Complex sampling in latent variable models

Complex surveys Latent variable models (LVM) Estimation of LVM under complex sampling Effect on LVM Conclusion
Complex sampling in latent variable models
Daniel Oberski
Department of methodology and statistics
Complex sampling in latent variable models Daniel Oberski

• When doing latent class analysis, factor analysis, IRT, or
structural equation modeling, should you use sampling
weights, stratification, and clustering variables?
• What is complex about surveys?
• What is ``pseudo'' about pseudo-maximum likelihood?
• What are design effects and what makes them so deft?

Outline
..1 Complex surveys
..2 Latent variable models (LVM)
..3 Estimation of LVM under complex sampling
..4 Effect on LVM
..5 Conclusion

Does it make a difference?

Unweighted regression
Weighted regression
Source: 1988 National Maternal and Infant Health Survey (Korn and Graubard, 1995).

Latent class analysis of eating vegetables
Unweighted LCA
Low High
Latent class 33% 77%
Recall 1 high 60% 80%
Source: The continuing Survey of Food Intakes by Individuals (Patterson et al., 2002).

Latent class analysis of eating vegetables
Unweighted LCA
Low High
LCA using weights
Low High
Source: The continuing Survey of Food Intakes by Individuals (Patterson et al., 2002).

Sample surveys, ``linear estimators''

Sample surveys
Purposes:
• Descriptive;
• Analytic.
Assessment of Health Status and Social Determinants
of Health (Padgol village, Gujarat, India).
Source: Boston U. India Research and Outreach Initiative.

Sample surveys
Idea of a sample survey: can generalize from a sample to a
population if the sample is ``like'' the population,
``representative method''.

Sample of people ``like'' the population?
• Neyman (1934) figured this would be true on average if
you draw a random sample;
• This is the theory we still use today.

``Linear estimator'':
Eπ

n−1
∑
i∈sample
yi

 = N−1
∑
i∈population
yi.
and generally
mn
d
→ N[µ, var(mn)]

``Linear estimator'':
Eπ

n−1
∑
i∈sample
yi

 = N−1
∑
i∈population
yi.
and generally
mn
d
→ N[µ, var(mn)]
``Design-consistent''

``Linear estimator''
• Most of the time when people talk about ``linear
estimators'', they are thinking about means and totals.
• But a proportion is a linear estimator too;

• for ex., proportion observed for response patterns:

Pattern Prop. Pattern Prop.
1111 0.226 0111 0.090
1110 0.087 0110 0.047
1101 0.092 0101 0.046
1100 0.049 0100 0.030
1011 0.085 0011 0.045
1010 0.048 0010 0.028
1001 0.049 0001 0.029
1000 0.029 0000 0.022

1111 0.226 0111 0.090
1110 0.087 0110 0.047
1101 0.092 0101 0.046
1100 0.049 0100 0.030
1011 0.085 0011 0.045
1010 0.048 0010 0.028
1001 0.049 0001 0.029
1000 0.029 0000 0.022
→

1111 0.226 0111 0.090
1110 0.087 0110 0.047
1101 0.092 0101 0.046
1100 0.049 0100 0.030
1011 0.085 0011 0.045
1010 0.048 0010 0.028
1001 0.049 0001 0.029
1000 0.029 0000 0.022
→
LCA
estimates:
Latent class
1 2
y1 0.77 0.56
y2 0.78 0.55
y3 0.76 0.55
y4 0.78 0.54

1111 0.226 0111 0.090
1110 0.087 0110 0.047
1101 0.092 0101 0.046
1100 0.049 0100 0.030
1011 0.085 0011 0.045
1010 0.048 0010 0.028
1001 0.049 0001 0.029
1000 0.029 0000 0.022
→
LCA
estimates:
Latent class
1 2
y1 0.77 0.56
y2 0.78 0.55
y3 0.76 0.55
y4 0.78 0.54
• Even the (co)variance is a linear estimator, if you redefine
d := (y − E(Y))(y − E(Y))T: then var(y) = (n − 1)−1
∑
d

Complications → ``complex surveys'':
• Clustering
• Stratification
• Selection with unequal probabilities πi
Equivalent: not independently and identically distributed (iid)

Clustering

Simple random sampling: a lot of driving
A simple random sample of voter locations in the US.
Source: Lumley (2010).

Source: Heeringa et al. (2010)

Sample clustering for several reasons:
• Geographic clustering of elements for household surveys
reduces interviewing costs by amortizing travel and
related expenditures over a group of observations. E.g.:
NCS- R, National Health and Nutrition Examination Survey
(NHANES), Health and Retirement Study (HRS)
• Sample elements may not be individually identified on the
available sampling frames but can be linked to aggregate
cluster units (e.g., voters at precinct polling stations,
students in colleges and universities). The available
sampling frame often identifies only the cluster groupings.
(Heeringa et al., 2010, p. 28)

Sample clustering for several reasons:
• Geographic clustering of elements for household surveys
reduces interviewing costs by amortizing travel and
related expenditures over a group of observations. E.g.:
NCS- R, National Health and Nutrition Examination Survey
(NHANES), Health and Retirement Study (HRS)
• Sample elements may not be individually identified on the
available sampling frames but can be linked to aggregate
cluster units (e.g., voters at precinct polling stations,
students in colleges and universities). The available
sampling frame often identifies only the cluster groupings.
• One or more stages of the sample are deliberately
clustered to enable the estimation of multilevel models
and components of variance in variables of interest (e.g.,
students in classes, classes within schools).

Stratification

Sample stratified by region

Stratified sampling serves several purposes:
• Relative to an SRS of equal size, smaller standard errors
• Disproportionately allocate the sample to subpopulations,
that is, to oversample specific subpopulations to ensure
sufficient sample sizes for analysis.

Unequal probabilities of selection

Common reasons for varying probabilities of case selection in
sample surveys include (Heeringa et al., 2010, p. 38--43):
• Disproportionate sampling within strata to
• achieve an optimally allocated sample

• deliberately increase precision for subpopulations

• Differentially sample subpopulations, e.g. NHANES
oversampling of people with disabilities.

• Subsampling of observational units within sample clusters,
e.g. selecting a single random respondent from the
eligible members of sample households.

• Sampling probability that can be obtained only in the
process of the survey data collection, e.g. in a random
digit dialing (RDD) telephone survey, number of distinct
landline telephone numbers

• Sampling probability that can be obtained only in the
process of the survey data collection, e.g. in a random
digit dialing (RDD) telephone survey, number of distinct
landline telephone numbers
• Nonresponse

Linear estimators in complex samples

Problems:
Bias: If some (types of) people have a differing chance πi
of being in the sample, usual sample statistics will not (on
average) equal the population quantities anymore.
Variance: Affected by clustering/stratification.

Problems:
If ˆµn := n−1
∑
i∈sample
1
πi
yi, notice:
Eπ

n−1
∑
i∈sample
1
πi
yi

 = N−1
∑
i∈population
πi
πi
yi = N−1
∑
i∈population
yi

Problems:
If ˆµn := n−1
∑
i∈sample
1
πi
yi, notice:
Eπ

n−1
∑
i∈sample
1
πi
yi

 = N−1
∑
i∈population
πi
πi
yi = N−1
∑
i∈population
yi
Solutions:
• weighted estimator ˆµn unbiased (Horvitz and Thompson, 1952);
• Can obtain variance of weighted estimate, var(ˆµn), under
clustering, stratification.

Latent variable modeling

Latent variable modeling (LVM)
• (Confirmatory) factor analysis (CFA);
• Structural Equation Modeling (SEM);
• Latent Class Analysis/Modeling (LCA/LCM);
• Latent trait modeling;
• Item Response Theory (IRT) models;
• Mixture models;
• Random effects/hierarchical/multilevel models;
• ``Anchoring vignettes'' models;
• ... etc.

• Proportions can be turned into an LC or IRT analysis;
• Covariances can be turned into a SEM analysis.

• Proportions can be turned into an LC or IRT analysis;
• Covariances can be turned into a SEM analysis.
Definition
Latent variable model estimation: a way of turning observed
covariances/proportions (``moments'') into LVM parameter
estimates.
LVM : mn → ˆθn

LVM : mn → ˆθn
Example: confirmatory factor analysis (CFA) with 1 factor, 3
indicators:
:
ˆλ11 =
√
cor(y1, y2)cor(y1, y3)/cor(y2, y3)
ˆλ21 =
√
ˆλ31 =
√

Inference in latent variable models under simple random
sampling

Data generating process:
→ →
Inference:
← ←
(Fuller, 2009).

Model
Superpopulation
→ →
Inference:
← ←
(Fuller, 2009).

Model
Superpopulation
→ Finite population →
Inference:
← ←
(Fuller, 2009).

Model
Superpopulation
→ Finite population → Sample
Inference:
← ←
(Fuller, 2009).

Model
Superpopulation
Inference:
← ← Sample
(Fuller, 2009).

Model
Superpopulation
Inference:
← Finite population ← Sample
(Fuller, 2009).

Model
Superpopulation
Inference:
Model
← Finite population ← Sample
(Fuller, 2009).

Superpopulation → Finite population of 100 subjects
Loadings: 0.707
→
y1
−2
0
2
−4 −2 0 2
Corr:
0.442
Corr:
0.475
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y20
2
−2 0 2
Corr:
0.321
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y3
0
2
−2 0 2
Loadings:
y1: 0.810
y2: 0.546
y3: 0.587

Simple random sample (SRS) of 20 from finite pop.
y1
−2
0
2
−4 −2 0 2
Cor : 0.442
1: 0.425
2: 0.568
Cor : 0.475
1: 0.361
2: 0.668
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y20
2
−2 0 2
Cor : 0.321
1: 0.258
2: 0.543
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y3
0
2
−2 0 2
●
(Superpopulation
loadings: 0.707)
SRS factor loading
estimates:
y1: 0.836
y2: 0.679
y3: 0.800

Superpopulation inference from SRS to
superpopulation
Superpopulation
← ←
Sample
y1
−2
0
2
−4 −2 0 2
Cor : 0.442
1: 0.425
2: 0.568
Cor : 0.475
1: 0.361
2: 0.668
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y20
2
−2 0 2
Cor : 0.321
1: 0.258
2: 0.543
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y3
0
2
−2 0 2
●
simple.random
1
2
λ11: 0.707
λ21: 0.707
λ31: 0.707
← ←
Avg. (sd) loading over
10,000 samples:
ˆλ11: 0.707 (0.125)
ˆλ21: 0.722 (0.127)
ˆλ31: 0.711 (0.122)

Complex sampling affects latent variable modeling

LVM : mn → ˆθn
This means that:
• bias in covariances/proportions (moments) leads to bias in
LVM parameter estimates;
• any across-sample variation in latent variable parameter
estimates is entirely due to variation in the sample
moments used to estimate them.
• With more observed variables (moments), use Maximum
Likelihood (ML) to get estimates, but above is still true.
• MLE: ˆθn = arg maxθ L(θ; ˆµn)

LVM: mn → ˆθn so bias in mn means bias in ˆθn
• One solution: modeling correctly all aspects of the
sampling design.
(Skinner et al., 1989, chapter 3)

sampling design.
• Another solution:
replacing the observed moments with
design-consistent moments will provide
design-consistent estimates =
``pseudo-maximum likelihood'' (PML).
ˆµn → ˆθn

sampling design.
• Another solution:
replacing the observed moments with
design-consistent moments will provide
design-consistent estimates =
``pseudo-maximum likelihood'' (PML).
ˆµn → ˆθn
• (A third solution: weighted least squares - less than
satisfactory results)

Variance of PMLE is obtained by sandwich (linearization)
estimate. In turn depends on variance of design-consistent
moment estimates (the ``meat'').
var(ˆθn) = (∆T
V∆)−1
∆T
V · var(ˆµn) · V∆(∆T
V∆)−1

Variance of PMLE is obtained by sandwich (linearization)
estimate. In turn depends on variance of design-consistent
moment estimates (the ``meat'').
var(ˆθn) = (∆T
V∆)−1
∆T
V · var(ˆµn) · V∆(∆T
V∆)−1
V: Depends on distributional assumptions (=ML)
∆: Depends on the specific model (=LVM)
var(ˆµn): Depends on variance of means/prop's/covar's under complex sampling

pseudo-
..1 supposed or purporting to be but not really so; false; not
genuine: pseudonym | pseudoscience.
..2 resembling or imitating: pseudohallucination |
pseudo-French.
ORIGIN from Greek pseudēs ‘false,’ pseudos ‘falsehood.’
Source: New Oxford American Dictionary

Pseudo-ML

Pseudo-ML
Why ML?
• Consistently estimate parameters aggregated over
clusters and strata;
• Estimates ``MLE that would be obtained by fitting the
model to the population data''.

Pseudo-ML
Why ML?
Why pseudo?
• Not exactly equal to the MLE obtained by correctly
modeling all aspects of the sampling design;
• Not asymptotically optimal.

Pseudo-ML
Why ML?
Why pseudo?
• Not exactly equal to the MLE obtained by correctly
modeling all aspects of the sampling design;
• Not asymptotically optimal.
Why PML?
• Aggregate parameters may be of interest;
• No assumptions/modeling on design necessary.

The effect of clustering

Cluster sample of 20 from finite population
y1
−2
0
2
−4 −2 0 2
Cor : 0.442
1: 0.412
2: 0.49
Cor : 0.475
1: 0.504
2: 0.455
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y20
2
−2 0 2
Cor : 0.321
1: 0.352
2: 0.205
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y3
0
2
−2 0 2
●
●
(Superpopulation
loadings: 0.707)
Cluster sample
loading estimates:
y1: 0.997
y2: 0.491
y3: 0.456

Superpopulation inference from SRS to
superpopulation
Superpopulation
← ←
Sample
y1
−2
0
2
−4 −2 0 2
Cor : 0.442
1: 0.425
2: 0.568
Cor : 0.475
1: 0.361
2: 0.668
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y20
2
−2 0 2
Cor : 0.321
1: 0.258
2: 0.543
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y3
0
2
−2 0 2
●
simple.random
1
2
λ11: 0.707
λ21: 0.707
λ31: 0.707
← ←
Avg. (sd) loading over
10,000 samples:
ˆλ11: 0.665 (0.157)
ˆλ21: 0.699 (0.140)
ˆλ31: 0.703 (0.145)

The effect of cluster sampling on factor analysis
ˆλ11
ˆλ21
ˆλ31
avg sd avg sd avg sd
Population: 0.707 0.707 0.707
SRS: 0.707 (0.125) 0.722 (0.127) 0.711 (0.122)
Cluster smp: 0.665 (0.157) 0.699 (0.140) 0.703 (0.145)
deft 1.26 1.10 1.19
deff 1.58 1.22 1.41
% Var. incr. 58% 22% 41%

Design effects' deftness
• ``Design effect'' or deff = varclus(ˆθ)/varsrs(ˆθ)(Kish, 1965);
• deff is increase in variance relative to a simple random
sampling design;
• deft is relative increase in standard errors;
• In practice deff/deft have to be estimated and we use the
sandwich estimator of variance.
Useful for:
• Seeing to what extent it makes a difference to take
complex sampling into account;

sampling design;
Useful for:
• Identifying parameters that are more or less affected;

sampling design;
Useful for:
• Identifying parameters that are more or less affected;
• Sample size and power calculations.

The effect of unequal probabilities of selection

Sampling with probability correlated with factor x

Sampling with probability correlated with x2

ˆλ11
ˆλ21
ˆλ31
Population: 0.707 0.707 0.707
SRS: 0.707 (0.125) 0.722 (0.127) 0.711 (0.122)
Selection probability proportional to latent factor x:
Unwghted: 0.679 (0.137) 0.683 (0.138) 0.692 (0.137)
Bias/deft -4% 1.13 -3% 1.12 -2% 1.12
Weighted: 0.687 (0.143) 0.698 (0.143) 0.703 (0.143)
Bias/deft -3% 1.18 -1% 1.16 -1% 1.17

ˆλ11
ˆλ21
ˆλ31
Population: 0.707 0.707 0.707
SRS: 0.707 (0.125) 0.722 (0.127) 0.711 (0.122)
Selection probability proportional to latent factor x:
Unwghted: 0.679 (0.137) 0.683 (0.138) 0.692 (0.137)
Bias/deft -4% 1.13 -3% 1.12 -2% 1.12
Weighted: 0.687 (0.143) 0.698 (0.143) 0.703 (0.143)
Bias/deft -3% 1.18 -1% 1.16 -1% 1.17
Selection probability proportional to x2:
Unwghted: 0.845 (0.060) 0.842 (0.061) 0.843 (0.061)
Bias/deft 20% 0.495 19% 0.492 19% 0.497
Weighted: 0.750 (0.139) 0.739 (0.141) 0.737 (0.137)
Bias/deft 6% 1.149 5% 1.145 4% 1.123

When does weighting make a difference for point
estimates of latent variable models?
• (Usually) when weights represent omitted variable(s) that
interact with observed or latent variables;
• (Sometimes, e.g. IRT, LCA) when selection is correlated
with a dependent variable.

• When the model is strongly misspeciﬁed:

• When the model is strongly misspeciﬁed:
0.5 1.0 1.5 2.0
-2-101
x
y1
True curve (black line),
Overall linear reg. line (green),
and reg. from unequal selection/weights

Should you weight?
..1 Purpose of the analysis: analytical versus descriptive;
..2 Anticipated bias from an unweighted analysis;
..3 If unweighted analysis is unbiased, relative magnitude of
inefficiency resulting from a weighted analysis;
..4 Whether variables are available and known to model the
sample design instead of weighting the analysis.
(Patterson et al., 2002, p. 727)

Conclusions
• Surveys are not usually simple random samples (or iid);
• Sample design may bias the results of latent variable
modeling (confidence intervals, significance tests, fit
measures, parameter estimates);
• Pseudo-maximum likelihood can take the design into
account without additional assumptions;
• Implemented in software. SEM: lavaan.survey in R
• Nonparametric correction for the design;
• ``Aggregate modeling'';
• Payment is in variance (efficiency);
• Alternative is modeling the effects of strata, clusters,
covariates behind; ``disaggregate modeling''.

Thank you for your attention!
Daniel Oberski
doberski@uvt.nl
http://daob.org/

References
References
Fuller, W. A. (2009). Sampling statistics. Wiley, New York.
Heeringa, S., West, B., and Berglund, P. (2010). Applied survey data analysis.
Horvitz, D. and Thompson, D. (1952). A generalization of sampling without
replacement from a finite universe. Journal of the American Statistical
Association, 47(260):663--685.
Kish, L. (1965). Survey sampling. New York: Wiley.
Korn, E. and Graubard, B. (1995). Examples of differing weighted and
unweighted estimates from a sample survey. The American Statistician,
49(3):291--295.
Lumley, T. (2010). Complex surveys: a guide to analysis using R. Wiley.
Neyman, J. (1934). On the two different aspects of the representative
method: the method of stratified sampling and the method of purposive
selection. Journal of the Royal Statistical Society, 97(4):558--625.
Patterson, B., Dayton, C., and Graubard, B. (2002). Latent class analysis of
complex sample survey data. Journal of the American Statistical
Association, 97(459):721--741.
Skinner, C., Holt, D., and Smith, T. (1989). Analysis of complex surveys. John
Wiley & Sons.

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