Bayes Inference for Surveys - American Statistical Association

Report
The Calibrated Bayes
Approach to Sample Survey
Inference
Roderick Little
Department of Biostatistics, University of Michigan
Associate Director for Research & Methodology,
Bureau of Census
Learning Objectives
1. Understand basic features of alternative modes of
inference for sample survey data.
2. Understand the mechanics of Bayesian inference for finite
population quantitities under simple random sampling.
3. Understand the role of the sampling mechanism in sample
surveys and how it is incorporated in a Calibrated
Bayesian analysis.
4. More specifically, understand how survey design features,
such as weighting, stratification, post-stratification and
clustering, enter into a Bayesian analysis of sample survey
data.
5. Introduction to Bayesian tools for computing posterior
distributions of finite population quantities.
Models for complex surveys 1: introduction
2
Acknowledgement and Disclaimer
• These slides are based in part on a short course on
Bayesian methods in surveys presented by Dr.
Trivellore Raghunathan and I at the 2010 Joint
Statistical Meetings.
• While taking responsibility for errors, I’d like to
acknowledge Dr. Raghunathan’s major
contributions to this material
• Opinions are my own and not the official position
of the U.S. Census Bureau
Models for complex surveys 1: introduction
3
Module 1: Introduction
• Distinguishing features of survey sample inference
• Alternative modes of survey inference
– Design-based, superpopulation models, Bayes
• Calibrated Bayes
Models for complex surveys 1: introduction
4
Distinctive features of survey inference
1. Primary focus on descriptive finite population
quantities, like overall or subgroup means or totals
– Bayes – which naturally concerns predictive
distributions -- is particularly suited to inference about
such quantities, since they require predicting the values
of variables for non-sampled items
– This finite population perspective is useful even for
analytic model parameters:
 = m odel param eter (m eaningful only in c ontext of the m odel)
 (Y ) = "estim ate" of  from fitting m odel to w hole population Y
(a finite population quantity, exists regardless of validity of m odel)
A good estim ate of  should be a good est im ate of 
(if not, then w hat's being estim ated?)
Models for complex surveys 1: introduction
5
Distinctive features of survey inference
2. Analysis needs to account for "complex" sampling
design features such as stratification, differential
probabilities of selection, multistage sampling.
• Samplers reject theoretical arguments suggesting such
design features can be ignored if the model is correctly
specified.
• Models are always misspecified, and model answers are
suspect even when model misspecification is not easily
detected by model checks (Kish & Frankel 1974, Holt, Smith
& Winter 1980, Hansen, Madow & Tepping 1983, Pfeffermann &
Holmes (1985).
• Design features like clustering and stratification can
and should be explicitly incorporated in the model to
avoid sensitivity of inference to model misspecification.
Models for complex surveys 1: introduction
6
Distinctive features of survey inference
3. A production environment that precludes detailed
modeling.
• Careful modeling is often perceived as "too much
work" in a production environment (e.g. Efron 1986).
• Some attention to model fit is needed to do any good
statistics
• “Off-the-shelf" Bayesian models can be developed that
incorporate survey sample design features, and for a
given problem the computation of the posterior
distribution is prescriptive, via Bayes Theorem.
• This aspect would be aided by a Bayesian software
package focused on survey applications.
Models for complex surveys 1: introduction
7
Distinctive features of survey inference
4. Antipathy towards methods/models that involve
strong subjective elements or assumptions.
• Government agencies need to be viewed as objective
and shielded from policy biases.
• Addressed by using models that make relatively weak
assumptions, and noninformative priors that are
dominated by the likelihood.
• The latter yields Bayesian inferences that are often
similar to superpopulation modeling, with the usual
differences of interpretation of probability statements.
• Bayes provides superior inference in small samples
(e.g. small area estimation)
Models for complex surveys 1: introduction
8
Distinctive features of survey inference
5. Concern about repeated sampling (frequentist)
properties of the inference.
• Calibrated Bayes: models should be chosen to have
good frequentist properties
• This requires incorporating design features in the model
(Little 2004, 2006).
Models for complex surveys 1: introduction
9
Approaches to Survey Inference
• Design-based (Randomization) inference
• Superpopulation Modeling
– Specifies model conditional on fixed parameters
– Frequentist inference based on repeated samples from
superpopulation and finite population (hybrid approach)
• Bayesian modeling
– Specifies full probability model (prior distributions on
fixed parameters)
– Bayesian inference based on posterior distribution of
finite population quantities
– argue that this is most satisfying approach
Models for complex surveys 1: introduction
10
Design-Based Survey Inference
Z  ( Z1 ,..., Z N )  design variables, known for population
I  ( I1 ,..., I N ) = Sample Inclusion Indicators
 1, unit included in sample
Ii  
0, otherwise
I Z
Y  (Y1 , ..., Y N ) = population values,
recorded only for sam ple
Yinc  Yinc ( I )  part of Y included in the survey
N ote: here I is random variable, ( Y , Z ) are fixed
Q  Q(Y , Z ) = target finite population quantity
qˆ  qˆ ( I , Yinc , Z ) = sample estimate of Q
Vˆ ( I , Y , Z ) = sample estimate of V
1
1
1
0
0
0
0
0
Y
Yinc
[Yexc ]
inc
 qˆ 1.96 Vˆ , qˆ  1.96 Vˆ   95% confidence interval for Q
Models for complex surveys 1: introduction
11
Random Sampling
• Random (probability) sampling characterized by:
– Every possible sample has known chance of being selected
– Every unit in the sample has a non-zero chance of being
selected
– In particular, for simple random sampling with replacement:
“All possible samples of size n have same chance of being
selected”
Z  {1, ..., N } = set of units in the sam ple fram e
 N N
N!
1/   ,  I i  n ,  N 
P r( I | Z )=   n  i 1
;  
 n  n !( N  n ) !

 0, otherw ise
E ( I i | Z )  Pr( I i  1 | Z )  n / N
Models for complex surveys 1: introduction
12
Example 1:N Mean for Simple Random Sample
1
Q Y 
N
N
 y , population mean
i 1
i
Random variable
qˆ ( I )  y   I i yi / n, the sample mean
i 1
Fixed quantity, not modeled


U nbiased for Y : E I   I i y i / n  
 i 1

N
N
E
N
I
( I i ) yi / n 
i 1
 (n / N ) y
i
/n Y
i 1
1 N
2
VarI ( y )  V  (1  n / N ) S / n, S 
(
y

Y
)
 i
N  1 i 1
(1  n / N )  finite population correction
N
1
2
2
2
Vˆ  (1  n / N ) s / n, s = sample variance =
I
(
y

y
)
i i
n  1 i 1
2
2

95% confidence interval for Y  y  1.96 Vˆ , y  1.96 Vˆ
Models for complex surveys 1: introduction

13
Example 2: Horvitz-Thompson estimator
Q(Y )  T  Y1 ...YN
 i  E ( I i | Y ) = inclusion probability  0
N
N
N
i 1
i 1
i 1
tˆHT   I iYi /  i , E I (tˆHT )   E ( I i )Yi /  i =   iYi /  i  T
vˆHT  Variance estimate, depends on sample design
 tˆ
HT
 1.96 vˆ H T , tˆH T  1.96 vˆ H T
 = 95% C I for T
• Pro: unbiased under minimal assumptions
• Cons:
– variance estimator problematic for some designs (e.g.
systematic sampling)
– can have poor confidence coverage and inefficiency
Models for complex surveys 1: introduction
14
Role of Models in Classical Approach
• Inference not based on model, but models are often
used to motivate the choice of estimator. E.g.:
– Regression model
regression estimator
– Ratio model
ratio estimator
– Generalized Regression estimation: model estimates
adjusted to protect against misspecification, e.g. HT
estimation applied to residuals from the regression estimator
(Cassel, Sarndal and Wretman book).
• Estimates of standard error are then based on the
randomization distribution
• This approach is design-based, model-assisted
Models for complex surveys 1: introduction
15
Model-Based Approaches
• In our approach models are used as the basis for the entire
inference: estimator, standard error, interval estimation
• This approach is more unified, but models need to be
carefully tailored to features of the sample design such as
stratification, clustering.
• One might call this model-based, design-assisted
• Two variants:
– Superpopulation Modeling
– Bayesian (full probability) modeling
• Common theme is “Infer” or “predict” about non-sampled
portion of the population conditional on the sample and
model
Models for complex surveys 1: introduction
16
Superpopulation Modeling
• Model distribution M:
Y ~ f (Y | Z ,  ), Z = design variables,   fixed parameters
• Predict non-sampled values Yˆexc :
yˆi  E ( yi | zi ,   ˆ), ˆ  model estimate of  I
R
S
T
yi , if unit sampled;
~ ~
q  Q(Y ), yi 
yi , if unit not sampled
 ( q), over distribution of I and M
v  mse
aq  1.96
v , q  1.96 v
f=
95% C I for Q
1
1
1
0
0
0
0
0
Z
Y
Yinc
Yˆexc
In the modeling approach, prediction of nonsampled values
is central
In the design-based approach, weighting is central: “sample
represents … units in the population”
Models for complex surveys 1: introduction
17
Bayesian Modeling
• Bayesian model adds a prior distribution for the parameters:
(Y ,  ) ~  ( | Z ) f (Y | Z ,  ),  ( | Z )  prior distribution
Inference about  is based on posterior distribution from Bayes Theorem:
I Z
Y
p ( | Z , Yinc )   ( | Z ) L( | Z , Yinc ), L = likelihood
Inference about finite population quantitity Q(Y ) based on 1
Yinc
1
p (Q(Y ) | Yinc )  posterior predictive distribution
1
0
of Q given sample values Yinc
p (Q(Y ) | Z , Yinc ) 
 p(Q(Y ) | Z , Y
inc
,  ) p( | Z , Yinc ) d
0
0
0
0
Yˆexc
(Integrates out nuisance parameters  )
In the super-population modeling approach, parameters are
considered fixed and estimated
In the Bayesian approach, parameters are random and integrated out
of posterior distribution – leads to better small-sample inference
Models for complex surveys 1: introduction
18
Bayesian Point Estimates
• Point estimate is often used as a single summary
“best” value for the unknown Q
• Some choices are the mean, mode or the median of
the posterior distribution of Q
• For symmetrical distributions an intuitive choice is
the center of symmetry
• For asymmetrical distributions the choice is not
clear. It depends upon the “loss” function.
Models for complex surveys: simple random sampling
19
Bayesian Interval Estimation
• Bayesian analog of confidence interval is posterior
probability or credibility interval
– Large sample: posterior mean +/- z * posterior se
– Interval based on lower and upper percentiles of
posterior distribution – 2.5% to 97.5% for 95% interval
– Optimal: fix the coverage rate 1-a in advance and
determine the highest posterior density region C to
include most likely values of Q totaling 1-a posterior
probability
Models for complex surveys: simple random sampling
20
Bayes for population quantities Q
• Inferences about Q are conveniently obtained by first
conditioning on  and then averaging over posterior of  .
In particular, the posterior mean is:
E ( Q | Yinc )  E  E ( Q | Yinc ,  ) | Yinc 
and the posterior variance is:
V ar ( Q | Yinc )  E V ar ( Q | Yinc ,  ) | Yinc   V ar  E ( Q | Yinc ,  ) | Yinc 
• Value of this technique will become clear in applications
• Finite population corrections are automatically obtained as
differences in the posterior variances of Q and 
• Inferences based on full posterior distribution useful in
small samples (e.g. provides “t corrections”)
Models for complex surveys: simple random sampling
21
Simulating Draws from Posterior Distribution
• For many problems, particularly with high-dimensional 
it is often easier to draw values from the posterior
distribution, and base inferences on these draws
(d )
• For example, if ( 1 : d  1,..., D )
is a set of draws from the posterior distribution for a scalar
parameter  1 , then
1  D
1
 d 1  1
D
s  ( D  1)
2
1
(d )

approxim ates posterior m ean
D
d 1
( 1
(d )
  1 ) approxim ates posterior variance
2
(  1  1.96 s ) or 2.5 th to 97.5th percentiles of draw s
approxim ates 95% posterior credibility i nterva l for 
G iven a draw 
(d )
of  , usually easy to draw non-sam pled
values of data, and hence finite population quantities
Models for complex surveys: simple random sampling
22
Calibrated Bayes
• Any approach (including Bayes) has properties in
repeated sampling
• We can study the properties of Bayes credibility
intervals in repeated sampling – do 95%
credibility intervals have 95% coverage?
• A Calibrated Bayes approach yields credibility
intervals with close to nominal coverage
• Frequentist methods are useful for forming and
assessing model, but the inference remains
Bayesian
• See Little (2004) for more discussion
Models for complex surveys 1: introduction
23
Summary of approaches
• Design-based:
– Avoids need for models for survey outcomes
– Robust approach for large probability samples
– Less suited to small samples – inference basically
assumes large samples
– Models needed for nonresponse, response errors, small
areas – this leads to “inferential schizophrenia”
Models for complex surveys 1: introduction
24
Summary of approaches
• Superpopulation/Bayes models:
– Familiar: similar to modeling approaches to statistics in
general
– Models needs to reflect the survey design
– Unified approach for large and small samples,
nonresponse and response errors.
– Frequentist superpopulation modeling has the limitation
that uncertainty in predicting parameters is not reflected
in prediction inferences:
– Bayes propagates uncertainty about parameters, making
it preferable for small samples – but needs specification
of a prior distribution
Models for complex surveys 1: introduction
25
Module 2: Bayesian models for
simple random samples
2.1 Continuous outcome: normal model
2.2 Difference of two means
2.3 Regression models
2.4 Binary outcome: beta-binomial model
2.5 Nonparametric Bayes
Models for complex surveys 1: introduction
26
Models for simple random samples
• Consider Bayesian predictive inference for population
quantities
• Focus here on the population mean, but other posterior
distribution of more complex finite population quantities
Q can be derived
• Key is to compute the posterior distribution of Q
conditional on the data and model
– Summarize the posterior distribution using posterior mean,
variance, HPD interval etc
• Modern Bayesian analysis uses simulation technique to
study the posterior distribution
• Here consider simple random sampling: Module 3
considers complex design features
Models for complex surveys: simple random sampling
27
Diffuse priors
• In much practical analysis the prior information is diffuse,
and the likelihood dominates the prior information.
• Jeffreys (1961) developed “noninformative priors” based
on the notion of very little prior information relative to the
information provided by the data.
• Jeffreys derived the noninformative prior requiring
invariance under parameter transformation.
• In general,
 ( )  | J ( ) |
1/ 2
w here
  2 log f ( y |  ) 
J ( )   E 

t






Models for complex surveys: simple random sampling
28
Examples of noninformative priors
Normal:  (  ,  )  
2
B inom ial:  ( )  
P oisson:  (  )  
 1/ 2
2
(1   )
 1/ 2
 1/ 2
N orm al regression w ith slopes  :  (  ,  )  
2
2
In simple cases these noninformative
priors result in numerically same answers
as standard frequentist procedures
Models for complex surveys: simple random sampling
29
2.1 Normal simple random sample
Yi ~ iid N (  ,  ); i  1, 2, ..., N
2
 ( , )  
2
2
sim ple random sam ple results in Yinc  ( y1 , ..., y n )
Q Y 
ny  ( N  n )Y exc
N
 f  y  (1  f )  Y exc
D erive posterior distribution of Q
Models for complex surveys: simple random sampling
30
2.1 Normal Example
Posterior distribution of (,2)
p (  ,
 ( )
2
2

1
2  n / 2 1
| Yinc )  (2  )
exp  
 2
 n / 2 1
1
2
exp    ( y i  y ) / 
2  iinc
2
2
2
( yi   ) 
 2 
iinc

2
2 
 n(  y ) /  

The above expressions imply that
(1) 
2

| Yinc ~
( y i  y ) /  n 1
2
2
i inc
(2)  | Yinc , 
2
~ N ( y ,
2
/ n)
Models for complex surveys: simple random sampling
31
2.1 Posterior Distribution of Q
Y exc |  , 
2
2



~ N  ,

N

n


Y exc |  , Yinc
2
2
2
2





~ N  y,



N

n
n
(1

f
)
n


Q  f  y  (1  f )  Y exc
Q |  , Yinc
2
Y exc | Yinc
Q | Yinc
 (1  f )
~ N  y,
n

2



2


s
~ t n 1  y ,

(1  f ) n 

2

(1  f ) s 
~ t n 1  y ,

n


Models for complex surveys: simple random sampling
32
2.1 HPD Interval for Q
Note the posterior t distribution of Q is symmetric
and unimodal -- values in the center of the
distribution are more likely than those in the tails.
Thus a (1-a)100% HPD interval is:
y  t n 1,1  a / 2
(1  f ) s
2
n
Like frequentist confidence interval, but
recovers the t correction
Models for complex surveys: simple random sampling
33
2.1 Some other Estimands
• Suppose Q=Median or some other percentile
• One is better off inferring about all non-sampled values
• As we will see later, simulating values of Yexc adds enormous
flexibility for drawing inferences about any finite population
quantity
• Modern Bayesian methods heavily rely on simulating values
from the posterior distribution of the model parameters and
predictive-posterior distribution of the nonsampled values
• Computationally, if the population size, N, is too large then
choose any arbitrary value K large relative to n, the sample
size
– National sample of size 2000
– US population size 306 million
– For numerical approximation, we can choose K=2000/f, for some
small f=0.01 or 0.001.
Models for complex surveys: simple random sampling
34
2.1 Comments
• Even in this simple normal problem, Bayes is
useful:
– t-inference is recovered for small samples by putting a
prior distribution on the unknown variance
– Inference for other quantities, like Q=Median or some
other percentile, is achieved very easily by simulating
the nonsampled values (more on this below)
• Bayes is even more attractive for more complex
problems, as discussed later.
Models for complex surveys: simple random sampling
35
2.2 Comparison of Two Means
• Population 1
• Population 2
P opulation size  N 1
P opulation size  N 2
Sam ple size  n1
Sam ple size  n 2
Y1i  ind N (  1 ,  )
Y 2 i  ind N (  2 ,  2 )
2
2
1
 ( 1 ,  )  
2
1
2
 ( 2 , 2 )   2
2
2
1
2
2
Sam ple Statistics : ( y 1 , s1 )
Sam ple Statistics : ( y 2 , s 2 )
P osterior distributions :
P osterior distributions :
( n1  1) s1 /  1 ~  n1 1
( n 2  1) s 2 /  2 ~  n 2 1
 1 ~ N ( y1 ,  1 / n1 )
 2 ~ N ( y2 ,  2 / n2 )
Y1i ~ N (  1 ,  1 ), i  exc
Y 2 i ~ N (  2 ,  2 ), i  exc
2
2
2
2
2
2
2
2
2
2
Models for complex surveys: simple random sampling
36
2.2 Estimands
• Examples
– Y1  Y 2 (Finite sample version of Behrens-Fisher Problem)
– Difference Pr(Y1  c )  Pr(Y2  c )
– Difference in the population medians
– Ratio of the means or medians
– Ratio of Variances
• It is possible to analytically compute the posterior
distribution of some these quantities
• It is a whole lot easier to simulate values of nonsampled Y in Population 1 and Y in Population 2
's
1
's
2
Models for complex surveys: simple random sampling
37
2.3 Ratio and Regression Estimates
• Population: (yi,xi; i=1,2,…N)
• Sample: (yi, iinc, xi, i=1,2,…,N).
For now assume SRS
Objective: Infer about the population mean
N
Q 

yi
i 1
Excluded Y’s are missing values
y1 x1
y2 x2
. .
. .
. .
yn xn
xn 1
xn  2
.
.
.
Models for complex surveys: simple random sampling
xN
38
2.3 Model Specification
(Yi | x i ,  ,  ) ~ ind N (  x i ,  x i )
2
2
2g
i  1, 2,..., N
g know n
2
P rior distribution:  (  ,  )  
2
g=1/2: Classical Ratio estimator. Posterior variance equals
randomization variance for large samples
g=0: Regression through origin. The posterior variance is nearly the
same as the randomization variance.
g=1: HT model. Posterior variance equals randomization variance
for large samples.
Note that, no asymptotic arguments have been used in deriving
Bayesian inferences. Makes small sample corrections and uses tdistributions.
Models for complex surveys: simple random sampling
39
2.3 Posterior Draws for Normal Linear
Regression
g
=
0
ˆ
(  , s )  ls estim ates of slopes and resid var iance
2

(d )2

(d )
 ( n  p  1) s /  n  p 1
2
T
 ˆ  A z
2
(d )
 n  p 1 = chi-squared deviate w ith n  p  1 df
2
z  ( z1 , ..., z p 1 ) , z i ~ N (0,1)
T
A  upper triangular C holesky factor of ( X X )
T
A A  (X X )
T
T
1
:
1
N onsam pled values y i | 
(d )
,
(d )
~ N (
(d )
xi , 
(d )2
)
• Easily extends to weighted regression
Models for complex surveys: simple random sampling
40
2.4 Binary outcome: consulting
example
• In India, any person possessing a radio, transistor
or television has to pay a license fee.
• In a densely populated area with mostly makeshift
houses practically no one was paying these fees.
• Target enforcement in areas where the proportion
of households possessing one or more of these
devices exceeds 0.3, with high probability.
Models for complex surveys: simple random sampling
41
2.4 Consulting example (continued)
N  P opulation S ize in particular area
 1, if household i has a device
Yi  
 0, otherw ise
N
Q 
Y
i
/ N P roportion of households w ith a devic e
i 1
Q uestion of Interest: P r( Q  0.3)
• Conduct a small scale survey to answer the question of
interest
• Note that question only makes sense under Bayes paradigm
Models for complex surveys: simple random sampling
42
2.4 Consulting example
srs of size n, Yinc  {Y1 ,..., Yn }, Yexc  {Yn1 ,..., YN }
Yi |  ~ iid Bernoulli( )
n
x   Yi
i 1
f ( x |  )  ( nx ) x (1   ) n x
Model for observable
 ( )  1   (0,1)
Prior distribution
N


Q   Yi / N   x   Yi  / N
i 1
i  n 1


N
Models for complex surveys: simple random sampling
Estimand
43
2.4 Beta Binomial model
The posterior distribution is
p ( | x ) 
p ( | x ) 
f ( x |  ) ( )

f ( x |  ) ( ) d 
( ) (1   )
n
x
x
n x
 ( x ) (1   )
n
x
n x
 f ( x |  ) ( )
1
d
 | x ~ B eta ( x  1, n  x  1)
Models for complex surveys: simple random sampling
44
2.4 Infinite Population
F or N   , Y N  
P r(Y N  0.3 | x )  P r(  0.3 | x )
C om pute using cum ulative distribution fu nction
of a beta distribution w hich is a standa rd function
in m ost softw are such as S A S , R
What is the maximum proportion of households in
the population with devices that can be said with
great certainty?
P r(   ? | x )  0.9
Inverse C D F of B eta D istribution
Models for complex surveys: simple random sampling
45
2.5 Bayesian Nonparametric
Inference
•
•
•
•
•
Population: Y1 , Y2 , Y3 , ..., Y N
All possible distinct values: d1 , d 2 ,..., d K
Model: Pr(Yi  d k )   k
Prior:  (1 , 2 ,..., k )   k1 if  k  1
k
k
Mean and Variance:
E (Yi |  )   
d
k
k
k
V ar (Yi |  )  
2


dkk  
2
2
k
Models for complex surveys: simple random sampling
46
2.5 Bayesian Nonparametric Inference
• SRS of size n with nk equal to number
of dk in the sample
• Objective is to draw inference about
the population mean:Q  f  y  (1  f )  Y
• As before we need the posterior
distribution of  and 2
exc
Models for complex surveys: simple random sampling
47
2.5 Nonparametric Inference
• Posterior distribution of  is Dirichlet:
 ( | Yinc ) 
 k
n k 1
if

k
 1 and
n
k
k
k
n
k
• Posterior mean, variance and covariance of 
E ( k | Yinc ) 
nk
n
, V ar ( k | Yinc ) 
C ov ( k ,  l | Yinc )  
nk (n  nk )
n ( n  1)
2
n k nl
n ( n  1)
2
Models for complex surveys: simple random sampling
48
2.5 Inference for Q
E (  | Yinc ) 

dk
k
nk
 y
n
s n 1
2
V ar (  | Yinc ) 
E (
2
| Yinc )  s
n n 1
2
;s 
2
1

n  1 i inc
( yi  y )
2
n 1
n 1
Hence posterior mean and variance of Q are:
E ( Q | Yinc )  f  y  (1  f ) E (  | Yinc )  y
s n 1
2
V ar ( Q | Yinc )  (1  f )
n n 1
Models for complex surveys: simple random sampling
49
Module 3: complex sample designs
• Considered Bayesian predictive inference for population
quantities
• Focused here on the population mean, but other
posterior distribution of more complex finite population
quantities Q can be derived
• Key is to compute the posterior distribution of Q
conditional on the data and model
– Summarize the posterior distribution using posterior mean,
variance, HPD interval etc
• Modern Bayesian analysis uses simulation technique to
study the posterior distribution
• Models need to incorporate complex design features like
unequal selection, stratification and clustering
Models for complex surveys: simple random sampling
50
Modeling sample selection
• Role of sample design in model-based (Bayesian)
inference
• Key to understanding the role is to include the sample
selection process as part of the model
• Modeling the sample selection process
– Simple and stratified random sampling
– Cluster sampling, other mechanisms
– See Chapter 7 of Bayesian Data Analysis (Gelman, Carlin,
Stern and Rubin 1995)
Models for complex sample designs
51
Full model for Y and I
p (Y , I | Z ,  ,  )  p (Y | Z ,  ) p ( I | Y , Z ,  )
Model for
Population
Model for
Inclusion
• Observed data: (Yinc , Z , I ) (N o m issing values)
• Observed-data likelihood:
L ( ,  | Yinc , Z , I )  p (Yinc , I | Z ,  ,  ) 
 p (Y , I | Z ,  ,  ) dY
exc
• Posterior distribution of parameters:
p ( ,  | Yinc , Z , I )  p ( ,  | Z ) L ( ,  | Yinc , Z , I )
Models for complex sample designs
52
Ignoring the data collection process
• The likelihood ignoring the data-collection process is
based on the model for Y alone with likelihood:
L ( | Yinc , Z )  p (Yinc | Z ,  ) 
 p (Y | Z ,  ) dY
exc
• The corresponding posteriors for  and Yexc are:
p ( | Yinc , Z )  p ( | Z ) L ( | Yinc , Z )
p (Y exc | Yinc , Z ) 
 p (Y
exc
Posterior predictive
distribution of Yexc
| Yinc , Z ,  ) p ( | Yinc , Z ) d 
• When the full posterior reduces to this simpler
posterior, the data collection mechanism is called
ignorable for Bayesian inference about  , Yexc .
Models for complex sample designs
53
Bayes inference for probability samples
• A sufficient condition for ignoring the selection mechanism is
that selection does not depend on values of Y, that is:
p ( I | Y , Z ,  )  p ( I | Z ,  ) for all Y .
• This holds for probability sampling with design variables Z
• But the model needs to appropriately account for relationship of
survey outcomes Y with the design variables Z.
• Consider how to do this for (a) unequal probability samples, and
(b) clustered (multistage) samples
Models for complex sample designs
54
Ex 1: stratified random sampling
• Population divided into J strata
• Z is set of stratum indicators:
Sample Population
Z Y
Z
1, if unit i is in stratum j ;
zi  
 0, otherw ise.
• Stratified random sampling: simple random sample
of n j units selected from population of N j units in stratum j.
• This design is ignorable providing model for
outcomes conditions on the stratum variables Z.
• Same approach (conditioning on Z works for poststratification, with extensions to more than one
margin.
Models for complex sample designs
55
Inference for a mean from a stratified sample
• Consider a model that includes stratum effects:
[ y i | z i  j ] ~ ind N ( j ,  j )
2
• For simplicity assume

2
j
is known and the flat prior:
p ( j | Z )  const .
• Standard Bayesian calculations lead to
[Y | Yinc , Z , { j }] ~ N ( y st ,  st )
2
where:
2
J
y st 
P
j
y j , P j  N j / N , y j  sam ple m ean in stratum j ,
j 1
J
 st 
2

P j (1  f j )
2
2
j
/ nj, fj  nj / N
j
j 1
Models for complex sample designs
56
Bayes for stratified normal model
• Bayes inference for this model is equivalent to
standard classical inference for the population mean
from a stratified random sample
• The posterior mean weights case by inverse of
inclusion probability:
y st  N
1
J
N
j 1
j
yj  N
1
J

yi /  j ,
j 1 i : x i  j
w here  j  n j / N j  selection probability in stratum j .
• With unknown variances, Bayes’ for this model with
flat prior on log(variances) yields useful t-like
corrections for small samples
Models for complex sample designs
57
Suppose we ignore stratum effects?
• Suppose we assume instead that:
[ y i | z i  j ] ~ ind N ( ,  ),
2
the previous model with no stratum effects.
• With a flat prior on the mean, the posterior mean of Y is then the
unweighted mean
J
E (Y | Yinc , Z ,  )  y 
2

pj yj, pj  nj / n
j 1
• This is potentially a very biased estimator
if the selection rates
 j  n j / N j vary across the strata
– The problem is that results from this model are highly sensitive
violations of the assumption of no stratum effects … and
stratum effects are likely in most realistic settings.
– Hence prudence dictates a model that allows for stratum
effects, such as the model in the previous slide.
Models for complex sample designs
58
Design consistency
• Loosely speaking, an estimator is design-consistent if
(irrespective of the truth of the model) it converges to the true
population quantity as the sample size increases, holding design
features constant.
• For stratified sampling, the posterior mean y st based on the
stratified normal model converges to Y , and hence is designconsistent
• For the normal model that ignores stratum effects, the posterior
mean y converges to
Y 

J
j 1
 j N j Y j /  j 1  j N
J
j
and hence is not design consistent unless  j  const .
• We generally advocate Bayesian models that yield designconsistent estimates, to limit effects of model misspecification
Models for complex sample designs
59
Ex 2. A continuous (post)stratifier Z
Consider PPS sampling, Z = measure of size
Standard design-based estimator is weighted
Horvitz-Thompson estimate
yHT
1  n

  yi /  i
N  i 1
Sample Population
Z Y
Z

 ;  i  selection prob (H T )

y H T  m odel-based prediction estim ate for
y i ~ N or (   i ,   i ) ("H T m odel")
2
2
When the relationship between Y and Z deviates a lot from the
HT model, HT estimate is inefficient and CI’s can have poor
coverage
Models for complex sample designs
60
Ex 4. One continuous (post)stratifier Z
y wt
1  n

  yi /  i
N  i 1

 ;  i  selection prob (H T )

Sample Population
Z Y
Z
A m odeling alternative to the H T estim at or is create
predictions from a m ore robust m odel relating Y to Z :
1  n
y m od =
  yi 
N  i 1

ˆ
 y i  , yˆ i predictions from :
i  n 1

N
y i ~ N or ( S ( i ),   i ); S ( i ) = penalized s pline of Y on Z
2
k
(Zheng and Little 2003, 2005)
Models for complex sample designs
61
Ex 3. Two stage sampling
• Most practical sample designs involve selecting a
cluster of units and measure a subset of units
within the selected cluster
• Two stage sample is very efficient and cost
effective
• But outcome on subjects within a cluster may be
correlated (typically, positively).
• Models can easily incorporate the correlation
among observations
Models for complex sample designs
62
Two-stage samples
• Sample design:
– Stage 1: Sample c clusters from C clusters
– Stage 2: Sample k i units from the selected cluster i=1,2,…,c
K i  P opulation size of cluster i
C
N 
K
i
i 1
• Estimand of interest: Population mean Q
• Infer about excluded clusters and excluded units within
the selected clusters
Models for complex sample designs
63
Models for two-stage samples
• Model for observables
Yij ~ N (  i ,  ); i  1, ..., C ; j  1, 2, ..., K i
2
 i ~ iid N ( ,  )
2
A ssum e  and  are know n
• Prior distribution
 ( )  1
Models for complex sample designs
64
Estimand of interest and inference strategy
• The population mean can be decomposed as
c
NQ 
 [k
C
i
y i  ( K i  k i )Yi ,exc ] 
i 1

K i Yi
i  c 1
• Posterior mean given Yinc
c
 [k
E ( N Q | Yinc ,  i , i  1, 2, ..., c ;  ) 
C
i
yi  ( K i  ki )  i ] 
i 1
 [k
C
i
y i  ( K i  k i ) E (  i | Yinc )] 
i 1
w here E (  i | Yinc ) 
ˆ  E ( | Yinc ) 

K i
i  c 1
c
E ( N Q | Yinc ) 


K i E ( | Yinc )
i  c 1
2
2
y i  ( k i /  )  ˆ  (1 /  )
ki / 
y i / (  
2
2
2
1/
2
/ ki )
i
 1 / (  
2
2
/ ki )
i
Models
for complex sample designs
65
Posterior Variance
• Posterior variance can be easily computed
c
V ar ( N Q | Yinc ) 

C
( K i  k i )(
2
 ( K i  k i ) ) 
2
i 1

K i (
2
 K i )
2
i  c 1
V ar (Yi ,exc | Yinc )  E [V ar (Yi ,exc | Yinc ,  i ) | Yinc ]  V ar [ E (Y i ,exc | Yinc ,  i ) | Yinc ]


2
  , i  1, 2,
2
K i  ki
,c
V ar (Yi | Yinc )  E [V ar (Y i | Yinc ,  i ) | Yinc ]  V ar [ E (Y i | Y inc ,  i ) | Y inc ]

2
/ K i   , i  c  1, c  2,
2
,C
Models for complex sample designs
66
Inference with unknown  and 
• For unknown  and 
– Option 1: Plug in maximum likelihood estimates. These can
be obtained using PROC MIXED in SAS. PROC MIXED
actually gives estimates of ,, and E(i|Yinc) (Empirical
Bayes)
– Option 2: Fully Bayes with additional prior
2
 ( ,  ,  )   
2
2
2  v
exp   b / (2 ) 
2
where b and v are small positive numbers
Models for complex sample designs
67
Extensions and Applications
• Relaxing equal variance assumption
Yil ~ N (  i ,  i )
2
(  i , log  i ) ~ iid B V N ( ,  )
• Incorporating covariates (generalization of ratio
and regression estimates)
Yil ~ N ( x il  i ,  i )
2
(  i , log  i ) ~ iid M V N ( ,  )
• Small Area estimation. An application of the
hierarchical model. Here the quantity of interest is
E (Yi | Yinc )  ( k i y i  ( K i  k i ) E (Yi ,exc | Yinc )) / K i
Models for complex sample designs
68
Extensions
• Relaxing normal assumptions
Yil |  i ~ G lim (  i  h ( x il  i ),  v (  i ))
2
v : a know n function
 i ~ iid M V N ( ,  )
• Incorporate design features such as stratification
and weighting by modeling explicitly the sampling
mechanism.
Models for complex sample designs
69
Summary
• Bayes inference for surveys must incorporate design
features appropriately
• Stratification and clustering can be incorporated in
Bayes inference through design variables
• Unlike design-based inference, Bayes inference is not
asymptotic, and delivers good frequentist properties
in small samples
Models for complex sample designs
70
Module 4: Short introduction to
Bayesian computation
• A Bayesian analysis uses the entire posterior distribution of the
parameter of interest.
• Summaries of the posterior distribution are used for statistical
inferences
– Means, Median, Modes or measures of central tendency
– Standard deviation, mean absolute deviation or measures of
spread
– Percentiles or intervals
• Conceptually, all these quantities can be expressed analytically
in terms of integrals of functions of parameter with respect to its
posterior distribution
• Computations
– Numerical integration routines
– Simulation techniques – outline here
Models for Complex Surveys: Bayesian Computation
71
Types of Simulation
• Direct simulation (as for normal sample, regression)
• Approximate direct simulation
– Discrete approximation of the posterior density
– Rejection sampling
– Sampling Importance Resampling
• Iterative simulation techniques
– Metropolis Algorithm
– Gibbs sampler
– Software: WINBUGS
Models for Complex Surveys: Bayesian Computation
72
Approximate Direct Simulation
• Approximating the posterior distribution by a normal
distribution by matching the posterior mean and
variance.
– Posterior mean and variance computed using numerical
integration techniques
• An alternative is to use the mode and a measure of
curvature at the mode
– Mode and the curvature can be computed using many
different methods
• Approximate the posterior distribution using a grid of
values of the parameter and compute the posterior
density at each grid and then draw values from the
grid with probability proportional to the posterior
density
Models for Complex Surveys: Bayesian Computation
73
Normal Approximation
P osterior density :  ( | x )
E asy to w ork w ith log-posterior density
l ( )  log( ( | x ))
A t the m ode, f ( )  l '( )  0
C urvature : f '( )  l ''( )
F o r lo g arith m o f th e n o rm al d en sity
M o d e is th e m ean an d
th e cu rvatu re at th e m o d e
is n eg ative o f th e p recisio n
(P recisio n :recip ro cal o f varian ce)
Models for Complex Surveys: Bayesian Computation
74
Rejection Sampling
• Actual Density from
which to draw from
• Candidate density from
which it is easy to draw
• The importance ratio is
bounded
• Sample  from g,
accept  with
probability p otherwise
redraw from g
 ( | data )
g ( ), w ith g ( )  0 for all 
w ith  ( | data )  0
 ( | data )
g ( )
p
 M
 ( | data )
M  g ( )
Models for Complex Surveys: Bayesian Computation
75
Sampling Importance Resampling
• Target density from which to
 ( | data )
draw
g ( ), such that g ( )  0
• Candidate density from
which it is easy to draw
for all  w ith  ( | data )  0
• The importance ratio
 ( | data )
w ( ) 
• Sample M values of  from g
• Compute the M importance
ratios and resample with
probability proportional to
the importance ratios.
g ( )
 1* ,  *2 , ... ,  *M
w ( i ); i  1,2 , ... , M
*
Models for Complex Surveys: Bayesian Computation
76
Markov Chain Simulation
• In real problems it may be hard to apply direct or
approximate direct simulation techniques.
• The Markov chain methods involve a random walk in the
parameter space which converges to a stationary
distribution that is the target posterior distribution.
– Metropolis-Hastings algorithms
– Gibbs sampling
Models for Complex Surveys: Bayesian Computation
77
Metropolis-Hastings algorithm
• Try to find a Markov Chain whose stationary distribution is
the desired posterior distribution.
• Metropolis et al (1953) showed how and the procedure was
later generalized by Hastings (1970). This is called
Metropolis-Hastings algorithm.
• Algorithm:
– Step 1 At iteration t, draw
y ~ p( y | x
(t )
)
y : C andidate P oint
p : C andidate D ensity
Models for Complex Surveys: Bayesian Computation
78
– Step 2: Compute the ratio
(t )

f ( y) / p( y | x ) 
w  M in 1,

(t )
(t )
 f (x ) / p(x | y) 
– Step 3: Generate a uniform random number, u
X
X
–
–
–
–
( t  1)
 y if u  w
( t  1)
 X
(t )
o th erw ise
This Markov Chain has stationary distribution f(x).
Any p(y|x) that has the same support as f(x) will work
If p(y|x)=f(x) then we have independent samples
Closer the proposal density p(y|x) to the actual density f(x),
faster will be the convergence.
Models for Complex Surveys: Bayesian Computation
79
Gibbs sampling
• Gibbs sampling a particular case of Markov Chain Monte Carlo
method suitable for multivariate problems
x  ( x1 , x 2 , ..., x p ) ~ f ( x )
f ( x i | x1 , x 2 , ..., x i 1 , x i 1 , ..., x p )
G ibbs sequence :
( t  1)
x1
( t  1)
x2
( t  1)
xi
( t  1)
xp
(t )
(t )
(t )
~ f ( x1 | x 2 , x 3 , ..., x p )
( t  1)
(t )
~ f ( x 2 | x1
( t  1)
~ f ( x i | x1
( t  1)
(t )
(t )
, ..., x i 1 , x i 1 , ..., x p )
( t  1)
~ f ( x p | x1
(t )
, x 3 , ..., x p )
( t 1)
1. This is also a Markov
Chain whose stationary
Distribution is f(x)
2. This is an easier
Algorithm, if the
conditional densities
are easy to work with
3. If the conditionals are
harder to sample from,
then use MH or Rejection
technique within the
Gibbs sequence
, ..., x p 1 )
Models for Complex Surveys: Bayesian Computation
80
Conclusion
• Design-based: limited, asymptotic
• Bayesian inference for surveys: flexible, unified,
now feasible using modern computational
methods
• Calibrated Bayes: build models that yield
inferences with good frequentist properties –
diffuse priors, strata and post-strata as covariates,
clustering with mixed effects models
• Software: Winbugs, but software targeted to
surveys would help.
• The future may be Calibrated Bayes!
Models for complex surveys 1: introduction
81

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