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Journal of the American Statistical Association

ISSN: 0162-1459 (Print) 1537-274X (Online) Journal homepage: http://www.tandfonline.com/loi/uasa20

Regression Analysis of Additive Hazards ModelWith Latent Variables

Deng Pan, Haijin He, Xinyuan Song & Liuquan Sun

To cite this article: Deng Pan, Haijin He, Xinyuan Song & Liuquan Sun (2015) RegressionAnalysis of Additive Hazards Model With Latent Variables, Journal of the American StatisticalAssociation, 110:511, 1148-1159, DOI: 10.1080/01621459.2014.950083

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Regression Analysis of Additive Hazards Model WithLatent Variables

Deng PAN, Haijin HE, Xinyuan SONG, and Liuquan SUN

We propose an additive hazards model with latent variables to investigate the observed and latent risk factors of the failure time of interest.Each latent risk factor is characterized by correlated observed variables through a confirmatory factor analysis model. We develop a hybridprocedure that combines the expectation–maximization (EM) algorithm and the borrow-strength estimation approach to estimate the modelparameters. We establish the consistency and asymptotic normality of the parameter estimators. Various nice features, including finite sampleperformance of the proposed methodology, are demonstrated by simulation studies. Our model is applied to a study concerning the riskfactors of chronic kidney disease for Type 2 diabetic patients. Supplementary materials for this article are available online.

KEY WORDS: Additive risk; Borrow-strength estimation; Factor analysis; Latent trait; Multiple correlated outcomes.

1. INTRODUCTION

The proportional and the additive hazards (AH) models aretwo popular frameworks in biomedical studies for investigatingthe association between risk factors and disease occurrence ordeath (Cox 1972; Cox and Oakes 1984; Aalen 1989; Huffer andMcKeague 1991; Lin and Ying 1994). The AH model providesa characterization of the effects of risk factors different from theproportional hazards (PH) model, and has remarkable featuresthat are not found in the latter model. In particular, the AH modelpertains to the risk difference, which is especially relevant andinformative in epidemiological and clinical studies. Similar tothe PH model, the AH model also has sound biological andempirical bases (Breslow and Day 1987).

There are many situations in medical, behavioral, and psy-chological research settings when outcomes or potential riskfactors cannot be measured by a single observed variable. In-stead, they should be characterized by several observed vari-ables from different perspectives. Examples include glycemiameasured by glycated hemoglobin (HbA1c) and fasting plasmaglucose (FPG); lipid characterized by total cholesterol (TC),high-density lipoprotein cholesterol (HDL-C), and triglycerides(TG); and behavioral problem assessed by antisocial, anxious,dependent, headstrong, and hyperactive behavior. The conven-tional PH and AH models manage latent factors by incorporat-ing one or more of their surrogates into the regression analysis.However, problems may exist with the separate inclusion of par-tial surrogates of the latent factors. First, important attributes of

Deng Pan is Lecturer, School of Mathematics and Statistics,Huazhong University of Science and Technology, Wuhan, China (E-mail:pand.whu@gmail.com). Haijin He (E-mail: s115500892@sta.cuhk.edu.hk) isPost-doctoral Fellow, and Xinyuan Song (E-mail: xysong@sta.cuhk.edu.hk)is Associate Professor, Department of Statistics, The Chinese University ofHong Kong, Hong Kong, China. Liuquan Sun is Professor, Institute of Ap-plied Mathematics, Chinese Academy of Science (CAS), Beijing 100190, China(E-mail: slq@amt.ac.cn). This research was supported by GRF 404711 fromthe Research Grant Council of the Hong Kong Special Administration Re-gion, the National Natural Science Foundation of China Grants (no. 11471277,11231010, 11171330, and 11021161), Key Laboratory of RCSDS, CAS (no.2008DP173182), and BCMIIS. The authors are thankful to the editor, the as-sociate editor, two anonymous reviewers, and Professor Sik-Yum Lee for theirvaluable comments and suggestions that improved the article substantially, toDr. Feng C. for pointing out an error in the proof and providing methods tocorrect it, and to Professor Juliana Chan from the Department of Medicine andTherapeutics, and Prince of Wales Hospital of the Chinese University of HongKong, for providing the data in the real example.

latent factors may be lost because of incomplete measurement,leading to bias or misleading results. For instance, systolic bloodpressure (SBP) and diastolic blood pressure (DBP) are usuallyused together to characterize blood pressure. Although highlycorrelated, SBP and DBP reflect blood pressure from differentangles and do not replace each other. If one only uses SBP (orDBP) to measure blood pressure, then the insufficient informa-tion cannot reveal the danger of too high (low) DBP (or SBP),leading to incorrect conclusion for people with normal SBP (orDBP) but abnormal DBP (or SBP). Second, multiple surrogatesof a latent factor are highly correlated because they share cer-tain similarity elicited by this latent factor. Such high correlationmay cause the multicollinearity problem, resulting in incorrectestimation or divergence of the computer program. The latterexample (see Table 4) shows that incorporating the individualhighly correlated observed variables WAIST, BMI, SBP, DBP,HbA1c, FPG, TC, HDL-C, and TG in a standard analysis causesmulticollinearity and produces confounding results. Finally, in-terpreting the effect of each surrogate separately may be tediousand incomprehensible. For instance, when examining the effectof lipid control on the development of chronic kidney disease(CKD), a standard analysis only tells how the individual ob-served variables TC, HDL-C, and TG influence the hazards ofCKD but never addresses the major concern on the overall im-pact of lipid on CKD. Further, the diverse effects caused by themulticollinearity (Table 4) makes the interpretation even harder.By contrast, the proposed joint analysis groups TC, HDL-C, andTG together into a latent variable “lipid” and provides scientificevidence for the overall effect of this latent variable and thusmakes the interpretation conceptually simple and comprehensi-ble. Simultaneously, the dimension of covariates is also reducedsubstantially.

A number of studies have assessed the interdependence, cau-sations, and correlations among multiple observed variables andlatent factors. The commonly used techniques include factoranalysis (Lawley and Maxwell 1971) and latent variable mod-els (Bollen 1989; Joreskog and Sorbom 1996; Lee 2007; Songand Lee 2012). The latent variable analysis originated in thefield of psychology to relate various mental test results to latent

© 2015 American Statistical AssociationJournal of the American Statistical Association

September 2015, Vol. 110, No. 511, Theory and MethodsDOI: 10.1080/01621459.2014.950083

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Pan et al.: Regression Analysis of AH Model with Latent Variables 1149

traits, such as intelligence. Additionally, it has recently beenused in medical studies to investigate the relationships betweencovariates (e.g., age) and a medical trait (e.g., overall treatmenteffect), or between the endpoint of interest (e.g., kidney fail-ure) and various potential latent risk factors (e.g., glycemia andlipid). A typical modeling framework for such latent variableanalysis consists of two models. The first model is the so-calledconfirmatory factor analysis (CFA) model (Bollen 1989, chap.7; Shi and Lee 2000; Lee 2007, chap. 2) for characterizing latentfactors through multiple correlated observed variables, whereasthe other one is a regression type model for relating observedand latent risk factors to quantities of interest. Sammel and Ryan(1996) measured the overall severity of birth defects using mul-tiple adverse effects and then regressed it on the exposure toanticonvulsants when analyzing the effects of anticonvulsantmedication during pregnancy. Roy and Lin (2000, 2002) sum-marized the effectiveness of treatment practices using three lon-gitudinal outcomes and related the overall treatment effect tocovariates through a linear mixed model when examining theeffectiveness of methadone treatment practice in reducing illicitdrug use.

In this study, we propose a joint model to analyze the sur-vival data with latent variables. The proposed model consistsof two components: a CFA model for characterizing the latentrisk factors, and an AH model to relate the observed and latentrisk factors to the hazard function of interest. To analyze suchkinds of joint models with latent variables, published reports(e.g., Sammel and Ryan 1996; Roy and Lin 2002) have mainlyused the expectation–maximization (EM) algorithm, in whichthe latent variables and/or random effects were treated as miss-ing data, and the maximum likelihood estimates (MLEs) of themodel parameters were obtained by maximizing the expectationof the complete-data log-likelihood function. In our proposedmodel, however, the AH model is nonlinear with an unspecifiedbaseline hazard function, which makes the complete-data log-likelihood function highly intractable. Directly applying the EMalgorithm to analyze the proposed model is challenging. Thus,to address such difficulty, we develop a borrow-strength esti-mation procedure in this study. The procedure is a two-stageapproach. At the first stage, latent variables are characterized bythe CFA model and estimated using the EM algorithm. At thesecond stage, the AH model with the estimated latent variablesis analyzed using the corrected pseudo-score method (Carroll,Ruppert, and Stefanski 1995; Kulich and Lin 2000; Song andHuang 2006). To the best of our knowledge, no study has ana-lyzed the proposed joint survival model with latent variables.

The present research is motivated by a study based on theHong Kong Diabetes Registry, which was established in 1995 aspart of a continuous quality improvement program at the Princeof Wales Hospital in Hong Kong. A 10-year prospective cohortof 3586 Chinese Type 2 diabetic patients who may have expe-rienced CKD, one of the most common diabetic complications,was analyzed. A primary interest of this study is to investigatethe potential risk factors of CKD to prevent diabetic compli-cations and improve the quality of life for diabetic patients.However, some risk factors, such as obesity, blood pressure,glycemia, and lipid, are latent traits that cannot be measuredby a single observed variable. Thus, how these latent risk fac-tors affect the overall development of CKD is of great interest

in medical research. To address this problem, we propose thecurrent joint model with latent variables. The empirical analysis(Section 5) shows that our model provides important insightsinto the overall effects of latent risk factors. Such kinds of overalleffects cannot be revealed by conventional AH models.

The rest of the article is organized as follows. In Section 2,we describe the proposed joint model and present the borrow-strength estimation procedure for regression parameters of in-terest. The asymptotic properties of the proposed estimators areestablished in Section 3. Section 4 presents simulation studiesto evaluate the empirical performance of the proposed methods.An application to the CKD data is reported in Section 5. Sec-tion 6 concludes the article with discussion. Proofs are providedin the Appendix and other details are given in supplementarymaterial.

2. MODEL AND INFERENCE PROCEDURES

2.1 Joint Models

We propose a joint model that involves two major compo-nents. The first component is the CFA model and the secondcomponent is the AH model. For i = 1, . . . , n, let Vi be a p × 1vector of the observed variables and ξ i be a q × 1 vector of latentvariables (factors), where p > q. The CFA model for charac-terizing latent variables through multiple observed variables isdefined as

Vi = Bξ i + εi , (1)

where B is a p × q matrix of factor loadings, ξ i ∼ N (0,�),and εi is a p × 1 vector of random errors independent of ξ i anddistributed as N (0,�ε) with a diagonal �ε . The elements of Breflect the associations between the observed variables and theircorresponding latent variables. In a CFA model, any element ofB can be fixed at preassigned values. This setup allows greatflexibility in selecting an appropriate structure of B to achievea clear interpretation of the latent variables. In substantive re-search, we usually already have a clear idea of the number oflatent variables and the structure of the factor loading matrix. Forexample, in the CKD study, obesity, blood pressure, glycemia,and lipid are important medical traits for assessing the healthprofile of diabetic patients. Our main objective is to examinetheir effects on the risk of CKD. Thus, the number of latentvariables in this study is clearly four. Experts in diabetes and theexisting medical literature (e.g., Song et al. 2009; Wang et al.2014) suggested appropriate observed variables for these latentvariables. Thus, the number of latent variables and the structureof B can be naturally decided based on the study objectives,the meaning of the observed variables suggested by subject ex-perts, and/or the existing literature. Moreover, an exploratoryfactor analysis (EFA) can be used to cross validate the decision.Well-known software in structural equation modeling, such asLISREL (Joreskog and Sorbom 1996), EQS (Bentler 2002), andMplus (Muthen and Muthen 1998–2007), can be used to con-duct the EFA. In the analysis of CKD data, we conducted anEFA using LISREL to verify the decision on the number offactors and the structure of B (see details in Section 5).

To investigate the potential risk factors of a failure time ofinterest, the latent variables characterized by the CFA model,along with some observed covariates, are further related to the

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1150 Journal of the American Statistical Association, September 2015

failure time through an AH model with latent variables. Let Tibe the failure time of interest, Zi be a s × 1 vector of covari-ates, and Ci be the censoring time assumed to be conditionallyindependent of Ti given Zi and ξ i . Let Xi = min(Ti, Ci) bethe observed time and�i = I (Ti ≤ Ci) be the failure indicator,where I (·) is the indicator function. The AH model specifiesthat, given Zi and ξ i , the hazard function of Ti takes the form

λ(t |Zi , ξ i) = λ0(t) + βTZi + γ T ξ i , (2)

where λ0(t) is an unspecified baseline hazard function, and β

and γ are the s × 1 and q × 1 vectors of unknown regressionparameters. Given that ξ i plays a role of risk factors in model(2), the observed variables in model (1) are actually multiplerisk factors that are related to a latent variable (another riskfactor). Thus, the proposed joint model is different from a typicalone in joint modeling literature, in which several post-exposureoutcome variables of interest exist.

Song and Huang (2006) developed a joint analysis to ac-commodate longitudinal covariates measured with error in anAH model. Their first model is essentially a growth curve modelwith measurement error. It evaluates a time-dependent covariatethrough a single surrogate with measurement error at intermit-tent occasions. By contrast, our first model emphasizes the useof “latent variables” to assess latent traits that should be mea-sured by several (usually highly correlated) observed variables.For example, the latent variable “blood pressure” is measured byobserved variables SBP and DBP, and the latent variable “lipid”is measured by observed variables TC, HDL-C, and TG. Thiskind of latent variables contributes an integral part in structuralequation models and has been widely used in the medical, be-havioral, and social sciences (see Bollen 1989; Lee 2007; Songand Lee 2012, and references therein). Our first model is a CFAmodel that is used to form the latent variables in ξ i via groupingappropriate observed variables through a factor loading matrixB. In the CFA analysis, the unknown parameters in B, which canbe regarded as data-driven weights to reflect how the observedvariables contribute to the characterization of a latent variable,are estimated along with the correlations of the latent variablesand the error variances. Clearly, Song and Huang’s model andother measurement error models do not involve such kind oflatent variables and the factor loading matrix. Thus, the CFAmodel in (1) is quite different from the growth curve measure-ment error model of Song and Huang (2006). In formulatingthe AH model, Song and Huang (2006) directly incorporatedmeasurement error covariates into the model. By contrast, wepropose the use of latent variables obtained from the CFA, ratherthan a large number of covariates, to model the hazard functionof interest.

While our model and that of Song and Huang (2006) are dif-ferent in terms of scientific goal, modeling of latent variables,and scope of applications, they share certain similarities. First,the method proposed by Song and Huang (2006) assumed mixedeffects model for time-dependent covariates. Time-independentcovariates with repeated measurements can be viewed as a spe-cial case with the slope equal to 0. Thus, when the B matrix isblock diagonal with each block being all 1’s vector, the CFAmodel can be viewed similarly as a measurement error model.Second, given that both methods involve variables measuredwith errors, they encounter similar difficulty induced by unob-

servable variables and bear the same spirit of using correctedpseudo-score approach in statistical inference.

The proposed model subsumes several statistical models asits special cases. For instance, if p = q = 0, then the proposedmodel reduces to a simple AH model with only fixed effects.If p = q and B = I (identity matrix), then the proposed modelreduces to an AH model with covariates measured with error.If one only considers model (2) and fixes the elements of γ at1, then the proposed model reduces to a frailty AH model withboth fixed and random effects. Compared with the conventionalAH model or frailty AH models (Yin and Ibrahim 2005; Caiand Zeng 2011), the proposed joint model has several appealingfeatures. First, the analytic power of the joint model is greatly en-hanced using the information from multiple indicators. Second,the joint model provides statistical evidence and attractive inter-pretation for the effects of latent traits that are well known buthard to measure directly. The frailty terms in conventional AHmodels only address the dependence among clustered or longitu-dinal observations, whereas the latent variables in the proposedmodel represent medical traits with specific meanings. Theselatent variables not only address the possible heterogeneity inthe data but also provide insights into their impacts on the failuretime of interest. Third, the multicollinearity problem in the con-ventional regression analysis can be eliminated in the proposedjoint model because the highly correlated variables have beengrouped into relatively independent latent factors through theCFA model. Finally, given that the number of latent variables,q, is often much smaller than the number of observed variables,p, the dimension of the risk factors in model (2) can be reducedsignificantly.

Remark 1. (Model Identification). Model (1) is not identi-fiable because for any nonsingular matrix L, we have Vi =Bξ i + εi = BLL−1ξ i + εi = B∗ξ ∗

i + εi , where B∗ = BL andξ ∗i = L−1ξ i ∼ N [0,L−1�(L−1)T ], indicating that parameters

B and � are not simultaneously estimable without imposingan identifiability constraint. In this study, we follow a commonpractice in the literature (Bollen 1989; Lee 2007) to (i) fix thediagonal elements of � to 1, such that � is a correlation matrix.This constraint unifies the scale of the latent variables. (ii) Fixadditional elements in B at preassigned values, such that theonly possible nonsingular matrix L that makes B∗ satisfy theconstraints is the identity matrix.

Remark 2. In model (1), we make a generic assumptionof E(ξ i) = 0. The extension to the case of E(ξ i) �= 0 isstraightforward. When E(ξ i) �= 0, model (2) can be writtenas λ(t |Zi , ξ i) = λ∗

0(t) + βTZi + γ T ξ i , where λ∗0(t) = λ0(t) +

γ T E(ξ i), and ξ i = ξ i − E(ξ i).

Let θ be a vector of the unknown parameters in model (1),and θ be the MLE of θ . Given that model (1) is a standard CFAmodel, the estimation of θ is relatively simple (Bollen 1989;Shi and Lee 2000; Lee and Song 2004). We first focus on theparameter estimation procedure for model (2), and then outlinethe EM algorithm to obtain θ in model (1).

2.2 Borrow-Strength Estimation

We define the counting process Ni(t) = I (Xi ≤ t,�i = 1)and the at risk process Yi(t) = I (Xi ≥ t), i = 1, . . . , n. If the

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latent variables ξ i are observable, α = (βT , γ T )T can be esti-mated based on the pseudo-score functions using the approachof Lin and Ying (1994):

U1(α) =n∑i=1

∫ τ

0{Zi − Z(t)}[dNi(t) − Yi(t)(β

TZi + γ T ξ i)dt],

and

U2(α) =n∑i=1

∫ τ

0{ξ i − ξ (t)}[dNi(t) − Yi(t)(β

TZi + γ T ξ i)dt],

where τ is a prespecified positive constant, such that P (Xi ≥τ ) > 0, Z(t) = S(1)

z (t)/S(0)(t), ξ (t) = S(1)ξ (t)/S(0)(t), S(1)

z (t) =n−1 ∑n

i=1 Yi(t)Zi , S(1)ξ (t) = n−1 ∑n

i=1 Yi(t)ξ i , and S(0)(t) =n−1 ∑n

i=1 Yi(t).However, the latent variables ξ i are unobservable in the pro-

posed joint model. Consequently, the above pseudo-score func-tions are not directly applicable. Thus, to address this problem,we first estimate ξ i based on model (1). For a given θ , an es-timator of ξ i , denoted by ξ i(θ ), can be written as (Lee 2007,chap. 2)

ξ i(θ) = (θ )Vi ,

where (θ ) = (BT�−1ε B)−1BT�−1

ε is a q × p matrix func-tion of θ . Although E{ξ i(θ )|ξ i} = ξ i , a simple replacementof ξ i by ξ i(θ) in the estimating functions U1(α) and U2(α)would lead to biased estimators because the resulting estimat-ing functions do not have zero mean because E{ξ i(θ)⊗2|ξ i} =D(θ) + ξ⊗2

i , where D(θ ) = (BT�−1ε B)−1 is a q × qmatrix func-

tion of θ and a⊗2 = aaT for a vector a. Thus, for a givenθ , we propose the corrected pseudo-score function U(α; θ ) =(U1(α; θ )T ,U2(α; θ )T )T for α as follows:

U1(α; θ ) =n∑i=1

∫ τ

0{Zi − Z(t)}[dNi(t)

−Yi(t){βTZi + γ T ξ i(θ)}dt],

U2(α; θ ) =n∑i=1

∫ τ

0{ξ i(θ) − ξ

∗(t ; θ)}[dNi(t) − Yi(t){βTZi

+γ T ξ i(θ)}dt] +n∑i=1

∫ τ

0Yi(t)D(θ)γ dt,

where ξ∗(t ; θ) = n−1 ∑n

i=1 Yi(t)ξ i(θ )/S(0)(t).For given θ , we estimate α by solving U(α; θ ) = 0. The re-

sulting estimator has an explicit expression:

α =(

n∑i=1

∫ τ

0Yi(t)

{[Zi − Z(t)

(θ)Vi − ξ∗(t ; θ)

]⊗2

−[

0s×s 0s×q0q×s D(θ)

]}dt

)−1

×n∑i=1

∫ τ

0

[Zi − Z(t)

(θ)Vi − ξ∗(t ; θ)

]dNi(t). (3)

Let �0(t) = ∫ t0 λ0(u)du denote the baseline cumulative haz-

ard function. The corresponding estimator of �0(t) is given by

�0(t) ≡ �0(t ; α, θ ) (0 ≤ t ≤ τ ), where

�0(t ; α, θ )

=∫ t

0

n−1 ∑ni=1[dNi(u) − Yi(u){βTZi + γ T(θ)Vi}du]

S(0)(u).

2.3 Inference of CFA Model

Let V = (V1, . . . ,Vn) and ξ = (ξ 1, . . . , ξn). Given that ξ i’sare unobservable, we employed the EM algorithm to obtain θ

in model (1) (e.g., Shi and Lee 2000; Lee and Song 2004). TheEM algorithm consists of two steps: an E-step for calculatingthe conditional expectation of the complete-data log-likelihoodfunction with respect to the distribution of ξ i given the observeddata and the current value of θ , as well as an M-step for updatingθ by maximizing the conditional expectation. The complete-datalog-likelihood function of model (1) is

l(θ |V, ξ ) = −1

2

{n(p + q) log(2π ) + n log |�ε | + n log |�|

+n∑i=1

(Vi − Bξ i)T�−1

ε (Vi − Bξ i) +n∑i=1

ξTi �−1ξ i

}.

The θ can be obtained by iterating the E-step and M-step asfollows. At the rth iteration with the current value θ (r):

E-step: Evaluate Q(θ |θ (r)) = E{l(θ |V, ξ )|V, θ (r)} =∫l(θ |V, ξ )p(ξ |V, θ (r))dξ , where p(ξ i |V, θ )

D= N [(�−1 +BT�−1

ε B)−1BT�−1ε Vi , (�−1 + BT�−1

ε B)−1]�= N (μξi ,�ξ ).

The conditional expectations of the sufficient statistics requiredin the following M-step have explicit forms

E{ξ i |Vi , θ} = μξi and E{ξ⊗2i |Vi , θ} = �ξ + μ⊗2

ξi.

M-step: We maximize Q(θ |θ (r)) with respect to θ to obtain anew value of θ . Let BTj be the jth row of B, φkj be the entry inthe kth row and jth column of �, ψj be the jth diagonal elementof �ε , and Vij be the jth element of Vi . The vector of unknownelements in Bj is denoted by Bj1 and the vector of fixed onesis denoted by Bj2. The subvectors of ξ i corresponding to Bj1

and Bj2 are denoted by ξ i,j1 and ξ i,j2, respectively. Simplecalculations show that the updated estimates of Bj1 and ψj takethe following explicit forms:

Bj1 =[ n∑i=1

E{ξ⊗2i,j1|Vi , θ

(r)}]−1

×n∑i=1

E{(Vij − BTj2ξ i,j2)ξ i,j1|Vi , θ

(r)} ,ψj = n−1

n∑i=1

E{(Vij − BTj ξ i

)2 |Vi , θ(r)}.

The updated estimates of unknown parameters in � do nothave closed forms, so we use the Newton–Raphson algorithmto calculate them iteratively. Let �1 be a vector of the unknownparameters in �,

�new1 = �

old1 −

{∂2Q(θ |θ (r))

∂�1∂�T1

}−1∂Q(θ |θ (r))

∂�1

∣∣∣∣∣�

old1

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1152 Journal of the American Statistical Association, September 2015

The entries in ∂Q(θ |θ (r))∂�1

and ∂2Q(θ |θ (r))∂�1∂�

T1

can be calculated as

∂Q(θ |θ (r))

∂φkj= 1

2tr

(�−1

{ n∑i=1

[E{ξ⊗2i |Vi , θ

(r)}

−�]

}�−1 ∂�

∂φkj

),

∂2Q(θ |θ (r))

∂φkj ∂φls= −1

2tr

(�−1 ∂�

∂φkj�−1

{ n∑i=1

[E{2ξ⊗2i |Vi , θ

(r)}

−�]

}�−1 ∂�

∂φls

).

The expectations required in this step are calculated in theE-step. The convergence of the EM algorithm can be monitoredusing a commonly used stopping criterion

|l(θ (r)|V) − l(θ (r−1)|V)| < δ, (4)

where δ is the predetermined small value (e.g., 0.001) and l(θ |V)is the observed-data log-likelihood function

l(θ |V) = −1

2

{np log(2π ) + n log |�(θ)| +

n∑i=1

VTi �(θ)−1Vi

},

in which �(θ) = B�BT + �ε .The asymptotic covariance matrix of θ can be calculated by

the inverse of the observed information matrix

I (θ) = − ∂2l(θ |V)

∂θ∂θT

∣∣∣∣θ=θ

. (5)

Given that the observed information matrix contains compli-cated matrix differentiation, we approximate it numerically. Letl(θ) = l(θ |V), and let θ j denote the jth element of θ , then

∂2l(θ)

∂θj ∂θk

∣∣∣∣θ=θ

≈ 1

4mjmk[l(θ +mjej +mkek)

−l(θ +mjej −mkek) − l(θ −mjej +mkek)

+l(θ −mjej −mkek)], (6)

where ej is the canonical basis vector that is one at the jthcoordinate, andmj is a constant that is small relative to θj (e.g.,θj × 10−3).

Remark 3. Shi and Lee (2000) and Lee and Song (2004) alsoused the EM algorithm to perform the statistical inference forthe CFA model. The EM algorithm they employed is actuallya Monte Carlo EM (MCEM) algorithm, in which the E-stepapproximates the conditional expectation via the simulated la-tent variables and other unknowns via the Gibbs sampling. Inthe current study, the Monte Carlo step is unnecessary becausethe conditional expectation can be expressed in a closed formof the parameters. Compared with those in Shi and Lee (2000)and Lee and Song (2004), the inference for the CFA model inthe current study is more efficient.

3. ASYMPTOTIC PROPERTIES

Given that the asymptotic properties of θ have been wellstudied in the literature (Amemiya, Fuller, and Pantula 1987;Anderson and Amemiya 1988; Lee 2007, pp. 41–43), we an-alyze the asymptotic properties of α, which is of our primary

interest in the proposed joint model. Let

K(θ ) = [�(θ){�(θ )−1 ⊗ �(θ )−1}�(θ )T ]−1

× �(θ ){�(θ )−1 ⊗ �(θ)−1},Ri(θ ) = [1(θ)K(θ )vec{V⊗2

i − �(θ )}, . . . , p(θ)

× K(θ )vec{V⊗2i − �(θ )}],

Pi(θ ) = [D1(θ )K(θ)vec{V⊗2i − �(θ)}, . . . , Dq(θ)

× K(θ )vec{V⊗2i − �(θ )}],

where ⊗ denotes the Kronecker product of two matrices,vec denotes the operation that converts a matrix into acolumn vector by stacking the rows sequentially, �(θ) =∂(vec�(θ))T /∂θ ,s(θ ) (s = 1, . . . , p) denote the derivativesof the sth column of (θ) with respect to θT , and Dr (θ ) (r =1, . . . , q) denote the derivatives of the rth column of D(θ ) withrespect to θT . Define

S(1)v (t) = n−1

n∑i=1

Yi(t)Vi , and V(t) = S(1)v (t)/S(0)(t),

dMi(t) = dNi(t) − Yi(t)[d�0(t) + {βTZi + γ T(θ)Vi}dt],

H1 = n−1n∑i=1

∫ τ

0{Zi − Z(t)}Yi(t)VT

i dt,

H2 = n−1n∑i=1

∫ τ

0Yi(t){(θ)Vi − ξ

∗(t ; θ)}VT

i dt,

H3 = n−1n∑i=1

∫ τ

0{Vi − V(t)}[dNi(t) − Yi(t){βTZi

+ γ T(θ)Vi}dt],

H4 = n−1n∑i=1

∫ τ

0Yi(t)dt.

Let α0 and θ0 be the true values of α and θ . Using the consistencyof θ , we obtain the consistency of (θ ) and D(θ) (Lemma A.1 ofthe Appendix). Then, we can establish the asymptotic propertiesof α. The results are summarized in the following theorem withthe proof given in the Appendix.

Theorem 1. Under the regularity conditions (C1)−(C3) asstated in the Appendix, α is consistent to α0, and n1/2(α − α0)has asymptotically a normal distribution with mean zero and co-

variance matrix that can be consistently estimated by A−1

�A−1

,

where � = n−1 ∑ni=1 U

⊗2i , Ui = (U

T

i1, UT

i2)T ,

Ui1 =∫ τ

0{Zi − Z(t)}dMi(t) − H1Ri(θ )T γ ,

Ui2 =∫ τ

0{(θ)Vi − ξ

∗(t ; θ)}dMi(t) +

∫ τ

0Yi(t)D(θ)γ dt

−H2Ri(θ )T γ + Ri(θ)H3 + Pi(θ)γ H4,

A =[

A11 A12

AT

12 A22

], A11 = n−1

n∑i=1

∫ τ

0Yi(t){Zi − Z(t)}⊗2dt,

A12 = n−1n∑i=1

∫ τ

0Yi(t){Zi − Z(t)}[(θ )Vi − ξ

∗(t ; θ)]T dt,

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Table 1. Results of Simulation 1—AH model, ξ i ∼ N (0,�), εi ∼ N (0, 0.3I)

Type (I) λ0(t) Type (II) λ0(t) Type (III) λ0(t)

n CR Para Bias SE SEE CP Bias SE SEE CP Bias SE SEE CP

500 20% β 0.015 0.357 0.350 0.945 0.016 0.427 0.418 0.939 0.013 0.393 0.386 0.941γ1 −0.000 0.198 0.188 0.950 −0.002 0.239 0.227 0.953 −0.002 0.219 0.209 0.950γ2 0.008 0.191 0.189 0.945 0.009 0.231 0.228 0.946 0.009 0.212 0.210 0.946

50% β 0.009 0.433 0.442 0.951 0.006 0.489 0.499 0.954 0.006 0.452 0.458 0.953γ1 −0.001 0.248 0.243 0.957 −0.002 0.278 0.275 0.959 0.000 0.257 0.252 0.957γ2 0.012 0.245 0.243 0.959 0.013 0.274 0.275 0.955 0.012 0.252 0.252 0.961

1000 20% β 0.008 0.246 0.246 0.945 −0.001 0.293 0.294 0.951 0.007 0.276 0.272 0.943γ1 −0.002 0.128 0.132 0.966 0.001 0.162 0.160 0.957 0.004 0.156 0.148 0.942γ2 0.005 0.134 0.133 0.946 0.001 0.164 0.160 0.945 0.004 0.149 0.148 0.954

50% β −0.003 0.306 0.311 0.961 0.007 0.361 0.351 0.948 −0.004 0.314 0.322 0.956γ1 −0.003 0.176 0.170 0.945 −0.004 0.194 0.193 0.952 −0.000 0.179 0.177 0.940γ2 0.005 0.170 0.171 0.954 −0.001 0.192 0.193 0.950 0.001 0.178 0.177 0.953

and

A22 = n−1n∑i=1

∫ τ

0Yi(t){(θ)Vi − ξ

∗(t ; θ)}⊗2dt

− n−1n∑i=1

∫ τ

0Yi(t)D(θ )dt.

Theorem 2. Under the regularity conditions (C1)−(C3) asstated in the Appendix, �0(t) converges in probability to�0(t) uniformly in t ∈ [0, τ ], and n1/2{�0(t) −�0(t)} con-verges weakly on [0, τ ] to a zero-mean Gaussian process whosecovariance function at (t, s) can be consistently estimated byϒ(t, s) = n−1 ∑n

i=1 Oi(t)Oi(s), where

Oi(t) =∫ t

0

dMi(u)

S(0)(u)− γ TRi(θ)

∫ t

0V(u)du− W(t)T A

−1Ui ,

and

W(t) =(∫ t

0Z(u)T du,

∫ t

0V(u)T du(θ)T

)T.

4. SIMULATION STUDIES

4.1 Simulation 1

This simulation was conducted to evaluate the finite sampleperformance of the proposed methodology. We considered ajoint model defined by (1) and (2) withp = 6, q = 2, and s = 1.The parameters in model (1) were set as

BT =[b11 b21 b31 0∗ 0∗ 0∗

0∗ 0∗ 0∗ b42 b52 b62

],� =

[1∗ φ12

φ21 1∗

]and �ε = diag(ψ1, ψ2, ψ3, ψ4, ψ5, ψ6), where the elementswith asterisk are fixed. The unknown parameters in the CFAmodel include the factor loadings bjks, the correlation parame-ters φ12 = φ21, and the error variances ψj , j = 1, . . . , 6. Thetrue population values were set as b11 = b21 = b31 = b42 =b52 = b62 = 0.8, φ12 = φ21 = 0.2, and ψ1 = · · · = ψ6 = 0.3.The nonoverlapping structure of B implies that latent variablesξ1 and ξ2 are measured by {V1, V2, V3} and {V4, V5, V6}, re-spectively. The diagonal elements of � were fixed at 1.0 foridentifiability.

In model (2), covariate Zi was independently drawn from theBernoulli distribution with success probability of 0.5. The truepopulation values of the parameters were set as β = 1 and γ =(0.2, 0.5)T . The failure time Ti was generated based on model(2) with three types of the baseline hazard function: (I) λ0(t) = 3(constant), (II) λ0(t) = 4t + 3 (linear), and (III) λ0(t) = 6t2 +3 (nonlinear). The censoring time Ci was generated from auniform distribution on (0, c), where c was selected as {1.5, 1.2,1.3} and {0.47, 0.40, 0.45} to yield two levels of censoring rate(CR), 20% and 50%, for the three types of λ0(t).

In implementing the EM algorithm to estimate θ , the conver-gence was monitored and claimed with δ = 0.001 (see (4)). Theα was then obtained using (3). The results presented below arebased on 1000 replications with sample sizes n = 500 and 1000.

Table 1 presents the simulation results on the estimate ofα = (β, γ T )T , whereas Table 2 presents the simulation resultson the estimate of θ . In these tables, Bias is the sampling meanof the estimate minus the true value, SE is the sampling stan-dard error of the estimate, SEE is the sampling mean of standarderror estimate, and CP is the 95% empirical coverage probabil-ity for the parameter based on the normal approximation. The

Table 2. Results of Simulation 1—CFA model: ξ i ∼ N (0,�),εi ∼ N (0, 0.3I)

n = 500 n = 1000

Para Bias SE SEE CP Bias SE SEE CP

b11 0.001 0.039 0.038 0.944 0.002 0.027 0.027 0.945b21 0.001 0.039 0.038 0.948 0.002 0.027 0.027 0.949b31 0.000 0.038 0.038 0.954 0.001 0.027 0.027 0.951b42 0.001 0.038 0.038 0.952 −0.000 0.027 0.027 0.938b52 −0.000 0.038 0.038 0.956 −0.001 0.026 0.027 0.951b62 0.000 0.037 0.038 0.962 0.000 0.027 0.027 0.953φ12 0.000 0.050 0.050 0.945 −0.000 0.036 0.035 0.937ψ1 −0.001 0.031 0.030 0.944 −0.001 0.021 0.021 0.945ψ2 −0.001 0.030 0.030 0.949 −0.001 0.021 0.021 0.945ψ3 −0.001 0.030 0.030 0.941 −0.000 0.021 0.021 0.957ψ4 −0.001 0.030 0.030 0.950 −0.001 0.021 0.021 0.945ψ5 −0.001 0.030 0.030 0.942 0.000 0.021 0.021 0.948ψ6 −0.002 0.029 0.030 0.956 0.000 0.021 0.021 0.951

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1154 Journal of the American Statistical Association, September 2015

Table 3. The factor loadings obtained based on EFA through LISREL

Variable Factor 1 Factor 2 Factor 3 Factor 4 Unique variance

WAIST 1.005 0.009 −0.057 −0.025 0.000BMI 0.830 0.021 0.029 −0.023 0.307SBP 0.005 0.839 0.038 −0.039 0.311DBP 0.024 0.737 0.007 0.001 0.449HbA1c 0.040 −0.040 −0.963 −0.047 0.102FPG 0.000 0.012 −0.723 0.019 0.470TC −0.072 0.151 −0.158 0.303 0.825HDL-C −0.245 0.052 −0.063 −0.271 0.843TG −0.033 −0.034 0.040 1.019 0.000

results indicate that our proposed method performed satisfacto-rily for all the situations under consideration. Specifically, theestimators produced were virtually unbiased. Good agreementexists between the estimated and the empirical standard errors.The coverage probabilities of the 95% confidence intervals wereclose to the nominal level. As expected, the performance of theproposed estimators was improved when the sample size wasincreased from n = 500 to 1000.

To investigate the performance of the proposed method underthe situation when the factor loading matrix B has an overlappingstructure, we reconducted the above simulation by using anoverlapping B as follows:

BT =[b11 b21 b31 b41 0∗ 0∗

b12 0∗ 0∗ b42 b52 b62

],

where the elements with asterisk are fixed and bjk’s are unknownfactor loadings whose true population values were set as 0.8. Theoverlapping structure of B implies that latent variables ξ1 and ξ2

are measured by {V1, V2, V3, V4} and {V1, V4, V5, V6}, respec-tively. Using the proposed estimation procedure, we obtainedthe results under different types of λ0(t) and sample sizes. Theresults with Type (I) λ0(t) and n = 500 are presented in TableS1 of supplementary material. Those obtained under other sce-narios are similar and therefore not reported. The results showthat the performance of parameter estimates does not changeregardless of the structure of B.

We also consider a setup that mimics the real data exampleof Section 5 to check the robustness of the proposed method tomisspecification of B. The model and parameter values are thesame as those of the CKD study except b81 = −0.245. Based onthe data generated from this new model, we conduct the analy-sis with a misspecified B, wherein nonzero b81 is misspecifiedas zero. Table S2 of supplementary material shows that the es-timation results related to the AH model are quite robust, butthose related to the CFA model are slightly influenced under thecase considered. Nevertheless, it is worthwhile to note that thestatistical inference for the CFA is sensitive to misspecificationof the number of latent factors. When fitting a wrong modelwith extra (nonexistent) or missing (existent) latent factors, wefound that the computer algorithm always diverged.

Further, we examined the empirical performance of the pro-posed method in which the error variance increases. We againused the previous simulation setting but increased the error vari-anceψj , j = 1, . . . , 6 from 0.3 to 1.0. The results obtained with

Type (1) λ0(t) and n = 500 are reported in Table S3 of supple-mentary material. Except for slightly increased standard errorestimates, the parameter estimates are still unbiased and closeto the ones obtained under smaller error variance.

4.2 Simulation 2

The main goal of this simulation is to investigate the sensitiv-ity of the proposed method to the normality assumption of thelatent variables and the random errors. As suggested by one ofthe anonymous reviewers, we consider several nonnormal cases,including the severely skewed distributions used in Huang andWang (2000). The model setup is the same as that of Simula-tion 1 except Case (1): ξ i ∼ 2

3N (μ1,�1) + 13N (μ2,�2), where

μ1 = (−0.5,−0.5)T , μ2 = (1.0, 1.0)T , and {σ11, σ12, σ22} in�1 = �2 are {0.5,−0.3, 0.5}; Case (2): ξ i ∼ {Gamma(4, 2) −2}I2, where Gamma(α1, α2) stands for the gamma distributionwith shape parameter α1 and rate parameter α2, and I2 is atwo-dimensional identity matrix; Case (3): ξ i ∼ beta(3, 1)I2,where beta(α1, α2) stands for the beta distribution with shapeparameters α1 and α2; Case (4): εij ∼ Gamma(5, 4) − 5/4 forj = 1, . . . , 6, where εij is the jth element of εi ; Case (5):εij ∼ t(3), where t(3) stands for the t distribution with degreesof freedom 3; and Case (6): εij ∼ beta(3, 1).

The results with Type (I) λ0(t) and n = 500 under Cases(1)–(6) are presented in Tables S4 and S5 of supplementarymaterial. Most of the parameter estimates, especially those re-lated to the AH model, are quite robust to the nonnormality ofξ i and εi . While nonnormal ξ i mainly influence the standarderror estimates of the factor loadings in B, nonnormal εi mostlyimpact on those of the error variances in �ε . We also checkedthe sensitivity of the estimation results when ξ i and εi followother nonnormal distributions. Similarly, the inference relatedto the AH model is very robust to the nonnormality of ξ i andεi , whereas the standard errors of some parameters in the CFAmodel are overestimated/underestimated. Considering that theprimary goal of the current study is to investigate the potentialrisk factors of hazards of interest, the inaccurate standard errorestimates of error variances in the CFA model would not causea problematic inference of the AH model.

5. APPLICATION TO CKD DATA

The proposed methodology was applied to a study of CKDfor Type 2 diabetic patients as discussed in Section 1. The mainobjective is to investigate the potential risk factors that affectthe hazards of CKD. The failure time of CKD was calculatedas the periods from enrollment to the date of the first clinicalendpoint or January 31, 2009, whichever came first. The failure(clinical endpoint) of CKD was defined by diabetic nephropathy(DNP) plus follow-up estimate glomerular filtration rate (eGFR)< 60 (Wang et al. 2014). The CR of CKD is approximately73%. Possible risk factors include those relevant to the patients’characteristics, such as age at enrollment, duration of diabetes,sex (1 = female, 0 = male), waist circumference (WAIST), bodymass index (BMI), SBP, DBP, HbA1c, FPG, TC, HDL-C, andTG. Based on the medical meaning of these observed variablesand the existing literature (Song et al. 2009; Wang et al. 2014),grouping WAIST and BMI into a latent variable “obesity,” SBPand DBP into a latent variable “blood pressure,” HbA1c and

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Pan et al.: Regression Analysis of AH Model with Latent Variables 1155

WAIST ( )

BMI ( )

SBP ( )

DBP ( )

HbA1c ( )

FPG ( )

TG ( )

TC ( )

HDL-C ( )Blood

pressure( )

Glycemia( )

Hazardfunc�on

Age atenrollment

( )

Dura�on ofdiabetes

( )

Sex( )

Lipid( )

Obesity( )

0.019(0.001)

0.014(0.002)

-0.003(0.002)

0.003(0.001)

0.004(0.001)

0.004(0.002)

0.007(0.002)

0.955(0.024)

0.865(0.023)

0.714(0.031)

0.861(0.036)

0.733(0.038)

0.937(0.046)

0.341(0.019)

-0.304(0.019)

0.983(0.048)

Figure 1. Path diagram of the proposed joint model, along with parameter estimates and their standard error estimates (in parentheses), inthe analysis of the CKD data. In the path diagram, the latent variables are enclosed by ellipses, whereas the observed variables are enclosed bysquares.

FPG into a latent variable “glycemia,” and TC, HDL-C, and TGinto a latent variable “lipid” is natural and acceptable. Hence, inbuilding a CFA model, we propose the number of latent variables(factors) to be four, a nonoverlapping factor loading matrixB, and a clear interpretation of the latent variables. To cross-validate the decision, we conducted an EFA using LISREL. Theestimated number of factors is four, and the estimated factorloadings reported in Table 3 clearly show that Factor 1 canbe interpreted as “obesity” (i.e., large loadings associated withWAIST and BMI, and small loadings associated with others).Meanwhile, Factors 2, 3, and 4 can be interpreted as “bloodpressure,” “glycemia,” and “lipid,” respectively. In determiningthe factor loading matrix, we fixed the parameters associatedwith small loadings at zero, consequently obtaining absolutelyclear interpretations of the latent variables. Therefore, a jointmodel with latent variables was proposed to examine how theobserved risk factors, such as age, duration of diabetes, andsex, as well as latent risk factors stipulated above, influence thehazards of CKD.

Let V = (V1, V2, . . . , V9)T = (WAIST, BMI, SBP, DBP,HbA1c, FPG, TC, HDL-C, TG)T , each of which is standard-ized prior to the analysis, ξ = (ξ1, ξ2, ξ3, ξ4)T = (obesity, bloodpressure, glycemia, lipid)T , and Z = (Z1, Z2, Z3)T = (age at en-rollment, duration of diabetes, sex)T . We employed models (1)

and (2) with p = 9, q = 4, s = 3, and the following nonover-lapping factor loading matrix to conduct the analysis:

BT =

⎡⎢⎢⎣b11 b21 0 0 0 0 0 0 00 0 b32 b42 0 0 0 0 00 0 0 0 b53 b63 0 0 00 0 0 0 0 0 b74 b84 b94

⎤⎥⎥⎦ ,

where the zeros are fixed elements.In estimating θ , we set δ = 0.001. The θ and α were then

obtained based on the EM algorithm and (3), and their stan-dard error estimates were obtained based on (5), (6), andTheorem 1.

The proposed joint model and the analytical results are re-ported in Figure 1. For clarity, the less important parametersin �ε and � are not reported. Several findings are obtainedfrom Figure 1. First, the age at enrollment has significantlypositive effect on the hazards of CKD, indicating that olderpatients have higher risks of developing CKD. Second, the du-ration of diabetes also has significantly positive effect on thehazards of CKD. Patients with longer duration of diabetes aremore predisposed to CKD. Third, different from age and du-ration of diabetes, the effect of sex on CKD is not significant.Thus, male and female diabetic patients have equal risks ofsuffering this complication. Fourth, all latent risk factors have

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Table 4. Comparison of the standard and joint analyses of the CKD data

Standard analysis Joint analysisλ(t |Zi ,Vi) = λ0(t) + βT Zi + γ T Vi λ(t |Zi , ξ i) = λ0(t) + βT Zi + γ T ξ i

Variable Est SEE Variable Est SEE

Age at enrollment 0.0152 0.0014 Age at enrollment 0.0191 0.0013Duration of diabetes 0.0115 0.0017 Duration of diabetes 0.0140 0.0017Sex −0.0024 0.0025 Sex −0.0032 0.0022WAIST 0.0041 0.0023BMI −0.0022 0.0022 Obesity 0.0028 0.0013SBP 0.0130 0.0019DBP −0.0062 0.0017 Blood pressure 0.0038 0.0016HbA1c 0.0085 0.0018FPG −0.0020 0.0019 Glycemia 0.0040 0.0015TC −0.0030 0.0014HDL-C −0.0020 0.0014 Lipid 0.0071 0.0017TG 0.0072 0.0017

significantly positive effects on the hazards of CKD, indicatingthat obesity, hypertension, higher level of blood glucose, andbad control of blood fat increase the risk of CKD. Finally, allthe estimated factor loadings are highly significant, indicatingstrong associations between each latent variable and its corre-sponding observed variables. These findings have public healthimplications, especially for the aggressive control of risk fac-tors to prevent CKD or other complications for Type 2 diabeticpatients and to improve their quality of life.

For comparison, we conducted a standard analysis by regard-ing the observed variables, V1, . . . , V9, as independent covari-ates in the AH model. The results are reported in Table 4, whichshows the diverse effects of the observed indicators of a latentrisk factor. For instance, the effect on the hazards of CKD issignificant for WAIST but nonsignificant for BMI, positive forSBP but negative for DBP, significant for HbA1c but nonsignif-icant for FPG, and negative for TC, nonsignificant for HDL-C,but positive for TG. A further check reveals high correlationsamong the observed variables. For instance, the sample corre-lation between WAIST and BMI, SBP and DBP, and Hb1Acand FPG, are 0.827, 0.615, and 0.687, respectively. These highcorrelations elicit multicollinearity and result in diverse ef-fects that are confounding and hard to interpret. By contrast,the proposed joint analysis avoids the multicollinearity prob-lem and provides important insights for the latent risk factorsof CKD.

Given that there is a slight disagreement between the LIS-REL analysis (b81 = −0.245) and our specification (b81 = 0)for the CFA model, we reanalyzed the dataset under the samesetup except allowing b81 �= 0. The parameter estimates, alongwith those obtained under b81 = 0, are reported in Table S6 ofsupplementary material. The two analyses yield very similarresults. In particular, the parameter estimates related to the AHmodel are identical up to the first three decimals. A sensitivityanalysis in Simulation 1 also demonstrated the robustness ofthe proposed method to such a misspecification. Although weadopted a nonoverlapping structure of B based on the sugges-tion of medical experts and the existing medical literature in thisstudy, the use of overlapping or nonoverlapping B in substantiveresearch should be decided on a problem-to-problem basis.

6. DISCUSSION

In this article, we propose a joint model to investigate howobserved and latent risk factors influence the hazard rate ofinterest. The proposed model is novel and advantageous overconventional AH models in terms of both analytic power andinterpretability. A hybrid procedure that combines the EM al-gorithm and a borrow-strength estimation method is proposedto conduct the statistical inference. The consistency and asymp-totic normality of the parameter estimators are established. Sim-ulation results demonstrate that the proposed method performedwell. The novel model and methodology were applied to a realproblem of identifying potential risk factors of diabetic kidneydisease. Important insights for preventing such complicationwere obtained.

The borrow-strength estimation procedure developed in thisarticle is essentially a two-stage approach. The first stage es-timates the latent variables and the parameters of the CFAmodel by the EM algorithm, whereas the second stage es-timates the regression coefficients of the AH model by us-ing the estimated latent variables with corrected pseudo-scorefunctions. A possible alternative estimation method is to imple-ment the EM algorithm based on the complete-data likelihoodfunction of the proposed joint model. However, this complete-data likelihood function involves high-dimensional integrationand unspecified baseline hazard function. The nonlinearity ofthe AH model makes the E-step extremely challenging becausef (ξ i |Vi , Xi,�i, θ ,α, λ0(t)) is very complicated, and thus sam-pling becomes difficult. In addition, the unspecified baselinehazard function leads to a tedious maximization in the M-step.Thus, the feasibility of such complete-data likelihood-basedmethod is uncertain and requires further investigation.

Our approach can be regarded as a generalization of the con-ventional AH model to a broader class of models through theincorporation of latent variables. To the best of our knowledge,our study is the first to introduce latent variables into the survivalmodeling framework. Further, our approach can be extended tomore generalizable models. First, the current study requiresthe normal assumption of the latent variables and the mea-surement error. However, officially assessing this assumption is

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impossible because the latent variables are not directly observ-able. Although the previous simulation demonstrates that theinference of the AH model is robust to the violation of thisnormal assumption, it is of interest to develop novel methodsthat do not require such assumption. One promising directionin relaxing this assumption is considering an asymptoticallydistribution-free approach (Browne 1984) to obtain the param-eter estimation of the CFA model and then modifying the cor-rected pseudo-score method accordingly to perform the statis-tical inference for the AH model. Second, our model can alsobe extended to a longitudinal setting, in which the CFA modelforms time-dependent latent variables, and the AH model re-veals the dynamic effects of observed covariates and latent riskfactors on the failure time of interest. Third, the current uni-variate AH model can be generalized to a multivariate versionto accommodate multivariate failure time data. The estimationprocedure proposed in this article can be extended to analyze themultivariate AH model with latent variables in a straightforwardmanner.

Finally, the proposed joint analysis can be applied to othermodel frameworks, such as the PH model

λ(t |Zi , ξ i) = λ0(t)exp(βTZi + γ T ξ i), (7)

where λ0(t), β, γ , Zi , and ξ i are defined similarly as in theCFA-AH joint model. A standard analysis of PH model is basedon estimating function Up(α) = (Up1(α)T , Up2(α)T )T with

Up1(α)

=n∑i=1

∫ τ

0

{Zi −

∑nj=1 Yj (t)exp(βTZj + γ T ξ j )Zj∑nj=1 Yj (t)exp(βTZj + γ T ξ j )

}dNi(t),

Up2(α)

=n∑i=1

∫ τ

0

{ξ i −

∑nj=1 Yj (t)exp(βTZj + γ T ξ j )ξ j∑nj=1 Yj (t)exp(βTZj + γ T ξ j )

}dNi(t).

However, Up(α) is not applicable because of the existence oflatent variables. Again, a simple replacement of ξ i by ξ i(θ)in Up(α) will lead to biased estimation. Here, the correc-tion of Up(α) pertaining to exponential functions of ξ i ismore difficult than that corresponding to quadratic functionsof ξ i . Inspired by the previous work (e.g., Nakumura 1992;Kong and Gu 1999) on PH model with covariate measurementerror, we propose corrected estimating function Up(α; θ ) =(Up1(α; θ )T ,Up2(α; θ )T )T with

Up1(α; θ) =n∑i=1

∫ τ

0

{Zi −

∑nj=1 Yj (t)exp

{βT Zj + γ T ξ j (θ)

}Zj∑n

j=1 Yj (t)exp{βT Zj + γ T ξ j (θ)

}}dNi (t),

Up2(α; θ)

=n∑i=1

∫ τ

0

{ξ i (θ) −

∑nj=1 Yj (t)exp

{βT Zj + γ T ξ j (θ)

}ξ j (θ)∑n

j=1 Yj (t)exp{βT Zj + γ T ξ j (θ)

}}dNi (t)

+n∑i=1

∫ τ

0D(θ)γ dNi (t).

The estimator of (α, θ ) can be obtained similarly. As expected,the derivation of asymptotic properties of the parameter esti-mates for CFA-PH joint model will be more technically in-volved. The details are not reported here. The same rationaleis applicable to the additive-multiplicative hazards model thatsubsumes both PH and AH models as special cases. As sug-gested by the associate editor, considering linear transforma-

tion models (Zeng, Yin, and Ibrahim 2005; Zeng, Lin, and Lin2008) is a promising attempt for formulating the approach in amore general way, but it requires new theoretical and compu-tational developments. The feasibility of obtaining a correctedestimating function analytically under the transformation modelframework requires further investigation.

APPENDIX: PROOFS OF ASYMPTOTIC RESULTS

We use the same notation defined earlier and take all limits at n →∞. Let s(1)

z (t), s(1)ξ (t), and s(0)(t) be the limits of S(1)

z (t), S(1)ξ (t), and

S(0)(t), respectively. To study the asymptotic properties of the proposedestimators, we need the following regularity conditions:

(C1) The matrix �(θ0) is positive definite; all partial derivatives ofthe first three orders of �(θ) with respect to the elements of θ arecontinuous and bounded in a neighborhood of θ0; the matrix �(θ )is of full rank in a neighborhood of θ0.

(C2) �0(τ ) < ∞, P (Xi ≥ τ ) > 0, and Zi is bounded almost surely.(C3) A is nonsingular, and

A =[

A11 A12

AT12 A22

], A11 = E

[∫ τ

0Yi(t) {Zi − ez(t)}⊗2 dt

],

A12 = E

[∫ τ

0Yi(t) {Zi − ez(t)}

{ξ i − eξ (t)

}Tdt

],

A22 = E

[∫ τ

0Yi(t)

{ξ i − eξ (t)

}⊗2dt

],

where ez(t) = s(1)z (t)/s(0)(t), and eξ (t) = s(1)

ξ (t)/s(0)(t).

Note that under model (1), Vi is distributed as N (0,�(θ)). LetS∗ = n−1

∑n

i=1 V⊗2i . The likelihood function of V = (V1, . . . ,Vn) is

L(θ |V) = (2π )−pn/2|�(θ )|−n/2 exp

{−1

2

n∑i=1

VTi �(θ )−1Vi

},

and the θ is the maximizer ofL(θ |V). Some straightforward calculationyields that θ is the minimizer of the following discrepancy function:

F ∗(θ ) = log |�(θ )| + tr{S∗�(θ )−1} + p log(2π ).

To prove Theorems 1 and 2, we need the following lemma.

Lemma A.1. Under the condition (C1), (θ ) and D(θ ) are consistentto (θ0) and D(θ0), respectively. Furthermore,

θ − θ0 = K(θ0)vec{S∗ − �(θ0)} + op(n−1/2), (A.1)

(θ ) − (θ0) = n−1n∑i=1

Ri(θ0) + op(n−1/2), (A.2)

D(θ ) − D(θ0) = n−1n∑i=1

Pi(θ0) + op(n−1/2), (A.3)

where K(θ ), Ri(θ ), and Pi(θ) are defined in the first paragraph ofSection 3.

Proof. The maximum likelihood estimation theory of the CFAmodel (Amemiya, Fuller, and Pantula 1987; Anderson and Amemiya1988; Lee 2007, pp. 41–43) indicates that θ is consistent to θ0, thatis, θ converges in probability to θ0. In addition, using the continu-ous mapping theorem and consistency of θ , we can obtain the con-sistency of (θ ) and D(θ ). Let F ∗(θ ) = ∂F ∗(θ)/∂θ , and F ∗(θ) =∂2F ∗(θ )/∂θ∂θT . A straightforward calculation shows that

F ∗(θ ) = −�(θ )(�(θ )−1 ⊗ �(θ )−1)vec(S∗ − �(θ)),

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1158 Journal of the American Statistical Association, September 2015

and

F ∗(θ ) = �(θ)(�(θ)−1 ⊗ �(θ )−1)�(θ)T

−2�(θ ){�(θ)−1 ⊗ �(θ )−1(S∗−�(θ ))�(θ )−1}�(θ)T

−�(θ){Iρ ⊗ (�(θ )−1 ⊗ �(θ )−1)vec(S∗ − �(θ))},where ρ is the dimension of θ . The uniform law of large numbersimplies that F ∗(θ ) converges in probability to a nonrandom matrixfunction uniformly in θ . In particular, F ∗(θ0) converges in probabilityto �(θ0)(�0(θ0)−1 ⊗ �0(θ0)−1)�(θ 0)T . For k = 1, . . . , ρ, let F ∗

k (θ)denote the kth element of F ∗(θ ). It follows from the Taylor expansionthat

F ∗k (θ ) = F ∗

k (θ0) + ∂F ∗k (θ )

∂θT

∣∣∣∣θ=θk∗

(θ − θ0),

where θ k∗ is on the line segment between θ and θ 0. Note that F ∗(θ ) = 0.By the consistency of θ , one can show that θ k∗ (k = 1, . . . , ρ) convergein probability to θ0. Then,(

∂F ∗1 (θ )

∂θ

∣∣∣∣θ=θ1∗

, . . . ,∂F ∗

ρ (θ )

∂θ

∣∣∣∣∣θ=θρ∗

)T

converges in probability to the limit of F ∗(θ0), which is nonsingu-lar by the regularity conditions. Note that by the multivariate centrallimit theorem, F ∗(θ0) = Op(n−1/2).Using the uniform convergence ofF ∗(θ ) and consistency of θ , we obtain (A.1). Similarly, by the Taylorexpansion,

(θ ) − (θ0) = [1(θ0)(θ − θ0), . . . , p(θ0)(θ − θ0)]

+ op(n−1/2),

which together with (A.1) gives (A.2). In a similar manner, we obtain(A.3). �

Proof of Theorem 1. Define

dMi(t) = dNi(t) − Yi(t){λ0(t) + βT0 Zi + γ T0 (θ0)Vi}dt.Note that E{(θ0)Vi |ξ i} = ξ i . Then, under model (2), Mi(t) are zero-mean stochastic processes, i = 1, . . . , n. Thus, it follows from LemmaA.1 of Lin and Ying (2001) that

U1(α0; θ0) =n∑i=1

∫ τ

0{Zi − Z(t)}dMi(t)

=n∑i=1

∫ τ

0{Zi − ez(t)}dMi(t) + op(n1/2). (A.4)

Using (A.2), we have

U1(α0; θ ) − U1(α0; θ0) = −nH1{(θ ) − (θ0)}T γ 0

= −n∑i=1

H1Ri(θ0)T γ 0 + op(n1/2),

(A.5)

where H1 is defined in Section 3, and H1 is the limit of H1. It followsfrom (A.4) and (A.5) that

U1(α0; θ ) = U1(α0; θ0) + {U1(α0; θ ) − U1(α0; θ0)}

=n∑i=1

Ui1 + op(n1/2), (A.6)

where

Ui1 =∫ τ

0{Zi − ez(t)}dMi(t) − H1Ri(θ0)T γ 0.

Following arguments similar to those in the proof of (A.4), we obtain

U2(α0; θ0) =n∑i=1

∫ τ

0{ξ i(θ0) − ξ

∗(t ; θ0)}dMi(t)

+n∑i=1

∫ τ

0Yi(t)D(θ0)γ 0dt

=n∑i=1

∫ τ

0{(θ0)Vi − eξ (t)}dMi(t)

+n∑i=1

∫ τ

0Yi(t)D(θ0)γ 0dt + op(n1/2). (A.7)

Some straightforward calculation yields

U2(α0; θ ) − U2(α0; θ0) = −nH∗2{(θ ) − (θ0)}T γ 0

+ n{(θ) − (θ0)}H∗3

+ n{D(θ) − D(θ0)}γ 0H4,

where

H∗2 = n−1

n∑i=1

∫ τ

0Yi(t){(θ0)Vi − ξ

∗(t ; θ0)}VT

i dt,

H∗3 = n−1

n∑i=1

∫ τ

0{Vi − V(t)}[dNi(t) − Yi(t){βT0 Zi

+ γ T0 (θ )Vi}dt],and H4 is defined in Section 3. Using Lemma 1, we obtain

U2(α0; θ ) − U2(α0; θ0) = −H2

n∑i=1

Ri(θ0)T γ 0 +n∑i=1

Ri(θ0)H3

+n∑i=1

Pi(θ0)γ 0H4 + op(n1/2), (A.8)

where H2, H3, and H4 are the limits of H∗2, H

∗3, and H4, respectively.

Thus, using (A.7) and (A.8), we obtain

U2(α0; θ ) = U2(α0; θ0) + {U2(α0; θ ) − U2(α0; θ0)}

=n∑i=1

Ui2 + op(n1/2), (A.9)

where

Ui2 =∫ τ

0{(θ0)Vi − eξ (t)}dMi(t) +

∫ τ

0Yi(t)D(θ0)γ 0dt

− H2Ri(θ0)T γ 0 + Ri(θ0)H3 + Pi(θ0)γ 0H4.

Let Ui = (UTi1,UT

i2)T . Then it follows from (A.6) and (A.9) that

U(α0; θ ) =n∑i=1

Ui + op(n1/2), (A.10)

which is a sum of n independent identically distributed zero-meanrandom vectors plus an asymptotically negligible term. The law oflarge numbers and the multivariate central limit theorem show thatn−1U(α0; θ ) → 0 in probability and n−1/2U(α0; θ ) converges in dis-tribution to a normal random vector with mean zero and covariancematrix � = E{U⊗2

i }. Note that

α − α0 = n−1A−1

U(α0; θ ), (A.11)

and A → A in probability by the consistency of (θ ) and D(θ ). Then,based on (A.11), α converges in probability to α0, and n1/2(α − α0)is asymptotically normal with mean zero and covariance matrix

A−1�A−1, which can be consistently estimated by A−1

�A−1

. �

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Pan et al.: Regression Analysis of AH Model with Latent Variables 1159

Proof of Theorem 2. Note that

dNi(u) − Yi(u){βT

Zi + γ T(θ)Vi

}du− Yi(u)λ0(u)du

= dMi(u) − Yi(u)[(β − β0)T Zi + γ T {(θ ) − (θ0)}Vi

+(γ − γ 0)T(θ0)Vi]du.

Thus,

�0(t) −�0(t) = n−1

∫ t

0

∑n

i=1 dMi(u)

S(0)(u)− γ T {(θ ) − (θ0)}

×∫ t

0V(u)du− W(t)T (α − α0),

where

W(t) =(∫ t

0Z(u)T du,

∫ t

0V(u)T du(θ0)T

)T.

Let s(1)v (u) denote the limit of S(1)

v (u), ev(u) = s(1)v (u)/s(0)(u), and

ew(t) =(∫ t

0ez(u)T du,

∫ t

0ev(u)T du(θ0)T

)T.

Then using the functional central limit theorem (Pollard 1990), togetherwith (A.2), (A.10), and (A.11), we have that uniformly on [0, τ ],

�0(t) −�0(t) = n−1n∑i=1

Oi(t) + op(n−1/2), (A.12)

where

Oi(t) =∫ t

0

dMi(u)

s(0)(u)− γ T0 Ri(θ0)

∫ t

0ev(u)du− ew(t)T A−1Ui .

In view of the consistency of α, from the uniform law of largenumbers (Pollard 1990) and the multivariate central limit theorem,�0(t) converges in probability to �0(t) uniformly in t ∈ [0, τ ], andn1/2{�0(t) −�0(t)} converges in finite-dimensional distributions to azero-mean Gaussian process. Since Oi(t) (i = 1, . . . , n) can be writ-ten as sums or products of monotone functions of t and are thus tight(van der Vaart and Wellner 1996). Thus, n1/2{�0(t) −�0(t)} is tightand converges weakly to a zero-mean Gaussian process with covari-ance function at (t, s) given byϒ(t, s) = E{Oi(t)Oi(s)}, which can beconsistently estimated by ϒ(t, s) defined in Theorem 2. �

SUPPLEMENTARY MATERIALS

Tables S1–S6 are presented in the online supplementary ma-terials.

[Received October 2013. Revised May 2014.]

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