survival analysis: from square one to square two yin bun cheung, ph.d. paul yip, ph.d. readings

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 analysis1. 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 curveDay(t)

Death /Cens.

Atrisk

Pt(d) Pt(surv) S(t)

1 4 0.00 1.00 1.002 1 d 4 0.25 0.75 0.753 1 c 3 0.00 1.00 0.754 1 d 2 0.50 0.50 0.385 1 c 1 0.00 1.00 0.38

5

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

KM survival curve

S(t

)

Day

0 1 2 3 4 50.00

0.25

0.50

0.75

1.00

No. of expected deathsExpected 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

X2 = (OA-EA)2/EA+ (OB-EB)2/EB,

where OA & EA are the observed & expected number of deaths in group A, etc.

Proportional hazard assumption

S(t

)

Time0 5 10 15 20

0

.2

.4

.6

.8

1

Log rank test preferred (PH true )

Breslow test preferred (non-PH)

S(t

)

Time0 5 10 15 20

0

.2

.4

.6

.8

1

Risk, conditional risk, hazard

Pro

port

ion

Month0 1 2 3

0.1.2.3.4.5

Another look of PH

Log rank test preferred (PH true )

Breslow test preferred (non-PH)

Haz

ard

Time0 5 10 15 20

Haz

ard

Time0 5 10 15 20

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) = hB(t) exp(BXi)

2. hj(t) = hB(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 treatmentperiod interaction

• Example: mortality of widows

Computer practice

A clinical trial of

stage I bladder tumor

Thiotepa vs Control

Data from StatLib

Data structure

Two most important variables:

• Time to recurrence (>0)

• Indicator of failure/censoring

(0=censored; 1=recurrence)

(coding depends on software)

KM estimates

S(t

)

Months0 20 40 60

0.00

0.25

0.50

0.75

1.00

Thiotepa

Control

Log rank test Recurrence

Group Observed ExpectedControl 29 24.9Thiotepa 18 22.1

chi2(1) = 1.52Pr>chi2 = 0.22

Cox regression modelsModel I

HRModel II

HRThiotepa

(vs Control)0.70

(P=0.23)0.60

(P=0.11)Number of

tumor1.26

(P<0.01)

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.