申訴理賠時間與醫療成本的二元分配函數

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國立臺灣大學理學院數學系碩士論文

Department of Mathematics College of Science

National Taiwan University Master Thesis

申訴理賠時間和醫療成本的二元分配函數

Bivariate Distribution of Claiming Time and Medicare Reimbursement Based on Incomplete Data

黃姿蓉

Tzu-Jung Huang

指導教授 ^: 江金倉博士

Advisor: Chin-Tsang Chiang, Ph.D.

中華民國九十七年七月

July, 2008

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口試委員審定書

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List of Tables

4.1 The averages and the standard deviations (SD) of 500 estimates FHL(t, u) and the averages of 500 standard errors (SE) at the selected points with the sample sizes (n) of 200 and 400, and the censoring rate of 30% . . . 21 4.2 The averages and the standard deviations (SD) of 500 estimates

F_{IP W}(t, u) and the averages of 500 standard errors (SE) at the se- lected points with the sample sizes (n) of 200 and 400, and the censoring rate of 30% . . . 22 4.3 The averages and standard deviations (SD) of 500 estimates F_IM(t, u)

at the selected points with the sample sizes (n) of 200 and 400, and the censoring rate of 30% . . . 23 4.4 The averages and standard deviations (SD) of 500 estimates F_HL(t, u)

and the averages of 500 standard errors (SE) at the selected points with the sample sizes (n) of 200 and 400, and the censoring rate of 50% . . . 24 4.5 The averages and standard deviations (SD) of 500 estimates F_{IP W}(t, u)

and the averages of 500 standard errors (SE) at the selected points with the sample sizes (n) of 200 and 400, and the censoring rate of 50% . . . 25 4.6 The averages and standard deviations (SD) of 500 estimates FIM(t, u)

at the selected points with the sample sizes (n) of 200 and 400, and the censoring rate of 50% . . . 26 4.7 The empirical coverage probabilities of FHL(t, u) at the selected points

with the sample sizes (n) of 200 and 400, and the censoring rates (c.r.) of 30% and 50% . . . 27 4.8 The empirical coverage probabilities of FIP W(t, u) at the selected

points with the sample sizes (n) of 200 and 400, and the censoring rates (c.r.) of 30% and 50% . . . 28

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4.9 The averages and standard deviations (SD) of 500 estimates F_{IP W}^∗ (t, u) and the averages of 500 standard errors (SE) at the selected points with the sample sizes (n) of 200 and 400, and the mixture rate of 30% 29 4.10 The averages and the standard deviations (SD) of 500 estimates

F_{IP W}^∗ (t, u) and the averages of 500 standard errors (SE) at the se- lected points with the sample sizes (n) of 200 and 400, and the mixture rate of 50% . . . 30 4.11 The empirical coverage probabilities of F_{IP W}^∗ (t, u) at the selected

points with two sample sizes (n) of 200 and 400, and the mixture rates of censoring and death (m.r.) of 30% and 50% . . . 31 5.1 The characteristics of the colorectal cancer data and subsample . . . 34 5.2 The estimates of P (D > Xô) and E(Yô) under different age layers

and disease stages . . . 35 5.3 The estimates of P (D > Xô) and E(Yô) under different age-disease

stage groups . . . 35

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List of Figures

5.1 The joint distributions of claiming time and reimbursement for different age layers and disease stages . . . 36 5.2 The joint distributions of claiming time and reimbursement for dif-

ferent age layers and disease stages . . . 37 5.3 The estimates of P (Xô ≤ t|D > Xô) and P (Yô ≤ u|D > Xô) for

patients with age layers of 61-70(solid line), 71-80(dashed line) and

> 80(dotted line) . . . 38 5.4 The estimates of P (Xô ≤ t|D > Xô) and P (Yô ≤ u|D > Xô) for

patients with disease stages of 0(solid line), 1(dashed line), 2(dotted line), 3(dotted-dash line) and 4(long-dashed line) . . . 39

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Acknowledgements

感謝江金倉老師的指導與啟發, 讓我從對做研究一無所知, 到能對生物統計相關的議題都有大致的了解。老師對做研究的熱誠與堅持, 讓人十分地感動。這篇論文的完成, 要感謝的人太多了, 以至於我未能一一細數。許多加油的隻字片語, 出現在不同的字裡行間; 更多令人會心溫暖的影像, 紛然雜陳。已忘記當初確切是如何的動力, 引領我走至今時今日, 就如同每一滴水, 都對於即將傾盆的暴雨有所貢獻。然而在每一場大雨後, 我們總是都看不清水滴的面目。唯一能說的是, 未來不論我在何處何地, 我都不會忘記身為一個統計人的快樂, 驕傲與責任:

”The joy and job of statisticians are to listen to carefully to the data, not to tell the data how to behave.”

最後, 僅將此論文獻給, 總是無怨無悔支持著我, 最最親愛的爸爸與媽媽。

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摘要

目前有關二元分配函數分析的研究, 都集中於處理雙存活時間的資料結構, 以及探討其相關的統計推論。有別於目前大部分的研究主題, 在這篇論文中, 我們關心存活時間(申訴理賠時間) 和指標變數(申訴產生的醫療理賠) 的二元分配函數。根據實際觀測的右設限資料, 我們利用機率倒數加權法(Inverse Probability Weighting Method) 和替換法 (Imputation Method) 提出了更具延伸性的估計式。同時, 我們也運用機率倒數加權法, 針對有終點事件 (Terminal

Events) 發生的情況, 提出相關的二元分配函數的估計式。進一步, 我們更建立上述估計式的

大樣本性質, 利用估計式的高斯過程逼近, 配合著變異矩陣的估計式, 建構其相對應的信賴區間。我們執行了一系列的模擬檢證這些估計式以及信賴區間在有限樣本下之特性, 此外, 應用所提出的估計法在 SEER-Medicare 資料庫有關結腸癌的資料上。

關鍵字: 二元分配函數; 指標變數; 醫療成本; 設限; 終點事件; 機率倒數加權法; 替換法; 高斯過程

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Abstract

Although numerous attempts have been made on bivariate failure times, however, there is little attention on the study of relation between failure time(claiming time) and mark variable(medicare reimbursement). Meanwhile, from the viewpoints of brokers and the management in insurance company, it is more attractive and en- grossing to capture the dynamic pattern so that our research interest would focus on the joint distribution of claiming time and the corresponding medicare reimbursement. Based on survival censored data, we propose two estimation procedures:

the inverse probability weighting (IPW) method and the imputation method. Fur- thermore, it is meaningful to accommodate terminal events occurring prior to the realization of failure time and the IPW method could lead to the resolution of this obstacle. Moreover, the limiting Gaussian processes of estimated distributions and the estimators of variance-covariance functions are developed and enable us to construct approximated regions. To investigate the ﬁnite sample properties of proposed estimators and the performance of inference procedures, a class of simulations would be conducted. An application to the colorectal cancer data retrieved from the Surveillance, Epidemiology, and End Results (SEER) Medicare database is also presented. In the end, we provide a brief discussion and further research topics of interest.

KEY WORDS: bivariate distribution, mark variable, medical cost, censoring, terminal events, inverse probability weighting (IPW), imputation, U-statistics, Gaussian processes

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Chapter 1 Introduction

Bivariate failure time data are often encountered in many biomedical contexts, for example, times to blindness in both eyes of diabetic retinopathy patients or gap times to successive stages in the progression of acquired immunodeﬁciency syndrome.

Estimation methods for dealing with parallel bivariate survival times can be found in Campbell (1981), Tsai, et al. (1986), Burke (1988), Dabrowska (1988), and Akritas, et al.(2003), among others. Statistical analyses for bivariate serial gap times include the works of Wang, et al. (1998) and Lin, et al. (1999). Distinguished from bivariate failure time data, an overwhelming emphasis is placed on health care cost controlling and ﬁscal accountability in medicine, and a brand new survival- type data structure emerge, called the cost data. The cost data consist of claiming time (time to the occurrence of medicare reimbursement) and the mark variable (medical cost or reimbursement), which is possibly correlated to claiming time and not observed until the corresponding claiming time fulﬁlls. Rather than censoring, the marker would be missing due to the loss of follow-up or drop-out. Thus, it is inappropriate to use the existing statistical methods for bivariate failure time data.

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Even though medical cost management has received immense attention, re- searchers merely highlighted the topic in two primary aspects over the past decades.

One is mean cost evaluation within the period of interest based on complete or censored data. The other is cost-effectiveness (CE) ratio, the ratio of the mean difference in costs to the mean difference in effectiveness. CE ratio is a practical measurement for the trade off between budget constraints and patient benefits. In contrast, from the viewpoints of brokers and the management in insurance company, it is more cru- cial and appealing to explore the joint distribution of the claiming time Xô versus the medicare reimbursement Yô. Few attempts have made at this sort of phenomena except for Huang and Louis (1998) and Hudgens, et al (2007). Based on the censored cost data {Xi, δi, Yi}ⁿ_i=1 with Xi = X_iô ∧ Ci, Yi = δiY_iô, δi = I(X_iô ≤ Ci) being the censoring status, and C being the censoring time, Huang and Louis obtained the estimator for joint distribution F (t, u) = P (Xô ≤ t, Yô ≤ u) by estimating cumulative mark-specific hazard function Λ(t, u). For this data setting, we propose an inverse probability weighting (IPW) estimator and an imputation estimator via using the induced binary responses B_i(t, u) = I(X_iô≤ t, Y_iô≤ u), i=1, · · · , n.

In a colorectal cancer cohort study, the clinical trial might be terminated before the study endpoint is reached. The terminal events, such as death, preclude any further medical costs and definitely relate with (Xô, Yô). For instance, it is instinctive to anticipate that poor health patients prone to death might raise larger medical costs. As long as the death arises before the first claim, the medicare reimbursements of the departed would be zero rather than missing. Instead of

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only considering the joint distrbution of claiming times and reimbursements, it is more meaningful to exploit the joint distribution for those who have medicare re- imbursement prior to death, say, F^∗(t, u) = P (Xô ≤ t, Yô ≤ u|D > Xô), where D denotes the terminal time. In this study, our research focuses on seeking for

an appropriate estimation method of F^∗(t, u) based on the transformed cost data {X_i^∗, δX_iô, δCi, Y_i^∗}ⁿ_i=1, where X_i^∗ = X_iô ∧ Ci ∧ Di, Y_i^∗ = Y_iôδX_i^∗, δX_iô = I(X_i^∗ = X_iô),

and δ_C_i = I(X_i^∗ = C_i). In addition, we provide the estimator for the probability of cost occurring before death P (D > X^o). and the mean cost of patients, which can be derived as E[Y^o] ={

ud_uF^∗(t, u)}P (D > X^o).

We divide the rest of this thesis into the following parts. In Chapter 2, we brieﬂy reviews the estimation method of Huang-Louis and propose the IPW and imputation estimators for F (t, u). Moreover, the asymptotic properties and inference procedures are developed in this chapter. In Chapter 3, we further deal with the occurrence of terminal events in estimation. In Chapter 4, a class of simulations to investigate the ﬁnite-sample properties of the estimators and the performance of our proposed procedures. We present an application of our methods to the colorectal cancer data retrieved from the Surveillance, Epidemiology, and End Results (SEER) Medicare database in Chapter 5. Finally, a concise discussion and a further research topic are provided in Chapter 6.

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Chapter 2 Estimation and Inference

Procedures without a Terminal Event

2.1 Review of Huang-Louis Estimator

Baed on the data {(X_i, δ_i), Y_i}ⁿ_i=1, Huang and Louis (1998) proposed a non- parametric estimator FHL(t, u) for F (t, u). Before introducing their estimation method, some concise notations are presented ﬁrst. Let SX^o(t) = 1−F (t, ∞), SX(t) = P (X > t), FXY(t, u) = P (X ≤ t, Y ≤ u, δ = 1), SC(t) = P (C > t), and Λ(t, u) =

_t

0S_X⁻¹o(s)dsF (s, u) be the cumulative cost-speciﬁc hazard function, where

dsF (s, u)

is a Lebesgue-Stieltjes integration over s for ﬁxed u. It was derived that F (t, u) can be expressed as

F (t, u) =

_t

0 S_X^o(s)d_sΛ(s, u) =

_t

0

[0,s)

{1 − d_vΛ_Xo(v)}d_sΛ(s, u), (2.1.1)

where

denotes the product integral and Λ_Xo(v) = Λ(v, ∞).

Under the assumption of random censorship (A1: C is independent of (X^o, Y^o)),

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one has

SX(t) = SX^o(t)SC(t) and dtFXY(t, u) = SC(t){dtF (t, u)}. (2.1.2)

Substituting the empirical estimators FXY(t, u) = n⁻¹_n

i=1δiBi(t, u) and SX(t) = n⁻¹_n

i=1I(Xi > t) for FXY(t, u) and SX(t), an estimator for Λ(t, u) was proposed by

Λ(t, u) = _t

0

dsFXY(s, u)

SX(s⁻) . (2.1.3)

It is straightforward to obtain an estimator from (2.1.1) and (2.1.3) as

F_HL(t, u) =

_t

0

[0,s)

{1 − d_vΛ_X^o(v)}d_sΛ(s, u), (2.1.4)

with Λ_Xo(v) = Λ₍v, ∞). In the following theorem, we state the uniform consis- tency and asymptotic Gaussian process of FHL(t, u). Further assumptions (A2:

No common jump points in the distributions SX^o(t) and SC(t)) and (A3: SC(t) and F (t, u) are absolutely continuous on Ω = {(t, u) : 0 < t ≤ τ, u > 0} with τ = sup{t : S_X(t) ≥ > 0} for some > 0) are made throught the thesis for the main results.

Theorem 2.1.1. Supposed that assumptions (A1)-(A3) are satisﬁed. Then,

sup

(t,u)∈Ω| F_HL(t, u) − F (t, u) |−→ 0,^p (2.1.5) and n^1/2( FHL(t, u) − F (t, u)) converges weakly to a mean zero Gaussian process with variance-covariance function Γ₁(w₁, w₂) = Cov(Z₁(w₁), Z₁(w₂)) for w_j = (t_j, u_j) ∈ Ω, j=1,2, where Z₁(t, u) =_t

0 F (s, u)d_sϕ(s, ∞)+_t

0 S_X^o(s⁻)d_sϕ(s, u)−F (t, u)ϕ(t, u) and ϕ(t, u) = S_X⁻¹(X⁻)δB(t, u) −_t

0 S_X⁻²(s⁻)I(X ≥ s)d_sF_XY(s, u).

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2.2 IPW Estimator

The IPW technique has been widely adopted in estimation for dealing with the biased sample due to censoring. The main idea is to use subjects with available Bi(t, u)’s and weight each observation by the inverse of the selection probability πt = P (Vt = 1|X^o), where Vt = V1t + V2t with V1t = I(X > t) and V2t = I(X ≤ t, δ = 1). Under the validity of random censorship (A1), the selection probability πt

is desired to be V1tSC(t) + V2tSC(X^o). By the property,

E[Vtπ_t⁻¹(B(t, u) − F (t, u))] = 0 (2.2.1)

and substituting a consistent estimator S_C(t) for S_C(t), an IPW estimator for F (t, u) can be constructed as

F_{IP W}(t, u) =

_n

i=1δ_iS_n_C⁻¹(X_i^o)B_i(t, u)

i=1Vitπ⁻¹_it , (2.2.2) where πit = Vi1tSC(t) + Vi2tSC(X_i^o). Naturally, the Kaplan-Meier estimator is used to estimate SC(t). The asymptotic Gaussian process of n^1/2( FIP W(t, u) − F (t, u)) is established below.

Theorem 2.2.1. Suppose that assumptions (A1)-(A3) hold. Then, n^1/2( F_{IP W}(t, u)

− F (t, u)) converges weakly to a Gaussian process with mean zero and variance-

covariance function Γ₂(w₁, w₂) = Cov(Z_i2(w₁), Z_i2(w₂)), where Z_i2(t, u) = E[ψ_ij(t, u)+

ψji(t, u)|((Xi, δi), Yi)] with MCi(t) = I(Xi ≤ t)(1 − δi) +_t

0 I(Xi ≥ v)d(lnSC(v)) and ψij(t, u) = { Vi1t

SC(t)(1 + 1 n

_t

0

dM_C_j(s)

SX(s⁻)) + Vi2t

SC(X_i^o)(1 + 1 n

_X_i^o

0

dM_C_j(s)

SX(s⁻))}(Bi(t, u)−

F (t, u)).

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Proof. From (2.2.2), one has

n^1/2( F_{IP W}(t, u) − F (t, u)) = n^−1/2_n

i=1Vitπ_it⁻¹(Bi(t, u) − F (t, u)) n⁻¹_n

i=1V_itπ⁻¹_it . (2.2.3) By the boundness of Vit’s, the uniform convergence of SC(t), and the Euclidean class of {V_itπ_it⁻¹ : 0 < t ≤ τ } (cf. Akrita (1994), Pakes and Pollard (1989), and Pollard (1990)), it is entailed that n⁻¹_n

i=1V_itπ_it⁻¹ uniformly converges to one. Thus,

n^1/2( FIP W(t, u) − F (t, u)) = n^−1/2

n i=1

Vitπ⁻¹_it (Bi(t, u) − F (t, u)) + r1n(t, u),

(2.2.4) where sup

(t,u)∈Ω|r1n(t, u)| = op(1). The ﬁrst order Taylor expansion of the dominating term in (2.2.4) with respect to SC(t) = SC(t) yields that

n^−1/2

n i=1

Vitπ_it⁻¹(Bi(t, u) − F (t, u)) = n^−1/2

n i=1

Vitπ⁻¹_it (Bi(t, u) − F (t, u))−

n^−1/2

n i=1

{ V_i1t

SC(t)(SˆC(t)

SC(t) − 1) + Vi2t

SC(X_i^o)(SˆC(X_i^o)

SC(X_i^o) − 1)}(Bi(t, u) − F (t, u)) + r2n(t), (2.2.5) where sup

t∈(0,τ]|r2n(t)| = op(n^−1/2). Since{ SC(t)/SC(t) − 1} can be uniformly approximated by {−n⁻¹_n

i=1

_t

0 dMCi(u)/SX(u⁻)} (cf. Fleming and Harrington (1991)).

It follows from (2.2.4)-(2.2.5) that

n^1/2( FIP W(t, u) − F (t, u)) = {n^1/2(n − 1)}⁻¹

i=j

ψij(t, u) + r3n(t, u), (2.2.6)

where sup

(t,u)∈Ω|r_3n(t, u)| = o_p(n^−1/2). By the Euclidean class of{ψ_ij(t, u) : (t, u) ∈ Ω}, the decomposition of a U-statistic into the sum of degenerate U-statistics, and Corollary 4 of Sherman (1994), n^1/2( FIP W(t, u) − F (t, u)) can be uniformly approximated by the term of independent and identically distributed random quantities

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n^−1/2_n

i=1Z_i2(t, u). By the functional central limit theorem (Pollard (1990)) and the uniform convergence of n⁻¹_n

i=1{Vit/πit} to one, the proof for Theorem 2.2.1 is completed.

2.3 Imputation Estimator

Applying the method of Buckley and James (1979), we propose a alternative estimator based on the considered data setting. In this estimation method, the unavailable statuses Bi(t, u)’s are substituted by the corresponding expectations E[B_i(t, u)|X_i, δ_i = 0]’s. Let B_i^∗(t, u) = I(V_it= 1)B_i(t, u)+I(V_it= 0)E[B_i(t, u)|X_i, δ_i = 0]. A direct calculation ensures that

E[B_i^∗(t, u)|Xi, δi] =

1 k=0

E[B_i^∗(t, u)|Xi, δi]I(Vit = k)

=

1 k=0

E[Bi(t, u)|Xi, δi]I(Vit = k) = E[Bi(t, u)|Xi, δi], (2.3.1)

which implies that E[B_i^∗(t, u)] = E[Bi(t, u)] = F (t, u). Under assumption (A1), we further derive that

E[B_i(t, u)|X_i = x, δ_i = 0] = S_X⁻¹o(x){F (t, u) − F (x, u)}. (2.3.2)

Following from (2.3.1)-(2.3.2), an estimation procedure for F (t, u) is proposed. The imputation estimator FIM(t, u) is obtained via solving

n⁻¹

n i=1

(B_i^∗(t, u) − F (t, u))

= n⁻¹

n i=1

{(Bi(t, u) − F (t, u))I(Vit= 1) + (1− SX^o(Xi))F (t, u) − F (Xi, u)

SX^o(Xi) I(Vit= 0)}

0, (2.3.3)

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where S_X^o(t) is the Kaplan-Meier estimator of S_X^o(t).

Note that F (Xi, u) in (2.3.3) is unknown when Vit = 0. Generally, a consistent estimator F (Xi, u) is substituted for F (Xi, u). Let At be the collection of ordered censoring times C(1) < C(2) < · · · < C(kt) ≤ t with size kt. The joint distribution F (C(1), u) is ﬁrst estimated by

FIM(C(1), u) = n⁻¹

n i=1

Bi(C(1), u). (2.3.4)

Subsequently, the estimator for F (C(j), u), j=2,· · · , kt, can be obtained as

FIM(C(j), u) =

_n

i=1B_i(C_(j), u)I(V_iC_(j) = 1)−_j−1

l=1 S_X⁻¹o(C_(l)) F (C_(l), u)

_n

i=1I(V_iC_(j) = 1)−_j−1

l=1 S_X⁻¹o(C_(l))(1− S_X^o(C_(l))) . (2.3.5) An explicit expression for the estimator of F (t, u) is then derived to be

FIM(t, u) =

_n

i=1Bi(t, u)I(Vit = 1)−_n

i=1S_X⁻¹o(Xi) F (Xi, u)I(Vit= 0)

_n

i=1I(Vit = 1)−_n

i=1S_X⁻¹^o(Xi)(1− SX^o(Xi))I(Vit= 0) . (2.3.6) Remark: Since the number of censoring times prior to t is random and might be

considerably large, it becomes cumbersome to develop the inference procedure for F (t, u) based on F_IM(t, u). Currently, there is still no existing statistical methodol- ogy to facilitate this obstacle.

2.4 Inferences Procedures

In this subsection, pointwise and simultaneous conﬁdence bands for F (t, u) are con- structed based on the asymptotic Gaussian processes of F_HL(t, u) and F_{IP W}(t, u). As shown in Theorem 2.1.1, the limiting process n^1/2( FHL(t, u) − F (t, u)) is uniformly asymptotically equivalent to n^−1/2_n

i=1Zi1(t, u). The variance-covariance function

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of F_HL(t, u) is suggested to be estimated by Γ₁(w₁, w₂) = n⁻¹_n

i=1Z_i1(w₁) Z_i1(w₂), where

Zi1(t, u) =

_t

0

FHL(s, u)ds ϕi(s) +

_t

0

SX^o(s)ds ϕi(s, u) − FHL(t, u) ϕi(t, u) (2.4.1)

with ϕ_i(t, u) = S_X⁻¹(X_i)δ_iB_i(t, u) −_t

0 S_X⁻²(s)I(X_i ≥ s)ds F_XY(s, u). Similarly, the limiting process of n^1/2( FIP W(t, u)−F (t, u)) is asymptotically equivalent to n^−1/2_n

i=1Zi2

(t, u). Let MCj(t) = I(Xj ≤ t)(1 − δj) +_t

0 I(Xj ≥ v)d(ln SC(v)). An estimator for the variance-covariance function Γ₂(w1, w2) is proposed to be

Γ₂(w₁, w₂) = n⁻¹

n i=1

Z_i2(w₁) Z_i2(w₂), (2.4.2)

where Zi2(t, u) = (n − 1)⁻¹

j=i( ψij(t, u) + ψji(t, u)) and ψij(t, u) = (1

n Vi1t

S_C(t)

_t

0

d MCj(s) S_X(s⁻) + 1

n Vi2t

S_C(X_i^o)

_X_i^o

0

d MCj(s)

S_X(s⁻) )(Bi(t, u) − FIP W(t, u)).

The uniform consistency of Γ₁(w₁, w₂) and Γ₂(w₁, w₂) are established in the following theorem.

Theorem 2.4.1. Supposed that assumptions (A1)-(A3) hold. Then,

sup

w1,w2∈Ω| Γ_l(w₁, w₂)− Γ_l(w₁, w₂)|−→ 0, as n −→ ∞, l = 1, 2.^p (2.4.3) Proof. By the uniform convergence of empirical estimators (Pollard (1990)), we can

show that FXY(t, u) and SX(t) converge to FXY(t, u) and SX(t) uniformly in (t, u).

Together with (2.4.1), Theorem 2.1.1, and the boundness of δiBi(t, u)’s, it is ensured that

sup

(t,u)∈Ω| ϕ_i(t, u) − ϕ_i(t, u) |−→ 0 as n −→ ∞, i = 1, · · · , n.^p (2.4.4)

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One further derives by the intergration by parts that

sup

(t,u)∈Ω| Zi1(t, u) − Zi1(t, u) |−→ 0 as n −→ ∞, i = 1, · · · , n.^p (2.4.5) From (2.4.8), Theorem 2.2.1, and the inequality

| Γ₁(w₁, w₂)− Γ₁(w₁, w₂)| ≤ | Γ₁(w₁, w₂)− n⁻¹ⁿ

i=1

Z_i1(w₁)Z_i1(w₂)|

+| n⁻¹

n i=1

Z_i1(w₁)Z_i1(w₂)− Γ₁(w₁, w₂)|, (2.4.6)

the uniform consistency of Γ₁(w1, w2) is obtained.

For the uniform consistency of Γ₂(w1, w2), it is implied from (2.4.2) and Theorem 2.2.1 that

Γ₂(w1, w2) = n⁻¹

n i=1

Zi2(w1) Zi2(w2)

={n(n − 1)²}⁻¹

i,j,k

( ψij(w1) + ψji(w1))( ψik(w2) + ψki(w2)). (2.4.7)

The expression of Γ₂(w₁, w₂) in (2.4.7) and the uniform convergence of S_X(t), S_C(t), and F_{IP W}(t, u) entail that Γ₂(w₁, w₂) is uniformly approximated by

{n(n − 1)(n − 2)}

i=j=k

(ψij(w1) + ψji(w1))(ψik(w2) + ψki(w2)). (2.4.8)

Since{ψij(t, u) : (t, u) ∈ Ω} is Euclidean, the term in (2.4.8) converges to E[(ψij(w1)+

ψji(w1))(ψik(w2) + ψki(w2))] = E[Zi2(w1)Zi2(w2)] uniformly in (w1, w2).

The limiting Gaussian processes in Theorems 2.1.1 and 2.2.1 as well as the estimated variance-covariance functions enable us to construct an approximated (1−α) pointwise conﬁdence interval for F (t, u) via either

FHL(w) ± z1−α/2Γ^1/2₁ (w, w) or FIP W(w) ± z1−α/2Γ^1/2₂ (w, w), (2.4.3)

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where z_1−α is the 100(1− α)th percentile of a univariate standard normal distribu- tion and w = (t, u). To draw inference on the pattern of F (t, u), a simultaneous conﬁdence band can also be established based on the quantities:

Kl= sup

w∈Υ| n^−1/2_n

i=1MiZil(w)

Γ^1/2_l (w) |, l = 1, 2, (2.4.4)

where Υ is a region of interest within Ω, and {Mi : i = 1, · · · , n} are independent realizations of standard normal variable and are independent of {(Xi, δi), Yi}ⁿ_i=1. An approximated (1− α) simultaneous conﬁdence band for F (t, u) is then constructed via either

F_HL(w) ± n^−1/2Q_1−α( K₁)Γ^1/2₁ (w, w) or F_{IP W}(w) ± n^−1/2Q_1−α( K₂)Γ^1/2₂ (w, w),

(2.4.5) where Q1−α( Kl) is the 100(1− α)th percentile of realizations of (2.4.4) computed based on a large number of generations of{M_i : i = 1, · · · , n}, l = 1, 2.

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Chapter 3 Estimation and Inference

Procedures with Terminal Events

By adapting the IPW method mentioned in Chapter 2 for the appearance of termi- nal events, we use subjects with available B_i(t, u)s and weight each subject with π_i = P (δ_X_iô = 1|X_iô)s. Under the assumption of random censorship (AA1: C is independent of (Xô, Yô, D)) and assumption (AA2: The distribution functions of Xô, C and D do not have jump points in common), πi is derived to be SC(X_iô).

Using

E[δX^oS_C⁻¹(X^o)(B(t, u) − F^∗(t, u))] = 0 (3.1.1)

and replacing SC(t) by a consistent estimator SC(t), F^∗(t, u) is proposed to be estimated by

F^∗(t, u) =

_n

i=1δ_X_i^oS_C⁻¹(X_i^o)B_i(t, u)

_n

i=1δX_iôS_C⁻¹(X_iô) . (3.1.2) Generally, it is reasonably to estimate S_C(t) by the Kaplan-Meier estimator. Based on (3.1.2), the probability P (D > Xô) and the mean cost E[Yô] are naturally

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estimated by

P (D > X ^o) = n⁻¹

n i=1

δX_i^oS_C⁻¹(Xi) and E[Y^o] = n⁻¹

n i=1

δX_iôS_C⁻¹(X_iô)Y_iô. (3.1.3)

To establish the asymptotic properties of n^1/2( F^∗(t, u) − F^∗(t, u)), the condition

(AA3: SC(t), SD(t), and F^∗(t, u) are absolutely continuous on (0, τ^∗] and Ω^∗ = {(t, u) : 0 < t ≤ τ^∗, u > 0 } with τ^∗ = sup{t : SX^∗(t) ≥ > 0} for some > 0. ) is

further made in the following theorem.

Theorem 3.1. Suppose that assumptions (AA1)-(AA3) are satisﬁed. n^1/2( F^∗(t, u)−

F^∗(t, u)) converges weakly to a mean-zero Gaussian process with variance-covariance function Γ^∗(w₁, w₂) = Cov(Z_i^∗(w₁), Z_i^∗(w₂)) for w₁, w₂ ∈ Ω^∗, where Z_i^∗(t, u) = E[ψ_ij^∗(t,

u) + ψ_ji^∗(t, u)|((X_i^∗, δX^o_i, δCi), Y_i^∗)], M_C^∗_i(t) = δCiI(X_i^∗ ≤ t) +_t

0 I(X_i^∗ ≥ v)d(lnSC(v)) and

ψ^∗_ij(t, u) = δ_X_i^o

{S_C(X_i^o)P (D > X^o)}(1 + 1 n

_Xo i

0 S_X⁻¹∗(s⁻)dM_C^∗_j(s))(Bi(t, u) − F^∗(t, u)).

Proof. From (3.1.2), n^1/2( F^∗(t, u) − F^∗(t, u)) can be re-expressed as

n^−1/2_n

i=1δ_X_i^oS_C⁻¹(X_i^o)(B_i(t, u) − F^∗(t, u)) n⁻¹_n

i=1δX_iôS_C⁻¹(X_iô) . (3.1.4) By the boundedness of δX_iô’s and the uniform convergence of SC(t), the denominator term in (3.1.4) is shown to be asymptotically equivalent to P (D > Xô). Thus,

n^−1/2( F^∗(t, u) − F^∗(t, u)) = n^−1/2_n

i=1δX_i^oS_C⁻¹(X_i^o)(Bi(t, u) − F^∗(t, u))

P (D > X^o) + r^∗_1n(t, u), (3.1.5)

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where sup

(t,u)∈Ω^∗|r^∗_1n(t, u)| = o_p(1). Applying the Taylor expansion theorem to the nu- merator term in (3.1.5) with respect to SC(t) = SC(t), it can be derived to be

n^−1/2

n i=1

δX_i^o

SC(X_i^o)(Bi(t, u) − F^∗(t, u)) = n^−1/2

n i=1

δX_i^o

SC(X_i^o)(Bi(t, u) − F^∗(t, u))−

n^−1/2

n i=1

{ δX_i^o

SC(X_i^o)(Sˆ_C(X_i^o)

SC(X_i^o)− 1)(Bi(t, u) − F^∗(t, u))} + r^∗_2n(t), (3.1.6)

where sup

t∈(0,τ^∗]|r^∗_2n(t)| = op(n^−1/2). Together with (3.1.5) and the proof argument for Theorem 2.2.1, we obtain that

n^1/2( F^∗(t, u) − F^∗(t, u)) = {n^1/2(n − 1)}⁻¹

i=j

ψ_ij^∗(t, u) + r_n^∗(t), (3.1.7)

where sup

t∈(0,τ^∗]|r^∗_n(t)| = op(n^−1/2). The asymptotic Gaussian process of F^∗(t, u) is developed.

Let M_C^∗_i(t) = δCiI(X_i^∗ ≤ t) + _t

0 I(X_i^∗ ≥ v)d(ln SC(v)), i = 1, · · · , n, and SX^∗(t) = n⁻¹_n

i=1I(X_i^∗ > t). The variance-covariance function Γ^∗(w1, w2) is suggested to be estimated by Γ^∗(w1, w2) = n⁻¹_n

i=1Z_i^∗(w₁) Z_i^∗(w₂) with Z_i^∗(t, u) = (n − 1)⁻¹

j=i

( ψ_ij^∗(t, u) + ψ_ji^∗(t, u)) and

ψ^∗_ij(t, u) = δ_X_iô{ S_C(X_iô) P (D > Xô)}⁻¹1 n

_X_i^o

0

d M_C^∗_j(s)

SX^∗(s⁻)(B_i(t, u) − F^∗(t, u)).

(3.1.8) The uniform consistency of Γ^∗(w1, w2) is given in the following theorem.

Theorem 3.2. Supposed that assumptions (AA1)-(AA3) hold. Then,

w1,wsup2∈Ω^∗| Γ^∗₁(w1, w2)− Γ^∗₁(w1, w2)|−→ 0, as n −→ ∞.^p (3.1.9)

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Proof. By using the techniques in the proof of Theorem 3.1, the uniform convergence

of P (D > X^o) is obtained. Paralleling the argument for the uniform consistency of

Γ₂(w1, w2) in Theorem 2.4.1, the uniform consistency of Γ^∗(w1, w2) is developed.

Similarly to the aforementioned procedure, approximated (1− α) pointwise and simultaneous conﬁdence intervals are seperately constructed via

F^∗(w) ± z_1−α/2Γ^∗1/2(w, w) and F^∗(w) ± n^−1/2Q_1−α( K^∗)Γ^∗1/2(w, w), (3.1.10)

where

K^∗ = sup

w∈Υ^∗ | n^−1/2_n

i=1M_i^∗Z_i^∗(w)

Γ^∗1/2(w) |

with Υ^∗is a region of interest within Ω^∗, and (M_i^∗ : i = 1, · · · , n) are independent re- alizations of standard normal variable and are independent of{(X_i^∗, δX_i^o, δCi), Y_i^∗}ⁿ_i=1.

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Chapter 4 Numerical Studies

In this chapter, we conduct two simulation scenarios to investigate the ﬁnite sample properties of proposed estimators and the performance of inference procedures. One is for the case of censoring data and the other accommodates the appearance of terminal events. In the simulation process, data are repeatedly generated 500 times with the sample sizes of 200 and 400, and the censoring rates of 30% and 50%. The estimators are evaluated at the selected grid points (t, u), where u takes values of 0.25, 0.5, 0.75, and 1, and t takes values of 0.2231, 0.5108, 0.9163, and 1.6094 with the cumulative probabilities of X^o being 0.2, 0.4, 0.6, and 0.8.

4.1 Simulation Setting of (X

^o

, Y

^o

)

The pair random vector (Xô, Yô) is specified from the Frank’s bivariate family (Gen- est (1987)), in which

F_X^(α)oY^o(t, u) =

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

log_α{1 + (α^F^Xo^(t)− 1)(α^F^{Y o}^(u)− 1)/(α − 1)}, α = 1

FX^o(t)FY^o(u), α = 1.

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Simulations are implemented with α = exp(−10), which implies a positive associa- tion between the claiming time and the medical cost.

4.2 Senario I - Without a terminal event

In the section, we examine the ﬁnite sample properties of FHL(t, u), FIP W(t, u), and FIM(t, u), and evaluate the inference procedures based on FHL(t, u) and FIP W(t, u).

The censoring time C is independently generated from an exponential distribution with diﬀerent scale parameters 0.5 and 1 for the expected censoring rates of 30%

and 50%.

Tables 4.1-4.6 exhibit the averages and standard deviations of 500 estimates, and the averages of 500 standard errors based on (2.1.4), (2.2.2), and (2.3.7) at the selected points. We detect that the averages of 500 estimates F_{IP W}(t, u) are more close to F (t, u) than those of FHL(t, u), especially for a higher censoring rate.

Furthermore, FIM(t, u) is found to have larger biases at points of (1.6094, 0.75) and (1.6094, 1.0) when the sample size is small and the censoring rate is high. The biases of these estimators are negligible when the sample size is large enough. The standard deviations of three estimates are almost the same. As expected, the standard deviations decrease with increasing sample size and decreasing censoring rate. It is revealed in these tables that the averages of 500 standard errors of F_HL(t, u) and F_{IP W}(t, u) are very close to the standard deviations of their 500 estimates. Note

that the averages of 500 standard errors of FHL(t, u) diverge from the standard deviations of estimates as the value of time is large, while those of FIP W(t, u) seem