MNAR

Statistical analysis of non-ignorable nonresponse:
Review of the existing methods
Jae-Kwang Kim 1
Department of Statistics, Iowa State University
June 8, 2017
1
Joint work with Kosuke Morikawa at Osaka University

1 Introduction
2 Full likelihood-based ML estimation
3 Partial Likelihood approach
4 GMM method
5 Exponential tilting method
6 Semiparametric eﬃcient estimation
7 Concluding Remarks
Jae-Kwang Kim (ISU) Nonignorable missing June 8, 2017 2 / 56

Lemma 1
Suppose that we can decompose the covariate vector x = (u, z) such that
g(δ|y, x) = g(δ|y, u) (1)
and, for any given u, there exist zu,1 and zu,2 such that
f (y|u, z = zu,1) = f (y|u, z = zu,2). (2)
Under some other minor conditions, all the parameters in f and g are identiﬁable.

Remark
• Condition (1) means
δ ⊥ z | y, u.
• That is, given (y, u), z does not help in explaining δ.
Figure: A DAG for understanding nonresponse instrumental variable Z
Z U
Y δ
• We may call z the nonresponse instrument variable.
• Rigorous theory developed by Wang et al. (2014).

Another condition for identification problem
• It is hard to find the instrumental variable Z by using only observed data.
• Also, if the dependence of Y on Z is weak, the estimator can be unstable.
• There is a necessary and sufficient condition for model identification by restricting
on the outcome model.

Lemma 2
• Let O(φ; x, y) = 1/P(δ = 1 | x, y; φ) − 1, f (y | x, δ = 1; γ0) be a parametric model
known up to γ; f (y | x, δ = 1; γ) is identifiable. Also let E1(· | x) = E(· | x, δ = 1).
• If E1{O(φ; x, Y ) | x; γ0} exists for almost all x and it is identifiable for φ, the
observed likelihood is identifiable.
• We can show a model is identifiable without using any instrumental variable.
• The proof is given in Morikawa and Kim (2017).

Identiﬁable Model
Example 1
• Suppose that
yi | xi , δi = 1 ∼ N(τ(xi ), σ2
).
Assume that xi is always observed but we observe yi only when δi = 1 where
δi ∼ Bernoulli [πi (φ)] and
πi (φ) =
exp (φ0 + φ1xi + φ2yi )
1 + exp (φ0 + φ1xi + φ2yi )
.
• Then, E1{O(φ, x, Y ) | x; γ0} = exp(−φ0 − φ1x − φ2τ(x) − φ2
2σ2
/2).
• Therefore, the model is identiﬁable unless τ(x) is constant or linear.

Parameter estimation under nonresponse nonresponse
• Full likelihood-based ML estimation
• Generalized method of moment (GMM) approach (Section 6.3 of KS)
• Conditional likelihood approach (Section 6.2 of KS)
• Pseudo likelihood approach (Section 6.4 of KS)
• Exponential tilting method (Section 6.5 of KS)
• Latent variable approach (Section 6.6 of KS)
Reference
Kim, J.K. and Shao, J. (2013). “Statistical Methods for Handling Incomplete Data”,
Chapman & Hall / CRC.

1 Introduction
4 GMM method

Full likelihood-based ML estimation
• Wish to find ˆη = (ˆθ, ˆφ), that maximizes the observed likelihood
Lobs (η) =
δi =1
f (yi | xi ; θ) g (δi | xi , yi ; φ)
×
δi =0
• Mean score theorem: Under some regularity conditions, finding the MLE by
maximizing the observed likelihood is equivalent to finding the solution to
¯S(η) ≡ E{S(η) | yobs , δ; η} = 0,
where yobs is the observed data. The conditional expectation of the score function
is called mean score function.

EM algorithm
• Interested in ﬁnding ˆη that maximizes Lobs (η). The MLE can be obtained by
solving Sobs (η) = 0, which is equivalent to solving ¯S(η) = 0 by the mean score
theorem.
• EM algorithm provides an alternative method of solving ¯S(η) = 0 by writing
¯S(η) = E {S(η) | yobs , δ; η}
and using the following iterative method:
ˆη(t+1)
← solve E S(η) | yobs , δ; ˆη(t)
= 0.

EM algorithm
Deﬁnition
Let η(t)
be the current value of the parameter estimate of η. The EM algorithm can be
deﬁned as iteratively carrying out the following E-step and M-steps:
• E-step: Compute
Q η | η(t)
= E ln f (y, δ; η) | yobs, δ, η(t)
• M-step: Find η(t+1)
that maximizes Q(η | η(t)
) w.r.t. η.

Monte Carlo EM
Motivation: Monte Carlo samples in the EM algorithm can be used as imputed values.
Monte Carlo EM
1 In the EM algorithm deﬁned by
• [E-step] Compute
Q η | η(t)
= E ln f (y, δ; η) | yobs, δ; η(t)
• [M-step] Find η(t+1)
that maximizes Q η | η(t)
,
E-step is computationally cumbersome because it involves integral.
2 Wei and Tanner (1990): In the E-step, ﬁrst draw
y
∗(1)
mis , · · · , y
∗(m)
mis ∼ f ymis | yobs, δ; η(t)
and approximate
Q η | η(t) ∼=
1
m
m
j=1
ln f yobs , y
∗(j)
mis , δ; η .

Monte Carlo EM
Example 2
• Suppose that
yi ∼ f (yi | xi ; θ)
Assume that xi is always observed but we observe yi only when δi = 1 where
δi ∼ Bernoulli [πi (φ)] and
πi (φ) =
exp (φ0 + φ1xi + φ2yi )
1 + exp (φ0 + φ1xi + φ2yi )
.
• To implement the MCEM method, in the E-step, we need to generate samples from
f (yi | xi , δi = 0; ˆθ, ˆφ) =
f (yi | xi ; ˆθ){1 − πi (ˆφ)}
f (yi | xi ; ˆθ){1 − πi (ˆφ)}dyi
.

Monte Carlo EM
Example 2 (Cont’d)
• We can use the following rejection method to generate samples from
f (yi | xi , δi = 0; ˆθ, ˆφ):
1 Generate y∗
i from f (yi | xi ; ˆθ).
2 Using y∗
i , compute
π∗
i (ˆφ) =
exp(ˆφ0 + ˆφ1xi + ˆφ2y∗
i )
1 + exp(ˆφ0 + ˆφ1xi + ˆφ2y∗
i )
.
Accept y∗
i with probability 1 − π∗
i (ˆφ).
3 If y∗
i is not accepted, then goto Step 1.

Monte Carlo EM
• Using the m imputed values of yi , denoted by y
∗(1)
i , · · · , y
∗(m)
i , and the M-step can
be implemented by solving
n
i=1
m
j=1
S θ; xi , y
∗(j)
i = 0
and
n
i=1
m
j=1
δi − π(φ; xi , y
∗(j)
i ) 1, xi , y
∗(j)
i = 0,
where S (θ; xi , yi ) = ∂ log f (yi | xi ; θ)/∂θ.

Remark
• Identiﬁability condition is needed to guarantee the convergence of EM sequence.
• The fully parametric model approach is known to be sensitive to the failure of
model assumptions: Little (1985), Kenward and Molenberghs (1988).
• Sensitivity analysis is often recommended: Scharfstein et al. (1999).

1 Introduction
4 GMM method

Partial Likelihood approach
• A classical likelihood-based approach for parameter estimation under non ignorable
nonresponse is to maximize Lobs (θ, φ) with respect to (θ, φ), where
Lobs (θ, φ) =
δi =1
f (yi | xi ; θ) g (δi | xi , yi ; φ)
×
δi =0
• Such approach can be called full likelihood-based approach because it uses full
information available in the observed data.
• On the other hand, partial likelihood-based approach (or conditional likelihood
approach) uses a subset of the sample.

Pseudo Likelihood approach
Idea
• Consider bivariate (xi , yi ) with density f (y | x; θ)h(x) where yi are subject to
missingness.
• We are interested in estimating θ.
• Suppose that Pr(δ = 1 | x, y) depends only on y. (i.e. x is nonresponse
instrument)
• Note that f (x | y, δ) = f (x | y).
• Thus, we can consider the following conditional likelihood
Lc (θ) =
δi =1
f (xi | yi , δi = 1) =
δi =1
f (xi | yi ).
• We can consider maximizing the pseudo likelihood
Lp(θ) =
δi =1
f (yi | xi ; θ)ˆh(xi )
f (yi | x; θ)ˆh(x)dx
,
where ˆh(x) is a consistent estimator of the marginal density of x.

Idea
• We may use the empirical density in ˆh(x). That is, ˆh(x) = 1/n if x = xi . In this
case,
Lc (θ) =
δi =1
f (yi | xi ; θ)
n
k=1 f (yi | xk ; θ)
.
• We can extend the idea to the case of x = (u, z) where z is a nonresponse
instrument. In this case, the conditional likelihood becomes
i:δi =1
p(zi | yi , ui ) =
i:δi =1
f (yi | ui , zi ; θ)p(zi |ui )
f (yi | ui , z; θ)p(z|ui )dz
. (3)

• Let ˆp(z|u) be an estimated conditional probability density of z given u. Substituting
this estimate into the likelihood in (3), we obtain the following pseudo likelihood:
i:δi =1
f (yi | ui , zi ; θ)ˆp(zi |ui )
f (yi | ui , z; θ)ˆp(z|ui )dz
. (4)
• The pseudo maximum likelihood estimator (PMLE) of θ, denoted by ˆθp, can be
obtained by solving
Sp(θ; ˆα) ≡
δi =1
[S(θ; xi , yi ) − E{S(θ; ui , z, yi ) | yi , ui ; θ, ˆα}] = 0
for θ, where S(θ; x, y) = S(θ; u, z, y) = ∂ log f (y | x; θ)/∂θ and
E{S(θ; ui , z, yi ) | yi , ui ; θ, ˆα} =
S(θ; ui , z, yi )f (yi | ui , z; θ)p(z | ui ; ˆα)dz
f (yi | ui , z; θ)p(z | ui ; ˆα)dz
.

• The Fisher-scoring method for obtaining the PMLE is given by
ˆθ(t+1)
p = ˆθ(t)
p + Ip
ˆθ(t)
, ˆα
−1
Sp(ˆθ(t)
, ˆα)
where
Ip (θ, ˆα) =
δi =1
E{S(θ; ui , z, yi )⊗2
| yi , ui ; θ, ˆα} − E{S(θ; ui , z, yi ) | yi , ui ; θ, ˆα}⊗2
.
• First considered by Tang et al. (2003) and further developed by Zhao and Shao
(2015).
• Model section with penalized validation criterion is proposed by Fang and Shao
(2016).

1 Introduction
4 GMM method

Basic setup
• (X, Y ): random variable
• θ: Deﬁned by solving
E{U(θ; X, Y )} = 0.
• yi is subject to missingness
δi =
1 if yi responds
0 if yi is missing.
• Want to ﬁnd wi such that the solution ˆθw to
n
i=1
δi wi U(θ; xi , yi ) = 0
is consistent for θ.

Basic Setup
• Result 1: The choice of
wi =
1
E(δi | xi , yi )
(5)
makes the resulting estimator ˆθw consistent.
• Result 2: If δi ∼ Bernoulli(πi ), then using wi = 1/πi also makes the resulting
estimator consistent, but it is less eﬃcient than ˆθw using wi in (5).

Parameter estimation : GMM method
• Because z is a nonresponse instrumental variable, we may assume
P(δ = 1 | x, y) = π(φ0 + φ1u + φ2y)
for some (φ0, φ1, φ2).
• Kott and Chang (2010): Construct a set of estimating equations such as
n
i=1
δi
π(φ0 + φ1ui + φ2yi )
− 1 (1, ui , zi ) = 0
that are unbiased to zero.
• May have overidentiﬁed situation: Use the generalized method of moments
(GMM).

GMM method
Example 3
• Suppose that we are interested in estimating the parameters in the regression
model
yi = β0 + β1x1i + β2x2i + ei (6)
where E(ei | xi ) = 0.
• Assume that yi is subject to missingness and assume that
P(δi = 1 | x1i , xi2, yi ) =
exp(φ0 + φ1x1i + φ2yi )
1 + exp(φ0 + φ1x1i + φ2yi )
.
Thus, x2i is the nonresponse instrument variable in this setup.

• A consistent estimator of φ can be obtained by solving
Û2(φ) ≡
n
i=1
δ
π(φ; x1i , yi )
− 1 (1, x1i , x2i ) = (0, 0, 0). (7)
Roughly speaking, the solution to (7) exists almost surely if E{∂ Û2(φ)/∂φ} is of
full rank in the neighborhood of the true value of φ. If x2 is vector, then (7) is
overidentified and the solution to (7) does not exist. In the case, the GMM
algorithm can be used.
• Finding the solution to Û2(φ) = 0 can be obtained by finding the minimizer of
Q(φ) = Û2(φ) Û2(φ) or QW (φ) = Û2(φ) W Û2(φ) where W = {V ( Û2)}−1
.

• Once the solution ˆφ to (7) is obtained, then a consistent estimator of
β = (β0, β1, β2) can be obtained by solving
ˆU1(β, ˆφ) ≡
n
i=1
δi
ˆπi
{yi − β0 − β1x1i − β2x2i } (1, x1i , x2i ) = (0, 0, 0) (8)
for β.

Asymptotic Properties
• The asymptotic variance of ˆβ obtained from (8) with ˆφ computed from the GMM
can be obtained by
V (ˆθ) ∼= ΓaΣ−1
a Γa
−1
(9)
where
Γa = E{∂ Û(θ)/∂θ}
Σa = V ( Û)
Û = ( Û1, Û2)
and θ = (β, φ).
• Rigorous theory developed by Wang et al. (2014).

1 Introduction
4 GMM method

Exponential tilting method
Motivation
• Parameter θ deﬁned by E{U(θ; X, Y )} = 0.
• We are interested in estimating θ from an expected estimating equation:
n
i=1
[δi U(θ; xi , yi ) + (1 − δi )E{U(θ; xi , Y ) | xi , δi = 0}] = 0. (10)
• The conditional expectation in (10) can be evaluated by using
f (y|x, δ = 0) = f (y|x)
P(δ = 0|x, y)
E{P(δ = 0|x, y)|x}
(11)
which requires correct speciﬁcation of f (y | x; θ). Known to be sensitive to the
choice of f (y | x; θ).

Exponential tilting method
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,
f0 (yi | xi ) = f1 (yi | xi ) ×
O (xi , yi )
E {O (xi , Yi ) | xi , δi = 1}
, (12)
where fδ (yi | xi ) = f (yi | xi , δi = δ) and
O (xi , yi ) =
Pr (δi = 0 | xi , yi )
Pr (δi = 1 | xi , yi )
(13)
is the conditional odds of nonresponse.

Remark
• If the response probability follows from a logistic regression model
π(xi , yi ) ≡ Pr (δi = 1 | xi , yi ) =
exp {g(xi ) + φyi }
1 + exp {g(xi ) + φyi }
, (14)
where g(x) is completely unspeciﬁed, the expression (12) can be simpliﬁed to
f0 (yi | xi ) = f1 (yi | xi ) ×
exp (γyi )
E {exp (γY ) | xi , δi = 1}
, (15)
where γ = −φ and f1 (y | x) is the conditional density of y given x and δ = 1.
• Model (15) states that the density for the nonrespondents is an exponential tilting
of the density for the respondents. The parameter γ is the tilting parameter that
determines the amount of departure from the ignorability of the response
mechanism. If γ = 0, the the response mechanism is ignorable and
f0(y|x) = f1(y|x).
• Kim and Yu (2011) proposed a method to estimate γ with validation sample.
• Shao and Wang (2016) developed a method to estimate γ without using the
validation sample.

Maximum Likelihood Estimation of tilting parameter
• We wish to solve
n
i=1
[δi Si (φ) + (1 − δi )E{Si (φ) | xi , δi = 0}] = 0
with respect to φ, where Si (φ) is the score function of φ evaluated at (xi , yi ).
• The conditional expectation can be written as
E{Si (φ) | xi , δi = 0} =
Si (φ)O(xi , y; φ)f1(y | xi )dy
O(xi , y; φ)f1(y | xi )dy
where
O(x, y; φ) =
1 − π(φ; x, y)
π(φ; x, y)
.
• Thus,
E{Si (φ) | xi , δi = 0} =
E1{Si (φ)O(xi,Y ; φ) | xi }
E1{O(xi,Y ; φ) | xi }
where
E1{Q(xi , Y ) | xi } = Q(xi , y)f1(y | xi )dy.

Riddles et al. (2016) method
• In practice, f1(y | x) is unknown and is estimated by ˆf1(y | x) = f1(y | x; ˆγ).
• Thus, given ˆγ, a fully eﬃcient estimator of φ can be obtained by solving
S2(φ, ˆγ) ≡
n
i=1
δi S(φ; xi , yi ) + (1 − δi ) ¯S0(φ | xi ; ˆγ, φ) = 0, (16)
where
¯S0(φ | xi ; ˆγ, φ) =
δj =1 S(φ; xi , yj )O(φ; xi , yj )f1(yj | xi ; ˆγ)/ˆf1(yj )
δj =1 O(φ; xi , yj )f1(yj | xi ; ˆγ)/ˆf1(yj )
and
ˆf1(y) = n−1
R
n
i=1
δi f1(y | xi ; ˆγ).
• Use EM algorithm to solve (16) for φ.
• Semiparametric extension (Morikawa et al., 2017): Use a nonparametric density for
f1(y | x).

A Toy Example: Categorical Data (All dichotomous)
Example (SRS, n = 10)
ID Weight x1 x2 y
1 0.1 1 0 1
2 0.1 1 1 1
3 0.1 0 1 M
4 0.1 1 0 0
5 0.1 0 1 1
6 0.1 1 0 M
7 0.1 0 1 M
8 0.1 1 0 0
9 0.1 0 0 0
10 0.1 1 1 0
M: Missing

A Toy 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 = ˆP(Y = j | X1 = 0, X2 = 1, δ = 0)
∝ ˆP(Y = j | X1 = 0, X2 = 1, δ = 1)
1 − ˆπ(0, j)
ˆπ(0, j)
with
w3,0 + w3,1 = 1

ID Weight x1 x2 y
6 0.1 · w6,0 1 0 0
0.1 · w6,1 1 0 1
7 0.1 · w7,0 0 1 0
0.1 · w7,1 0 1 1
8 0.1 1 0 0
9 0.1 0 0 0
10 0.1 1 1 0
w6,j ∝ ˆP(Y = j | X1 = 1, X2 = 0, δ = 1)
1 − ˆπ(1, j)
ˆπ(1, j)
w7,j ∝ ˆP(Y = j | X1 = 0, X2 = 1, δ = 1)
1 − ˆπ(0, j)
ˆπ(0, j)
with
w6,0 + w6,1 = w7,0 + w7,1 = 1.

Example (Cont’d)
• E-step: Compute the conditional probability using the estimated response
probability ˆπab.
• M-step: Update the response probability using the fractional weights. For fully
nonparametric model,
ˆπab =
δi =1 I(x1i = a, yi = b)
δi =1 I(x1i = a, yi = b) + δi =0
1
j=0 wi,j I(x1i = a, y∗
ij = b)
• The solution from the proposed method is ˆπ11 = 1, ˆπ10 = 3/4, ˆπ01 = 1/3, ˆπ00 = 1.

Example (Cont’d)
• The method can be viewed as a fractional imputation method of Kim (2011).
• On the other hand, GMM method is more close to nonresponse weighting
adjustment.

Example GMM method
ID Wgt 1 Wgt2 x1 x2 y
1 0.1 0.1ˆπ−1
11 1 0 1
2 0.1 0.1ˆπ−1
11 1 1 1
3 0.1 0.0 0 1 M
4 0.1 0.1ˆπ−1
10 1 0 0
5 0.1 0.1ˆπ−1
01 0 1 1
6 0.1 0.0 1 0 M
7 0.1 0.0 0 1 M
8 0.1 0.1ˆπ−1
10 1 0 0
9 0.1 0.1ˆπ−1
00 0 0 0
10 0.1 0.1ˆπ−1
10 1 1 0
M: Missing

• GMM method: Calibration equation
i
δi
ˆπi
I(x1i = a, x2i = b) =
i
I(x1i = a, x2i = b).
1 X1 = 1, X2 = 1: ˆπ−1
11 + ˆπ−1
10 = 2
2 X1 = 1, X2 = 0: ˆπ−1
11 + ˆπ−1
10 + ˆπ−1
10 = 4
3 X1 = 0, X2 = 1: ˆπ−1
01 = 3
4 X1 = 0, X2 = 0: ˆπ−1
00 = 1.
• The solution of GMM method does not exist.

1 Introduction
4 GMM method

Setup
Motivation
• For simplicity, assume that θ = E(Y ) is of our interest.
• A response model is assumed P(δ = 1 | x, y; φ) = π(φ; x, y).
• Therefore, our restriction is only on the response model (semiparametric model).
• What is the best way to estimate (φ, θ) in terms of the asymptotic variance?

Semiparametric Efficient Estimation
Morikawa and Kim (2017): The solution of the following estimating equations achieves
the semiparametric efficiency bound for (φ, θ):
n
i=1
1 −
δi
π(φ; xi , yi )
geff (φ; xi ) = 0,
n
i=1
δi (θ − yi )
π(φ; xi , yi )
+ 1 −
δi
π(φ; xi , yi )
(θ − Y (xi )) = 0,
(17)
where, geff (x; φ) = E { ˙π(φ; x, Y )/{1 − π(φ; x, Y )} | x}, ˙π(φ) = ∂π(φ)/∂φ,
Y (x) = E (Y | x),
E {g(x, Y ) | x} =
E{O(x, Y )g(x, Y ) | x}
E{O(x, Y ) | x}
,
and O(x, y) = 1/π(x, y) − 1.

• The two functions geﬀ (φ; x) and Y (x) in (17) are to be estimated.
• By using Bayes’s formula, for any function g(x, y), we have
E {g(x, Y ) | x} =
E{O(x, Y )g(x, Y ) | x}
E{O(x, Y ) | x}
=
E1{π−1
(x, Y )O(x, Y )g(x, Y ) | x}
E1{π−1(x, Y )O(x, Y ) | x}
, (18)
where E1(· | x) := E(· | x, δ = 1).
• Hence, if the distribution of f1(y | x) := f (y | x, δ = 1) is speciﬁed, (18) can be
computed; thus, the estimating equations (17) can be solved.

• Morikawa and Kim (2017) proposed two types of semiparametric estimators.
• First estimator uses a parametric f1(y | x; γ) model and solve (17) with estimated
ˆγ. The estimator has consistency and asymptotic normality even if the
specification of f1 is wrong. Also, if the specification is correct, it attains the
semiparametric efficiency bound.
• Second estimator uses a nonparametrc model on f1 model and solve (17). The
estimator always attains the semiparametric efficiency bound, but it does not work
well when the dimension of x is high.

Remark
• The concept of the semiparametric eﬃciency bound and the semiparametric theory
are rigorously developed: Bickel et al. (1998).
• Extension to parameters which may restrict the data generation such as the
regression parameter: Robins et al. (1999).
• The estimator for φ in (17) is a special case of Kott and Chang (2010): set
geﬀ (φ; xi ) as the calibration condition.

1 Introduction
4 GMM method

Concluding remarks
• Uses a model for the response probability.
• Parameter estimation for response model can be implemented using the idea of
maximum likelihood method or method of moments.
• Instrumental variable or model restriction on f1 is needed for identiﬁability of the
response model.
• Less tools for model diagnostics or model validation

REFERENCES
Bickel, P. J., C. A. J. Klaassen, Y. Ritov, and J. A. Wellner (1998). Eﬃcient and
Adaptive Estimation for Semiparametric Models. Springer.
Fang, F. and J. Shao (2016). Model selection with nonignorable nonresponse.
Biometrika 103(1), 1–14.
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