1. Copyright © 2013, SAS Institute Inc. All rights reserved. АНАЛИЗ ВЫЖИВАЕМОСТИ SAS/STAT

2. Copyright © 2013, SAS Institute Inc. All rights reserved. АНАЛИЗ ВЫЖИВАЕМОСТИ  Что такое Анализ Выживаемости и для решения каких задач его стоит применять  Математические основы метода  Какие инструменты Анализа Выживаемости вы можете найти в SAS/STAT  Примеры, примеры, примеры...

3. Copyright © 2013, SAS Institute Inc. All rights reserved. Анализ выживаемости – набор статистических методов для предсказания как факта наступления события, так и времени до него ИСТОРИЧЕСКИЙ ОБЗОР Появился около века назад (lifetime tables) Новый импульс - Cox (proportional hazards model) в журнале JRSSB-1972: на сегодняшний день - самая цитируемая статья по статистике в истории Главным образом применялся в клинических исследованиях и производственном контроле С большой скоростью набирает популярность в телекоме и кредитном скоринге

4. Copyright © 2013, SAS Institute Inc. All rights reserved. CRM СФЕРЫ ПРИМЕНЕНИЯ Оценка эффективности маркетинговых кампаний Кредитный скоринг Определение ключевых факторов риска Анализ выживаемости Планирование маркетинговых кампаний Медицина Predictive Maintenance Предсказание оттока

5. Copyright © 2013, SAS Institute Inc. All rights reserved. АНАЛИЗ ВЫЖИВАЕМОСТИ VS ТРАДИЦИОННЫЙ DATA MINING Модели точнее и функциональнее Анализ выживаемости В чем отличие от традиционных методов Data Mining? Используется информация обо всех объектах Наблюдения с неизвестным исходом не отбрасываются Помимо самих факторов, включаем в модель и их прогнозы (курсы валют, динамика поведения)

6. Copyright © 2013, SAS Institute Inc. All rights reserved. 1)Крупный частный латиноамериканский банк Система управления рисками Получение информации о динамике покупательной способности клиентов во времени 2)NHS Blood and Transplant Более эффективное использование скудной и ценной информации о выживаемости клиентов после пересадки органов Аккуратный подбор донора и реципиента продлевает срок жизни клиентов и существенно улучшает её качество APPLICATIONS & RESEARCH SAS НЕЗАВИСИМЫЕ ЭКСПЕРТЫ 1)Jonathan Crook Professor of Business Economics & Director, MSc Banking & Risk, Edinburgh 2)Christophe Mues Senior Lecturer of Southampton Management School, Southampton 3).....и многие, многие другие активно исследуют применении Анализа Выживаемости в кредитном скоринге и CRM

7. Copyright © 2013, SAS Institute Inc. All rights reserved. ОСНОВНЫЕ ОПРЕДЕЛЕНИЯ Событие: некий триггер, сработавший на «клиенте» Цензурирование: выбывание из наблюдаемой выборки под действием сторонних факторов  переезд в другой город, окончание эксперимента до наступления события, смерть Ковариаты: характеристики «клиента», влияющие на его «отток»  возраст, пол, город, а также динамика дохода, динамика курсов валют,... ФУНКЦИЯ ВЫЖИВАЕМОСТИ ФУНКЦИЯ РИСКА

8. Copyright © 2013, SAS Institute Inc. All rights reserved. ФУНКЦИЯ ВЫЖИВАЕМОСТИ

9. Copyright © 2013, SAS Institute Inc. All rights reserved. ЦЕНЗУРИРОВАНИЕ Начало наблюдений Начало наблюдений Конец наблюдений Конец наблюдений А что случится с ними? Этого никто не знает

10. Copyright © 2013, SAS Institute Inc. All rights reserved. EXPLORATORY DATA ANALYSIS USING SURVIVAL CURVES

11. Copyright © 2013, SAS Institute Inc. All rights reserved. KAPLAN-MEIER MODEL Количество выбывших в интервал времени T ( number at death ) Количество под угрозой выбывания ( number at risk )

12. Copyright © 2013, SAS Institute Inc. All rights reserved. KAPLAN-MEIER MODEL : COMPARING SURVIVAL CURVES Different Statistical Tests -Log Rank -Wilcoxon -Likelihood-Ratio Confidence Limits

13. Copyright © 2013, SAS Institute Inc. All rights reserved. KAPLAN-MEIER MODEL : DIFFERENT STATISTICAL TESTS Log Rank Wilcoxon Likelihood-Ratio (parametric) Distribution of Event times  Exponential

14. Copyright © 2013, SAS Institute Inc. All rights reserved. PROC LIFETEST

15. Copyright © 2013, SAS Institute Inc. All rights reserved. PROC LIFETEST: COMPARING SURVIVAL CURVES

16. Copyright © 2013, SAS Institute Inc. All rights reserved. Are Hazard Functions proportional?Does Likelihood-Ratio test applicable? PROC LIFETEST: COMPARING SURVIVAL CURVES YES NO

17. Copyright © 2013, SAS Institute Inc. All rights reserved. PROC LIFETEST: COMPARING MULTIPLE SURVIVAL CURVES proc lifetest data=sasuser.methadone plots=(survival(cb=hw)) notable; time time*status(0); strata dose(50 70) / test=logrank adjust=scheffe nodetail; run;

18. Copyright © 2013, SAS Institute Inc. All rights reserved. PROC LIFETEST: COMPARING MULTIPLE SURVIVAL CURVES Dose < 50 and Dose =60 differ?  NO Dose > 70 and Dose =60 differ?  YES Dose > 70 and Dose <50 differ?  YES Dose < 50 and Dose =60 differ?  NO Dose > 70 and Dose =60 differ?  YES Dose > 70 and Dose <50 differ?  YES proc lifetest data=sasuser.methadone plots=(survival(cb=hw)) notable; time time*status(0); strata dose(50 70) / test=logrank adjust=scheffe nodetail; run;

19. Copyright © 2013, SAS Institute Inc. All rights reserved. ALTERNATIVE TO KAPLAN-MEIER: LIFE TABLE METHODS LARGE SAMPLES LIFE TABLE the same as Kaplan- Meier Estimate, but … GROUP OBSERVATIONS INTO BINS CENSORED OBS ARE CENSORED IN THE MIDDLE OF INTERVAL

20. Copyright © 2013, SAS Institute Inc. All rights reserved. ALTERNATIVE TO KAPLAN-MEIER: LIFE TABLE METHODS proc lifetest data=sasuser.methadone plots=(survival(failure) hazard) method=life intervals=183 365 548; time time*status(0); strata clinic / test=(all) nodetail; run; proc lifetest data=sasuser.methadone plots=(survival(failure) hazard) method=life intervals=183 365 548; time time*status(0); strata clinic / test=(all) nodetail; run;

21. Copyright © 2013, SAS Institute Inc. All rights reserved. COX’S PROPORTIONAL HAZARDS MODEL

22. Copyright © 2013, SAS Institute Inc. All rights reserved. SURVIVAL MODELS  Models in Survival Analysis are written in terms of Hazard Functions  They assess the relationship of covariates to survival times  Models can be parametric or semi-parametric  Models in Survival Analysis are written in terms of Hazard Functions  They assess the relationship of covariates to survival times  Models can be parametric or semi-parametric SEMI-PARAMETRIC PROC PHREG SEMI-PARAMETRIC PROC PHREG PARAMETRIC PROC LIFEREG PARAMETRIC PROC LIFEREG 1.Distribution of Event Times is specified 2.Hazard function is completely specified (except for params) 1.Distribution of Event Times is unknown 2.Hazard function is unspecified Cox Proportional Hazards Model OK for ! Cox Proportional Hazards Model OK for ! Exp Hazards Weibull Hazards Usually a poor choice! Exp Hazards Weibull Hazards Usually a poor choice!

23. Copyright © 2013, SAS Institute Inc. All rights reserved. COX PROPORTIONAL HAZARDS MODEL 1.The model provides the primary information desired from a survival analysis 2.Minimum of assumptions 3.Robust regression estimates of the influence of covariates 4.Thus, the model is extremely popular

24. Copyright © 2013, SAS Institute Inc. All rights reserved. PROPORTIONAL HAZARDS ASSUMPTION

25. Copyright © 2013, SAS Institute Inc. All rights reserved. DERIVING COEFFICIENTS: PARTIAL LIKELIHOOD MAXIMIZATION ILLUSTRATION

26. Copyright © 2013, SAS Institute Inc. All rights reserved. DERIVING COEFFICIENTS: PARTIAL LIKELIHOOD MAXIMIZATION

27. Copyright © 2013, SAS Institute Inc. All rights reserved. TIED OBSERVATIONS Tied observations They must be taken into account in Partial Likelihood calculation! SAS/STAT PROC PHREG does it automatically! (Breslow approximation)

28. Copyright © 2013, SAS Institute Inc. All rights reserved. PROC PHREG

29. Copyright © 2013, SAS Institute Inc. All rights reserved. PROC PHREG: FIT COX REGRESSION MODEL TO METHADONE DATA COEFFICIENT ESTIMATE COEFFICIENT not equal to 0?

30. Copyright © 2013, SAS Institute Inc. All rights reserved. PROC PHREG: ADJUST SURVIVAL CURVES

31. Copyright © 2013, SAS Institute Inc. All rights reserved. COX PH MODEL ASSESSMENT COX MODEL ASSUMPTIONS 1.Proportional Hazards The effect of the predictor is the same over all values of time 2.Linearity Log Hazard linearly depends on predictors 3.Additivity The joint effect of predictors equals the sum of their separate effects COX MODEL ASSUMPTIONS 1.Proportional Hazards The effect of the predictor is the same over all values of time 2.Linearity Log Hazard linearly depends on predictors 3.Additivity The joint effect of predictors equals the sum of their separate effects TIME-VARIABLE DEPENDENCE CUMULATIVE RESIDUALS PLOT

32. Copyright © 2013, SAS Institute Inc. All rights reserved. ASSESS PH USING TIME-VARIABLE DEPENDENCE

33. Copyright © 2013, SAS Institute Inc. All rights reserved. ASSESS PH USING CUMULATIVE RESIDUALS PLOT RESIDUAL Simulated Observed SIMULATE IT!

34. Copyright © 2013, SAS Institute Inc. All rights reserved. MODELS WITH NON-PROPORTIONAL HAZARDS

35. Copyright © 2013, SAS Institute Inc. All rights reserved. MODELING NON-PROPORTIONAL HAZARDS WAYS to HANDLE NON- PROPORTIONAL HAZARDS 1.Stratified Cox PH Vary Baseline hazard 2.Cox PH with time-dependent vars Model non-proportionality using interactions with functions of time 3.Piecewise Cox PH The effect of variable is assessed separately for different times WAYS to HANDLE NON- PROPORTIONAL HAZARDS 1.Stratified Cox PH Vary Baseline hazard 2.Cox PH with time-dependent vars Model non-proportionality using interactions with functions of time 3.Piecewise Cox PH The effect of variable is assessed separately for different times

36. Copyright © 2013, SAS Institute Inc. All rights reserved. STRATIFIED COX MODEL

37. Copyright © 2013, SAS Institute Inc. All rights reserved. STRATIFIED COX MODEL 1. Dose*Clinic & Clinic*Prison 2. Clinic*Prison DROP Dose*Clinic DROP Clinic*Prison

38. Copyright © 2013, SAS Institute Inc. All rights reserved. STRATIFIED COX MODEL 3. No interactions STAY at this model complexity 4. Try to adjust Baseline Hazard by Clinic

39. Copyright © 2013, SAS Institute Inc. All rights reserved. MODELS WITH INTERACTIONS WITH TIME Change the effect β of the variable 2 WAYS of INTRODUCING TIME INTO PARAMETER ESTIMATES Change the variable itself

40. Copyright © 2013, SAS Institute Inc. All rights reserved. MODELS WITH INTERACTIONS WITH TIME KEEP

41. Copyright © 2013, SAS Institute Inc. All rights reserved. PIECEWISE COX MODEL CREATE INTERACTION with HEAVISIDE FUNCTION!

42. Copyright © 2013, SAS Institute Inc. All rights reserved. PIECEWISE COX MODEL

43. Copyright © 2013, SAS Institute Inc. All rights reserved. ADVANCED TOPICS

44. Copyright © 2013, SAS Institute Inc. All rights reserved. TIME-DEPENDENT COVARIATES New time-dependent covariates must be specified inside PROC PHREG proc phreg data=sasuser.methadone; class Clinic (param=ref ref='2'); model Time*Status(0)=Clinic Dose Prison Drink / ties=exact rl=pl; Drink=(0 <= DrinkStart < Time); run;

45. Copyright © 2013, SAS Institute Inc. All rights reserved. MODELING THE EFFECT OF TIME-DEPENDENT PREDICTORS «Drink» is time dependent and it’s important! Coefficients are the same for the whole survey period

46. Copyright © 2013, SAS Institute Inc. All rights reserved. REPEATED EVENTS Some events are intrinsically repeatable: pregnancy, infection  One should account for this in survival analysis

47. Copyright © 2013, SAS Institute Inc. All rights reserved. 1. 2. 3. Drop 4. Drop REPEATED EVENTS: DIFFERENT MODELS FOR SUCC EVENTS Build different survival models for successive events Model men’s muscle soreness in 4 intervals depending on age and treatment