Non-proportional Hazards

Hajime Uno, Ph.DDept. Biostat. & Comp. Biol. Dana-Farber Cancer Institute

Kaplan-Meier

Logrank test

Cox’s PH estimate

1) Description of survival curves

2) Test of no difference

3) Estimation of treatment effect

A standard methodThree key components in report

Critical issues on the PH estimator

• When the PH assumption is not correct, the PH estimator is estimating a quantity that cannot easily be interpreted. Also, the limiting quantity depends on study-specific underlying censoring distributions (Any model-based treatment contrast estimator has such issues)

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Survival functions Hazard Ratio

Censoring pattern

(1) (2) (3)

Non-PH(1) HR=0.77(2) HR=0.71(3) HR=0.82

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Survival functions Hazard Ratio

Censoring pattern

(1) (2) (3)

PH(1) HR=0.80(2) HR=0.80(3) HR=0.80

A cancer example• ECOG E4A03: A phase III randomized trial

to compare low- and high-dose dexamethasone for newly diagnosed multiple myeloma

• N=445 (223 on high-dose, 222 on low-dose) • One of the endpoints was overall survival

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Rajkumar et al. (2009)Low-doseHigh-Dose

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HR= 0.87 (0.60 - 1.27)

How hazard functions looked like?

Alternative measures

Model-free measures• Median survival time• t-year survival probability• Restricted mean survival time

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Median Survival Time

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t-year survival probability

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1-year? 2-year?3-year?

Restricted mean survival time (RMST)

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- The area under the survival curve up to t*- “t*-year life expectancy” – (e.g., your life

expectancy with low-dose treatment over the next 40 months is 35.4 months”

t*

t*=4035.4 months (low-dose) 33.3 months (high-dose)

• Difference: 2.1 (0.1 - 4.2)

• Ratio: 35.4/33.3 1.06 (1.00 - 1.13)

• Ratio of time lost =6.7/4.6 1.46 (1.02 - 2.13)

Question #1

Have you ever experienced any problem with non-proportional hazards in practice?

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Question #2

How do you handle the potential violation of the PH assumption in a post hoc setting? (How do you find the violation? What do you do when you find it?)

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Kaplan-Meier

Logrank test

Cox’s PH estimate

1) Description of survival curves

2) Test of no difference

3) Estimation of treatment effect

A standard methodThree key components in report

Logrank test

• The most powerful test if HR is constant• Equivalent to testing HR=1 with Cox regression

Negatives• May not be so powerful when HR is not

constant• Especially, it will fail when hazard functions

cross, even under stochastic ordering alternatives

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Logrank: p=0.233

The cancer example

A new test: p=0.005

Alternatives to logrankFor stochastic ordering alternatives

• Tests based on difference between the KM curves– Weighted KM test (Pepe-Fleming, 1989)

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Alternatives to logrankFor various patterns of alternatives• Combinations of multiple weighted logrank test

statistics or weighted KM test statistics– Linear combination: Gastwirth (1985), Lee (1996)– Maximum: Tarone (1981), Shen & Cai (2001)

• Sup-versions of weighted logrank statistics– Fleming, Harrington, O’Sullivan (1987)

• Adaptively weighed logrank (or weighted KM) test in a class of weighted statistics– Yang and Prentice (2010)

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Question #3

Should we consider employing a robust test as the primary analysis, instead of the logrank test at the design stage?

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A versatile test for equality of two survival functions based on weighted differences of Kaplan-Meier curves

Hajime Uno, Lu Tian, Brian Claggett, LJ Wei

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DifferenceSurvival curves Standardized difference

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The proposed procedure gives “a bona-fide p-value,” automatically adjusting for multiple choices of c.

Questions1. Have you ever experienced any problem with

non-proportional hazards in practice?2. How do you handle the potential violation of

the PH assumption in a post hoc setting? (How do you find the violation? What do you do when you find it?)

3. Should we consider employing a robust test as the primary analysis, instead of the logranktest at the design stage?

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