Accounting for Complex Sample Designs via Mixture Models Michael Elliott August 2008

Background
Robust Models
Simulation
Discussion
Accounting for Complex Sample Designs via
Mixture Models
Michael Elliott1
1 University
of Michigan School of Public Health
August 2008
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Talk Outline
1
Background
Design-Based Inference
Model-Based Inference
2
Robust Models
Finite Normal Mixture Models
Bayesian Density Estimation
3
Simulation
4
Discussion
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Design-Based Inference
Model-Based Inference
Design-Based Inference
Randomization or “design-based” inference is standard for
sample survey data.
Treat population values Y = (Y1 , ..., YN ) as fixed, and
sampling indicators I = (I1 , ..., IN ) as random.
Goal is to make inference about a population quantity Q(Y).
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Design-Based Inference
Model-Based Inference
Design-Based Inference
Randomization or “design-based” inference is standard for
sample survey data.
Treat population values Y = (Y1 , ..., YN ) as fixed, and
sampling indicators I = (I1 , ..., IN ) as random.
Goal is to make inference about a population quantity Q(Y).
Consider estimator qˆ(y, I) where
EI|Y (ˆ
q (y, I)) ≈ Q(Y)
and variance estimator of qˆ(y, I) vˆ (Yinc , I) where
EI|Y (ˆ
v (y, I)) ≈ VarI|Y (ˆ
q (y, I))
(Hansen and Hurwitz 1943; Kish 1965; Cochran 1977.)
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Design-Based Inference
Model-Based Inference
Model-Based Inference
Model-based approach posits p(Y | θ).
Superpopulation: θ fixed
Bayesian: θ ∼ p(θ)
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Design-Based Inference
Model-Based Inference
Bayesian Survey Inference
Focus on inference about Q(Y) based on p(Ynobs | y):
p(Ynobs | y) =
R
R
p(Y)
=
p(y)
p(Y | θ)p(θ)dθ
=
p(y)
p(Ynobs | y, θ)p(y | θ)p(θ)dθ
=
p(y)
Z
p(Ynobs | y, θ)p(θ | y)dθ
(Ericson 1969; Scott 1977; Rubin 1987).
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Design-Based Inference
Model-Based Inference
Design Inference vs. Bayesian Inference
Randomization approach has substantial advantages.
Y treated as fixed → no need for distributional assumptions.
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Design-Based Inference
Model-Based Inference
Design Inference vs. Bayesian Inference
Randomization approach has substantial advantages.
Y treated as fixed → no need for distributional assumptions.
In scientific surveys the distribution of I is (largely) under the
control of the investigator.
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Design-Based Inference
Model-Based Inference
Design Inference vs. Bayesian Inference
Randomization approach has substantial advantages.
Y treated as fixed → no need for distributional assumptions.
In scientific surveys the distribution of I is (largely) under the
control of the investigator.
“Automatically” account for sample design in inference.
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Design-Based Inference
Model-Based Inference
Design Inference vs. Bayesian Inference
Randomization approach does not always work well.
Inefficient (Basu 1971)
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Design-Based Inference
Model-Based Inference
Design Inference vs. Bayesian Inference
Randomization approach does not always work well.
Inefficient (Basu 1971)
Small-area estimation (Ghosh and Lahiri 1988)
Non-response (Little and Rubin 2002)
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Design-Based Inference
Model-Based Inference
Design Inference vs. Bayesian Inference
Randomization approach does not always work well.
Inefficient (Basu 1971)
Small-area estimation (Ghosh and Lahiri 1988)
Non-response (Little and Rubin 2002)
Lack of consistent reference distribution (Little 2004)
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Design-Based Inference
Model-Based Inference
Design Inference vs. Bayesian Inference
Bayesian approach avoids “inferential schizophrenia” (Little 2004)
Doesn’t rely on asymptotics
Focus on prediction of unsampled elements
Does require noninformative sampling: P(I | Y) = P(I) or,
more generally, unconfounded sampling P(I | Y) = P(I | y)
(Rubin 1987). Maintaining this assumption requires:
Probability sample
Model p(Y) attentive to design features and robust enough to
sufficiently capture all aspects of the distribution of Y relevant
to Q(Y).
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Design-Based Inference
Model-Based Inference
Accommodating Survey Weights in a Model
Stratify data by probabilities of inclusion h = 1, ..., H and allow
interaction between model quantities of interest and probabilities
of inclusion
yih ∼ N(µh , σ 2 )
X
Y | y ∼ N(N −1
{nh yh +(Nh −nh )yˆ h }, (1−n/N)σ 2 /n), yˆ h = E (µh | y)
h
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Design-Based Inference
Model-Based Inference
Accommodating Survey Weights in a Model
Stratify data by probabilities of inclusion h = 1, ..., H and allow
interaction between model quantities of interest and probabilities
of inclusion
yih ∼ N(µh , σ 2 )
X
Y | y ∼ N(N −1
{nh yh +(Nh −nh )yˆ h }, (1−n/N)σ 2 /n), yˆ h = E (µh | y)
h
Flat prior on µh → yˆ h = y h recovers fully-weighted estimator.
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Design-Based Inference
Model-Based Inference
Accommodating Survey Weights in a Model
Stratify data by probabilities of inclusion h = 1, ..., H and allow
interaction between model quantities of interest and probabilities
of inclusion
yih ∼ N(µh , σ 2 )
X
Y | y ∼ N(N −1
{nh yh +(Nh −nh )yˆ h }, (1−n/N)σ 2 /n), yˆ h = E (µh | y)
h
Flat prior on µh → yˆ h = y h recovers fully-weighted estimator.
Degenerate prior on µh at µ → yˆ h = y recovers unweighted
estimator.
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Design-Based Inference
Model-Based Inference
Accommodating Survey Weights in a Model
Stratify data by probabilities of inclusion h = 1, ..., H and allow
interaction between model quantities of interest and probabilities
of inclusion
yih ∼ N(µh , σ 2 )
X
Y | y ∼ N(N −1
{nh yh +(Nh −nh )yˆ h }, (1−n/N)σ 2 /n), yˆ h = E (µh | y)
h
Flat prior on µh → yˆ h = y h recovers fully-weighted estimator.
Degenerate prior on µh at µ → yˆ h = y recovers unweighted
estimator.
Assigning a proper prior µh ∼ N(µ, τ 2 ) (Holt and Smith 1979)
compromises between fully-weighted and unweighted
2
estimator: yˆ h = wh y h + (1 − wh )˜
y , wh = n nτh2τ+σ2 ,
h
P
−1 P
nh
nh
y˜ =
y
.
h n τ 2 +σ 2
h n τ 2 +σ 2 h
h
h
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Design-Based Inference
Model-Based Inference
Accommodating Survey Weights in a Model
Elliott and Little (2000) extend to consider µ ∼ N(f (h, β), Σ):
Adding structure to the mean and variance increases
robustness of estimation of population mean when stratum
mean strongly associated with probability of selection, though
efficiency gains over design-based estimator of Y when
stratum means are weakly associated with probability of
selection are reduced.
A Bayesian smoothing spline estimator of the mean is quite
robust but can still can yield efficiency gains.
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Design-Based Inference
Model-Based Inference
Accommodating Survey Weights in a Model
Developing models to accommodate survey weights for more
complex population quantities such as population regression
parameters more challenging.
Elliott (2007) and Huang and Elliott (2008) extend weight stratum
models to linear and generalized linear regression models by
allowing for interactions between weight strata and regression
parameters.
Efficiency gains are possible over design-based regression
estimators
Proliferation of parameters can make practical implementation
difficult, if number of covariates large and sample size modest.
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Finite Normal Mixture Models
Bayesian Density Estimation
Finite Normal Mixture Models
A simple finite normal mixture model without covariates:
Yi | Ci = c, µc , σc2 ∼ N(µc , σc2 ), C = 1, ..., K
Ci = c | π1 , ..., πK ∼ MULTI (1; π1 , ..., πK )
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Finite Normal Mixture Models
Bayesian Density Estimation
Finite Normal Mixture Models
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Finite Normal Mixture Models
Bayesian Density Estimation
Application to Complex Sample Data
Maintain robustness of design-based approach?
Use of models that include a large number of classes to model
highly non-normal data.
Increased efficiency of model-based approach?
If the data suggest that a small number (or single) class of
normal data is sufficient.
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Finite Normal Mixture Models
Bayesian Density Estimation
Normal Regression Mixture Model for Complex Sample
Design Data
Yi | xi , Ci = c, β c , σ 2 ∼ N(x0i β c , σ 2 ), C = 1, ..., K
Ci = c | α, γ, πi ∼ MULTI (1; p1 , ..., pK ), ηij = Φ(γj −f (πi , α)) for
ηij =
j
X
pk , j = 1, ..., K − 1
k=1
where γ1 = 0 to avoiding aliasing with the α parameters.
Accounts for regression model misspecification and skewness
and overdispersion in the residual errors term
Fits simple, highly efficient models when the data allow.
f (πi , α) could be simple parametric form (e.g., linear in π), or
non-parametric (e.g., linear P-spline).
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Finite Normal Mixture Models
Bayesian Density Estimation
Normal Mixture Model Priors
To ensure a proper posterior, we utilize conjugate priors of the form
p(β c ) ∼ N(β 0 , Σ0 )
p(σc2 ) ∼ Inv − χ2 (a, s)
p(α) ∼ N(α0 , Ω0 )
p(γj ) ∼ UNI (0, A)
By choosing relatively non-informative values for the prior
parameters, we should be able to avoid influencing the results of
the inference to an untoward degree.
Draws from p(β, σ 2 , α, γ | y) are obtained using a Gibbs sampling
algorithm.
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Finite Normal Mixture Models
Bayesian Density Estimation
Normal Mixture Model Posterior Predictive Distribution
Using simulations (β rep , (σ 2 )rep , αrep , γ rep ) from
p(β, σ 2 , α, γ | y, x), we obtain
PNfrom p(B | y, x) where
PN a simulation
0
−1
B is the population slope ( i=1 xi xi )
i=1 xi yi :
rep
picrep = Φ(γcrep − f (πi , αrep )) − Φ(γc−1
− f (πi , αrep ))
rep
p rep φ((yi − x0i β rep
c )/σc )
p˜icrep = PK ic rep
rep
0 rep
c=1 pic φ((yi − xi β c )/σc )
yˆirep =
K
X
p˜icrep x0i β rep
c
c=1
B rep = (X 0 WX )−1 X 0 W yˆ
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Finite Normal Mixture Models
Bayesian Density Estimation
Normal Mixture Model Shortcomings
A shortcoming of the above mixture model is that the number of
classes K must be known in advance.
Could let K be a random variable and either obtain yi | xi , K
analytically or include draw of K in Gibbs sampler via
reversible jump algorithm (Green 1995).
Alternatively, could use a Bayesian non-parametric approach
that avoids pre-specification of the number of mixture classes.
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Finite Normal Mixture Models
Bayesian Density Estimation
Bayesian Density Estimation Model
We can generalize the normal mixture model as
Z
f (yi | xi ) = N(yi | φi )Gxi (φi )
Previously Gxi was multinomial, but in Dunson et al. (2007) Gx is
an element in an uncountable collection of probability measures
Gx ∼ DP(αG0,x ), where DP denotes a Dirichlet process (Ferguson
1973) centered at base measure G0 with precision α.
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Finite Normal Mixture Models
Bayesian Density Estimation
Dirichlet process mixture models
In standard DP mixture models (MacEachren 1994)
Gx ≡ G ∼ DP(αG0 ). Expressing the Dirichlet process in “stick
breaking” form , we have
∞
X
πh
∼ BETA(1, α)
G=
πh δθh , Qh−1
π
h
l=1
h=1
where δθ is degenerate at θ and {θh } are atoms generated from
G0 . Use of a Polya urn scheme (Blackwell and MacQueen 1973)
integrates out the infinite dimensional G to obtain
X
α
1
φi | φ(i) , α ∼
G0 +
δφj
α+n−1
α+n−1
j6=i
Thus DP mixture models cluster subjects into K ≤ n classes for
which θ = (θ1 , ..., θK ), where θh are sampled independently from
G0 . The induced prior on K grows with n and α.
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Finite Normal Mixture Models
Bayesian Density Estimation
Dirichlet process mixture models
Under the normal model, the posterior predictive distribution of
yi | xi at a given draw of φ = (β, σ 2 ), K , and S denoting the
configuration of φ into K distinct values is given by
K
X
α
N(yi | xi , β0 , σ02 ) +
πh N(yi | xi , βh , σh2 )
α+n
h=1
where πh = nh /(α + n) and β0 and σ02 are further independent
draws from G0 . This forces the conditional posterior predictive
distribution of y to be linear in x:
E (yirep
| xi , φ, K , S) =
K
X
h=0
Michael Elliott
πh xi βh = xi β, β =
K
X
πh βh .
h=0
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Finite Normal Mixture Models
Bayesian Density Estimation
Dirichlet process mixture models
Dunson et al. (2007) thus extends the DP mixture model to allow
to DP prior to depend on covariates:
Gx =
n
X
j=1
γj exp(−ψ || x − xj ||)
bj (x)Gx∗j , bj (x) = Pn
, Gx∗j ∼ DP(αG0 )
γ
exp(−ψ
||
x
−
x
||)
l
l
l
DP prior is now itself a mixture of DP-distributed random
basis measures at each covariate value, with the weights given
by b(x).
Subjects with xi close to x will have basis distributions with a
high weight in Gx , with γ controlling the degree to which Gx
loads across multiple draws from DP(αG0 )
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Finite Normal Mixture Models
Bayesian Density Estimation
Dirichlet process mixture models
Priors on the parameters governing the DP mixture prior are as
follows
γj | κ ∼ GAMMA(κ, nκ), log κ ∼ N(µκ , σκ2 )
log ψ ∼ N(µψ , σψ2 )
Assuming a constant variance σ 2 across the mixture components,
we have
G0 ≡ N(β, Σ), β ∼ N(β0 , Vβ0 ), Σ−1 ∼ W (ν0 , (ν0 Σ0 )−1 )
σ −2 ∼ GAMMA(a, b)
An MCMC algorithm can be implemented as an Dunson et al.
(2007).
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Finite Normal Mixture Models
Bayesian Density Estimation
Posterior Predictive Distribution of Regression Parameters
Posterior predictive distribution at xi conditional on a draw of all
other model components given by
wi0 (xi )N(yirep , x0i β, σ 2 + x0i Σxi ) +
k
X
wih (xi )N(yirep , x0i βh , σ 2 )
h=1
wi0 (xi ) will be larger for larger α and when the ith subject is
currently assigned to a cluster with relatively few members.
wih (xi ) will be larger for smaller α and when the ith subject
covariates are closer in Euclidian distance to other subjects
current assigned to cluster h.
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Finite Normal Mixture Models
Bayesian Density Estimation
Posterior Predictive Distribution of Regression Parameters
Posterior predictive distribution of the population regression slope
given by multivariate normal with mean
(X 0 W ∗ X )−1 X 0 W ∗ y˜ , y˜i = wi0 x0i β +
K
X
wih x0i βh
h=1
and variance
(X 0 W ∗ X )−1
(
X
i
K
X
∗
2
2 0
xi Σxi )x0i
wi xi (σ (
wih2 ) + wi0
)
(X 0 W ∗ X )−1
h=0
where wi∗ are the survey case weights.
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Simulation Model
Yi | Xi , σ 2 ∼ N(α0 +
10
X
αh (Xi − h)+ , σ 2 ),
h=1
Xi ∼ UNI (0, 10), i = 1, . . . , N = 20000.
P(Ii = 1 | Hi ) = πh ∝ (1 + Hi )Hi
Hi = dXi e
Elements (Yi , Xi ) had ≈1/55th the selection probability when
0 ≤ Xi ≤ 1 as when 9 ≤ Xi ≤ 10.
n = 200 elements were sampled without replacement for each
of 50 simulations.
αC = (0, 0, 0, 0, .5, .5, 1, 1, 2, 2, 4): bias important for σ 2
small.
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Estimation procedures
Fully-weighted, unweighted, crude trimming weight
(maximum normalized value of 3).
2-class and 3-class mixture model.
ˆβ , βˆ = (X 0 X )−1 X 0 y and V
ˆβ = σ
ˆ 2 (X 0 X )−1
β0 = βˆ Σ0 = n2 V
P
2
−1
0ˆ 2
for σ
ˆ = (n − p)
i (yi − xi β) .
α0 = 0, Ω0 = diag (1000), and A = 10.
f (πi , α) = α0 + α1 wi , wi = πi−1 .
Bayes density model
α = .1 (limit mixture components)
log κ ∼ N(−2.5, 1), log ψ ∼ N(log(30), .5)
ˆβ ), Σ−1 ∼ W (2, .5I )
β ∼ N(0, V
−2
σ ∼ GAMMA(.1, .1)
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Posterior predictive median of yi under 3-class model
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−10
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2
4
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40
80
Variance=100
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−40
10
y
30
Variance=10
10
2
4
6
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200
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Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Posterior predictive median of yi under Bayesian density
model
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10
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Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Simulation Results
Estimator
UNWT
FWT
TWT3
MWT2
MWT3
BDWT
RMSE relative to FWT
Variance log10
1
2
3
4
4.09 2.16 1.03 0.54
1
1
1
1
1.61 1.06 0.69 0.69
3.22 1.69 0.86 0.58
1.25 1.31 0.99 0.73
0.97 0.92 0.87 0.84
Michael Elliott
True Coverage
Variance log10
1
2
3
4
0
2
92
96
76 80 94
84
40 66 96
86
22 44 84
90
42 50 88
86
88 96 100 100
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Simulation Results
Despite similarity in plots of posterior medians of population
regression slopes, 2- and 3-class model not sufficiently robust.
Bayes density estimator is robust and has moderate efficiency
gains over design-based estimator when model
misspecification is largely absent.
Bayes density estimator has substantial coverage gains over
design-based estimator model misspecification is present.
DP much slower and more difficult to implement.
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Discussion
Extensions to generalized linear models possible either by
embedding normal model in a latent variable context (e.g.,
probit modeling), or via alternative base distributions.
Quantile estimation or quantile regression may yield greater
dividends, since heteroscedasticity easily accounted for in
mixture models.
Valuable if partial covariate information available for entire
population.
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
Discussion
Take-home message:
Advances in statistical modeling during the past 10-15 years are
beginning to allow development of models sufficiently robust to
compete with design-based approaches.
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
References
Basu, D. (1971). An Essay on the Logical Foundations of Survey
Sampling, Foundations of Statistical Inference, eds. V.P. Godambe
and D.A. Sproot, Toronto: Holt, Rinehard, and Winston.
Blackwell, D., MacQueen, J.B. (1973). Ferguson distributions via
Polya urn schemes. The Annals of Statistics, 1, 353-355.
Cochran, W.G. (1977). Sampling Techniques, 3rd Ed., New York:
Wiley.
Dunson, D.B., Pillai, N., Park, J-H (2007). Bayesian density
regression. Journal of the Royal Statistical Society, B69, 163-183.
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
References
Elliott, M.R., and Little, R.J.A. (2000). Model-Based Alternatives
to Trimming Survey Weights. Journal of Official Statistics, 16,
191-209.
Elliott, M.R. (2007). Bayesian Weight Trimming for Generalized
Linear Regression Models. Survey Methodology, 33, 23-34.
Ericson, W.A. (1969). Subjective Bayesian Models in Sampling
Finite Populations. Journal of the Royal Statistical Society, B31,
195-234.
Ferguson,T.S. (1973). A Bayesian analysis of some non-parametric
problems. The Annals of Statistics, 1, 209-230.
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
References
Ghosh, M., and Lahiri, P. (1988). Bayes and Empirical Bayes
Analysis in Multistage Sampling Statistical Decision Theory and
Related Topics, 1, 195-212.
Green, P.J. (1995). Reversible Jump Markov Chain Monte Carlo
Computation and Bayesian model determination. Biometrika, 82,
711-732.
Hansen, M.H., and Hurwitz, W.N. (1943). On the Theory of
Sampling from Finite Populations. The Annals of Mathematical
Statistics, 14, 333-362.
Holt, D., Smith, T.M.F. (1979). Poststratification. Journal of the
Royal Statistical Society, A142, 33-46.
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
References
Kish, L. (1965). Survey Sampling, New York: Wiley.
Little, R.J.A., Rubin, D.B. (2002). Statistical Analysis with
Missing Data, 2n d Ed., New York: Wiley.
Little, R.J.A. (2004). To Model or Not to Model? Competing
Modes of Inference for Finite Population Sampling. Journal of the
American Statistical Association, 99, 546-556.
MacEachern, S.N. (1994). Estimating Normal Means with a
Conjugate Style Dirichlet Process Prior. in Statistics: Simulation
and Computation, 23, 727-741
Rubin, D.B. (1987). Multiple Imputation for Non-response in
Surveys, New York: Wiley.
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models
Background
Robust Models
Simulation
Discussion
References
Scott, A.J. (1977). Large Sample Posterior Distributions in Finite
Populations. The Annals of Mathematical Statistics, 42,
1113-1117.
Zheng, H., and Little, R. J. A. (2003). Penalized Spline
Model-based Estimation of the Finite Population Total from
Probability-proportional-to-size Samples. Journal of Official
Statistics, 19, 99-117
Zheng, H., and Little, R. J. A. (2005). Inference for the Population
Total from Probability-proportional-to-size Samples Based on
Predictions from a Penalized Spline Nonparametric Model. Journal
of Official Statistics, 21, 1-20.
Michael Elliott
Accounting for Complex Sample Designs via Mixture Models