ESTIMATION OF A LOGISTIC REGRESSION MODEL WITH MISMEASURED OBSERVATIONS
K. F. Cheng and H. M. Hsueh
National Central University and National Chengchi University
Abstract: We consider the estimation problem of a logistic regression model. We assume the response observations and covariate values are both subject to measure-ment errors. We discuss some parametric and semiparametric estimation methods using mismeasured observations with validation data and derive their asypmtotic distributions. Our results are extentions of some well known results in the litera-ture. Comparisons of the asymptotic covariance matrices of the studied estimators are made, and some lower and upper bounds for the asymptotic relative efficiencies are given to show the advantages of the semiparametric method. Some simulation results also show the method performs well.
Key words and phrases: Kernel estimation, estimated likelihood, logistic regression, measurement error, misclassification.
1. Introduction
Logistic regression is the most popular form of binary regression; see Cox (1970) and Pregibon (1981). Researchers often use logistic regression to estimate the effect of various predictors on some binary outcome of interest. Basically, the model assumes that the log of the odds of the outcome is a linear function of the predictors. That is, suppose that the variables (Xi, Yi) follow the model
P r(Yi = 1 | Xi = xi) = exp(xTi β0) {1 + exp(xT i β0)} ≡ F (x T i β0),
where the Xi are random d-vector predictors and the Yi are Bernoulli response
variables. Usually, the maximum likelihood (ML) method is used to estimate the regression coefficients β0. Under some regularity conditions, the maximum likelihood estimator of β0 is asymptotically normal.
The ML method requires that the data consist of precise measurements for the binary outcomes and predictors. However, the data are often not measured perfectly. For instance, Golm, Holloran and Longini (1998, 1999) mentioned that collecting information on exposure to infection for estimating vaccine efficacy may be mismeasured. On the other hand, Albert, Hunsberger and Bird (1997) and Bollinger and David (1997) gave examples showing that the binary outcome of
interest may also be misclassified. It is generally true that the usual analyses based on the mismeasured observations lead to inconsistent estimation.
The topic of binary regression when predictors Xi are measured with error
has been the subject of several recent papers, see Carroll, Spiegelman, Lan, Bai-ley and Abbott (1984), Carroll and Wand (1991), Reilly and Pepe (1995), and Lawless, Kalbfleisch and Wild(1999), etc. When the binary responses Yiare
sub-ject to misclassification, Pepe(1992) and Cheng and Hsueh (1999) discussed bias correction methods in the estimation of logistic regression parameters. In this paper, we study the estimation problem of β0when both observations Yiand
pre-dictors Xi are measured with error. Parametric and semiparametric methods are
discussed. We find that the proposed semiparametric estimation method is a gen-eralization of the pseudolikelihood method of Carroll and Wand (1991) and the estimated likelihood method of Pepe (1992). Many other estimating approaches have been proposed in the literature. A method based on the mean score was proposed by Pepe, Reilly and Fleming (1994) and Reilly and Pepe (1995) for the mismeasured outcome data problem and mismeasured covariate data prob-lem, respectively. However, Golm et al. (1999) and Lawless et al. (1999) argued that this semiparametric method is less efficient. Robins, Rotnitzky and Zhao (1994) proposed a class of semiparametric efficient estimators for the model with mismeasured predictors based on the inverse probability weighted estimating equation approach. When both response variable and predictors are subject to measurement error, the semiparametric efficient estimator has not been formally derived yet. In this paper we only emphasize various imputation approaches. The weighting methods, such as the semiparametric efficient estimation derived by Robins et al. (1994), will not be discussed further.
We assume in this paper that the complete data set consists of a primary sample plus a smaller validation subsample which is obtained by double sampling scheme. Extension of the selection probabilities for the validation data set to depend on the observed surrogate covariates is discussed briefly in Section 3. Asymptotically, we also suppose that the validation subsample size is a fraction of the major sample size. The estimators under discussion will be formally defined in Section 2. In Section 3, some asymptotic results are derived for semiparametric and parametric estimators of β0. Comparisons of their asymptotic covariance matrices are given in Section 4. Further, finite sample properties are explored through a simulation study in Section 5.
2. Estimation Methods
Suppose the true random variables (Y, X) are subject to mismeasurement and the surrogate observations are represented by (Y0, W ), here X = (1, X1)T
a smaller validation subsample is also observed in order to understand the mis-measurement structure. The sampling scheme is to randomly select k units from the primary sample and at the selected units the true measurement devices are used to obtain the validation data. Here the simple random sampling scheme is considered and for the sake of simplicity, the first k units of the primary sample are assumed to be the selected units. Thus we have the validation subsample {(Yi0, Yi, Xi, Wi), i = 1, . . . , k}. Later in Section 3 the theoretical results will be
given for the case that the sampling scheme depends on the value of the surrogate predictors W .
From the primary sample, we denote the regression function of Y0 on W = w
as π0(w) = P r(Y0 = 1 | w). For Y and W being assumed mutually
inde-pendent given X, and Y0 and X being mutually independent given Y, W , the
above regression function can be rewritten as π0(w) = π(w, β0), where π(w, β) =
{1 − θ0(w)}E{F (XTβ) | w} + φ0(w)E{ ¯F (XTβ) | w} ≡ 1 − ¯π(w, β). Here the ex-pectation is taken with respect to the conditional density fX|W0 , ¯F (·) = 1 − F (·), and the misclassification probability functions θ0(w) and φ0(w) are defined by
θ0(w) = P r(Y0 = 0 | Y = 1, w) and φ0(w) = P r(Y0 = 1 | Y = 0, w). Note that
the surrogate W is often a fallible measurement of X and corresponding to a coarser partition in the sample space. The conditional independence of Y0 and
X given Y , W would be implied if the misclassification scheme depends on X only through the value of W . It is clear that the expectation E[{Y0− F (WTβ0)}WT]
is not necessarily zero. Hence it is inappropriate to apply the usual likelihood method to the primary sample for inference about β0. In the following, we discuss some consistent estimates based on different imputation methods.
Suppose first that fX|W0 and θ0(w), φ0(w) are known a priori. Then for the observation (y0
j, wj), j = k + 1, . . . , n, in the primary sample, the corresponding
log-likelihood and score function are
L∗(β | y0j, wj)=yj0ln π(wj, β) + (1 − y0j) ln ¯π(wj, β), S∗(β | y0j, wj)≡ ∂L∗(β | y0j, wj) ∂β = y 0 j−π(wj, β) π(wj, β)¯π(wj, β){1−θ 0(w j)−φ0(wj)}E{F (XjTβ) ¯F (XjTβ)Xj | wj}. Since, y0 j − π(wj, β) π(wj, β)¯π(wj, β) = 2y 0 j − 1 yj0π(wj, β) + (1 − y0j)¯π(wj, β) , 1 − θ0(wj) − φ0(wj) = {1 − θ0(wj) − π(wj, β)} E{F (X T j β) | wj} E{F (XT j β) | wj}E{ ¯F (XjTβ) | wj} ,
we have y0 j − π(wj, β) π(wj, β)¯π(wj, β){1 − θ 0(w j) − φ0(wj)} = E(Yj | yj0, wj) − E{F (XjTβ) | wj} E{F (XT j β) | wj}E{ ¯F (XjTβ) | wj} . Consequently, S∗(β | yj0, wj) = E(Yj | y0j, wj) − E{F (XjTβ) | wj} E{F (XT jβ) | wj}E{ ¯F (XjTβ) | wj} EnF (XjTβ) ¯F (XjTβ)Xj | wj o = {A 0 1(y0j, wj) − A02(y0j, wj)}E{F (XjTβ) ¯F (XjTβ)Xj | wj} h A0 1(y0j, wj)E{F (XjTβ) | wj} + A20(y0j, wj)E{ ¯F (XjTβ) | wj} i,
where A01(y0j, wj) = P r(Yj0 = yj0 | Yj = 1, wj) = θ0(wj) (1−y
0 j) 1 − θ0(wj) y 0 j , and A02(y0j, wj) = P r(Yj0 = yj0 | Yj = 0, wj) = φ0(wj) y 0 j 1 − φ0(wj) (1−y 0 j). Using this result, we easily see that the MLE ˆβf of β0can be obtained by solving
the likelihood equations
k X i=1 S(β | yi, xi) + n X j=k+1 S∗(β | y0j, wj) = 0, (1) where S(β | yi, xi) = {yi− F (xTi β)}xi. Unfortunately, in applications f0
X|W(·), θ0(·) and φ0(·) are rarely known and
ˆ
βf can not be obtained. If the subsample size k is large enough, a simple
esti-mate ˆβscan be obtained by using only the validation subsample, i.e., ˆβs satisfies
Pk
i=1S(β | yi, xi) = 0. Such a simple estimate is in general not efficient; see
Cheng and Hsueh (1999). The basic reason is that the information contained in the primary sample is not properly exploited. To do this, assume there ex-ist parametric functions and parameters (γ0, α0), independent of β0, such that θ0(W ) = θ(W ; α0), φ0(W ) = φ(W ; α0) and fX|W0 (X | W ) = f(X | W ; γ0) al-most surely. See Carroll and Wand (1991) and Cheng and Hsueh (1999) for related discussions. Then one can employ the usual approach to obtain the joint MLE ( ˆβm, ˆαm, ˆγm). On the other hand, the maximum pseudolikelihood estimate
(MPLE) ˆβp can be obtained by solving (1) with α0 and γ0 in S∗(β | yj0, wj) being
replaced by their MLE ˆαp and ˆγp derived from the validation data.
In general, ˆβm can be shown to be more efficient than ˆβp asymptotically.
However, in solving the equations for finite sample cases, the computational com-plexity grows (and numerical stability deteriorates) with the number of unknown parameters, so that ˆβm may not have proper performance, see the simulation
The last approach to be discussed is a semiparametric method. Assuming fX|W0 (· | w), θ0(w) and φ0(w) in S∗(β | y0j, wj) of (1) are unknown functions,
a nonparametric method is used to estimate them. Formally, we propose the MPLE ˆβsp solving the following pseudolikelihood equations
k X i=1 S(β | yi, xi) + n X j=k+1 ˆ S∗(β | yj0, wj) = 0, ˆ S∗(β | y0, w) = { ˆA1(y 0, w) − ˆA 2(y0, w)} ˆE[F (XTβ) ¯F (XTβ)X | w] h ˆ A1(y0, w) ˆE{F (XTβ) | w} + ˆA2(y0, w) ˆE{ ¯F (XTβ) | w} i.
Here, for any integrable function g(x; β), the estimated expectation ˆE{g(X; β) | w} is given by ˆE{g(X; β) | w} = {Pk
i=1Kh(w − wi)g(xi; β)}/{Pki=1Kh(w −
wi)}. Moreover ˆA1(y0, w) and ˆA2(y0, w) are estimates of A01(y0, w) and A02(y0, w),
with θ0(w) and φ0(w) being replaced by their nonparametric estimates ˆθ(w) = {Pk
i=1Kh(w − wi)yi(1 − yi0)}/{
Pk
i=1Kh(w − wi)yi} and ˆφ(w) = {Pki=1Kh(w −
wi)(1 − yi)yi0}/{Pki=1Kh(w − wi)(1 − yi)}. The above kernel function K(t) is
taken to be a density function, Kh(t) = h−1K(t/h), the bandwidth h depends
on n and tends to zero as n → ∞.
We remark that the proposed semiparametric estimation method generalizes the methods of Carroll and Wand (1991) and Pepe (1992). If Y = Y0 with prob-ability one, i.e., only measurement error occurs in the predictor, ˆβsp is the MPLE
of Carroll and Wand (1991). On the other hand, if X = W with probability one, i.e., only misclassification occurs in the response, ˆβsp reduces to the maximum
estimated likelihood estimate of Pepe (1992). 3. Asymptotic Distributions
The asymptotic distributions of the estimators proposed in the preceding section will be presented here for the case d = 1. Extension to general d is simple and the related results will be given in a remark. The asymptotic properties depend on certain regularity conditions.
A.1 β0 ∈ Λ, an open set in R2.
A.2 E(X2) < ∞.
A.3 The misclassification probability functions θ0(w) and φ0(w) ∈ (0, 1), and
the density function fW0 (w) of W are strictly positive in the space of W . Furthermore, these functions and their l-th derivatives, l = 1, 2, are in L4
and also satisfy a weighted Lipschitz condition. (A function η(·) is said to satisfy a weighted Lipschitz condition if there exists a constant c and a bounded function ψ in L4 such that |η(x) − η(y)| < ψ(x)|x − y| for all x, y
A.4 The parametric functions θ(w; α),φ(w; α) and f (x | w; γ), and their partial derivatives ∂θ(w; α)/∂α, ∂φ(w; α)/∂α and ∂f (x | w; γ)/∂γ, are uniformly continuous at (α0, γ0) for all x and w in the support of X and W .
A.5 The function K(·) is a second-order kernel (see Gasser and M¨uller (1979)). A.6 h = hn → 0 with nh2 → ∞ and nh4 → 0, as n → ∞. Also, as n → ∞,
k = rn{1 + O(h2)}, where r ∈ (0, 1].
In the following, we give the basic asymptotic results under the assumption that α = (α0, α1)T and γ = (γ0, γ1)T. Except for the asymptotic normality of
ˆ
βsp, whose proof is given in the Appendix, the proofs for other estimators are
standard and thus are omitted.
Theorem 1. Suppose conditions (A.1)−(A.6) hold and let ˜β be any estimator discussed in Section 2. Then as n → ∞, n1/2( ˜β − β0) converges in distribution
to a normal with mean zero. Their asymptotic covariance matrices are lim n→∞Cov {n 1/2( ˆβ f − β0)} = Σf = {rIv+ (1 − r)Ivc}−1, lim n→∞Cov {n 1/2( ˆβ s− β0)} = Σs= (rIv)−1, lim n→∞Cov {n 1/2( ˆβ m− β0)} = Σm= Σf +(1 − r) 2 r ΣfImΣf, lim n→∞Cov {n 1/2( ˆβ p− β0)} = Σp= Σf+ (1 − r) 2 r ΣfIpΣf, lim n→∞Cov {n 1/2( ˆβ sp− β0)} = Σsp= Σf +(1 − r) 2 r ΣfIspΣf. Here Iv = E{F (XTβ0) ¯F (XTβ0)X⊗2} and
Ivc= E {1 − θ0(W ) − φ0(W )}2hE{F (XTβ0) ¯F (XTβ0)X | W }i⊗2 π0(W ){1 − π0(W )} , Ip= E {1−θ 0(W )−φ0(W )}E{F (XTβ0) ¯F (XTβ0)X| W } π0(W ){1 − π0(W )} ˙π0α(W ; α0, γ0) ˙πγ0(W ; α0, γ0) !T × E F (XTβ0 ) ˙θ⊗2(W ;α0 ) θ0 (W ){1−θ0 (W )} + ¯ F (XTβ0 ) ˙φ⊗2(W ;α0 ) φ0 (W ){1−φ0 (W )} 0 0 E{ ˙f (X | W ; γ0)}⊗2 −1 ×E {1−θ 0(W )−φ0(W )}E{F (XTβ0) ¯F (XTβ0)X| W } π0(W ){1 − π0(W )} ˙π0α(W ; α0, γ0) ˙πγ0(W ; α0, γ0) !T T ,
Isp= E {1 − θ0(W ) − φ0(W )}hE{F (XTβ0) ¯F (XTβ0)X | W }i⊗2 [π0(W ){1 − π0(W )}]2 ×E{F (XTβ0)| W }θ0(W ){1−θ0(W )}+E{ ¯F (XTβ0)| W }φ0(W ){1−φ0(W )} + {1 − θ0(W ) − φ0(W )}2[E{F2(XTβ0) | W } − E2{F (XTβ0) | W }]o, Im= r (1 − r)2∆(β0,α0,γ0){∆(α0,γ0)− ∆ T (β0 ,α0 ,γ0 )Σf∆(β0 ,α0 ,γ0 )}−1∆T(β0 ,α0 ,γ0 ), ∆(β0 ,α0 ,γ0 )= (1 − r)E " {1 − θ0(W ) − φ0(W )}E{F (XTβ0) ¯F (XTβ0)X | W } π0(W ){1 − π0(W )} × ˙π 0 α(W ; α0, γ0) ˙π0 γ(W ; α0, γ0) !T # , ∆(α0,γ0)= r E F (XTβ0 ) ˙θ⊗2(W ;α0 ) θ0 (W ){1−θ0 (W )} + ¯ F (XTβ0 ) ˙φ⊗2(W ;α0 ) φ0 (W ){1−φ0 (W )} 0 0 E{ ˙f (X | W ; γ0)}⊗2 +(1 − r)E 1 π0(W ){1 − π0(W )} ˙π0α(W ; α0, γ0) ˙π0 γ(W ; α0, γ0) !⊗2 . In this, ˙πα(W ; α0, γ0) = ∂π0(W ) ∂α (α0 ,γ0 ) = E{ ¯F (XTβ0) | W } ˙φ(W ; α0) − E{F (XTβ0) | W } ˙θ(W ; α0), ˙πγ(W ; α0, γ0) = ∂π 0(W ) ∂γ (α0,γ0) = {1 − θ0(W ) − φ0(W )}E{F (XTβ0) ˙f (X | W ; γ0) | W }, ˙θ(W ; α0) = ∂θ(W ; α) ∂α α0 , φ(W ; α˙ 0) = ∂φ(W ; α) ∂α α0 , ˙ f(X | W ; γ0) = ∂ ln f (X | W ; γ) ∂γ γ0 , where, for any column vector A, A⊗2= AAT.
Remark 1. It can be seen that Im, Ip and Isp are nonnegative definite matrices,
and thus (1−r)r 2ΣfImΣf, (1−r)
2
r ΣfIpΣf, (1−r)
2
addi-tional variations due to estimating the unknown functions θ0(W ), φ0(W ) and fX|W0 by parametric and nonparametric methods.
Remark 2. It is well known that the information matrix Iv is the variance
of the score; that is, Iv = Var {S(β0 | Y, X)} = E[Var {S(β0 | Y, X) | X}].
We see that the information matrices Ic
v and Isp also have similar expressions:
Ivc = E[Var {S∗(β0 | Y0, W ) | W }] = E[ρ(Y, Y0 | W )Var {S∗(β0 | Y, W ) | W }],
and Isp = E(ρ(Y, Y0 | W )[1−ρ(Y, Y0| W ){1−ρ(Y, F (XTβ0) | W )}]Var {S∗(β0|
Y, W ) | W }), where S∗(β | Y, W ) = ∂ ln P r(Y | W )/∂β, and ρ(Y, Y0 | W ) =
Corr2(Y, Y0 | W ) = {{1 − θ0(W ) − φ0(W )}2E{F (XTβ0) | W }E{ ¯F (XTβ0) |
W }}/[π0(W ){1 − π0(W )}],
ρ(Y, F (XTβ0) | W ) = Corr2(Y, F (XTβ0) | W ) =E{F
2(XTβ0) | W } − [E{F (XTβ0) | W }]2
E{F (XTβ0) | W }E{ ¯F (XTβ0) | W } .
Thus these information matrices depend not only on the score S∗(β0 | Y, W ) but
also on the squared correlation functions.
Remark 3. For obtaining the validation subsample, a simple random sampling design is used. However, the selection probabilities may depend on the value of W . For example, suppose g(w) is the selection probability. Then the asymptotic covariance matrix of ˆβsp becomes Σ∗sp= Σ∗+ Σ∗Isp∗Σ∗, where
Σ∗−1=Ehg(W )E{F (XTβ0) ¯F (XTβ0)X⊗2 | W }i +E{1−g(W )}{1−θ 0(W )−φ0(W )}2hE{F (XTβ0) ¯F (XTβ0)X| W }i⊗2 π0(W ){1 − π0(W )} , Isp∗ =En{1−g(W )} 2 g(W ) {1−θ0(W )−φ0(W )}hE{F (XTβ0) ¯F (XTβ0)X | W }i⊗2 [π0(W ){1 − π0(W )}]2 ×E{F (XTβ0)| W }θ0(W ){1−θ0(W )}+E{ ¯F (XTβ0)| W }φ0(W ){1−φ0(W )} +{1 − θ0(W ) − φ0(W )}2[E{F2(XTβ0) | W } − E2{F (XTβ0) | W }]o. Other asymptotic covariance matrices in Theorem 1 can be modified accordingly. Remark 4. It is clear that Σsp reduces to the asymptotic covariance matrix
of Carroll and Wand’s (1991) estimator provided Y = Y0 with probability one. Suppose X = W with probability one, then Σsp is the asymptotic covariance
Remark 5. The results in Theorem 1 can be extended to d > 1 vector predic-tors, provided the function K is a pth order kernel, p > d, and the bandwidth parameter satisfies nh2d→ ∞ and nh2p→ 0 as n → ∞.
Remark 6. All the asymptotic covariance matrices can be estimated by moment type estimates.
4. Comparisons of Asymptotic Covariance Matrices
In this section, we discuss the behaviors of different estimators by comparing their asymptotic covariance matrices. The general results basically agree with our expectation, but some of their proofs are not trivial. For any matrices A and B, we write A ≥ B if A − B is a nonnegative definite matrix.
4.1. Results under general conditions
First, from Remark 1 of Section 3, we have Σm ≥ Σf, Σp ≥ Σf and
Σsp ≥ Σf. Further, suppose we let {∆v(α0
,γ0
)}−1 be the asymptotic covariance
matrix of the MLE (ˆαp, ˆγp) which is obtained from the validation subsample.
Then one can rewrite Σp as Σp = Σf[Σ−1f + ∆(β0,α0,γ0){∆v
(α0,γ0)}−1∆(βT 0,α0,γ0)]Σf, and consequently we have
Σp− Σm =Σf∆(β0 ,α0 ,γ0 ) h {∆v(α0 ,γ0 )}−1−{∆(α0 ,γ0 )−∆T(β0 ,α0 ,γ0 )Σf∆(β0 ,α0 ,γ0 )}−1 i ∆T(β0 ,α0 ,γ0 )Σf. Since {∆v(α0 ,γ0 )}−1 and {∆(α0 ,γ0 )− ∆T(β0 ,α0 ,γ0 )Σf∆(β0 ,α0 ,γ0 )}−1 are respectively,
the asymptotic covariance matrices of (ˆαp, ˆγp) and (ˆαm, ˆγm), the above results
yield the following theorem.
Theorem 2. Given conditions (A.1)−(A.6), we have Σs ≥ Σf, Σsp ≥ Σf and
Σp≥ Σm≥ Σf.
4.2. Results under special constraints
Cheng and Hsueh (1999) compared the asymptotic covariance matrices when X = W with probability one. Here the comparisons focus on the case when Y = Y0 with probability one.
As a consequence of Theorem 1, if Y = Y0 with probability one, the matrices
Σf and Isp simplify to Σf = h rE{F (XTβ0) ¯F (XTβ0)X⊗2} +(1 − r)E [E{F (X Tβ0) ¯F (XTβ0)X | W }]⊗2 E{F (XTβ0) | W }E{ ¯F (XTβ0) | W } i−1 ,
Isp= E
ρ(Y, F (XTβ0) | W )Eh{S∗(β0 | Y, W )}⊗2 | W
i
. Then Corollary 1 follows easily from
Σs− Σsp= Σf (1 − r)E{S∗(β0| Y, W )}⊗2 −(1 − r) 2 r E ρ(Y, F (XTβ0) | W )E[{S∗(β0| Y, W )}⊗2 | W ] +(1−r) 2 r E{S∗(β 0 | Y, W )}⊗2hE{S(β0| Y, X)}⊗2i−1E{S∗(β0| Y, W )}⊗2 ! Σf.
Corollary 1. Suppose conditions (A.1)−(A.6) are satisfied, that Y = Y0 with probability one, and ρ(Y, F (XTβ0) | W ) ≤ min(r/(1 − r), 1). Then Σs≥ Σsp.
Accordingly, the semiparametric estimator ˆβspis always better than the
sim-ple MLE ˆβsunder our conditions. Similar conclusion can be derived for the case
that X = W with probability one. Now suppose β0 is a scalar. Then under the conditions of Theorem 1 and ρ(Y, F (XTβ0) | W ) ≤ ρ∗, with probability one for
some constant ρ∗, the ARE of ˆβs with respect to ˆβsp is always smaller than
e1(ρ∗, I, r) = 1 −(1 − r)
2I[I − {ρ∗− r/(1 − r)}]
{r + (1 − r)I}2 ,
where the coefficient of reliability I = E{S∗(β0| Y, W )}2/E{S(β0 | Y, X)}2 ≤ 1.
This can be used to measure the quality of surrogate predictors W . Some curves of e1 are given in Figure 1 for ρ∗ = 0.3. The results clearly show that if the
coefficient I is large enough, then even for a smaller sampling fraction r, the semi-parametric estimator ˆβsp is still much more efficient than the simple MLE ˆβs.
0.0 0.2 0.4 0.6 0.8 1.0 r 0.0 0.2 0.4 0.6 0.8 1.0 B o u n d o f A R E I=0.1 I=0.3 I=0.5 I=0.7 I=0.9 I=1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Figure 1. For ρ∗= 0.3, curves of e
Similar results can be derived for the ARE of ˆβsp with respect to ˆβf. We
note that under the same conditions, a lower bound for the ARE of ˆβsp with
respect to ˆβf is
e2(ρ∗, I, r) = 1 − (1 − r) 2Iρ∗
r2+ r(1 − r)I + (1 − r)2Iρ∗.
Some curves of e2 are also given in Figure 2 for ρ∗= 0.3. The ARE is at least
0.60 for r ≥ 0.3. Further, for fixed r, e2 increases as the coefficient of reliability
I decreases. 0.0 0.2 0.4 0.6 0.8 1.0 r 0.0 0.2 0.4 0.6 0.8 1.0 B o u n d o f A R E I=0.1 I=0.3 I=0.5 I=0.7 I=1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Figure 2. For ρ∗= 0.3, curves of e
2(ρ∗, I, r) at various I.
5. Simulation Studies
In order to study the finite sample performance of the estimates, some em-pirical studies were carried out. The logistic regression model considered was F (xTβ0) = exp(x)/{1 + exp(x)} so (β0
0, β10) = (0, 1). A linear error model
for the predictor X was assumed: Wi = Xi + v · Ui, for some v. Xi and Ui
were pseudo independent N (0, 0.25) random variables. Thus given Wi = wi,
Xi is a N (wi/(1 + v2), (0.25)v2/(1 + v2)) variate. We also assumed that given
Wi = wi, the misclassification probabilities were independent of Xi and Yi, i.e.,
θ0(wi) = φ0(wi), for all wi, where θ0(wi) = θ(wi, α0) and φ0(wi) = φ(wi, α0).
Four different models were considered.
(1a) Wi = Xi+ 0.1Ui and Yi = Yi0 with probability one. In this case, no
misclas-sification occurs, and hence the only functional form needed for estimating ˆ
βm and ˆβp is f (x|w; γ) = N(w/(1 + 4γ), γ/(1 + 4γ)), γ > 0.
(1b) Wi = Xi + 0.1Ui and θ0(wi) = φ0(wi) = 0.1. In this case, the functional
forms needed for computing ˆβm, ˆβp are θ(w, α) = α0, φ(w, α) = α1, and
(2a) Wi = Xi+ Ui and Yi = Yi0 with probability one. In this case, no
misclas-sification occurs, and hence the only functional form needed for estimating ˆ
βm and ˆβp is f (x|w; γ) = N(w/(1 + 4γ), γ/(1 + 4γ)), γ > 0.
(2b) Wi = Xi + Ui and θ0(wi) = φ0(wi) = exp(wi− 2.5)/{1 + exp(wi − 2.5)}.
In this case, the misclassification probabilities follow a logit model with regression coefficients α00 = −2.5, α01 = 1. The functional forms needed for estimating ˆβm and ˆβp are θ(w, α) = exp(α0 + α1w)/{1 + exp(α0 +
α1w)}, φ(w, α) = exp(α0 + α1w)/{1 + exp(α0 + α1w)}, and f(x|w; γ) =
N (w/(1 + 4γ), γ/(1 + 4γ)), γ > 0.
Model (1b) features small measurement error for both responses and covariates. On the other hand, Model (2b) has more severe measurement error. The primary sample size used was n = 150, and the sampling fractions for validation subsam-ples were r = 0.2, 0.4 and 0.6. One thousand pseudo data sets were generated to compute the simulated mean squared errors.
The function K(·) used in computing the nonparametric regression estimates was the Epanechnikov kernel, that is, K(t) = (3/4)(1 − t2)I
[−1,1](t); see Eubank
(1988). Several bandwidths h = aˆσwk−1/3 were used in our simulations. Here
ˆ
σw is the sample standard deviation of W based on the validation data set, and
a is some constant value. Such choice of bandwidth was justified by Sepanski, Knickerbocker and Carroll (1994); see also Carroll and Wand (1991) and Wang and Wang (1997).
First, we investigate the performance of ˆβsp for various choices of a. The
performance of ˆβsp = ( ˆβsp,0, ˆβsp,1)T is measured by its simulated total mean
squared error (TMSE), defined to be M SE( ˆβsp,0) + M SE( ˆβsp,1). Table 1 reports
the simulation results. It is seen that the performance of ˆβsp is not sensitive to
the choice of a. This agrees with many findings for semiparametric estimation using kernel regression. Here, however, a better choice is a = 1 and hence we use this value for h in remaining simulations.
Table 1. Simulated TMSE of ˆβspfor different a
W = X + .1U W = X + U θ0(W ) = 0 θ0(W ) = .1 θ0(W ) = 0 θ0(W ) = exp(W −2.5) {1+exp(W−2.5)} a r = .2 r = .4 r = .6 r = .2 r = .4 r = .6 r = .2 r = .4 r = .6 r = .2 r = .4 r = .6 0.5 .2005 .1860 .1742 .2270 .1972 .1748 .3418 .2444 .2011 .3810 .2584 .2110 1.0 .1899 .1863 .1625 .2112 .1813 .1843 .3091 .2121 .1989 .3158 .2470 .2046 1.5 .1947 .1900 .1715 .1991 .1901 .1870 .3592 .2604 .2182 .3612 .2496 .2238 2.0 .2027 .1900 .1767 .2453 .1867 .1832 .3645 .2665 .2254 .3374 .2505 .2268
Next, we compare the simulated mean squared error of different estimates for β = (β0, β1)T. The results are tabulated in Table 2. Obviously, ˆβsp has the best
overall performance among the competing estimates. Moreover, the behavior of the simulated TMSE for ˆβsp seems insensitive to the selection of r values.
This is particularly true when the model has only minor measurement errors. Unreported calculations also show that the simulated standard errors (SE) for
ˆ
βsp,0 and ˆβsp,1 are stable. Under all cases considered, the simulated SE’s range
from 0.166 to 0.188 for ˆβsp,0 and from 0.366 to 0.506 for ˆβsp,1. In general, ˆβsp
has much better performance than the other estimates, especially when r ≤ 0.4. From Table 2, we also see that the estimates ˆβf and ˆβp are quite competitive and
better than the simple estimate ˆβs. The latter statement is especially clear when
r = 0.2. However, the simulated SE’s for ˆβf,0, ˆβp,0and ˆβs,0are not very different.
These values range from 0.219 to 0.421. On the other hand, the simulated SE’s for ˆβf,1, ˆβp,1 are smaller than those of ˆβs,1, particularly when r = 0.2. The largest
simulated SE for ˆβf,1 and ˆβp,1 is 0.894 compared with the corresponding value
1.292 for ˆβs,1.
Table 2. Simulated TMSE of different estimators ˆ βs βˆm βˆp βˆf βˆsp W = X + .1U θ0(W ) = 0. r = .2 1.2917 14.6527 .8332 .8318 .1899 r = .4 .4917 6.4017 .4144 .4570 .1863 r = .6 .2840 1.5523 .2808 .2799 .1625 θ0(W ) = 0.1 r = .2 1.2917 13.6781 .8721 .8039 .2112 r = .4 .4917 4.7618 .4284 .4488 .1813 r = .6 .2840 2.4016 .2967 .2987 .1843 W = X + U θ0(W ) = 0. r = .2 1.2917 17.0013 .8311 .8945 .3091 r = .4 .4917 5.3543 .4606 .4339 .2121 r = .6 .2840 1.4972 .2697 .2476 .1989 θ0(W ) = exp(W −2.5) 1+exp(W −2.5) r = .2 1.2917 36.7846 .8622 .8203 .3158 r = .4 .4917 7.0655 .3740 .3992 .2470 r = .6 .2840 2.2060 .2754 .2530 .2046
Finally, we comment on the comparison between ˆβmand ˆβp. In general, ˆβp is
better than ˆβm unless r = 0.6. This happens because there are more parameters
to be estimated simultaneously in computing ˆβm, and the validation subsample
size k is not large enough. The largest simulated SE is 3.103 for ˆβm,1and 1.769 for
ˆ
βm,0, showing that the estimates ˆβm are not very stable. The performance of ˆβm
and ˆβp are expected to become better if k is sufficiently large and the parametric
functions θ(w, α), φ(w, α) and fX|W(x|w; γ) are correctly modeled. Cheng and Hsueh (1999) reported that the performances of ˆβm and ˆβp depend heavily on
the correct choice of the models for misclassification probabilities when X = W with probability one. Therefore, ˆβm and ˆβp are not very robust and extra care
needs to be taken to apply the parametric methods. Acknowledgements
This research was supported in part by the National Science Council, R.O.C. The authors thank the Editor, an associate editor and referees for their useful comments and suggestions which greatly improve the presentation of this paper. Appendix
Proof of Theorem 1. By Taylor’s Theorem and for n → ∞, we have √ n( ˆβsp−β0) = " 1 n ( −∂ 2ˆl(β0) ∂β2 ) −2n1 ( −∂ 3ˆl(β∗) ∂β3 ) ( ˆβsp−β0) #−1 1 √n ( ∂ˆl(β0) ∂β ) +op(1),
where β∗ is some quantity such that |β∗− β0| ≤ | ˆβ
sp− β0|. Here ˆl is the
pseudo-likelihood, and ∂ˆl(β0) ∂β = k X i=1 ( ∂l1,i(β0) ∂β ) + n X j=k+1 ( ∂ˆl2,j(β0) ∂β ) , ∂l1,i(β0) ∂β = {Yi− F (X T i β0)}Xi, ∂ˆl2,j(β0) ∂β = [ ˆA1(Yj0, Wj) − ˆA2(Yj0, Wj)] ˆE[F (XjTβ0) ¯F (XjTβ0)Xj | Wj] ˆ E{F (XT j β0) | Wj} ˆA1(Yj0, Wj) + ˆE{ ¯F (XjTβ0) | Wj} ˆA2(Yj0, Wj) . Further, −∂ 2ˆl(β0) ∂β2 = k X i=1 ( −∂ 2l 1,i(β0) ∂β2 ) + n X j=k+1 ( −∂ 2ˆl 2,j(β0) ∂β2 ) , −∂ 2l 1,i(β0) ∂β2 = F (X T i β0) ¯F (XiTβ0)XiXiT, −∂ 2ˆl 2,j(β0) ∂β2 = [ ˆA1(Y 0 j , Wj) − ˆA2(Yj0, Wj)] ˆE[F (XjTβ0) ¯F (XjTβ0)Xj| Wj] ˆ E{F (XjTβ0) | Wj} ˆA1(Yj0, Wj) + ˆE{ ¯F (XjTβ0) | Wj} ˆA2(Yj0, Wj) !⊗2 −[ ˆA1(Y 0 j, Wj) − ˆA2(Yj0, Wj)] ˆE[{1 − 2F (XjTβ0)}F (XjTβ0) ¯F (XjTβ0)Xj | Wj] ˆ E{F (XT j β0) | Wj} ˆA1(Yj0, Wj) + ˆE{ ¯F (XjTβ0)| Wj} ˆA2(Yj0, Wj) .
By the Law of Large Numbers, 1 n ( −∂ 2ˆl(β0) ∂β2 ) = 1 n ( −∂ 2l(β0) ∂β2 ) + " 1 n ( −∂ 2ˆl(β0) ∂β2 ) −n1 ( −∂ 2l(β0) ∂β2 )# = rE[F (XTβ0) ¯F (XTβ0)}XXT] +(1 − r)E ( {1 − θ0(W ) − φ0(W )}2 π0(W ){1 − π0(W )} (E[F (X Tβ0) ¯F (XTβ0)X | W ])⊗2 ) +op(1) + Op( 1 n√h+ 1 √n+ h2) = Σ−1f + op(1), n → ∞, h → 0 and n2h → ∞. Then as n → ∞, h → 0, n2h → ∞,√n( ˆβsp− β0) = {Σf+ op(1)} ∂ˆl(β0 ) ∂β /√n + op(1). Further, by Taylor’s Theorem again, as n → ∞,
n X i=k+1 ( ∂ˆl2,j(β0) ∂β ) = n X i=k+1 ( ∂l2,j(β0) ∂β + Hn,j ) + Op( 1 nh32 + nh4+ 1 h + √ nh32), where Hn,j= k1Pki=1hi,j, and
hi,j = Kh(Wi− Wj) f (Wj) ( ∂2l2,j(β0) ∂β∂θ0(W j) ) Yi{1 − Yi0− θ0(Wj)} E{F (XjTβ0) | Wj} + ( ∂2l 2,j(β0) ∂β∂φ0(W j) ) (1 − Yi){Yi0− φ0(Wj)} E{ ¯F (XT j β0) | Wj} + ( ∂2l 2,j(β0) ∂β∂E{F (XT j β0) | Wj} ) h F (XiTβ0) − E{F (XjTβ0) | Wj} i + ( ∂2l2,j(β0) ∂β∂E[F (XjTβ0) ¯F (XT j β0)Xj | Wj] ) ×hF (XiTβ0) ¯F (XiTβ0)Xi− E[F (XjTβ0) ¯F (XjTβ0)Xj | Wj] i ! . As a consequence, for n → ∞, 1 √ n ( ∂ˆl(β0) ∂β ) = √ n k(n−k) k X i=1 n X j=k+1 " k n ( ∂l1,i(β0) ∂β ) +n−k n ( ∂l2,j(β0) ∂β ) +n−k n hi,j # +Op( 1 n32h 3 2 + n12h4+ 1 n12h + h32).
Define two i.i.d. random vectors Zi = (Xi, Wi, Yi, Yi0), i = 1, . . . , k and Zj∗ = (Wj, Yj0), j = k + 1, . . . , n and let Qn(Zi; Zj∗) = k n ( ∂l1,i(β0) ∂β ) + n − k n ( ∂l2,j(β0) ∂β ) +n − k n hi,j. Then k−1(n−k)−1Pk
i=1Pnj=k+1Qn(Zi; Zj∗) is a generalized U-statistic. Applying
the Central Limit Theorem for generalized U-statistics (see Serfling(1980)), we have, for n → ∞, √ n k(n − k) k X i=1 n X j=k+1 Qn(Zi; Zj∗) = √n 1 k k X i=1 E{Qn(Zi; Zj∗) | Zi} + 1 n − k n X i=k+1 E{Qn(Zi; Zj∗) | Zj∗} + op(1) d → N 0, Σ−1f +(1 − r) 2 r Isp ! , E{Qn(Zi; Zj∗) | Zi} = k n ( ∂l1,i(β0) ∂β ) +n − k n {1 − θ0(W i) − φ0(Wi)}E[F (XiTβ0) ¯F (XiTβ0)Xi| Wi] π0(W i){1 − π0(Wi)} ×Yi{1 − Yi0− θ0(Wi)} − (1 − Yi){Yi0− φ0(Wi)} −{1 − θ0(Wi) − φ0(Wi)}[F (XiTβ0) − E{F (XiTβ0) | Wi}] + Op(h2), E{Qn(Zi; Zj∗) | Zj∗} = n − kn ( ∂l2,j(β0) ∂β ) . Consequently, as n → ∞,√n( ˆβsp− β0)→ Nd 0, Σf +(1−r) 2 r ΣfIspΣf . References
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Graduate Institute of Statistics, National Central University, Chungli, Taiwan, R.O.C. E-mail: kfcheng@cc.ncu.edu.tw
Department of Statistics, National Chengchi University, Taipei, Taiwan, R.O.C. E-mail: hsueh@nccu.edu.tw
