Research Article Analysis of Heart Transplant Survival Data Using Generalized Additive Models

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1 Hindawi Pubishing Corporation Computationa and Mathematica Methods in Medicine Voume 2013, Artice ID , 7 pages Research Artice Anaysis of Heart Transpant Surviva Data Using Generaized Additive Modes Masaaki Tsujitani 1 and Yusuke Tanaka 2 1 Department of Engineering Informatics, Osaka Eectro-Communication University, Osaka , Japan 2 Cinica Information Division Data Science Center, EPS Corporation, Osaka , Japan Correspondence shoud be addressed to Masaaki Tsujitani; ekaaf900@ricv.zaq.ne.jp Received 13 March 2013; Accepted 7 May 2013 Academic Editor: Frank Emmert-Streib Copyright 2013 M. Tsujitani and Y. Tanaka. This is an open access artice distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the origina work is propery cited. The Stanford Heart Transpant data were coected to mode surviva in patients using penaized smoothing spines for covariates whose vaues change over the course of the study. The basic idea of the present study is to use a ogistic regression mode and a generaized additive mode with B-spines to estimate the surviva function. We mode surviva time as a function of patient covariates and transpant status and compare the resuts obtained using smoothing spine, partia ogistic, Cox s proportiona hazards, and piecewise exponentia modes. 1. Introduction Cox s proportiona hazards mode has been proposed based on the reationship between surviva and the patient characteristics observed when the patient entered the study [1]. When the vaues of covariates change over the course of the study, however, a number of theoretica probems with respect to the baseine surviva function and the baseine cumuative hazard function need to be soved [2]. Severa prognostic modes [3 6]have become as widey used as Cox s proportiona hazards mode for the anaysis of surviva data having time-dependent covariates. The present study examines the noninear effects of the evoution of the covariates over time using penaized smoothing spines. Cox sproportionahazardsmodepostuatesthatthe hazard at time t is the product of two components: h (t) =h 0 (t) exp [ I i=1 b i x i ], (1) where b =(b 1,...,b I ) is a vector of coefficients. The proportiona hazards assumption is that the baseine hazard h 0 (t) is a function of t but does not invove the vaues of covariates. Severa prognostic modes for heart transpant surviva data have been deveoped using Cox s regression anaysis, and the vaues of a covariates are determined at the time when thepatiententeredthestudy[7 9]. However, situations may exist in which the vaues of covariates change over the course of the study. A time-dependent mode uses the foow-up data to estimate the effect of the evoution of the covariates over time. The reative hazard h(t)/h 0 (t) then depends on time t, and thus the proportiona hazards assumption is no onger satisfied [6, 10]. The time-dependent covariates X d =(x d 1,...,x d I ) are provided for patient no. d, wherex d 1 is the midpoint of the th time interva. Given the continuous survivor time, piecewise modes arise from the partition of the time axis into disjointed intervas. Biganzoi et a. [11, 12] showthatby treating the time interva as an input variabe in a feedforward neura network, it is possibe to estimate smoothed discrete hazards as conditiona probabiities of faiure. To appy a generaized additive mode (GAM), discretization of one-month or one-week intervas must be appied for the continuous survivor time with time-fixed covariates. However we cannot determine which discretization, one-month or one-week, shoud be appied; that is, the discretization is not initiay unique. In the case of time-dependent covariates X d, x d 1 is initiay determined as the midpoint of the th time interva for patient no. d. It is fairy straightforward

2 2 Computationa and Mathematica Methods in Medicine to extend the mode to survivor data with time-dependent covariates. Furthermore, by regarding a GAM as an extension of a partia ogistic mode (PLM), the unknown parameters can be estimated by maximizing the partia og-ikeihood [13, 14]. We use the Stanford Heart Transpant data, which has been coected to mode surviva in patients. Athough Cox s proportiona hazards mode is not appicabe in the case of time-dependent covariates, the survivor function can be estimated by taking h 0 (t) in (1) to be the piecewise exponentia hazard. Crowey and Hu [7], Aitkin et a. [8], and Lawess [9] used piecewise exponentia modes and potted the surviva function. Lagakos [15] aso examined a graphica technique for assessing covariates in Cox s proportiona hazards mode basedonapermutationoftheobservedrankstatistic.most previous studies compared the hazard functions to assess the effect of transpantation on surviva by fitting pretranspant and posttranspant data separatey. The difficuty is that there is no easiy used measure of the difference between the transpanted and nontranspanted groups. Inferences must be based on a comparison of the estimated function. As Aitkin et a. [8] pointedout, there are aways dangers in making inferences about the effect of treatment without adequate contro groups. We thus provide an anaysis that incudes pretranspant and posttranspant data simutaneousy as time-dependent covariates. It shoud be emphasized that patients who are not transpanted constitute a contro group reative to patients who have undergone heart transpantation by the same covariates. We use the 1977 version of the data, as given in Crowey and Hu [7], which is for 103 patients. As four of the transpanted patients have incompete data on the mismatch score, ouranaysisisbasedon99patientstoassessforwhatvaues of these covariates, if any, transpantation is ikey to proong surviva. More than 30 percent of cases are censored. In these data, surviva times are the number of days unti death foowing a heart transpant, as in Lagakos [15]. A distinctive feature of the present probem is that some of the covariates are time-dependent (and possiby random). For exampe, Tabe 1 shows the vaues of covariates for transpant status (i.e., waiting time), age at transpant (in years), mismatch score (as time-dependent covariates), and previous openheart surgery for patient no. 18. The previous surgery status does not change with time. In order to extend this setting, the covariate for transpant status is taken as an indicator (coded as 0 before the point of transpant and 1 after transpant). A the other time-dependent covariates are treated as being zero before transpant but changing from zero to the actua vaue of the particuar covariate at the time of transpant. Patient no. 18 generated six observations. The proposed methods aow for simutaneous investigation of severa covariates and provide estimates of the surviva function as we as the significance. 2. Generaized Additive Modes By extending the PLM for the grouped data based on partia ikeihood as introduced by Cox [16] andefron[17], a PLM can be proposed for ungrouped data [13, 14] having timedependent covariates for the discrete hazard rate h d of patient no. d at the time interva : PLM: n ( h d 1 h d )=β 0 +β 1 x d 1 +β 2 x d 2 +β 3 x d 3 + +β I x d I. In recent years, a variety of powerfu techniques have been deveoped for exporing the functiona form of effects. Here, GAM with smoothing spines proposed by Hastie et a. [18, 19] wi be used by extending the generaized inear mode (GLM) in McCuagh and Neder [20], where the inear predictor in (2) is specified as a sum of smooth functions s(x) with twice continuousy differentiabe functions of some or a of the covariates: GAM: n ( h d 1 h d )=β 0 +s 1 (x d 1 ) +s 2 (x d 2 )+s 3 (x d 3 ) + +s I (x d I ). The smooth functions in (3) can be represented as s 0 (x) = s 1 (x) =. s I (x) = q 0 j=1 q 1 j=1 q I j=1 β j b 0j (t), β q0 +jb 1j (x), β qi 1 +jb Ij (x), where q 1,q 2,...,q I are the numbers of knots, and β = (β 0,β 1,...,β q0,β q0 +1,β q0 +2,...,β q 0 +q 1,...,β q0 +q 1 + +q I 1 +1,β q0 +q 1 + +q I 1 +2,...,β q0 +q 1 + +q I ). For time interva of patient no. d,wehavethefoowing definitions: δ d = δ d = 1: patient no. ddied at time interva for the th cinic visit, 0: otherwise, 1: patient no. dwas censored at time interva, 0: o.w, (2) (3) (4) (5)

3 Computationa and Mathematica Methods in Medicine 3 Time interva Midpoint x 1 (days) Tabe1:Covariatevauesforpatientno.18. Transpant status x 2 Age at transpant x 3 (years) Mismatch score x 4 Previous surgery x k d k d = (δ d 1,δ d 1,δ d 2,δ d 2, =(k d...,δ d 1,δ d 1 )=(0,0,...,0),,δ d )=(0,0,...,0,δ d ), where k d is the history of defauts and is censored for the first 1time intervas of patient no. d and k d =(k d,δ d ) is the same history extended to incude δ d. Using the above mode and notation, Tsujitani and Sakon [13] derived the fu og-ikeihood for a patients n L=n L(β)+ n d=1 with the partia og-ikeihood n L(β) = n d=1 d 1 =1 d =1 n (1 h d n p(δ d )+δ d d (6) k d ) (7) n h d d +(1 δ d d ) n (1 h d d )}. Athough n L(β) is not a og-ikeihood in the usua sense, it possesses the usua asymptotic properties under fairy broad conditions, as proven in Andesen and Gi [21]. To avoid overfitting, such modes are estimated by penaized maximum ikeihood n L(β) = n d= d 1 =1 I i=1 n (1 h d ) +δ d d n h d d λ i s i (t)} 2 dt, (8) +(1 δ d d ) n (1 h d d )} where λ i are smoothing parameters that contro the trade-off between the fit and the smoothness. The functions s i (x) in (9) are represented by the B-spine basis functions b i (x);see,for detais, Tsujitani et a. [14]. (9) Two mode-fitting issues remain. The first concerns the seection of smoothing parameter λ i in (9). The optimum smoothing parameter choice is outweighed by the easy identification of a covariate s functiona form as we as the appicabiity of estabished inferentia methods to shortterm surviva prediction. In order to seect the smoothing parameters, the agorithm deveoped by Wood [22 24] can be appied by minimizing generaized cross vaidation (GCV) as an approximation to eave-one-out CV [23]. It shoud be noted that the eaving-one-out CV is to aow the deetion of ony one observation. On the other hand, the ordina V-fod CV divides the data randomy into V groups so that their sizes are as neary equa as possibe. This partition shoud be made to avoid possibe biases, as described in Zhang [25]. In many probems, the ordina V-fod CV is, thus, unsatisfactory in severa respects for time-dependent covariates. Appying this kind of data structure to the CV agorithm, we obtain insights into how the partition of data shoud be carried out. A natura extension of the V-fod CV agorithm by setting V =nis to aow the deetion of the patient with severa observations; see, for detais, Tsujitani et a. [14]. A second issue is the goodness-of-fit test of the mode. After choosing the optimum smoothing parameters via the variant V-fod CV agorithm, the deviance aows us to test the goodness-of-fit: Dev =2(n L max n L c ), (10) where n L c denotes the maximized partia og-ikeihood under some current GAM and the og-ikeihood for the maximum (fu) mode n L max iszero.thedeviance(10) is, however, not even approximatey an χ 2 distribution for the case in which ungrouped binary responses are avaiabe; see, for exampe, Coett [26], Landwehr et a. [27], and Tsujitani and Sakon [13]. The number of degrees of freedom required for the test for significance using the assumed χ 2 distribution for the deviance is a contentious issue. No adequate distribution theory yet exists for the deviance. The reason for this is somewhat technica; for detais, see Section 3.8 in Coett [2]. Consequenty, the deviance on fitting a mode to binary response data cannot be used as a summary measure of the goodness-of-fit of the mode. Thus, bootstrapping is appied to the deviance(10) in order to obtain the goodness-of-fit; see, for detais, Efron and Tibshirani [28] and Tsujitani et a.[14].

4 4 Computationa and Mathematica Methods in Medicine Tabe 2: Optimum smoothing parameters. Covariates Variant V-fod CV GCV Midpoint (x 1 ) Age (x 3 ) Mismatch score (x 4 ) Exampe As an initia mode for the Stanford Heart Transpant data, we empoy Mode I: s (x 1 ) +x 2 +s(x 3 ) +s(x 4 ) +x 5. (11) GCV is ony an approximation of eaving-one-out CV. Aternativey the variant V-fod CV is eaving-one-out CV based on each of n = 99 patients to aow the deetion of the patient with severa observations. By using variant V-fod CV and GCV for the initia mode, the optimum smoothing parameters for GAM are determined as shown in Tabe 2.By using a backward eimination procedure, we obtain Frequency Deviance Figure 1: Histogram of the bootstrapped Dev(b) for B = Mode II: s(x 1 )+x 2 +s(x 3 )+x 5. (12) The ikeihood ratio (LR) statistic based on deviance can be computed to test the significance of spine effects (i.e., noninearity). For exampe, the spine effect of Midpoint(x 1 ) can aso be tested by using Mode III: x 1 +x 2 +s(x 3 ) +x 5. (13) s(v1, 2.85) By comparing Mode II with III, the reduction in the vaue of deviance is Δ = 4.59 with 1.85d.f. This is significant at the 10%eve.Thespineeffectfor Age(x 3 ) isnotsignificant.we thus obtain the fina optimum GAM Mode IV: s(x 1 )+x 2 +x 3 +x 5 (14) with a variantv-fod score of Figure 1 shows a histogram of the bootstrapped Dev(b) for the optimum mode. The bootstrap estimate of the 95th percentie Dev is Dev = ThecomparisontoDev= of (10)suggeststhatthemodefitsthedata. Figure 2 shows the estimated contribution s(x 1 ) of Midpoint(x 1 ) tonĥ d /(1 ĥ d )}, together with the ±2 standard deviation (SD) curves for the fina optimum Mode IV.Thespineeffectsofx 1 are visuaized in Figure 2. Figure 2 nicey shows that the spine function s(x 1 ) of dying decreases initiay as the midpoint x 1 increases. Subsequenty, however, s(x 1 ) is staby maintained after 500 midpoint. For the purpose of comparison, Figure 3 shows the estimated contribution s(x 1 ) for GCV. From Figure 3, it is cear that the estimated s(x 1 ) of x 1 is fat unti 1500 and then tumbes because of too sma smoothing parameter (i.e., overfitting), as shown in Tabe 2.SovariantV-fod CV is superior to GCV. The anayses in this exampe are carried out using ibrarymgcv} in R. The surviva function for our discretized situation is Pr () = (1 h d i ). (15) 1 i Figure 2: Estimated contribution s(x 1 ) of x 1 (soid curve) and s(x 1 )±2SD[ s(x 1 )] (dashed curves). s(v1, 8.96) V Figure 3: Estimated contribution s(x 1 ) of x 1 (soid curve) and s(x 1 )±2SD[ s(x 1 )] (dashed curves) for GCV. V1

5 Computationa and Mathematica Methods in Medicine 5 Tabe 3: P vaues for the significance test of covariates. Covariates GAM Partia ogistic Proportiona hazard Piecewise exponentia Transpant status (x 2 ) < Age (x 3 ) Previous surgery (x 5 ) The average probabiity of surviva at time interva for patient no. d in group g canbeestimatedas S g () = 1 n [g] n [g] Pr [g] d=1 d (), g=1,2, =1,2,..., (16) where n [g] is the tota number of patients at time interva in group g and Pr [g] d () is the surviva function Pr() of patient no. d at time interva in group g; see, for exampe, Thomsen et a. [29]. Thedataareanayzedtodiscoverwhichvauesofthe covariates are ikey to be of benefit. We compare the resuts obtained using smoothing spine, partia ogistic, Cox s proportiona hazards, and piecewise exponentia modes [7, 8]. The resuts of fitting the various modes are summarized in Tabe 3.ItiscearfromTabe 3 that (i) a covariates for the smoothing spine mode are strongy significant (in particuar, Crowey and Hu [7] suggestedaquadraticeffectsofage)and (ii) there is itte difference between Cox s proportiona hazard mode and the piecewise exponentia mode. It shoud be noted that binary covariates in the mode remain inear. AsshowninAitkineta.[8, Figure2],itismoreappropriate to compare survivorship functions if the hazards are not proportiona. One point of interest is a comparison of surviva experience of transpanted and nontranspanted patients. Our proposa for comparing the surviva function is to use the estimated surviva function for ony 41 heart transpanted patients who died to assess the efficacy of transpantation and the effects of covariates by means of modeing the change in hazard at transpantation by using (15) and(16). Our particuar interest is the effect of waiting time on posttranspant surviva according to severa modes. In Figure 4, two time periods are used (group 1: up to 20 days; group 2:onger than 20 days).figure 4 shows a comparison of the estimated surviva function. The estimated surviva functions based on the smoothing spine suggest that patients with a short waiting time face a greater eary risk than those who had a onger waiting time. However the estimated surviva functions based on piecewise exponentia modes cannot revea the difference between a short and ong waiting times. Our method provides an aternative to Arjas [10] suggestion of comparing separate estimates of cumuative hazardbasedontheevesofthewaitingtime.athough Arjas [10] did not incude waiting time as a covariate in Cox s proportiona hazard mode because of nonproportionaity issues,weusedtranspantstatus(i.e.,waitingtime),whichis Probabiity of surviva Time Spine (<21 days) Spine (>20 days) Piecewise exponentia (<21 days) Piecewise exponentia (>20 days) Figure 4: Surviva function describing the effect of the waiting time for 41 heart transpanted patients who died. strongy significant for the smoothing spine mode according to theresutsshownintabe 3. A fundamentay different type of anaysis was suggested by Crowey and Hu [7] to investigate the effect of transpantation with a ow mismatch score. They pointed out that transpantation may be beneficia for younger patients ony basedonregressioncoefficientsforcox sproportionahazards mode, but our concusion can be derived by graphica anaysis as we as significance testing of covariates. Defining a ow mismatch score as ess than or equa to one for a 29 heart transpanted patients [7], Figure 5 shows a graphica comparison of the estimated surviva function for two groups, namey, the younger patients (ess than 50 years od at acceptance) and oder patients (greater than or equa to 50 at acceptance). From Figure 5, it is cear that oder patients face a greater eary risk than younger patients; see, for detais, Crowey and Hu [7,Chapter5]withrespecttothecutpoints for ow mismatch score as ess than or equa to one and the younger patients as ess than 50 years od. Kabfeish and Prentice [30, Section 4.6.3] estimated the cutpoint for age, based on a 65 transpanted patients, as Figure 6 shows a graphica comparison of the estimated surviva function for two groups, namey, the younger patients (ess than or equa 46 years od at acceptance) and oder patients (greater than 46 at acceptance). As Kabfeish and Prentice point out, transpantation is beneficia for younger patients.

6 6 Computationa and Mathematica Methods in Medicine Probabiity of surviva Time Spine (<50 years) Spine (>50 years) Piecewise exponentia (<50 years) Piecewise exponentia (>50 years) Figure 5: Surviva function describing the effect of the age of transpantation for patients with a ow mismatch score. Probabiity of surviva Time Spine (<47 years) Spine (>46 years) Piecewise exponentia (<46 years) Piecewise exponentia (>46 years) Figure 6: Surviva function describing the effect of the age for a 65 transpanted patients. 4. Concusion We confined our attention to time-dependent covariates. Aowing covariates to vary over the duration of the study not ony enabed us to study time-varying risk factors, but aso provided a fexibe way for modeing censored surviva data using penaized smoothing spines. We iustrated the procedures using data of the Stanford Heart Transpant data. By introducing the maximum ikeihood principe into GAM, (i) we coud visuaize the spine effects of the midpoint of the time interva; (ii) the smoothing parameters coud be seected by using variantv-fod CV; (iii) the goodness-of-fit of GAM coud be tested based on bootstrapping; (iv) the estimated average probabiities of surviva enabed us to investigate the effect of transpantation with a ow mismatch score for two groups, namey, the younger and oder patients. References [1] J. P. Kein and M. L. Moeschberger, Surviva Anaysis, Springer, New York, NY, USA, 2nd edition, [2] D. Coett, Modeing Surviva Data in Medica Research,Chapman and Ha, London, UK, [3] P. A. Murtaugh, E. R. Dickson, G. M. van Dam et a., Primary biiary cirrhosis: prediction of short-term surviva based on repeated patient visits, Hepatoogy, vo. 20, no. 1I, pp , [4] E.Christensen,P.Schichting,P.K.Anderseneta., Updating prognosis and therapeutic effect evauation in cirrhosis with Cox s mutipe regression mode for time-dependent variabes, Scandinavian Journa of Gastroenteroogy, vo.21,pp , [5]E.Christensen,D.G.Atman,J.Neubergereta., Updating prognosis in primary biiary cirrhosis using a time-dependent Cox regression mode, Gastroenteroogy, vo.105,no.6,pp , [6] D. G. Atman and B. L. de Stavoa, Practica probems in fitting a proportiona hazards mode to data with updated measurements of the covariates, Statistics in Medicine, vo. 13, no. 4, pp , [7] J. Crowey and M. Hu, Covariance anaysis of heart transpant surviva data, Journa of the American Statistica Association, vo. 72, pp , [8] M.Aitkin,N.Laird,andB.Francis, AreanaysisoftheStanford Heart Transpant data, Journa of the American Statistica Association,vo.78,pp ,1983. [9] J. F. Lawess, Statistica Modes and Methods for Lifetime Data, John Wiey, New York, NY, USA, 2nd edition, [10] E. Arjas, A graphica method for assessing goodness of fit in Cox s proportiona hazards mode, Journa of the American Statistica Association, vo. 83, pp , [11] E. Biganzoi, P. Boracchi, L. Mariani, and E. Marubini, Feed forward neura networks for the anaysis of censored surviva data: a partia ogistic approach, Statistics in Medicine, vo.17, pp , [12] E. Biganzoi, P. Boracchi, and E. Marubini, A genera framework for neura network modes on censored surviva data, Neura Networks,vo.15,no.2,pp ,2002. [13] M. Tsujitani and M. Sakon, Anaysis of surviva data having time-dependent covariates, IEEE Transactions on Neura Networks,vo.20,no.3,pp ,2009. [14] M. Tsujitani, Y. Tanaka, and M. Sakon, Surviva data anaysis with time-dependent covariates using generaized additive modes, Computationa and Mathematica Methods in Medicine,vo.2012,ArticeID986176,9pages,2012. [15]S.W.Lagakos, Thegraphicaevauationofexpanatoryvariabes in proportiona hazard regression modes, Biometrika, vo.68,no.1,pp.93 98,1981.

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