Propensity-score-adjustment method for nonignorable
nonresponse
Jae Kwang Kim
1
Joint work with Minsun Riddles and Jongho Im
1
Motivating Example
Exit Poll: (2012 legislative election in Korea, Gangdong-Gap district )
Gender
Male
Female
Total
Truth
Kim (ISU)
Age
20-29
30-39
40-49
5020-29
30-39
40-49
50-
A
93
104
146
560
106
129
170
501
1,809
62,489
Party
B
115
233
295
350
159
242
262
218
1,874
57,909
Other
4
8
5
3
8
5
5
7
45
1,624
Refusal
28
82
49
174
62
70
69
211
745
Total
240
427
495
1,087
335
446
506
937
4,473
122,022
Propensity-score-adjustment method for nonignorable nonresponse
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Comparison of the methods (%)
Method
No adjustment
Adjustment (Age*Sex)
Truth
Kim (ISU)
A
48.5
49.0
51.2
B
50.3
49.8
47.5
Other
1.2
1.2
1.3
Propensity-score-adjustment method for nonignorable nonresponse
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Introduction
1
Introduction
2
Basic Theory
3
Proposed method
4
An Example
5
Statistical Properties
6
Simulation studies
7
Concluding Remarks
Kim (ISU)
Propensity-score-adjustment method for nonignorable nonresponse
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Introduction
Basic Setup
(X , Y ): random variable
θ: Defined by solving
E {U(θ; X , Y )} = 0.
yi is subject to missingness
1
δi =
0
if yi responds
if yi is missing.
Want to find wi such that the solution θ̂w to
n
X
δi wi U(θ; xi , yi ) = 0
i=1
is consistent for θ.
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Introduction
Basic Setup
Two preliminary results:
Result 1: The choice of
wi =
1
E (δi | xi , yi )
(1)
makes the resulting estimator θ̂w consistent.
Result 2: If δi ∼ Bernoulli(πi ), then using wi = 1/πi also makes the
resulting estimator consistent, but it is less efficient than θ̂w using wi
in (1).
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Introduction
Basic Setup
May assume that δi ∼ Bernoulli(πi ) where πi = π(xi , yi ; φ).
For example, logistic regression model
π(xi , yi ; φ) =
exp(φ0 + φ1 xi + φ2 yi )
1 + exp(φ0 + φ1 xi + φ2 yi )
Propensity-score-adjusted (PSA) estimator of θ: solve
n
X
δi
i=1
π(xi , yi ; φ̂)
U(θ; xi , yi ) = 0
for θ, where φ̂ is a consistent estimator of φ.
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Introduction
Basic Setup
Questions:
1
2
3
Under what conditions, parameter φ is identifiable (or estimable) ?
How to estimate the parameter ?
What is the limiting distribution of the PSA estimator θ̂w using
wi = {π(xi , yi ; φ̂)}−1 ?
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Basic Theory
1
Introduction
2
Basic Theory
3
Proposed method
4
An Example
5
Statistical Properties
6
Simulation studies
7
Concluding Remarks
Kim (ISU)
Propensity-score-adjustment method for nonignorable nonresponse
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Basic Theory
Observed likelihood
f (y | x; θ): model of y on x
g (δ | x, y ; φ): model of δ on (x, y )
Observed likelihood
Lobs (θ, φ) =
Y
f (yi | x i ; θ) g (δi | x i , yi ; φ)
δi =1
×
YZ
f (yi | x i ; θ) g (δi | x i , yi ; φ) dyi
δi =0
Under what conditions the parameters are identifiable?
Kim (ISU)
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Basic Theory
Lemma
Suppose that we can decompose the covariate vector x into two parts, x 1
and x 2 , such that
g (δ|y , x) = g (δ|y , x 1 )
(2)
(a)
(b)
and, for any given x 1 , there exist x 2 6= x 2
(a)
such that
(b)
f (y |x 1 , x 2 = x 2 ) 6= f (y |x 1 , x 2 = x 2 )
(3)
almost surely. Under some other minor conditions, all the parameters in f
and g are identifiable.
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Basic Theory
Remark
Condition (2) means
δ ⊥ x2 | y , x 1 .
That is, given (y , x 1 ), x 2 does not help in explaining δ.
Thus, x 2 plays the role of instrumental variable in econometrics:
f (y ∗ | x ∗ , z ∗ ) = f (y ∗ | x ∗ ), Cov (z ∗ , x ∗ ) 6= 0.
Here, y ∗ = δ, x ∗ = (y , x 1 ), and z ∗ = x 2 .
We may call x 2 the nonresponse instrument variable.
Rigorous theory developed by Wang et al. (2014).
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Basic Theory
Parameter estimation : GMM method
Kott and Chang (2010)
May assume
P(δ = 1 | x, y ) = π(φ0 + φ1 x 1 + φ2 y )
for some (φ0 , φ1 , φ2 ).
Construct a set of estimating equations such as
n X
i=1
δi
− 1 (1, x 1i , x 2i ) = 0
π(φ0 + φ1 x 1i + φ2 yi )
that are unbiased to zero.
May have overidentified situation: Use the generalized method of
moments (GMM).
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Proposed method
1
Introduction
2
Basic Theory
3
Proposed method
4
An Example
5
Statistical Properties
6
Simulation studies
7
Concluding Remarks
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Proposed method
Motivation
The GMM approach is based on moment conditions for δ:
n X
i=1
δi
− 1 h(x i ) = 0
π(φ; x 1i , yi )
for some function h(x) of x.
The choice of h(x) determines the statistical efficiency and also the
computational stability.
Idea: Let’s consider the maximum likelihood approach and see
whether it works better.
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Proposed method
Motivation (Cont’d)
Recall, assuming for now that f (y | x) is known, the observed score
function is
Y
Lobs (φ) =
f (yi | x i ) g (δi | x 1i , yi ; φ)
δi =1
×
YZ
f (y | x i ) g (δi | x 1i , y ; φ) dy .
δi =0
How to obtain the MLE of φ?
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Proposed method
Motivation (Cont’d)
Mean score theorem: The MLE can be obtained by solving
S̄(φ) ≡ E {Scom (φ) | X , Yobs , δ} = 0
where Scom (φ) is the complete-sample score function of φ and Yobs is
the observed part of Y .
The above conditional expectation of the complete-sample score
function is called mean score function. First discussed by R.A. Fisher
in his seminal 1922 paper.
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Proposed method
Motivation (Cont’d)
Under the IID setup,
Scom (φ) =
n
X
Si (φ),
i=1
where Si (φ) = ∂ log g (δi | x1i , yi ; φ)/∂φ.
The mean score function becomes
S̄(φ) =
n
X
[δi Si (φ) + (1 − δi )E {Si (φ) | xi , δi = 0}] ,
i=1
where
R
E {Si (φ) | xi , δi = 0} =
Kim (ISU)
Si (φ)f (y | xi ){1 − π(x1i , y ; φ)}dy
R
.
f (y | xi ){1 − π(x1i , y ; φ)}dy
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Proposed method
Motivation (Cont’d)
How to compute the conditional expectation?
Classical approach (Greenlees and Zieschang (1982), Baker and Laird
(1988), Ibrahim et al. (1999)): Assume a parametric model on
f (y | x) = f (y | x; θ) and use the EM to solve the mean score
equation of the parameters in the full joint distribution.
R
E {Si (θ, φ) | xi , δi = 0} =
Si (φ)f (y | xi ; θ){1 − π(x1i , y ; φ)}dy
R
.
f (y | xi ; θ){1 − π(x1i , y ; φ)}dy
Two problems
1
2
Requires correct specification of f (y | x; θ). Known to be sensitive to
the choice of f (y | x; θ).
Computationally heavy: Often uses Monte Carlo computation.
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Proposed method
Proposed method
Remedy (for Problem One)
Idea
Instead of specifying a parametric model for f (y | x), consider specifying a
parametric model for f (y | x, δ = 1), denoted by f1 (y | x). In this case,
R
Si (φ)f1 (y | xi )O(x1i , y ; φ)dy
R
E {Si (φ) | xi , δi = 0} =
f1 (y | xi )O(x1i , y ; φ)dy
where
O(x1 , y ; φ) =
Kim (ISU)
1 − π(φ; x1 , y )
.
π(φ; x1 , y )
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Proposed method
Remark
Based on the following identity
f (y | x, δ = 0) = f (y | x, δ = 1)
O(x1 , y ; φ)
.
E {O(x1 , y ; φ) | x, δ = 1}
Kim and Yu (2011) considered a Kernel-based nonparametric
regression method of estimating f (y | x, δ = 1) to obtain
E (Y | x, δ = 0).
Kim (ISU)
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Proposed method
Proposed method
Problem Two
Given that f1 (y | xi ) is known, how to compute
R
Si (φ)O(x1i, y ; φ)f1 (y | xi )dy
E {Si (φ) | xi , δi = 0} = R
O(x1i , y ; φ)f1 (y | xi )dy
without relying on Monte Carlo computation?
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Proposed method
Proposed method
Express
E {Si (φ) | xi , δi = 0} =
where
E1 {Si (φ)O(x1i, Y ; φ) | xi }
E1 {O(x1i, Y ; φ) | xi }
Z
E1 {Q(xi , Y ) | xi } =
Q(xi , y )f1 (y | xi )dy .
How to estimate E1 {Q(xi , Y ) | xi } without using numerical
integration or Monte Carlo methods?
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Proposed method
Proposed method
Idea
If xi were null, then we would approximate the integration by the
empirical distribution among δ = 1.
Use
Z
f1 (y | xi )
f1 (y )dy
f1 (y )
X
f1 (yj | xi )
∝
Q(xi , yj )
f1 (yj )
Z
Q(xi , y )f1 (y | xi )dy
=
Q(xi , y )
δj =1
where
Z
f1 (y ) =
f1 (y | x)f (x | δ = 1)dx ∝
X
f1 (y | xi ).
δi =1
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Proposed method
Proposed method
In practice, f1 (y | x) is unknown and is estimated by
fˆ1 (y | x) = f1 (y | x; γ̂).
Thus, given γ̂, a fully efficient estimator of φ can be obtained by
solving
n
X
S2 (φ, γ̂) ≡
δi S(φ; xi , yi ) + (1 − δi )S̄0 (φ | xi ; γ̂, φ) = 0, (4)
i=1
where
P
S̄0 (φ | xi ; γ̂, φ) =
δj =1 S(φ; xi , yj )f1 (yj
P
δj =1 f1 (yj
and
fˆ1 (y ) = nR−1
| xi ; γ̂)O(φ; x 1i , yj )/fˆ1 (yj )
| xi ; γ̂)O(φ; x 1i , yj )/fˆ1 (yi )
n
X
δi f1 (y | xi ; γ̂).
i=1
May use EM algorithm to solve (4) for φ.
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Proposed method
Proposed method
Step 1: Use the responding part of (xi , yi ), obtain γ̂ in the model
f1 (y | x; γ).
X
S1 (γ) ≡
S1 (γ; xi , yi ) = 0.
(5)
δi =1
Step 2: Given γ̂ from Step 1, obtain φ̂ by solving (4):
S2 (φ, γ̂) = 0.
Step 3: Using φ̂ computed from Step 2, the PSA estimator of θ can
be obtained by solving
n
X
δi
U(θ; xi , yi ) = 0,
π̂i
(6)
i=1
where π̂i = πi (φ̂).
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Proposed method
Proposed method
In many cases, x is categorical and f1 (y | x) can be fully
nonparametric.
If x has a continuous part, nonparametric Kernel smoothing can be
used.
The proposed method seems to be robust against the failure of the
assumed model on f1 (y | x; γ) .
Asymptotic normality of PSA estimator can be obtained &
Linearization method can be used for variance estimation.
By augmenting the estimating function, we can also impose a
calibration constraint such as
n
n
X
X
δi
xi =
xi .
π̂i
i=1
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i=1
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An Example
1
Introduction
2
Basic Theory
3
Proposed method
4
An Example
5
Statistical Properties
6
Simulation studies
7
Concluding Remarks
Kim (ISU)
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An Example
Categorical Data (All dichotomous)
Example (SRS, n = 10)
ID Weight x1
1
0.1
1
2
0.1
1
3
0.1
0
4
0.1
1
5
0.1
0
6
0.1
1
7
0.1
0
8
0.1
1
9
0.1
0
10
0.1
0
M: Missing
Kim (ISU)
x2
0
1
1
0
1
0
1
0
1
0
y
1
1
M
0
1
M
M
0
1
0
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An Example
Categorical Data (All dichotomous)
Example (Cont’d) Assume P(δ = 1 | x1 , x2 , y ) = π(x1 , y )
ID Weight x1 x2 y
1
0.1
1 0 1
2
0.1
1 1 1
3 0.1 · w3,0 0 1 0
0.1 · w3,1 0 1 1
4
0.1
1 0 0
5
0.1
0 1 1
w3,j
= P̂(Y = j | X1 = 0, X2 = 1, δ = 0)
∝ P̂(Y = j | X1 = 0, X2 = 1, δ = 1)
Kim (ISU)
1 − π̂(0, j)
π̂(0, j)
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An Example
Categorical Data (All dichotomous)
Example (Cont’d)
ID Weight
6 0.1 · w6,0
0.1 · w6,1
7 0.1 · w7,0
0.1 · w7,1
8
0.1
9
0.1
10
0.1
w6,j
w7,j
Kim (ISU)
x1
1
1
0
0
1
0
0
x2
0
0
1
1
0
1
0
y
0
1
0
1
0
1
0
1 − π̂(1, j)
π̂(1, j)
1 − π̂(0, j)
∝ P̂(Y = j | X1 = 0, X2 = 1, δ = 1)
π̂(0, j)
∝ P̂(Y = j | X1 = 1, X2 = 0, δ = 1)
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An Example
Categorical Data (All dichotomous)
Example (Cont’d)
E-step: Compute the conditional probability using the estimated
response probability π̂ab .
M-step: Update the response probability using the fractional weights.
For fully nonparametric model,
P
δi =1 I (x1i = a, yi = b)
π̂ab = P
P
P1
∗
δi =1 I (x1i = a, yi = b) +
δi =0
j=0 wi,j I (x1i = a, yij = b)
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An Example
Categorical Data (All dichotomous)
Example (Cont’d)
The proposed method can be viewed as a fractional imputation
method of Kim (2011).
On the other hand, Kott and Chang method is more close to
nonresponse weighting adjustment.
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An Example
Categorical Data (All dichotomous)
Example Kott and Chang method
ID Wgt 1 Wgt2 x1 x2
−1
1
0.1
0.1π̂11
1 0
−1
2
0.1
0.1π̂11
1 1
3
0.1
0.0
0 1
−1
4
0.1
0.1π̂10
1 0
−1
5
0.1
0.1π̂01
0 1
6
0.1
0.0
1 0
7
0.1
0.0
0 1
−1
8
0.1
0.1π̂10
1 0
−1
9
0.1
0.1π̂01
0 1
−1
10
0.1
0.1π̂00
0 0
M: Missing
Kim (ISU)
y
1
1
M
0
1
M
M
0
1
0
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An Example
Categorical Data (All dichotomous)
Kott and Chang (2010) method: Calibration equation
X
X δi
I (x1i = a, x2i = b) =
I (x1i = a, x2i = b).
π̂i
i
i
1
2
3
4
X1
X1
X1
X1
= 1, X2
= 1, X2
= 0, X2
= 0, X2
= 1:
= 0:
= 1:
= 0:
−1
π̂11
=1
−1
−1
−1
π̂11
+ π̂10
+ π̂10
=4
−1
−1
π̂01 + π̂01 = 4
−1
π̂00
= 1.
The solution of Kott and Chang method is π̂11 = 1, π̂10 = 2/3,
π̂01 = 1/2, π̂00 = 1.
The solution from the proposed method is π̂11 = 0.2, π̂10 = 0.3,
π̂01 = 0.4, π̂00 = 0.1.
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Statistical Properties
1
Introduction
2
Basic Theory
3
Proposed method
4
An Example
5
Statistical Properties
6
Simulation studies
7
Concluding Remarks
Kim (ISU)
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Statistical Properties
Asymptotic properties
Writing
Up (θ, φ) =
n
X
i=1
δi
U(θ; xi , yi ),
πi (φ)
we can write (θ̂PS , φ̂, γ̂) as the solution to

  
S1 (γ)
0
 S2 (φ, γ)  = 0 ,
Up (θ, φ)
0
(7)
where S1 (γ) and S2 (φ, γ) are defined in (5) and (4), respectively.
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Statistical Properties
Asymptotic properties (Cont’d)
Thus, we can use the sandwich formula
is, we have



γ̂
V  φ̂  = T −1 V 
θ̂PS
where
to obtain the linearization. That

S1 (γ)
0
S2 (φ, γ)  T −1
Up (θ, φ)
(8)


0
0
 E (∂S1 /∂γ 0 )

0
T = E (∂S2 /∂γ 0 ) E (∂S2 /∂φ0 )
.


E (∂Up /∂γ 0 ) E (∂Up /∂φ0 ) E (∂Up /∂θ0 )
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Statistical Properties
Alternative expression
Writing η = (φ, γ) and
S(η) =
S1 (γ)
S2 (φ, γ)
,
we can write θ̂PS to be solution to
Up (θ; η̂) = 0
where η̂ satisfies S(η̂) = 0.
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Statistical Properties
Alternative expression (Cont’d)
Taylor linearization
−1
∂Up
∂S
∗
∼
UP (θ, η̂) = UP (θ, η ) − E
S(η ∗ )
E
∂η 0
∂η 0
= UPS (θ, η ∗ ) − Cov Up , S 0 {V (S)}−1 S(η ∗ ),
by the property of zero-mean function.
(i.e. If E (U) = 0, then E (∂U/∂η 0 ) = −Cov (U, S 0 ).)
So, we have
V {θ̂PS (η̂)} ∼
= V {θ̂PS (η ∗ ) | S ⊥ } ≤ V {θ̂PS (η ∗ )},
where
V {θ̂ | S ⊥ } = V (θ̂) − Cov (θ̂, S 0 ) {V (S)}−1 Cov (S, θ̂).
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Statistical Properties
Alternative expression (Cont’d)
The solution to Up (θ, η̂) = 0 can be obtained by minimizing
QPS (θ) = Up (θ, η̂)0 [Var {Up (θ, η̂)}]−1 Up (θ, η̂)
with respect to θ.
The solution can also be obtained by minimizing
∗
Q (θ, η) =
UP (θ, η)
S(η)
0 Var
UP (θ, η)
S(η)
−1 UP (θ, η)
S(η)
with respect to (θ, η).
Kim (ISU)
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Statistical Properties
Alternative expression (Cont’d)
Justification
0 −1 UP (θ, η)
V11 V12
UP (θ, η)
Q (θ, η) =
S(η)
V21 V22
S(η)
= Q1 (θ | η) + Q2 (η)
∗
where
0 −1 −1
−1
ÛP − V12 V22
S
V UP | S ⊥
ÛP − V12 V22
S
n
o−1
Q2 (η) = S(η)0 V̂ (S)
S(η)
Q1 (θ | η)
=
For the MLE η̂, we have Q2 (η̂) = 0 and Q1 (θ | η̂) = QPS (θ).
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Statistical Properties
Alternative expression (Cont’d)
The PS estimator θ̂PS is obtained by the solution to an optimization
problem:
θ̂PS = arg min QPS (θ) = arg min Q ∗ (θ, η).
In computing Q ∗ (θ, η), we can use
Pn
Pn
V̂11 V̂12
δi π̂i−2 (1 − π̂i )Ûi Ûi0
δi π̂i−1 Ûi Ŝi0
i=1
i=1
Pn
= Pn
−1
0
0
V̂21 V̂22
i=1 δi π̂i Ŝi Ûi
i=1 Ŝi Ŝi
where Ŝi = Si (η̂), Ûi = U(θ̂PS ; xi , yi ),
S(η) =
n
X
i=1
n X
δi S1i (γ)
Si (η) =
S2i (φ, γ)
i=1
and, by (4),
S2i (φ, γ) = δi S(φ; xi , yi ) + (1 − δi ) S̄0 (φ | xi ; γ, φ) .
Kim (ISU)
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Statistical Properties
Variance estimation
Linearized variance estimator
n
o
0
−1
V̂ (θ̂) = τ̂2−1 V̂11 − V̂12 V̂22
V̂21 τ̂2−1 ,
(9)
where
τ̂2 =
n
X
δi π̂i−1 U̇(θ̂; xi , yi )
i=1
and U̇(θ; x, y ) = ∂U(θ; x, y )/∂θ0 .
P
Instead of using V̂22 = ni=1 Si Si0 , one can use V̂22 = −∂S(η)/∂η
evaluated at η = η̂, the observed information using the formula of
Oakes (1999).
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Propensity-score-adjustment method for nonignorable nonresponse
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Simulation studies
1
Introduction
2
Basic Theory
3
Proposed method
4
An Example
5
Statistical Properties
6
Simulation studies
7
Concluding Remarks
Kim (ISU)
Propensity-score-adjustment method for nonignorable nonresponse
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Simulation studies
Simulation Study (Cont’d)
Simulation setup
Case 1:
Case 2:
√
y = −1 + 2(xi + 1) + ei
y = −1 + xi2 + ei
ei ∼ N(0, 1),
ei ∼ N(0, 1),
where xi ∼ N(0, 1).
2,000 MC samples of size n = 500.
Response mechanism: logistic regression model,
π(x, y ; φ) ≡ P(δ = 1 | x, y ) = {1 + exp(−φ0 − φ1 y )}−1 ,
where (φ0 , φ1 ) = (1, −0.2).
Average response rates for all cases are about 73%.
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Simulation studies
Simulation Study (Cont’d)
Compare three estimation methods:
1
2
MAR: PSA method assuming MAR.
GMM: Kott and Chang (2010)’s method.
φ̂ck : a solution to the following calibration condition,
n
X
{δi /π(φck ) − 1} zi = 0,
i=1
where zi = (1, xi ) for Case 1 and zi = (1, xi , xi2 ) for Case 2.
3
NEW: Proposed method, using
f1 (y | x) ∼ N(β0 + β1 x, σ 2 ) for Case 1.
f1 (y | x) ∼ N(β0 + β1 x + β1 x 2 , σ 2 ) for Case 2.
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Propensity-score-adjustment method for nonignorable nonresponse
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Simulation studies
Simulation Study (Cont’d)
Table: Monte Carlo Bias, Variance, and Mean Squared Error (MSE) of the
estimators of θ = E (Y )
Case 1
Case 2
Kim (ISU)
Method
MAR
GMM
NEW
MAR
GMM
NEW
Bias
-0.055
-0.001
-0.001
-0.167
-0.000
-0.001
Variance
0.0063
0.0068
0.0068
0.0072
0.0095
0.0085
MSE
0.0092
0.0068
0.0068
0.0352
0.0095
0.0085
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Simulation studies
Simulation Study (Cont’d)
Table: Monte Carlo Bias and Variance of the parameter estimates and Relative
bias of variance estimates
Parameter
Case1
Case2
θ
φ0
φ1
θ
φ0
φ1
Kim (ISU)
Bias
-0.00
0.01
-0.00
0.01
0.01
-0.01
GMM
Variance
0.0068
0.0111
0.0059
0.0095
0.0147
0.0082
R.Bias
0.05
0.01
-0.03
-0.02
-0.05
-0.08
Bias
-0.00
0.01
-0.00
-0.00
0.00
0.00
NEW
Variance
0.0068
0.0111
0.0057
0.0085
0.0110
0.0055
Propensity-score-adjustment method for nonignorable nonresponse
R.Bias
0.06
0.01
-0.00
-0.03
-0.01
-0.03
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Simulation studies
Back to Exit Poll Example
Table: Prediction Result (Gangdong-Gap)
Method
No adjustment
Adjustment (Age * Sex)
New Method
Truth
Kim (ISU)
A
48.5
49.0
51.0
51.2
B
50.3
49.8
47.7
47.5
Other
1.2
1.2
1.2
1.3
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Simulation studies
Back to Exit Poll Example
Table: Predicted Seats in Seoul
Method
No adjustment
Adjustment (Age* Sex)
New Method
Truth
Kim (ISU)
A
10
10
15
16
B
36
36
29
30
Other
2
2
4
2
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Concluding Remarks
1
Introduction
2
Basic Theory
3
Proposed method
4
An Example
5
Statistical Properties
6
Simulation studies
7
Concluding Remarks
Kim (ISU)
Propensity-score-adjustment method for nonignorable nonresponse
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Concluding Remarks
Concluding Remarks
Maximum likelihood (ML) approach to propensity score model
parameter estimation under non-ignorable missing.
Instrumental variable needed for identifiability of the response model.
Unlike the full joint ML approach, the proposed method uses a model
for f (y | x, δ = 1) and the result is robust against the failure of the
assumption on f (y | x, δ = 1).
More efficient than the method based on the method of moments.
Directly applicable to complex survey sampling.
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Concluding Remarks
Current and Future work
AUC estimation through propensity-score-adjustment approach under
nonignorable verification bias.
Semiparametric estimation of θ based on Kernel regression method
for f1 (y | x).
Score test and pre-test estimation of φ.
Fractional imputation under nonignorable missing data.
Extension to longitudinal missing data.
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References
REFERENCES
Baker, S. G. and N. M. Laird (1988), ‘Regression analysis for categorical
variables with outcome subject to nonignorable nonresponse’, Journal of
the American Statistical Association 83, 62–69.
Greenlees, W. S., Reece J. S. and K. D. Zieschang (1982), ‘Imputation of
missing values when the probability of response depends on the variable
being imputed’, Journal of the American Statistical Association
77, 251–261.
Ibrahim, J. G., S. R. Lipsitz and M. H. Chen (1999), ‘Missing covariates in
generalized linear models when the missing data mechanism is
non-ignorable’, Journal of the Royal Statistical Society, Series B
61, 173–190.
Kim, J. K. (2011), ‘Parametric fractional imputation for missing data
analysis’, Biometrika 98, 119–132.
Kim, J. K. and C. L. Yu (2011), ‘A semi-parametric estimation of mean
functionals with non-ignorable missing data’, Journal of the American
Statistical Association 106, 157–165.
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Concluding Remarks
Kott, P. S. and T. Chang (2010), ‘Using calibration weighting to adjust for
nonignorable unit nonresponse’, Journal of the American Statistical
Association 105, 1265–1275.
Oakes, D. (1999), ‘Direct calculation of the information matrix via the em
algorithm’, Journal of the Royal Statistical Society: Series B
61, 479–482.
Wang, S., J. Shao and J. K. Kim (2014), ‘An instrumental variable
approach for identifiability and estimation in problems with nonignorable
nonresponse’, Statistica Sinica 24, 1097–1116.
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