NASSER DAVARZANI DEPARTMENT OF KNOWLEDGE ENGINEERING MAASTRICHT UNIVERSITY, 6200 MAASTRICHT, THE NETHERLANDS 22 OCTOBER 2012 Introduction to Survival Analysis

Contents Introduction to Survival Analysis Censored Data Terminologies and Notations Kaplan-Meier Method Modeling in Survival Data Parametric Regression Models Cox Proportional Regression Model 2 13 September 2015 Survival Analysis (N. Davarzani)

Contents Introduction to Survival Analysis Censored Data Terminologies and Notations Kaplan-Meier Method Modeling in Survival Data Parametric Regression Models Cox Proportional Regression Model 3 13 September 2015 Survival Analysis (N. Davarzani)

What is survival analysis? Outcome variable: Time until an event occurs Time: years, months, weeks, or days Start follow-up Event Event: death, disease, relapse, recovery Time 4 13 September 2015 Survival Analysis (N. Davarzani)

Time ≡ survival time It gives the time that an individual has survived over some follow up period Event ≡ failure The event of interest usually is death, disease or any other negative individual experience. Maybe failure is a positive event 5 13 September 2015 Survival Analysis (N. Davarzani)

Examples 1) Leukemia patients/time in remission (weeks) Event: Going out of remission Outcome: Time in weeks until a person goes out of remission 2) Disease-free cohort/time until heart disease (years) Event: Developing heart disease Outcome: time in years until a person develops heart disease September 2015 Survival Analysis (N. Davarzani)

Examples 13 September 2015 Survival Analysis (N. Davarzani) 7 3) Elderly (60+) population/time until death (years) Event: Death Outcome: Time in years until death 4) Parolees (recidivism study)/time until get rearrested Event: Getting rearrested Outcome: Time in weeks until rearrest

Contents Introduction to Survival Analysis Censored Data Terminologies and Notations Kaplan-Meier Method Modeling in Survival Data Parametric Regression Models Cox Proportional Regression Model 8 ? 13 September 2015 Survival Analysis (N. Davarzani)

Censored data We don’t know the survival time exactly September 2015 Survival Analysis (N. Davarzani)

Why censoring may occur? 1) A person does not experience the event before the study ends. 2) A person is lost to follow-up during the study period. 3) A person withdraws from the study because of death (if death is not the event of interest) or some other reason September 2015 Survival Analysis (N. Davarzani)

Examples 13 September 2015 Survival Analysis (N. Davarzani) 11

Table of Survival Time 13 September 2015 Survival Analysis (N. Davarzani) 12

Right Censored Data 13 September 2015 Survival Analysis (N. Davarzani) 13

Contents Introduction to Survival Analysis Censored Data Terminologies and Notations Kaplan-Meier Method Modeling in Survival Data Parametric Regression Models Cox Proportional Regression Model September 2015 Survival Analysis (N. Davarzani)

Terminology and Notation T = Survival time (T ≥ 0) t = Specific value for T Survives > 5 years? T > t = September 2015 Survival Analysis (N. Davarzani)

Survival Function September 2015 Survival Analysis (N. Davarzani)

Survival Curve 13 September 2015 Survival Analysis (N. Davarzani) 17

Hazard function The hazard function gives the instantaneous potential per unit time for the event to occur, given that the individual has survived up to time t Hazard function focuses on failing September 2015 Survival Analysis (N. Davarzani)

Different type of hazard functions 13 September 2015 Survival Analysis (N. Davarzani) 19

Relationship of S(t) and h(t) If you know one, you can determine the other September 2015 Survival Analysis (N. Davarzani)

13 September 2015 Survival Analysis (N. Davarzani) 21

Appropriate Distributions Some popular distributions for estimating survival curves are: Weibull Exponential log-normal log-logistic September 2015 Survival Analysis (N. Davarzani)

13 September 2015 Survival Analysis (N. Davarzani) 23 2: To compare survivor and/or hazard functions. 1: To estimate and interpret survivor and/or hazard functions from survival data. 3: To assess the relationship of explanatory variables to survival time.

How to Draw the Survival Curves? 13 September 2015 Survival Analysis (N. Davarzani) 24

Contents Introduction to Survival Analysis Censored Data Terminologies and Notations Kaplan-Meier Method Modeling in Survival Data Parametric Regression Models Cox Proportional Regression Model September 2015 Survival Analysis (N. Davarzani)

Kaplan – Meier Method The Kaplan – Meier (KM) estimator is the most widely used for estimating survival function product-limit estimator nonparametric maximum likelihood estimator. When there are no censored data, the KM estimator is simple estimator September 2015 Survival Analysis (N. Davarzani)

Practical Example 13 September 2015 Survival Analysis (N. Davarzani) 27

How to estimate Survival Curve? 13 September 2015 Survival Analysis (N. Davarzani) 28 Suppose there are k distinct event times At each time there are individuals who are said to be at risk of an event. At risk, means they have not experienced an event nor have they been censored prior to time.

Lifetime Table 13 September 2015 Survival Analysis (N. Davarzani) 29

KM Estimator 13 September 2015 Survival Analysis (N. Davarzani) 30 The KM estimator is defined as Where number of failures at time

Lifetime Table 13 September 2015 Survival Analysis (N. Davarzani) 31

Are KM curves statistically equavalent? 13 September 2015 Survival Analysis (N. Davarzani) 32

H: Survival functions of all groups are the same Log-Rank test Wald test Score test LR test Wilcoxon test Peto test Taron-Ware test September 2015 Survival Analysis (N. Davarzani)

Weakness of KM method 13 September 2015 Survival Analysis (N. Davarzani) 34 Teatment V.S Placebo Continuous variables Effect of several variables

Contents Introduction to Survival Analysis Censored Data Terminologies and Notations Kaplan-Meier Method Modeling in Survival Data Parametric Regression Models Cox Proportional Regression Model September 2015 Survival Analysis (N. Davarzani)

Modeling Survival Data with Regression September 2015 Survival Analysis (N. Davarzani)

Parametric Regression Models are the regression coeffcients of interest.  is a scale parameter  is the random disturbance term September 2015 Survival Analysis (N. Davarzani)

The Cox Proportional Hazard Model September 2015 Survival Analysis (N. Davarzani)

Semi parametric Model is unspecified September 2015 Survival Analysis (N. Davarzani)

Proportional Hazard September 2015 Survival Analysis (N. Davarzani) b b h1 h2

13 September 2015 Survival Analysis (N. Davarzani) 41

13 September 2015 Survival Analysis (N. Davarzani) 42

13 September 2015 Survival Analysis (N. Davarzani) 43