Introduction to Survival Analysis Seminar in Statistics 1 Presented by: Stefan Bauer, Stephan Hemri

Definition Survival analysis:  method for analysing timing of events;  data analytic approach to estimate the time until an event occurs. Historically survival time refers to the time that an individual „survives“ over some period until the event of death occurs. Event is also named failure. 2

Areas of application Survival analysis is used as a tool in many different settings: proving or disproving the value of medical treatments for diseases; evaluating reliability of technical equipment; monitoring social phenomena like divorce and unemployment. 3

Examples Time from... marriage to divorce; birth to cancer diagnosis; entry to a study to relapse. 4

Censoring The survival time is not known exactly! This may occur due to the following reasons: a person does not experience the event before the study ends; a person is lost to follow-up during the study period; a person withdraws from the study because of some other reason. 5

Right censored 6

Left censored 7

Outcome variable 8

Survival time ≠ calendar time (e.g. follow- up starts for each individual on the day of surgery) Correct starting and ending times may be unknown due to censoring 9

Survivor function 10 tS(t) 1S(1) = P(T > 1) 2S(2) = P(T > 2) 3S(3) = P(T > 3)......

Survivor function 11

Survivor function 12

Hazard function 13

Example: Hazard function Assume having a huge follow-up study on heart attacks: 600 heart attacks (events) per year; 50 events per month; 11.5 events per week; events per minute. h(t) = rate of events occurring per time unit 14

Relation between S(t) and h(t) 15

Sketch of Proof i)Find relationship between density f(t) and S(t) ii)Express relationship between h(t) and S(t) as a function of density f(t) i in ii → h(t) as a function of S(t) and vice versa 16

Example: Relationship 17

Types of hazard functions 18

Hazard ratio 19

Example: Hazard ratio 20

Basic descriptive measures 21

Goals (of survival analysis) to estimate and interpret survivor and or hazard function; to compare survivor and or hazard function; to assess the relationship of explanatory variables to survival times -> we need mathematical modelling (Cox model). 22

Computer layout 23 individualt (in weeks) δ (failed or censored) ,

Computer layout Layout for multivariate data with p explanatory variables: 24 individualt (in weeks) δ (failed or censored) X1X1 X2X2....XpXp 1t1t1 1 X 11 X 12...X 1p 2t2t2 2 X 21 X 22...X 2p... … ntntn n X n1 X n2....X np

25 Notation & terminology Ordered failures: Frequency counts: m i = # individuals who failed at t (i) q i = # ind. censored in [t (i),t (i+1) ) Risk set R(t (i) ): Collection of individuals who have survived at least until time t (i) unordered censored t’s failed t’s ordered (t (i) )

Manual analysis layout 26 Ordered failure times # of failures m i # censored in [t (i),t (i+1) ) Risk set R(t (i) ) t (0) =0mimi q0q0 R(t (0) ) t (1) m1m1 q1q1 R(t (1) ) t (n) mnmn qnqn R(t (n) )

Manual analysis layout 27 Ordered failure times # of failures m i # censored in [t (i),t (i+1) ) Risk set R(t (i) ) t (0) = persons survive ≥ 0 weeks t (1) = persons survive ≥ 3,5 weeks t (2) = 5134 persons survive ≥ 5 weeks

Example: Leukaemia remission Extended Remission Data containing: two groups of leukaemia patients: treatment & placebo; log WBC values of each individual; (WBC: white blood cell count) Expected behaviour: The higher the WBC value is the lower the expected survival time. 28

Example: Analysis layout 29 Analysis layout for treatment group: t (j) mimi qiqi R(t (j) ) t (0) = persons t (1) = persons t (2) = persons t (3) = persons t (4) = persons t (5) = persons t (6) = persons t (7) = persons

Example: Confounding 30

Example: Confounding 31

Example: Interaction 32

Example: Conclusion Need to consider confounding and interaction; basic problem: comparing survival of the two groups after adjusting for confounding and interaction; problem can be extended to the multivariate case by adding additional explanatory variables. 33

Summary Survival analysis encompasses a variety of methods for analyzing the timing of events; problem of censoring: exact survival time unknown;  mixture of complete and incomplete observations  difference to other statistical data 34

Summary 35

References A Conceptual Approach to Survival Analysis, Johnson, L.L., Downloaded from on Applied Survival Analysis: Regression Modelling of Time to event Data, Hosmer, D.W., Lemeshow, S., Wiley Series in Probability and Statistics Lesson 6: Sample Size and Power - Part A, The Pennsylvania State University, Downloaded from courses/stat509/06_sample/09_sample_hazard.htm on Survival analysis: A self-learning text, Kleinbaum, D.G. & Klein M., Springer Survival and Event History Analysis: A Process Point of View (Statistics for Biology and Health), Aalen, O., Borgan, O. & Gjessing H., Springer