Parthasarathy, S

Vallinayagam, V

Venkatesan, P

NIRT Institutional Repository

344  X INDIAN JOURNAL OF APPLIED RESEARCHVolume : 4 | Issue : 1  | Jan 2014 | ISSN - 2249-555XRESEARCH PAPER MathematicsA Comparative Study of Life Time Models in the Analysis of Survival DataV. Vallinayagam S. Parthasarathy P. VenkatesanDepartment of Mathematics, St.Josephes Engineering College, ChennaiDepartment of Mathematics, SRM University-Ramapuram Campus, Ramapuram, Chennai -89Department of Statistics, National Institute for Research in Tuberculosis, ChennaiKEYWORDS Bone-Morrow transplantation, Deviance, MLE, Weibull, Log-normal, Proportional Hazard, Kaplan-MeierABSTRACT In this paper, an empirical comparison is made between two parametric models namely, Weibull and Log-normal and the semi-parametric Cox model in the analysis of survival data. The bone-morrow transplantation data is used for the comparison. The Lognormal model gave better fit than the other models in terms of deviance.1. INTRODUCTIONSurvival analysis can be described as a branch of statistics which handles with death in biological organisms and failure in mechanical systems. Survival methods are applied for a vast array of social phenomena including births, marriages, divorces, job terminations, promotions, arrests, migrations, and revolutions [1]. That is used to describe, explain, or pre-dict the occurrence and timing of events. This is called as reli-ability theory or reliability analysis in engineering [1]. Survival analysis focuses on time to event data. In the most general way, it contains techniques of  positive valued random vari-ables, such as, time to death, time to onset (or relapse) of a disease etc,. Some methods of survival analysis are purely de-scriptive (e.g.,Kaplan-Meier estimation of survival functions), but most applications involve estimation of regression mod-els, which come in a wide variety of forms [3]. These models are typically very similar to linear or logistic regression mod-els, except that the dependent variable is a measure of the timing or rate of event occurrence. Traditionally only a single event is considered in survival analysis. Recurring event or repeated event models relax that assumption. The study of recurring events is relevant in systems reliability and in areas of social sciences and medical research.  A key feature of all methods of survival analysis is the ability to handle right censoring, a phenomenon that is almost always present in longitudinal data. Right censoring occurs when some indi-viduals do not experience any events, implying that an event time cannot be measured. Introductory treatments of survival analysis for social scientists can be found in Teachman (1983), Allison (1984, 1995), Tuma and Hannan (1984), Kiefer (1988), Blossfeld and Rohwer (2001), and Box-Steffensmeier and Jones (2004). For a biostatistical point of view, see Collett (2003), Hosmer and Lemeshow (2003), Kleinbaum and Klein (2005), or Klein and Moeschberger (2003).    2. MODELS AND METHODSIn this section we discussed about the parametric and semi-parametric models.  2.1: WEIBULL MODELThe Weibull distribution is mainly used in connection with lifetime applications. It can be used to represent many dis-tributions as a function of the shape parameter . The density function is Greater significance is attached to the distribution function, however, in practical applications:  where t = variable, T = Characteristic life and b= Shape parameter, F(t) = frequency, f(t) = probability density for “moment” t.2.2: LOG-NORMAL MODELThe Log-normal distribution is a distribution that is asym-metrical on one side and which exhibits only positive values. Many interrelationships in nature have a positive skew, left steep and right flat distribution.  An illustrated explanation of a feature with non-symmetrical distribution is that the feature cannot undershoot or overshoot a certain boundary value. A significant example is the distribution of time values that can-not be negative. Logarithms are used to achieve values with approximately normal distribution particularly in the case of distributions that are limited to the left by the value 0. The creation of a Log-normal distribution may be attributed to the fact that many random variables interact multiplicatively. In contrast, the normal distribution is created by the additive interaction of many random variables. The Log-normal distri-bution is of particular significance in biology and economics applications.  The probability density is 2.3: PROPORTIONAL HAZARD MODELThe proportional hazards model was introduced in 1972 by D. R. Cox in order to estimate the effects of different covari-ates influencing the times to the failures of a system. This model has been employed for different applications in life-time data analysis. Because of its generality and flexibility, this model was quickly and widely adopted in various fields like biomedical, reliability and economics. Cox’s proportional hazard is expressed as where (t) h0 is the hazard function which is dependent on time only and without influence of covariates.  The positive functional (γz)  is dependent on the effects of different factors, which have multiplicative effect on the hazard function. The proportionality assumptions is. INDIAN JOURNAL OF APPLIED RESEARCH  X 345 Volume : 4 | Issue : 1  | Jan 2014 | ISSN - 2249-555XRESEARCH PAPERThe hazard at different z values are in constant proportion for all t > 0, hence the name for proportional hazard.  2.4: KAPLAN-MEIER ESTIMATEKaplan-Meier estimate is one of the best options to be used to measure the survival fraction (1958). This estimate is also called as “product limit estimate”. It involves computing of probabilities of occurrence of event at a certain point of time. Kaplan-Meier method is a nonparametric approach for sur-vival analysis. It incorporates information from all of the observations, both censored and uncensored by consider-ing survival to any point in time as a series of steps defined by the observed survival and censored times (Hosmer and Lemeshow, 1999) [18].The survival probability at any particular time is calculated by the formula given below:where   = Number of patients living at the start,  = Number of patients died.For each time interval, survival probability is calculated as the number of patients surviving divided by the number of patients at risk. Patients who have died, dropped out, or move out are not counted as “at risk” i.e., patients who are lost are considered “censored” and are not counted in the denominator. Total probability of survival till that time interval is calculated by multiplying all the probabili-ties of survival at all time intervals preceding that time (by applying law of multiplication of probability to calculate cumulative probability). Although the probability calcu-lated at any given interval is not very accurate because of the small number of events, the overall probability of surviving to each point is more accurate. There are three important SAS procedures available for analyzing survival data: LIFEREG, LIFETEST and PHREG. Procedure LIFEREG is a parametric regression procedure for modeling the dis-tribution of survival time with set of variables. Procedure LIFETEST is a non-parametric procedure for estimating the survivor function, comparing survival curves, and testing the association of survival time with other variables. Pro-cedure PHREG is semi-parametric procedure that fits the proportional hazard model.3. DATA BASEIn this section, we have considered the Bone-Morrow trans-plantation data for empirical comparison. The SAS (Statistical Analysis Software) package was used for calculation [2]. The bone-morrow transplantation data involves 137.  The follow-ing variables are considered for modeling whose descrip-tions are given in the table 1.Table 1: List of variable names Age-pt Patient age in yearsAge-don Donors age in yearsSex-pt Patient sex (1-Male, 0-Female)Sex-don Donors sex (1-Male, 0-Female)Pat-cmv Patients CMV status (1-Positive, 0-Nega-tive)Don-cmv Donors CMV status (1-Positive, 0-Negative)FAB It is a way of classification ruleHosp Hospital name (1-The Ohio state Uviversity, 2-Alferd, 3St.Vincent, 4-Hahnemann)MTX It is a modified classification of  FABAcut-indi Acute GVHD indicatorChro-indi Chronic indicatorPlate-indi Platelet recovery indicatorTime Time t0Courtesy: Survival Analysis by John P. Klein and L. Moe-schbegerTable 2: Parameter Estimates of the Models346  X INDIAN JOURNAL OF APPLIED RESEARCHVolume : 4 | Issue : 1  | Jan 2014 | ISSN - 2249-555XRESEARCH PAPERTable 3: The Mean and Median Estimates4. RESULTSThe Kaplan-Meier curves for the hospital sex and FAB for bone-morrow data are given in Fig 4.1 to 4.3. The deviance of Weibull distribution is 407.18, Log-normal is 389.58 and proportional Hazard is 665.49. If we compare between the two parametric models namely Weibull and Log-normal, Log-normal model is the best fit for this data because the deviance value of Log-normal is less than the deviance of Weibull. If we compare between the parametric and semi-parametric models, the semi-parametric model that is pro-portional hazard is not fit for this data because the deviance of proportional hazard is higher than the two other models namely Weibull and Log-normal. We notice that, among the four hospitals, Hahnemann hos-pital’s  estimated value is higher than other three hospitals (see figure 4.2). Also we observed that, Male patient’s sur-vival time is better than female patient’s survival time (see figure4.3). Figure4.1: survival time curve for Hahneman Hospital3000.002500.002000.001500.001000.00500.000.00Time_Death1.00.80.60.40.20.0Cum Survival4.00-censored3.00-censored2.00-censored1.00-censored4.003.002.001.00HospSurvival FunctionsFigure 4.2: Survival time curves for Hospitals       3000.002500.002000.001500.001000.00500.000.00Time_Death1.00.80.60.40.20.0Cum Survival1.00-censored.00-censored1.00.00sex_ptSurvival FunctionsFigure 4.3: Survival time curves for sex3000.002500.002000.001500.001000.00500.000.00Time_Death1.00.80.60.40.20.0Cum Survival1.00-censored.00-censored1.00.00FABSurvival FunctionsFigure 4.4: Survival time curves for FABThe regression coefficients for the covariate along with devi-ance are given in table 2 for the Weibull, lognormal and Cox models 4. CONCLUSIONWe have referred an article “Cure Models for Estimating Hos-pital-Based Breast Cancer Survival” they have documented the utility of a mixture model to estimate the cure fraction and compare it with other approaches [18]. The variables an-alyzed were tumor stage, postoperative pathology of patho-logic tumor residue (TR: negative or positive) and pathologic nodal status (PN: negative or positive). Lognormal kernel’s deviance was least when compared with exponential and Weibull distributions. The deviance of the non PH cure model was the least of all the models in this study.Also we have referred an article “Comparison of  Five Sur-vival Models: Breast Cancer Registry Data from Ege Univer-sity Cancer Research  Center”[7], in that article, Gompertz distribution is the best fit distribution based on the lowest AIC value, by comparing Weibull, Gamma, Log-logistic, Log-INDIAN JOURNAL OF APPLIED RESEARCH  X 347 Volume : 4 | Issue : 1  | Jan 2014 | ISSN - 2249-555XRESEARCH PAPERREFERENCE [1]. Allison P.D, Hardy M, Bryman A (2013), “Event History Analysis”. SAGE Publications, 46 | [2]. Allison P.D. (2010), “Survival Analysis Using SAS A Practical Guide.” 2nd Edition, Cary NC: SAS institute. | [3]. Breslow, N. E. (1975), “Analysis Of Survival Data Under The Proportional Hhazards Models.” International Statistical Review, 43; 45-7, | [4]. Cox, David R. (1972), “Regression models and life-tables.” Journal of the Royal Statistical Society, Series B, 34 187-220. | [5]. Curt- Ronniger. (2012), “Reliability Analysis With Weibull.” 12th Edition. | [6]. David D. Hanagal and Alok D. Dabade. (2013), “A Comparative Study At Shared Frailty Models For Kidney Infection Data With Generalized Exponential Baseline Distribution.” Journal of Data Science, 11, 109-142. | [7]. David G. Kleinbaum and Michael Kline. (2005), “Survival analysis: A Self Learning Text.” Springer, 2nd Edition. | [8]. Elvan Akturk Hayat, Aslisurer, Burak Uyar, Omar Dursun, Mehmet N. Orman, Gul Kitapcioglu, MD. (2010), “Comparison Of Five Survival Models: Breast Cancer Registry Data From Ege University Cencer Research Center.”Turkiye Klinikleri Journal of Medical Sciences, 30, 1665-74. | [9]. Hosmer DW, Lemeshow S (1999). “Applied Survival Analysis: Regression Modeling of Time to Event Data.” New York, Wiley | [10]. John P. Klein and Melvin L. Moeschberger. (2003), “Survival analysis: Techniques For Censored And Truncated Data.” Springer 2nd Edition. | [11]. Johnson, Norman L, Kotz, Samuel, Balakrishnan N. (1994), “14: Lognormal distributions, Continuous univariate distributions.” Wiley series in Probability and Mathematical Statistics 1011: Applied Probability and Statistics New York, 2nd Edition, John Wiley and Sons. | [12] John Fox. (2002),“Cox proportional-Hazards regression for survival data Appendix to An R and S-plus companion to applied regression.”, Will Rios. | [13]. Khard Limpert EC, Werner A. Stahel, and Markus ABBT. (2001), “Log-normal Distributions Across The Sciences: Keys and Clues.” Biosciences, 51(5). | [14]. Manish Kumar Goel, Pradeep Khanna, Jugal kishore. (2010), “Understanding Survival Analysis: Kaplan-Meier Estimate.” International Journal Of Ayurveda Research, 1; 274-278. | [15]. Mara Tableman and Jong sung Kim. (2003), “Survival analysis using S: Analysis Of Time To Event Data.” CRC press. | [16]. Papoulis Pillai. (2002), “Probability Random Variables, And Stochastic Process.” Tata Mc Graw-Hill 4th Edition. | [17]. Rockette H, Antle C, Klimbko LA. (1974), “Maximum Liklihood estimation with the Weibull model.”Journal Of The American Statistical Association, 69, 246-249. | [18]. Ranganathan Rama, Rajaraman Swaminathan, Perumal Venkatesan. (2010), “Cure Models for Estimating Hospital-Based Breast Cancer Survival.” Asian Pacific journal of Cancer Prevention, 11. | [19]. Turnbull B.W and Weiss, L. (1978), “A Likelihood Ratio Statistic For Testing Goodness Of Fit With Randomly Censored Data.” Biometrics 34, 364-375. | [20]. Vilijandas Bagdonavicius, Mikhail Nikulin. (2010), “Accelerated Life Models Modeling And Statistical Analysis.” CRC press. | [21]. Vaupel, J. W, Manton, K. G, and Stallard. E. (1979), “The Impact Of Heterogeneity In Individual Frailty On The Dynamics Of Mortality.” Demography, 16, 439-452. | [22].Wenge Guo.(2011),“Survival Analysis.” Springer. | [23]. Klein JP, JD Rizzo, M.J Zhang and N Keiding. (2001), “Statistical Methods for the Analysis and Presentation of the Results of Bone Marrow Transplants. Part 1: unadjusted analysis.”Division of Biostatistic, Medical College of Wisconsin, Milwauukee, Wi, USA, 28: 909-15 | normal and Gompertz distributions.   In our article, in the parametric models, Log-normal distribu-tion has the lowest deviance value than the deviance value of Weibull distribution. So we can conclude that, among the parametric models, Log-normal distribution is best fit model for this data. If we compare the deviance of the semi-para-metric model proportional hazard with the deviance of the parametric models Weibull and Log-normal distributions, proportional hazard has the highest deviance value so we can conclude that this model is not fit for this data.  From the Kaplan-Meier estimator for the variables Hospital, patient sex and FAB. From the Table 3, we can conclude that, Hahnemann Hospital’s patient survival time is more than the other three hospitals. Hahnemann hospital’s patient survival time is three times of Alferd hospital’s patient survival time. If we compare patient sex wise, we can conclude that male patient’s survival time is higher than female patient’s survival time. If we compare FAB wise FAB grade 4 or 5 and AML is lower survival time than other FAB classifications survival time.

A comparative study of life time models in the analysis of survival data

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A comparative study of life time models in the analysis of survival data

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