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

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

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Page 1: 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

Page 2: 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

Page 3: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

What’s in a name?

• time-to-event data• failure-time data• censored data

(unobserved outcome)

Page 4: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

Types of censoring

– loss to follow-up during the study period

– study closure

Page 5: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

Examples of survival analysis1. Marital status & mortality

2. Medical treatments & tumor recurrence & mortality in cancer patients

3. Size at birth & developmental milestones in infants

Page 6: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

Why survival analysis ?

• Censoring (time of event not observed)

• Unequal follow-up time

Page 7: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

What is time?What is the origin of time?

In epidemiology:

•Age (birth as time 0) ?

•Calendar time since a baseline survey ?

Page 8: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

What is the origin of time?

In clinical trials:

• Since randomisation ?

• Since treatment begins ?

• Since onset of exposure ?

Page 9: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

The choice of origin of time

• Onset of continuous exposure

• Randomisation to treatment

• Strongest effect on the hazard

Page 10: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

Types of survival analysis

1. Non-parametric method

Kaplan-Meier analysis

2. Semi-parametric method

Cox regression

3. Parametric method

Page 11: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

Square 1 to square 2

This lecture focuses on two commonly used methods

• Kaplan-Meier method

• Cox regression model

Page 12: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

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

Page 13: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

KM survival curve

S(t

)

Day

0 1 2 3 4 50.00

0.25

0.50

0.75

1.00

Page 14: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

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

Page 15: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

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.

Page 16: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

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.

Page 17: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

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

Page 18: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

Risk, conditional risk, hazard

Pro

port

ion

Month0 1 2 3

0.1.2.3.4.5

Page 19: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

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

Page 20: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

Cox regression model• Handles 1 exposure variables.

• Covariate effects given as Hazard Ratios.

• Semi-parametric: only assumes proportional hazard.

Page 21: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

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

Page 22: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

Test of proportional hazard assumption

• Scaled Schoenfeld residuals

• Grambsch-Therneau test

• Test for treatmentperiod interaction

• Example: mortality of widows

Page 23: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

Computer practice

A clinical trial of

stage I bladder tumor

Thiotepa vs Control

Data from StatLib

Page 24: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

Data structure

Two most important variables:

• Time to recurrence (>0)

• Indicator of failure/censoring

(0=censored; 1=recurrence)

(coding depends on software)

Page 25: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

KM estimates

S(t

)

Months0 20 40 60

0.00

0.25

0.50

0.75

1.00

Thiotepa

Control

Page 26: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

Log rank test Recurrence

Group Observed ExpectedControl 29 24.9Thiotepa 18 22.1

chi2(1) = 1.52Pr>chi2 = 0.22

Page 27: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

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)

Page 28: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

Test of PH assumption

Grambsch-Therneau test for PH in model II

• Thiotepa P=0.55• Number of tumor P=0.60

Page 29: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

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.

Page 30: Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings

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.