 2
M
m
M
m
N
nMmNn
NiMM
mm
ijj|iIi
i
i
0
000
0
0
/ ,/
,...,1 ,
2 stageat sizes sample equal -
ssi sj ij yMym
Mt
I i
ˆ
Self-weighting sample: equal inclusion probabilities for all units in the population
132
Unequal cluster sizes. PPS – SRS sampling• In social surveys: good reasons to have equal inclusion
probabilities (self-weighting sample) for all units in the population (similar representation to all domains)
• Stage 1: Select PSUs with probability proportional to size Mi
• Stage 2: SRS (or systematic sample) of SSUs• Such that sample is self-weighting
Mm
MmM
Mn
ijIiij
iiiji
Ii
/ such that
/ and
|
|
mi = m/n = equal sample sizes in all selected PSUs
syMt ˆ
133
Remarks
• Usually one interviewer for each selected PSU• First stage sampling is often stratified PPS• With self-weighting PPS-SRS:
– equal workload for each interviewer– Total sample size m is fixed
134
II. Likelihood in statistical inference and survey sampling
• Problems with design-based inference
• Likelihood principle, conditionality principle and sufficiency principle
• Fundamental equivalence
• Likelihood and likelihood principle in survey sampling
135
Traditional approach
Design-based inference
• Population (Target population): The universe of all units of interest for a certain study: U = {1,2, …, N}
– All units can be identified and labeled
– Variable of interest y with population values
– Typical problem: Estimate total t or population mean t/N
• Sample: A subset s of the population, to be observed
• Sampling design p(s) is known for all possible subsets;
– The probability distribution of the stochastic sample
),...,,( 21 Nyyyy
136
Problems with design-based inference
• Generally: Design-based inference is with respect to hypothetical replications of sampling for a fixed population vector y
• Variance estimates may fail to reflect information in a given sample
• If we want to measure how a certain estimation method does in quarterly or monthly surveys, then y will vary from quarter to quarter or month to month – need to assume that y is a realization of a random vector
• Use: Likelihood and likelihood principle as guideline on how to deal with these issues
137
Problem with design-based variance measure Illustration 1
N
i is ss
s
yN
y)s(p)y(E
y
1 2
1
2
1 :Unbiased
mean populationfor estimator as Usec)
2
1
22
2
1
2
1
:variance-Design
~N
)y()y(E)y(VarN
i iss
a) N +1 possible samples: {1}, {2},…,{N}, {1,2,…N}
b) Sampling design: p({i}) =1/2N , for i = 1,..,N ; p({1,2,…N})= 1/2
d) Assume we select the “sample” {1,2,…,N}. Then we claim that the “precision” of the resulting sample (known to be without error) is22 /~
138
Problem with design-based variance measureIllustration 2
N/nf,)y(N
n)f
y
N
i i
s
1
1
-(1by measured isPrecision
estimate and SRS :1Expert a)
1
22
2
n/~ys
2by precision measures
estimate andt replacemen with SRS :2Expert b)
Both experts select the same sample, compute the same estimate, but give different measures of precision…
139
The likelihood principle, LPgeneral model
• LP: The likelihood function contains all information about the unknown parameters
• More precisely: Two proportional likelihood functions for , from the same or different experiments, should give identically the same statistical inference
model in the parametersunknown theare ; ),(~ :Model xfX
• The likelihood function, with data x: )()( xflx
l is quite a different animal than f !!
Measures the likelihood of different values in light of the data x
140
• Maximum likelihood estimation satisfies LP, using the curvature of the likelihood as a measure of precision (Fisher)
• LP is controversial, but hard to argue against because of the fundamental result by Birnbaum, 1962:
• LP follows from sufficiency (SP) and conditionality principles (CP) that ”no one” disagrees with.
• SP: Statistical inference should be based on sufficient statistics
• CP: If you have 2 possible experiments and choose one at random, the inference should depend only on the chosen experiment
141
Illustration of CP• A choice is to be made between a census og taking a sample of size 1. Each with probability ½.
• Census is chosen
• Unconditional approach:
.N
iPP
iPcensusPi
2
11
2
11/2
1) size of sample|selected is (1) size of (sample 1/2
selected) is and 1 size of (sample)(
142
The Horvitz-Thompson estimator:
! 22 tytiUHT
Conditional approach: iand HT estimate is t
143
LP, SP and CP
model in the parametersunknown theare ; ),(~ :Model xfX
xE E,xI
,f,,XE
n observatiowith experiment in the about Inference : )(
}}{{ triplea is Experiment
))()(( oft independen ),()(
Assume . }}{{ and }}{{Let
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f,,XEf,,XE
x,x,
Likelihood principle:
)()( :Then 2211 x,EIx,EI
This includes the case where E1 = E2 and x1 and x2 are two different observations from the same experiment
144
Sufficiency principle: Let T be a sufficient statistics for in the experiment E. Assume T(x1) = T(x2). Then I(E, x1) = I(E, x2).
Conditionality principle:
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then is in n observatio The observed. is and 1/2y probabilit
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145
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147
Consequences for statistical analysis
• Statistical analysis, given the observed data: The sample space is irrelevant
• The usual criteria like confidence levels and P-values do not necessarily measure reliability for the actual inference given the observed data
• Frequentistic measures evaluate methods
– not necessarily relevant criteria for the observed data
148
Illustration- Bernoulli trials
successes ofnumber observe
and s)(0' failures 3get we until trialsContinue :
observeand nsobservatio 12:
:about n informatiogain tosexperiment Two
y probabilitwith (success) 1
,..,...,
2
2
12111
1
Y
E
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X
XX
i i
i
i
9 are results theSuppose 21 yy
149
The likelihood functions:
binomial negative )1()()(
binomial )1()()(3911
9)2(
9
39129
)1(9
l
l
Proportional likelihoods: )()4/1()( )1(9
)2(9 ll
LP: Inference about should be identical in the two cases
Frequentistic analyses give different results:
0.0327 value-P :)9,( 0.0730 value-P :)9,(
2/1:against 2/1: test F.ex.
21
10
EE
HH
because different sample spaces: (0,1,..,12) and (0,1,...)
150
Frequentistic vs. likelihood
• Frequentistic approach: Statistical methods are evaluated pre-experimental, over the sample space
• LP evaluate statistical methods post-experimental, given the data
• History and dicussion after Birnbaum, 1962: An overview in ”Breakthroughs in Statistics,1890-1989, Springer 1991”
151
Likelihood function in design-based inference
)...,,( 21 Nyyyy
}:),{( , siyix iobs
}for :{ , siyy iobsix y
• Unknown parameter:
• Data:
• Likelihood function = Probability of the data, considered as a function of the parameters
• Sampling design: p(s)
• Likelihood function:
otherwise 0
if )()( x
x
spl
yy
• All possible y are equally likely !!
152
• Likehood principle, LP : The likelihood function contains all information about the unknown parameters
• According to LP: – The design-model is such that the data contains no
information about the unobserved part of y, yunobs
– One has to assume in advance that there is a relation between the data and yunobs :
• As a consequence of LP: Necessary to assume a model
– The sampling design is irrelevant for statistical inference, because two sampling designs leading to the same s will have proportional likelihoods
153
Let p0 and p1 be two sampling designs. Assume we get the same sample s in either case. Then the data x are the same and x is the same for both experiments.
The likelihood function for sampling design pi , i = 0,1:
otherwise 0
if )()(,
xixi
spl
yy
)()(
)()(
: for then and
if )(/)()(/)(
,00
1,1
01,0,1
yy
y
yyy
xx
xxx
lsp
spl
all
spspll
154
• Same inference under the two different designs. This is in direct opposition to usual design-based inference, where the only stochastic evaluation is thru the sampling design, for example the Horvitz-Thompson estimator
• Concepts like design unbiasedness and design variance are irrelevant according to LP when it comes to do the actual statistical analysis.
• Note: LP is not concerned about method performance, but the statistical analysis after the data have been observed
• This does not mean the sampling design is not important. It is important to assure we get a good representative sample. But once the sample is collected the sampling design should not play
a role in the inference phase, according to LP
155
Model-based inference
• Assumes a model for the y vector• Conditioning on the actual sample• Use modeling to combine information• Problem: dependence on model
– Introduces a subjective element– almost impossible to model all variables in a
survey• Design approach is “objective” in a perfect world
of no nonsampling errors
156
III. Model-based inference in survey sampling
• Model-based approach. Also called the prediction approach– Assumes a model for the y vector– Use modeling to construct estimator
– Ex: ratio estimator • Model-based inference
– Inference is based on the assumed model– Treating the sample s as fixed, conditioning on the actual sample
• Best linear unbiased predictors• Variance estimation for different variance measures
157
Model-based approach
si i
N
i si ii yyyt1
N
N
YYY
yyy
,..., variablesrandom
of valuesrealized are ,...,,
21
21
We can decompose the total t as follows:
Treat the sample s as fixed
Two stochastic elements:
fYYYps N ~),...,( 2) )(~ sample )1 21
[Model-assisted approach: use the distribution assumption of Y to construct estimator, and evaluate according to distribution of s, given the realized vector y]
158
si isi i
si i
YZyz
y
of valuerealized the,
estimate tois problem theknown, is Since
• The unobserved z is a realized value of the random variable Z, so the problem is actually to predict the value z of Z.
Can be done by predicting each unobserved yi: siyi ,ˆ
zz
zyyytsi isi isi ipred
for predictor a is ˆ
ˆˆˆ :Estimator
• The prediction approach, the prediction based estimator
modelingby Determine iy
159
Remarks:
1. Any estimator can be expressed on the “prediction form:
si it
tsi i
ytz
zyt
ˆˆ letting
ˆˆ
ˆ
ˆ
2. Can then use this form to see if the estimator makes any sense
160
Ex 1.
si sisisisis yyynNyyNt )(ˆ
siyyyz sisi s allfor ,ˆ and ˆ Hence,
Ex.2
HTsi i
siisi
sx
ix
i
isi i
sii
xisi isi
i
ixHT
zyxxnt
nxt
x
y
ny
nx
tyy
nx
ytt
HT
ˆ1
1ˆ
ˆ
Reasonable sampling design when y and x are positively correlated
N
i ixxiisi iiHT xt/tnx π /πyt1
,and
tcoefficien regression unusualrather a is ˆ
ˆˆˆ
HT
si isi iHTHT yxz
161
Three common models
0),( and )( ,0)( with 2 jiiiiiii CovxVarExY
I. A model for business surveys, the ratio model:
• assume the existence of an auxiliary variable x for all units in the population.
0),( and )( , )( 2 jiiiii YYCovxYVarxYE
162
II. A model for social surveys, simple linear regression:
0),( and )( ,0)( , 221 jiiiiii CovVarExY
III. Common mean model:
eduncorrelat are ' theand )( , )( 2 sYYVarYE iii
• Ex: xi is a measure of the “size” of unit i, and yi tends to increase with increasing xi. In business surveys, the regression goes thru the origin in many cases
163
Model-based estimators (predictors)
)|)ˆ(()|ˆ( 2 sTTEsTTVar
1. Predictor: ZYTsi i
ˆˆ
N
i iYTsTTET1
, 0)|ˆ( if unbiased-model is ˆ .3
2. Model parameters:
4. Model variance of model-unbiased predictor is the variance of the prediction error, also called the prediction variance
5. From now on, skip s in the notation: all expectations and variances are given the selected sample s, for example
)|ˆ()ˆ(
)|ˆ()ˆ(
sTTVarTTVar
sTTETTE
164
Prediction variance as a variance measure for the actual observed sample
TYNT s totalpopulation for theestimator theas ˆ Use
0)0()ˆ( VarTTVar
N +1 possible samples: {1}, {2},…,{N}, {1,2,…N}
Assume we select the “sample” {1,2,…,N}.
Prediction variance:
Illustration 1, slide 137
TYNT ˆThen
Illustration 2, slide 138: Exactly the same prediction variance for the two sampling designs
165
:predictorslinear unbiased-model
all among varianceprediction minimumuniformly has ˆ )2
unbiased-model is ˆ 1)
if for predictor (BLU) unbiasedlinear best theis ˆ
0
0
0
T
T
TT
6. Definition:
Linear predictor:
si ii YsaT )(ˆ
allfor )ˆ()ˆ( 0 TTVarTTVar
Tpredictor linear unbiased-modelany For
166
0),( , eduncorrelat are ,...,
)()( and 0)( ,
:
1
2
jiN
iiiiii
CovYY
xvVarExY
Model
Suggested Predictor:
of (BLUE)estimator unbiasedlinear best theis ˆ where
ˆˆ
opt
si ioptsi ipred xYT
si ii
si iiiopt xvx
xvYx
)(/
)(/ˆ2
2g0 ,)( Usually, gxxv
167
1)(,)()ˆ(
)(ˆ
si iisi ii
si ii
xscxscE
Ysc
si ii xvcVar )()ˆ( 22
method Lagrange using
1 osubject t )( Minimize 2
si iiisi i xcxvc
)()2/(
0)(2/
)1()(2
i
ii
iiii
si iisi ii
xv
xc
xxvccQ
xcxvcQ
168
1)(/2/
:1such that )2/( Determine2
si ii
si ii
xvx
xc
si ii xvx )(//1)2/( 2
sj jj
si iii
si ioptiopt
sj jj
iiopti
xvx
xvYxYc
xvx
xvxc
)(/
)(/ˆ and
)(/
)(/ and
2,
2,
This is the least squares estimate based on )(/ ii xvY
169
predictorlinear and unbiased-model a be ˆLet T
si isi i
si isi i
xYT
xZYTZ
ˆˆ
./ˆˆ and ˆˆLet
• We shall show that
TTpred for predictor (BLU) unbiasedlinear best theis ˆ
)ˆ( unbiased-model ˆ and
),(in linear is ˆ predictor linear ˆ
ET
siYT i
170
si isi isi i
si isi i
xExxE
YxETTE
])ˆ([]ˆ[
)ˆ()ˆ( since
)ˆ(0)ˆ(such that ETTE
The prediction variance of model-unbiased predictor:
si isi i
si isi i
si isi i
xvVarx
YVarxVar
YxVarTTVar
)()ˆ()(
)()ˆ(
)ˆ()ˆ(
22
To minimize the prediction variance is equivalent to minimizing )ˆ(VarGiving us predictor BLU theas ˆ
predT
171
The prediction variance of the BLU predictor:
si i
si ii
si i
si i
si iisi i
si ioptsi ipred
xvxvx
x
xvxvx
x
xvVarxTTVar
)()(/
)(
)()(/
)(
)()ˆ()()ˆ(
2
2
2
22
22
22
A variance estimate is obtained by using the model-unbiased estimator for
si ioptii
xYxvn
22 )ˆ()(
1
1
1ˆ
172
The central limit theorem applies such that for large n, N-n we have that
)1,0(ely approximat is )ˆ(ˆ/)ˆ( NTTVTTpred
Approximate 95% confidence interval for the value t of T:
)ˆ(ˆ96.1ˆ TTVt pred
si i
si ii
si i
pred xvxvx
xTTV )(
)(/
)(ˆ)ˆ(ˆ
2
2
2
Also called a 95% prediction interval for the random variable T
173
Three special cases: 1) v(x) = x, the ratio model, 2) v(x)= x2 and 3) xi =1 for all i, the common mean model
1. v(x) = x
ratio sample usual the, ˆ)(/
)(/ˆ2
Rx
Y
xvx
xvYx
si i
si i
si ii
si iiiopt
xsi isi i
si isi ipred
tRxRxR
xRYT
ˆˆˆ
ˆˆ
the usual ratio estimator
si isi isi ipred xxxTTVar )/()()ˆ( 22
N
i isi ir
s
r
xxnNxxNnf
x
xx
n
fN
1
22
and )/(,/
,1
174
2. v(x) =x2
ratios theofmean sample the,/
)(/
)(/ˆ2 n
xY
xvx
xvYxsi ii
si ii
si iiiopt
si isii
isi i
si ioptsi ipred
xx
Y
nY
xYT
)1
(
ˆˆ
si isi i
si i
si ii
si i
pred
xn
x
xvxvx
xTTVar
2
2
2
2
2
2
)(
)()(/
)()ˆ(
175
:/when estimator T-H theResembles xii tnx
sxsii
ixHT Rt
nx
YtT ˆ
) (ˆissi isxsi issi ipred xRYRtxRYT
si isiii nRRxYR / and /Let
When the sampling fraction f is small or when the xi values vary little, these two estimators are approximately the same. In the latter case:
si iisi ssi is
s YxRYxn
R and 1
Also model-unbiased
176
3. xi =1
0),( , eduncorrelat are ,...,
)( and 0)( ,
:
1
2
jiN
iiii
CovYY
VarEY
Model
mean sample the,1
)(/
)(/ˆ2
si si
si ii
si iiiopt YY
nxvx
xvYx
ssi ssi ipred YNYYT ˆ
nfNnN
n
nN
xvxvx
xTYNVar
si i
si ii
si i
s
22
22
2
2
2
)1()()(
)()(/
)()(
This is also the usual, design-based variance formula under SRS
177
We see that the variance estimate is given by
variancesample the
)(1
1ˆ 22
si si yy
n
nfN
22 ˆ
)1(
Exactly the same as in the design-approach, but the interpretation is different
178
Simple Linear regression model
eduncorrelat are ,...,
)( ,0)( ,
1
221
N
iiiii
YY
VarExY
BLU predictor:
ss
si si
si isi
si si
si sisi
si isi ipred
xY
xx
Yxx
xx
YYxx
xYT
21
222
21
21
ˆˆ
)(
)(
)(
))((ˆ
,estimators LS theare ˆ and ˆ
where
)ˆˆ(ˆ
179
)(ˆ
))((ˆ)(
)ˆˆ(ˆ
2
2
21
sxs
ssi iss
si isi ipred
xNtYN
xnNxYnNYn
xYT
)](ˆ[ˆ2 sspred xxYNT
)()}()(1
{)ˆ(
and
)()()(
:unbiased-model is ˆ Clearly,
21221
211 21
xNxxxn
NTE
xNxTE
T
sisipred
N
i i
pred
180
We shall now show that this predictor is BLU
)./()/ˆ(let and predictor, unbiased-model
linear, a be ˆLet . first that Assume
ss
s
xxYNTb
Txx
)]([ˆ)(ˆ1ssss xxbYNTxxbYT
N
Hence, any predictor can be expressed on this form and the predictor is linear if and only if b is linear in the Yi’s
).()()(
)()]()([
)()()ˆ(
:)( unbiased-model is ˆ Also,
222
2121
21
2
sss
ss
xxxxbExx
xNbExxxN
xNTETE
bET
181
Prediction variance:
: ofestimator unbiased ,)(
)()()()ˆ(
2
2
si ii
ss
Yscb
nNxxNbYnNVarTTVar
si iisi i
isi i
xcc
xcbE
221
2212 )()(
si iisi i xccbE 1 )2( and 0 )1()( 2
So we need to minimize the prediction variance with respect to the ci’s under (1) and (2)
182
])(
)(2)([
)(
)()()(
minimize i.e.
22222
22
si si isis
si is
si isiss
n
nNcxxN
n
nNcxxN
cxxNn
nN
cxxNn
nNYVarxxNbYnNVar
(2) and (1) conditionsunder minimize enough to isit
,0 Since2
si i
si i
c
c
183
iiiii
si iisi isi i
xcxccQ
xcccQ
2121
212
0222/
)1(2)(2
11 )2(
00 )1(2
21
21
si issi ii
ssi i
xxnxc
xc
si si
si si
s
xx
xnx
x
22
22
22
21
)(/1
1 :)2( from
)1(
184
222
2221
ˆ)(
)(
)( and
)()(
sj sj
si isi
sj sj
sisi i
sj sj
sisiii
xx
Yxx
xx
xxYb
xx
xxxxxc
The prediction variance is given by
2
22
2
2
22
2
))(ˆ(2
1ˆ
with estimatingby obtained is estimate varianceand
)(
)()1()ˆ(
si sisi
si si
spred
xxYYn
xx
xxn
N
n
n
NTTVar
185
predictor. BLU theis and ˆThen
? if What . far, So
spred
ss
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xxxx
)]()1([
)(])1([)ˆ(
ˆ predictor,linear any For
22
2
nNa
nNYaVarTTVar
YaT
si i
si ii
si ii
si isi ia anaaYaT / ,ˆLet
)]()1([
)]()1([)ˆ(22
22
nNan
nNaTTVarsia
186
22 )1()1( ana
si i
.ˆˆ
:/ and unbiased-model ˆ and
)ˆ()ˆ(
predsa
si i
a
TYNT
nNaNaT
TTVarTTVar
187
Anticipated variance (method variance)
sthisTTVar
Ts
sample particular for y uncertaint themeasures )ˆ(
:ˆ unbiased-model with , sample on the lConditiona .1
We want a variance measure that tells us about the expected uncertainty in repeated surveys
)(on distributi sampling over the )},ˆ({
:surveys repeatedfor y uncertaint expected The .2
pTTVarEp
3. This is called the anticipated variance.
4. It can be regarded as a variance measure that describes how the estimation method is doing in repeated surveys
188
error squaremean danticipate they,uncertaintfor criterion a as
})ˆ({
use weunbiased,-modelnot is ˆ If2TTEE
T
p
)ˆ()ˆ()|)ˆ(
and
)}|)ˆ({})ˆ({
thenunbiased-design is ˆ If :Note
22
22
tVarttETTE
TTEETTEE
T
ppp
pp
yY
Y
And the anticipated MSE becomes the expected design-variance, also called the anticipated design variance
)}ˆ({})ˆ({ 2 TVarETTEE pp
189
Example: Simple linear regression and simple random sample
N
i i
N
i ispsp
s
Nn
fN
YYN
En
fNYNVarETYNEE
YN
1
222
1
222
})(1
1{
1
})(1
1{
1)}({})({
:unbiased-design isbut
unbiased,-modelnot isIt :used is mean sample If
xxYE iii 2121 ,)(
N
i ix
xsp
xxN
S
Sn
fNYNVarE
1
22
222
22
})(1
1
}{1
)}({
190
Let us now study the BLU predictor.( It can be shown that it is approximately design-unbiased )
si sip
sp
si si
sppredp
si si
spred
xxE
xxnE
N
n
n
N
xx
xxnE
N
n
n
NTTVarE
xx
xxn
N
n
n
NTTVar
2
22
2
2
22
2
2
22
2
)(
})({)1(
)(
)()1()}ˆ({
)(
)()1()ˆ(
22 )1()()( xspsp SfxnVarxxnE 22 )1()( xsi sip SnxxE
191
22
22
22
)1()1(1
1
1)1()}ˆ({
fn
Nf
n
N
n
ff
n
NTTVarE predp
}{1
)}({
tocompared
222
22xsp S
n
fNYNVarE
s
xpred
YN
ST
than efficient moremuch is and
term theeliminates ˆ 222
192
Remarks
• From a design-based approach, the sample mean based estimator is unbiased, while the linear regression estimator is not
• Considering only the design-bias, we might choose the sample mean based estimator
• The linear regression estimator would only be selected over the sample mean based estimator because it has smaller anticipated variance
• Hence, difficult to see design-unbiasedness as a means to choose among estimators
193
Robust variance estimation
• The model assumed is really a “working model”• Especially, the variance assumption may be
misspecified and it is not always easy to detect this kind of model failure– like constant variance
– variance proportional to size measure xi
• Standard least squares variance estimates is sensitive to misspecification of variance assumption
• Concerned with robust variance estimators
194
Variance estimation for the ratio estimator
0),( , eduncorrelat are ,...,
)( and 0)( ,
1
2
jiN
iiiiii
CovYY
xVarExY
22 ˆ1
)ˆ(ˆ s
rxR x
xx
n
fNTtRV
Working model:
Under this working model, the unbiased estimator of the prediction variance of the ratio estimator is
ss
si iii
xYR
xRYxn
/ˆ
)ˆ(1
1
1ˆ 22
195
This variance estimator is non-robust to misspecification of the variance model.
Suppose the true model has
)()( and )( 2iiii xvYVarxYE
Ratio estimator is still model-unbiased but prediction variance is now
si isi is
r
si i
si i
si i
si i
si isi ix
xvxvxn
xnN
xvx
xvx
xvRVarxTtRVar
)()()(
)()(
)()(
)()ˆ()()ˆ(
22
222
22
22
22
196
rsrs
rss
rx
vfxxvfn
fN
vnNvxn
xnNTtRVar
222
2
222
)/()1(1
)()(
)ˆ(
si irsi is nNxvvnxvv )/()( and /)(
Moreover,
si iisssss
si iii
xxvn
xvxvxvn
xv
xRYExn
E
/)(1
)/( , }/)/{(1
1)/(
)ˆ(1
1
1)ˆ(
2
22
:)ˆ( 22 E
197
Robust variance estimator for the ratio estimator
varianceprediction in the termleading the
, )/(1
})/({)/(1
)/()1(1
)ˆ(
222
2222
222
srs
srsrsrs
rsrsx
xxn
fNv
xxvvfxxvn
fN
vfxxvfn
fNTtRVar
si isi is YVarn
xvn
v )(1
)(1
:and 22
})(1
{)(1
222
si iisi iis xY
nExYE
nv
198
Suggests we may use:
si iisrob xRYn
v 22 )ˆ(1
1
Leading to the robust variance estimator:
si iisrxrob xRYnn
fNxxTtRV 222 )ˆ(
1
11)/()ˆ(ˆ
Almost the same as the design variance estimator in SRS:
si iisxSRS xRYnn
fNxxtRV 222 )ˆ(
1
11)/()ˆ(ˆ
199
)ˆ(1
)/()ˆ(ˆ 222 TtRVvn
fNxxTtRVE xssrxrob
Can we do better?
Require estimator to be exactly unbiased under ratio model, v(x) = x:
si sixs
xs
sis
iisi ii
si ii
xxn
sx
s
nx
xn
xx
nxRYE
n
xRYn
Exxv
222
22
22
2
)(1
1 ,
11
)1( 1
1 )ˆ(
1
1
})ˆ(1
1{:)(When
200
So a robust variance estimator that is exactly unbiased under the working model , v(x) = x:
The prediction variance when v(x) = x:
22 1)ˆ(
s
rx x
xx
n
fNTtRV
2
2222 1
1)/(1
)}ˆ(ˆ{s
xsrxrob x
s
nxx
n
fNTtRVE
)ˆ(ˆ11)}ˆ(ˆ
1
2
2
, xrobs
x
rxrobR tRV
x
s
nx
xTtRV
si iisrsx xRYnn
fNxxxxsn 2221221 )ˆ(
1
11)/()}/(1{
)ˆ(ˆ)/()}/(1{ 1221xSRSrsx tRVxxxsn
201
General approach to robust variance estimation
)()()1()ˆ(
ˆ .2
2
si isi iis
si iis
YVarYVarwTTVar
YwT
1. Find robust estimators of Var(Yi), that does not depend on model assumptions about the variance
model under true )( estimate ˆ
)ˆ()(ˆ :For 3. 2
ii
iii
YE
YYVsi
4. Estimate only leading term in the prediction variance, typically dominating, or estimate the second term from the more general model
202
• Reference to robust variance estimation:
• Valliant, Dorfman and Royall (2000):
Finite Population Sampling and Inference. A Prediction Approach, ch. 5
203
Model-assisted approach
population wholefor theknown is Here, .population in the
eachfor estimate"" based-regression a is ˆˆ Suppose
ii
ii
xy
xy
• Design-based approach
• Use modeling to improve the basic HT-estimator. Assume the population values y are realized values of random Y
• Assume the existence of auxiliary variables, known for all units in the population• Basic idea:
estimator-HTby estimated becan and estimate, easier tomuch is
),ˆ( where , and )ˆ(ˆ111 iii
N
i i
N
i ii
N
i i yyeeeyyyt
204
sii
iHT
ee
ˆ
Final estimator, the regression estimator:
N
i HTireg ext1
ˆˆˆ
Alternative expression:
N
i ixsii
ixsi
i
ireg xt
xt
yt
1 , )(ˆˆ
)ˆ(ˆˆˆ,, HTxxHTyreg tttt
205
Simple random sample
estimator ratio the,ˆ
/ˆ :estimator unbiasedlinear Best
)( and 0)( ,
andt independen are ' The :
)(ˆˆ
2
s
sxsx
s
ssreg
ss
iiiiii
i
sxsreg
x
ytyNt
x
yyNt
xy
xVarExY
sY
xNtyNt
Model
206
In general with this “ratio model”, in order to get approximately design-unbiased estimators:
si ii
si iiHTxHTy
si iiHTx
si iiHTy
N
i i
N
i i
x
ytt
xt
yt
xy
/
/ˆ/ˆˆ use
/ˆby estimated isr Denominato
/ˆby estimated isNumerator
/ of estimatean as estimate- regardCan
,,
,
,
11
ii
N
i ixreg
xy
ytt
ˆˆ where
ˆˆˆ1
207
Reference: Sarndal, Swensson and Wretman : Model Assisted Survey Sampling (1992, ch. 6), Wiley
• Regression estimator is approximately unbiased
Variance and variance estimation
• Variance estimation:
where
, :residuals ample The
xy
siyyes
ii
iii
si ijsj
j
j
i
i
ij
ijjireg
eetV
ns
,
2)(
)ˆ(ˆ
:advancein fixed , || If
208
Approximate 95% CI, for large n, N-n:
)ˆ(ˆ96.1ˆregreg tVt
• Remark: In SSW (1992,ch.6), an alternative variance estimator is mentioned that may be preferable in many cases
209
Common mean model
eduncorrelat are ' theand )( , )( 2 sYYVarYE iii
N
ty
ytt HTy
s
si i
si iiHTxHTy ˆ
ˆ~/1
/ˆ/ˆˆ ,
,,
sxreg yNNtt ~ˆˆˆ
The ratio model with xi =1.
This is the modified H-T estimator (slide 73,74)Typically much better than the H-T estimator when different
210
si ijsj
j
j
i
i
ij
ijjis
sii
eeyNV
yye
,
2
)~(ˆ
~
Alternatively,
2
,2 )(
)ˆ/()~(ˆ
j
j
i
isi ij
sj
ij
ijjis
eeNNyNV
211
1. The model-assisted regression estimator has often the form
N
i ireg yt1
ˆˆ
2. The prediction approach makes it clear: no need to estimate the observed yi
Remarks:
3. Any estimator can be expressed on the “prediction form:
si it
tsi i
ytz
zyt
ˆˆ letting
ˆˆ
ˆ
ˆ
4. Can then use this form to see if the estimator makes any sense