Survival Analysis From Square One to Square Two

Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph. D. Paul Yip, Ph. D. Readings

Lecture structure • Basic concepts • Kaplan-Meier analysis • Cox regression • Computer practice

What’s in a name? • time-to-event data • failure-time data • censored data (unobserved outcome)

Types of censoring – loss to follow-up during the study period – study closure

Examples of survival analysis 1. Marital status & mortality 2. Medical treatments & tumor recurrence & mortality in cancer patients 3. Size at birth & developmental milestones in infants

Why survival analysis ? • Censoring (time of event not observed) • Unequal follow-up time

What is time? What is the origin of time? In epidemiology: • Age (birth as time 0) ? • Calendar time since a baseline survey ?

What is the origin of time? In clinical trials: • Since randomisation ? • Since treatment begins ? • Since onset of exposure ?

The choice of origin of time • Onset of continuous exposure • Randomisation to treatment • Strongest effect on the hazard

Types of survival analysis 1. Non-parametric method Kaplan-Meier analysis 2. Semi-parametric method Cox regression 3. Parametric method

Square 1 to square 2 This lecture focuses on two commonly used methods • Kaplan-Meier method • Cox regression model

KM survival curve * d=death, c=censored, surv=survival

KM survival curve

No. of expected deaths Expected death in group A at time i, assuming equality in survival: EAi =no. at risk in group A i death i total no. at risk i Total expected death in group A: EA = EAi

Log rank test • A comparison of the number of expected and observed deaths. • The larger the discrepancy, the less plausible the null hypothesis of equality.

An approximation The log rank test statistic is often approximated by X 2 = (OA-EA)2/EA+ (OB-EB)2/EB, where OA & EA are the observed & expected number of deaths in group A, etc.

1 1 . 8 . 6 S(t) Proportional hazard assumption . 4 . 2 0 0 0 5 10 Time 15 20 Log rank test preferred (PH true ) 0 5 10 Time 15 20 Breslow test preferred (non-PH)

Risk, conditional risk, hazard

Hazard Another look of PH 0 5 10 15 20 Time Log rank test Breslow test preferred (PH true ) preferred (non-PH)

Cox regression model • Handles 1 exposure variables. • Covariate effects given as Hazard Ratios. • Semi-parametric: only assumes proportional hazard.

Cox model in the case of a single variable 1. hi(t) = h. B(t) exp(BXi) 2. hj(t) = h. B(t) exp(BXj) 3. hi(t)/hj(t) =exp[B(Xi-Xj)] exp(B) is a Hazard Ratio

Test of proportional hazard assumption • Scaled Schoenfeld residuals • Grambsch-Therneau test • Test for treatment period interaction • Example: mortality of widows

Computer practice A clinical trial of stage I bladder tumor Thiotepa vs Control Data from Stat. Lib

Data structure Two most important variables: • Time to recurrence (>0) • Indicator of failure/censoring (0=censored; 1=recurrence) (coding depends on software)

KM estimates Thiotepa Control

Log rank test chi 2(1) = 1. 52 Pr>chi 2 = 0. 22

Cox regression models

Test of PH assumption Grambsch-Therneau test for PH in model II • Thiotepa P=0. 55 • Number of tumor P=0. 60

Major References (Examples) Ex 1. Cheung. Int J Epidemiol 2000; 29: 93 -99. Ex 2. Sauerbrei et al. J Clin Oncol 2000; 18: 94 -101. Ex 3. Cheung et al. Int J Epidemiol 2001; 30: 66 -74.

Major References (General) • Allison. Survival Analysis using the SAS® System. • Collett. Modelling Survival Data in Medical Research. • Fisher, van Belle. Biostatistics: A Methodology for the Health Sciences.