Survival Analysis and Cox Regression for Cancer Trials

Presented at PG Department of Statistics,

Sardar Patel University January 29, 2013

Dr. Bhaswat S. ChakrabortySr. VP & Chair, R&D Core Committee

Cadila Pharmaceuticals Ltd., Ahmedabad

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Part 2: Cox Regression Analysis of Cancer CTs

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Clinical Trials Organized scientific efforts to get direct answers from

relevant patients on important scientific questions on (doses and regimens of) actions of drugs (or devices or other interventions).

Questions are mainly about differences or null Modern trials (last 40 years or so) are large, multicentre,

often international and co-operative endeavors Ideally, primary objectives are consistent with mechanism

of action Results can be translated to practice Would stand the regulatory and scientific scrutiny

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Cancer Trials (Phases I–IV) Highly complex trials involving cytotoxic drugs, moribund patients,

time dependent and censored variables Require prolonged observation of each patient Expensive, long term and resource intensive trials Heterogeneous patients at various stages of the disease Prognostic factors of non-metastasized and metastasized diseases are

different Adverse reactions are usually serious and frequently include death Ethical concerns are numerous and very serious Trial management is difficult and patient recruitment extremely

challenging Number of stopped trials (by DSMB or FDA) is very high Data analysis and interpretation are very difficult by any standard

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Source: WHO6

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India: 2010 7137 of 122 429 study deaths were due to cancer, corresponding to 556 400 national

cancer deaths in India in 2010. 395 400 (71%) cancer deaths occurred in people aged 30—69 years (200 100 men

and 195 300 women). At 30—69 years, the three most common fatal cancers were oral (including lip and

pharynx, 45 800 [22·9%]), stomach (25 200 [12·6%]), and lung (including trachea and larynx, 22 900 [11·4%]) in men, and cervical (33 400 [17·1%]), stomach (27 500 [14·1%]), and breast (19 900 [10·2%]) in women.

Tobacco-related cancers represented 42·0% (84 000) of male and 18·3% (35 700) of female cancer deaths and there were twice as many deaths from oral cancers as lung cancers.

Age-standardized cancer mortality rates per 100 000 were similar in rural (men 95·6 [99% CI 89·6—101·7] and women 96·6 [90·7—102·6]) and urban areas (men 102·4 [92·7—112·1] and women 91·2 [81·9—100·5]), but varied greatly between the states.

Cervical cancer was far less common in Muslim than in Hindu women (study deaths 24, age-standardized mortality ratio 0·68 [0·64—0·71] vs 340, 1·06 [1·05—1·08]).

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Survival Analysis Survival analysis is studying the time between entry

to a study and a subsequent event (such as death). Also called “time to event analysis” Survival analysis attempts to answer questions such

as: which fraction of a population will survive past a certain

time ? at what rate will they fail ? at what rate will they present the event ? How do particular factors benefit or affect the probability of

survival ?

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What kind of time to event data? Survival Analysis typically focuses on time to event data. In the most general sense, it consists of techniques for

positive-valued random variables, such as time to death time to onset (or relapse) of a disease length of stay in a hospital money paid by health insurance viral load measurements

Kinds of survival studies include: clinical trials prospective cohort studies retrospective cohort studies retrospective correlative studies

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Definition and Characteristics of Variables Survival time (t) random variables (RVs) are always non-

negative, i.e., t ≥ 0. T can either be discrete (taking a finite set of values, e.g.

a1, a2, …, an) or continuous [defined on (0,∞)]. A random variable t is called a censored survival time RV

if x = min(t, u), where u is a non-negative censoring variable.

For a survival time RV, we need: (1) an unambiguous time origin (e.g. randomization to clinical

trial) (2) a time scale (e.g. real time (days, months, years) (3) defnition of the event (e.g. death, relapse)

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Sample of Target Population

Randomize

Control

Test

Time to Event

Event

Non-Event

Non-Event

Event

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Illustration of Survival Data

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Why Regression for Survival Data? Survival, in the form of hazard function, and one or more

explanatory co-variables can be very interesting research investigation

The relation with risk factors can be studied using group-specific Kaplan-Meier estimates, together with Logrank and/or Wilcoxon tests

Investigating the relation with covariates, requires a regression-type model

Relating the outcome to several factors and/or covariates simultaneously requires multiple regression, ANOVA, or ANCOVA models

The most frequently used model is the Cox (proportional hazards) model

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Understanding the Effect of Co-variables

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Cox Proportional Hazards Regression Most common Cox are linear-like models for the log hazard

For example, a parametric regression model based on the exponential distribution:

loge hi(t) = α + β1xi1 + β2xi2 + … + βkxik

or, equivalently,

hi(t) = exp (α + β1xi1 + β2xi2 + … + βkxik)

= eα x eβ1xi1 x eβ2xi2 x … x eβkxik

Where i indexes subjects and xi1, xi2, …, xik are the values of the co-variates for the ith

subject

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Cox Model contd.. This is therefore a linear model for the log-hazard or a multiplicative

model for the hazard itself

The model is parametric because, once the regression parameters α, β1, … βk are specified, the hazard function hi(t) is fully characterized by the model

The regression constant α represents a kind of baseline hazard, since loge hi(t) = α, or equivalently, hi(t) = eα, when all of the x’s are 0

Other parametric hazard regression models are based on other distributions commonly used in modeling survival data, such as the Gompertz and Weibull distributions.

Parametric hazard models can be estimated with standards softwares

Source: John Fox19

Cox Regression is a Proportional Hazards Model Consider two observations, h1(t): hazard for the experimental group and

h0(t): hazard for the control group

h1(t)/h0(t) = exp(β)

exp (β) indicates how large (small) is the hazard in experimental group with the respect to the hazard in the reference group

and it is constant, does not depend on time. Hence, it is called “proportional hazards” over time

Other qualities: Usually provides better estimates of survival probabilities and

cumulative hazard than those provided by the Kaplan-Meier function when assumptions are met

The coefficients in a Cox regression relate to hazard a positive coefficient indicates a worse prognosis a negative coefficient indicates a protective effect of the variable with which it is

associated

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Exploring Co-variables by Cox Regression

Source: Yesilda Balavarca, Internet21

Interpretation of Resultsh1 (t,X) = h0(t) exp (β1 gender + β2 treatment)

Gender: 1 = male, 0 = female; treament: 1 = experimental, 0 = control

h1 (t,X) = h0(t) exp (−0.51 gender + 0.69 treatment) and

exp(β1 ) = exp(−0.51 ) = 0.6 and exp(β2 ) = exp(0.69 ) = 2.0

This means a reduction of hazards for males, i.e., males have larger probabilities of survival than females

The experimental treatment increases hazard, i.e., patients receiving the new experimental treatment have lower survival probabilities than patients on the control (standard) treatment

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Checking Proportionality of Hazards Check to see if the estimated survival curves cross

If they do, then this is evidence that the hazards are not proportional

More formal test: e.g., scaled Schoenfeld Residuals show interactions between covariates and time Testing the time dependent covariates is equivalent to testing

for a non-zero slope in a generalized linear regression of the scaled Schoenfeld residuals on functions of time

A non-zero slope is an indication of a violation of the proportional hazard assumption.

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Proportionality of Hazards: Schoenfeld Residuals

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Cox Regression is a Proportional Hazards Model Cox regression (or proportional hazards regression) is method

for investigating the effect of several variables upon the time a specified event takes to happen

When an outcome is death this is known as Cox regression for survival analysis

Assumptions: the effects of the predictor variables upon survival are constant over time are additive in one scale

Usually provides better estimates of survival probabilities and cumulative hazard than those provided by the Kaplan-Meier function when assumptions are met

The coefficients in a Cox regression relate to hazard a positive coefficient indicates a worse prognosis a negative coefficient indicates a protective effect of the variable with

which it is associated25

Remember the Survival Data in Part 1?

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Organized Input DataGroup Surv Time SurvCensor Surv

2 142 11 143 12 157 12 163 11 165 11 188 11 188 11 190 11 192 12 198 12 204 02 205 11 206 11 208 11 212 11 216 01 216 11 220 11 227 11 230 1

Group Surv Time Surv Censor Surv

2 232 12 232 12 232 12 233 12 233 12 233 12 233 11 235 12 239 12 240 11 244 01 246 12 261 11 265 12 280 12 280 12 295 12 295 11 303 12 323 12 344 0

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Hazard Rate Plot

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Log Hazard Plot

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Cox Hazard Analysis

Coefficient95% Conf.

(±) Std.Error P Hazard =

Exp(Coef.)

Group Surv -0.5861172 0.6726008 0.343165 0.0876 0.55648

The significance test for the coefficient b1 tests the null hypothesis that it equals zero and thus that its exponent equals one

The confidence interval for b1 is therefore the confidence interval for the relative death rate or hazard ratio

What is your conclusion of this analysis?30

And the Case Study Data in Part 1?

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Case Study: Results Cox proportional hazards:

Factors associated with increased mortality risk were male sex, poor KPS (< 80), presence of liver metastases, high serum lactate dehydrogenase, and low serum albumin.

Adjusted for these variables, there was no statistically significant difference in survival rates between patients treated with gemcitabine and marimastat 25 mg, but patients receiving either marimastat 10 or 5 mg were found to have a significantly worse survival rate than those receiving gemcitabine

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Bad or Wrong Methods of Analysis Comparison of life tables at one point in time ignoring their structure

elsewhere (except very rapid processes) If a few patients are at risk for more than a certain time but do not die, this

should not be taken as evidence of cure. Look at all the data of all the patients

Median survival times are not very reliable unless the death rate around that median is very high

A simple count of number of death in each group is inefficient as it ignores the rate of death

The best estimate of the probability of survival for a certain time (say 5 years), is given by the life table value at that time. Other simplistic calculations may be misleading

Randomized controls are always better than historical controls

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Bad or Wrong Methods of Analysis contd.

Estimation of survival is best done from randomization time. If it is done from the time of 1st treatment it can be misleading (as initiating time for two treatments can be different)

Superficial comparison of the slopes of survival graphs as it biases the proportion surviving at each given time

Declaring ITT is better than per protocol analysis or the reverse Check all the data carefully especially the P values associated with either

type of analysis When you get an overall non-significant treatment effect, do not insist that

a sub-stratum can still benefit from the treatment even if that stratum analysis is significant

Realistically not checking the actual number of survivors on the last day of the study (follow up)

Be sure of your reason to use and report one-sided vs. two-sided t-tests

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Overall Conclusions Survival time is measured for each patient from his/her date of

randomization The life table is a table or graph estimating the proportion of

surviving patients at different times after randomization The Log Rank test is a comparison of observed and expected

death in each experimental group P value of Log Rank can be estimated by a chi square (2 ) test.

A patients are divided into strata (prospectively or retrospectively), K-M life tables or Log Rank can be used to compare prognosis in each stratum, for testing heterogeneity, etc.

Usually Cox regression yields slightly better analysis of cancer trial data provided assumptions are met

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Notes 1

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Notes 2

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End of Part 2

Your ?s

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Thank You Very Much

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