Survival Analysis: From Square One to Square Two

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Survival Analysis From Square One to Square Two

• Yin Bun Cheung, Ph.D.
• Paul Yip, Ph.D.

Readings
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Lecture structure

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

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Whats in a name?

• time-to-event data
• failure-time data
• censored data
• (unobserved outcome)

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Types of censoring

• loss to follow-up during the study period
• study closure

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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

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Why survival analysis ?

• Censoring (time of event not observed)
• Unequal follow-up time

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What is time?What is the origin of time?

• In epidemiology
• Age (birth as time 0) ?
• Calendar time since a baseline survey ?

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What is the origin of time?

• In clinical trials
• Since randomisation ?
• Since treatment begins ?
• Since onset of exposure ?

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The choice of origin of time

• Onset of continuous exposure
• Randomisation to treatment
• Strongest effect on the hazard

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

• This lecture focuses on two commonly used methods
• Kaplan-Meier method
• Cox regression model

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KM survival curve
ddeath, ccensored, survsurvival
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KM survival curve
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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

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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.

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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.

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Proportional hazard assumption
Log rank test preferred (PH true )
Breslow test preferred (non-PH)
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Risk, conditional risk, hazard
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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)
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Cox regression model

• Handles ?1 exposure variables.
• Covariate effects given as Hazard Ratios.
• Semi-parametric only assumes proportional hazard.

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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) expB(Xi-Xj)
• exp(B) is a Hazard Ratio

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Test of proportional hazard assumption

• Scaled Schoenfeld residuals
• Grambsch-Therneau test
• Test for treatment?period interaction
• Example mortality of widows

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Computer practice

• A clinical trial of
• stage I bladder tumor
• Thiotepa vs Control
• Data from StatLib

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Data structure

• Two most important variables
• Time to recurrence (gt0)
• Indicator of failure/censoring
• (0censored 1recurrence)
• (coding depends on software)

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KM estimates
Thiotepa
Control
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Log rank test
chi2(1) 1.52 Prgtchi2 0.22
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Cox regression models
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Test of PH assumption

• Grambsch-Therneau test
• for PH in model II
• Thiotepa P0.55
• Number of tumor P0.60

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Major References (Examples)

• Ex 1. Cheung. Int J Epidemiol 20002993-99.
• Ex 2. Sauerbrei et al. J Clin Oncol
  20001894-101.
• Ex 3. Cheung et al. Int J Epidemiol 20013066-74.

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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.