Fractional Imputation in Survey Sampling:
A Comparative Review
Shu Yang
Jae-Kwang Kim
Iowa State University
Joint Statistical Meetings, August 2015
Outline
Introduction
Fractional imputation
Features
Numerical illustration
Conclusion
Yang & Kim (Iowa State University )
Fractional Imputation
2015 JSM
2 / 29
Introduction
Basic Setup
U = {1, 2, · · · , N}: Finite population
A ⊂ U: sample (selected by a probability sampling design).
Under complete response, suppose that
X
η̂n,g =
wi g (yi )
i∈A
is an unbiased estimator of ηg = N −1
known function.
PN
i=1 g (yi ).
Here, g (·) is a
For example, g (y ) = I (y < 3) leads to ηg = P(Y < 3).
Yang & Kim (Iowa State University )
Fractional Imputation
2015 JSM
3 / 29
Introduction
Basic Setup (Cont’d)
A = AR ∪ AM , where yi are observed in AR . yi are missing in AM
Ri = 1 if i ∈ AR and Ri = 0 if i ∈ AM .
yi∗ : imputed value for yi , i ∈ AM
Imputed estimator of ηg
η̂I ,g =
X
wi g (yi ) +
i∈AR
X
wi g (yi∗ )
i∈AM
Need E {g (yi∗ ) | Ri = 0} = E {g (yi ) | Ri = 0}.
Yang & Kim (Iowa State University )
Fractional Imputation
2015 JSM
4 / 29
Introduction
ML estimation under missing data setup
Often, find x (always observed) such that
Missing at random (MAR) holds: f (y | x, R = 0) = f (y | x)
Imputed values are created from f (y | x).
Computing the conditional expectation can be a challenging problem.
1
Do not know the true parameter θ in f (y | x) = f (y | x; θ):
E {g (y ) | x} = E {g (yi ) | xi ; θ} .
2
Even if we know θ, computing the conditional expectation can be
numerically difficult.
Yang & Kim (Iowa State University )
Fractional Imputation
2015 JSM
5 / 29
Introduction
Imputation
Imputation: Monte Carlo approximation of the conditional
expectation (given the observed data).
M
1 X ∗(j) E {g (yi ) | xi } ∼
g yi
=
M
j=1
1
2
Bayesian approach: generate yi∗ from
Z
f (yi | xi , yobs ) = f (yi | xi , θ) p(θ | xi , yobs )dθ
Frequentist approach: generate yi∗ from f yi | xi ; θ̂ , where θ̂ is a
consistent estimator.
Yang & Kim (Iowa State University )
Fractional Imputation
2015 JSM
6 / 29
Comparison
Model
Computation
Prediction
Parameter update
Parameter est’n
Imputation
Variance estimation
Yang & Kim (Iowa State University )
Bayesian
Posterior distribution
f (latent, θ | data)
Data augmentation
I-step
P-step
Posterior mode
Multiple imputation
Rubin’s formula
Fractional Imputation
Frequentist
Prediction model
f (latent | data, θ)
EM algorithm
E-step
M-step
ML estimation
Fractional imputation
Linearization
or Resampling
2015 JSM
7 / 29
Fractional Imputation: Basic Idea
Approximate E {g (yi ) | xi } by
E {g (yi ) | xi } ∼
=
Mi
X
∗(j)
wij∗ g (yi
)
j=1
∗(j)
where wij∗ is the fractional weight assigned to yi
value of yi .
The fractional weights satisfy
Yang & Kim (Iowa State University )
PMi
∗
j=1 wij
Fractional Imputation
, the j-th imputed
= 1.
2015 JSM
8 / 29
Fractional Imputation for categorical data
If yi is a categorical variable, we can use
Mi
∗(j)
yi
∗(j)
wij
= total number of possible values of yi
= the j-th possible value of yi
∗(j)
= P(yi = yi
| xi ; θ̂),
where θ̂ is the pseudo MLE of θ.
Yang & Kim (Iowa State University )
Fractional Imputation
2015 JSM
9 / 29
Parametric Fractional Imputation (Kim, 2011)
More generally, we can write yi = (yi1 , · · · , yip ) and yi can be
partitioned into (yi,obs , yi,mis ).
1
2
More than one (say M) imputed values of ymis,i , denoted by
∗(M)
∗(1)
ymis,i , · · · , ymis,i , are generated from some density h (ymis,i | yobs ).
Create weighted data set
wi wij∗ , yij∗ ; j = 1, 2, · · · , M; i ∈ A
where
PM
j=1
∗(j)
wij∗ = 1, yij∗ = (yobs,i , ymis,i )
∗(j)
wij∗ ∝ f (yij∗ ; θ̂)/h(ymis,i | yi,obs ),
3
θ̂ is the (pseudo) maximum likelihood estimator of θ, and f (y ; θ) is the
joint density of y .
The weight wij∗ are the normalized importance weights and can be
called fractional weights.
Yang & Kim (Iowa State University )
Fractional Imputation
2015 JSM
10 / 29
EM algorithm using PFI
EM algorithm by fractional imputation
1
2
∗(j)
Initial imputation: generate ymis,i ∼ h (yi,mis | yi,obs ).
E-step: compute
∗(j)
∗
∝ f (yij∗ ; θ̂(t) )/h(yi,mis | yi,obs )
wij(t)
3
PM
∗
= 1.
where j=1 wij(t)
M-step: update
θ̂(t+1) : solution to
M
XX
∗
wi wij(t)
S θ; yij∗ = 0,
i∈A j=1
4
where S(θ; y ) = ∂ log f (y ; θ)/∂θ is the score function of θ.
Repeat Step2 and Step 3 until convergence.
∗
We may add an optional step that checks if wij(t)
is too large for
some j. In this case, h(yi,mis | yi,obs ) needs to be changed.
Yang & Kim (Iowa State University )
Fractional Imputation
2015 JSM
11 / 29
Approximation: Calibration Fractional imputation
In large scale survey sampling, we prefer to have smaller M.
Two-step method for fractional imputation:
1
2
Create a set of fractionally imputed data with size nM, (say
M = 1000).
Use an efficient sampling and weighting method to get a final set of
fractionally imputed data with size nm, (say m = 10).
Thus, we treat the step-one imputed data as a finite population and
the step-two imputed data as a sample. We can use efficient sampling
technique (such as systematic sampling or stratification) to get a final
imputed data and use calibration technique for fractional weighting.
Yang & Kim (Iowa State University )
Fractional Imputation
2015 JSM
12 / 29
Why Fractional Imputation?
1
To improve the efficiency of the point estimator (vs. single
imputation)
2
To obtain valid frequentist inference without congeniality condition
(vs. multiple imputation)
3
To handle informative sampling mechanism
We will discuss the third issue first.
Yang & Kim (Iowa State University )
Fractional Imputation
2015 JSM
13 / 29
Informative sampling design
Let f (y | x) be the conditional distribution of y given x.
x is always observed but y is subject to missingness.
A sampling design is called noninformative (w.r.t f ) if it satisfies
f (y | x, I = 1) = f (y | x)
(1)
where Ii = 1 if i ∈ A and Ii = 0 otherwise.
If (1) does not hold, then the sampling design is informative.
Yang & Kim (Iowa State University )
Fractional Imputation
2015 JSM
14 / 29
Missing At Random
Two versions of Missing At Random (MAR)
1
PMAR (Population Missing At Random)
Y ⊥R|X
2
SMAR (Sample Missing At Random)
Y ⊥ R | (X , I )
Fractional imputation assumes PMAR while multiple imputation
assumes SMAR
Under noninformative sampling design, PMAR=SMAR
Yang & Kim (Iowa State University )
Fractional Imputation
2015 JSM
15 / 29
Imputation under informative sampling
Two approaches under informative sampling when PMAR holds.
1
2
Weighting approach: Use weighted score equation to estimate θ in
f (y | x; θ). The imputed values are generated from f (y | x, θ̂).
Augmented model approach: Include w into model covariates to get
the augmented model f (y | x, w ; φ). The augmented model makes the
sampling design noninformative in the sense that
f (y | x, w ) = f (y | x, w , I = 1). The imputed values are generated
from f (y | x, w ; φ̂), where φ̂ is computed from unweighted score
equation.
Yang & Kim (Iowa State University )
Fractional Imputation
2015 JSM
16 / 29
Berg, Kim, and Skinner (2015)
Figure : A Directed Acyclic Graph (DAG) for a setup where PMAR holds but
SMAR does not hold. Variable U is latent in the sense that it is never observed.
R
Y
W
X
I
U
f (y | x, R) = f (y | x) holds but f (y | x, w , R) 6= f (y | x, w ).
Yang & Kim (Iowa State University )
Fractional Imputation
2015 JSM
17 / 29
Imputation under informative sampling
Weighting approach generates imputed values from f (y | x, R = 1)
and
X
θ̂I =
wi {Ri yi + (1 − Ri )yi∗ }
(2)
i∈A
is unbiased under PMAR.
The augmented model approach generates imputed values from
f (y | x, w , I = 1, R = 1) and (2) is unbiased when
f (y | x, w , I = 1, R = 1) = f (y | x, w , I = 1, R = 0)
(3)
holds. PMAR does not necessarily imply SMAR in (3).
Fractional imputation is based on weighting approach.
Yang & Kim (Iowa State University )
Fractional Imputation
2015 JSM
18 / 29
Numerical illustration
A pseudo finite population constructed from a single month data in
Monthly Retail Trade Survey (MRTS) at US Bureau of Census
N = 7, 260 retail business units in five strata
Three variables in the data
h: stratum
xhi : inventory values
yhi : sales
Yang & Kim (Iowa State University )
Fractional Imputation
2015 JSM
19 / 29
Box plot of log sales and log inventory values by strata
Box plot of inventory data by strata
17
Box plot of sales data by strata
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strata
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Fractional Imputation
2015 JSM
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Imputation model
log (yhi ) = β0h + β1 log(xhi ) + ehi
where
ehi ∼ N(0, σ 2 )
Yang & Kim (Iowa State University )
Fractional Imputation
2015 JSM
21 / 29
Residual plot and residual QQ plot
Regression model of log(y) against log(x) and strata indicator
Normal Q−Q
3
Residuals vs Fitted
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−5
0
−1
−2
Residuals
1
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−3
● 6658
4●
● 6658
●4
12
13
14
15
16
−4
Fitted values
Yang & Kim (Iowa State University )
−2
0
2
4
Theoretical Quantiles
Fractional Imputation
2015 JSM
22 / 29
Stratified random sampling
Table : The sample allocation in stratified simple random sampling.
Strata
Strata size Nh
Sample size nh
Sampling weight
Yang & Kim (Iowa State University )
1
352
28
12.57
2
566
32
17.69
Fractional Imputation
3
1963
46
42.67
4
2181
46
47.41
5
2198
48
45.79
2015 JSM
23 / 29
Response mechanism: PMAR
Variable xhi is always observed and only yhi is subject to missingness.
PMAR
Rhi ∼ Bernoulli(πhi ), πhi = 1/[1 + exp{4 − 0.3 log(xhi )}].
The overall response rate is about 0.6.
Yang & Kim (Iowa State University )
Fractional Imputation
2015 JSM
24 / 29
Simulation Study
Table 1 Monte Carlo bias and variance of the point estimators.
Parameter
θ = E (Y )
Estimator
Complete sample
MI
FI
Bias
0.00
0.00
0.00
Variance
0.42
0.59
0.58
Std Var
100
134
133
Table 2 Monte Carlo relative bias of the variance estimator.
Parameter
V (θ̂)
Yang & Kim (Iowa State University )
Imputation
MI
FI
Relative bias (%)
18.4
2.7
Fractional Imputation
2015 JSM
25 / 29
Discussion
Rubin’s formula is based on the following decomposition:
V (θ̂MI ) = V (θ̂n ) + V (θ̂MI − θ̂n )
where θ̂n is the complete-sample estimator of θ. Basically, WM term
estimates V (θ̂n ) and (1 + M −1 )BM term estimates V (θ̂MI − θ̂n ).
For general case, we have
V (θ̂MI ) = V (θ̂n ) + V (θ̂MI − θ̂n ) + 2Cov (θ̂MI − θ̂n , θ̂n )
and Rubin’s variance estimator ignores the covariance term. Thus, a
sufficient condition for the validity of unbiased variance estimator is
Cov (θ̂MI − θ̂n , θ̂n ) = 0.
Meng (1994) called the condition congeniality of θ̂n .
Congeniality holds when θ̂n is the MLE of θ (self-efficient estimator).
Yang & Kim (Iowa State University )
Fractional Imputation
2015 JSM
26 / 29
Discussion
For example, there are two estimators of θ = E (Y ) when log(Y )
follows from N(β0 + β1 x, σ 2 ).
1
Maximum likelihood method:
θ̂MLE = n−1
n
X
exp{β̂0 + β̂1 xi + 0.5σ̂ 2 }
i=1
2
Method of moments:
θ̂MME = n−1
n
X
yi
i=1
The MME of θ = E (Y ) does not satisfy the congeniality and Rubin’s
variance estimator is biased (R.B. = 58.5%)
Rubin’s variance estimator is essentially unbiased for MLE of θ (R.B.
= -1.9%) but MLE is rarely used in practice.
Yang & Kim (Iowa State University )
Fractional Imputation
2015 JSM
27 / 29
Summary
Fractional imputation is developed as a frequentist imputation.
Multiple imputation is motivated from a Bayesian framework. The
frequentist validity of multiple imputation requires congeniality.
Fractional imputation does not require the congeniality condition and
works well for Method of Moments estimators.
For informative sampling, augmented model approach does not
necessarily achieve SMAR. Fractional imputation uses weighting
approach for informative sampling.
Yang & Kim (Iowa State University )
Fractional Imputation
2015 JSM
28 / 29
The end
Yang & Kim (Iowa State University )
Fractional Imputation
2015 JSM
29 / 29