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

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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
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)

What’s in a name?
examples of survival analysis
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

what is time what is the origin of 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
What is the origin of time?

In clinical trials:

  • Since randomisation ?
  • Since treatment begins ?
  • Since onset of exposure ?
the choice of origin of time
The choice of origin of time
  • Onset of continuous exposure
  • Randomisation to treatment
  • Strongest effect on the hazard
types of survival analysis
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
Square 1 to square 2

This lecture focuses on two commonly used methods

  • Kaplan-Meier method
  • Cox regression model
km survival curve
KM survival curve

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

no of expected deaths
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
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
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

1

1

.8

.8

.6

.6

S(t)

S(t)

.4

.4

.2

.2

0

0

0

5

10

15

20

0

5

10

15

20

Time

Time

Proportional hazard assumption

Log rank test preferred (PH true )

Breslow test preferred (non-PH)

another look of ph
Another look of PH

Hazard

Hazard

0

5

10

15

20

0

5

10

15

20

Time

Time

Log rank test preferred (PH true )

Breslow test preferred (non-PH)

cox regression model
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
Cox model in the case of a single variable
  • . hi(t) = hB(t)  exp(BXi)
  • . hj(t) = hB(t)  exp(BXj)
  • . hi(t)/hj(t) =exp[B(Xi-Xj)]
  • exp(B) is a Hazard Ratio
test of proportional hazard assumption
Test of proportional hazard assumption
  • Scaled Schoenfeld residuals
  • Grambsch-Therneau test
  • Test for treatmentperiod interaction
  • Example: mortality of widows
computer practice
Computer practice

A clinical trial of

stage I bladder tumor

Thiotepa vs Control

Data from StatLib

data structure
Data structure

Two most important variables:

  • Time to recurrence (>0)
  • Indicator of failure/censoring

(0=censored; 1=recurrence)

(coding depends on software)

km estimates
KM estimates

Thiotepa

Control

log rank test26
Log rank test

chi2(1) = 1.52

Pr>chi2 = 0.22

test of ph assumption
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
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
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.