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 1: Survival 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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Characterization of Survival There are several equivalent ways to characterize the

probability distribution of a survival random variable. We will use the following terms:

The density function f(t) The survival or survivor function S(t) The hazard function H(t) The cumulative hazard function Λ(t)

Median survival, called τ, is defined as S(τ ) = 0.5 Similarly, any other percentiles could be defined

Practically, we estimate median survival as the smallest time τ such that Ŝ (τ) ≤ 0.5

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Estimation of Survival Function from Censored Data Hazard shapes can be increasing (e.g. aging after 65),

decreasing (e.g. survival after surgery), bathtub (e.g. age-specifc mortality) or constant (e.g. survival of patients with advanced chronic disease)

If there are censored observations, then Ŝ(t) by a parametric method is not a good estimate of the true S(t), so non-parametric methods must be used to account for censoring (life-table methods, Kaplan-Meier estimator)

Kaplan-Meier estimator is the most popular

However, theoretically (or when conditions are met) survival data can be fit to parametric models as well

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Source: Ibrahim J (2005), Amer Stat Assoc 18

Survival Rate by Kaplan Meier This function estimates survival rates and hazard from data that may be incomplete.

The survival rate is expressed as the survivor function (S):

where t is the survival time e.g. 2 years in the context of 5 year survival rates

Sometimes S is estimated as the probability of surviving to time t for those alive just before t multiplied by the proportion of subjects surviving to t

Calculated by product limit (PL) method 1 or the simpler Nelson-Aalen estimate2, estimate of which is always less than a Peterson estimate

Peterson PL Method Nelson-Aalen Method

where ti is duration of study at point i, di is number of deaths up to point i and ni is number of individuals at risk just prior to ti.

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Data Following example is data of death from a cancer after

exposure to a particular carcinogen in two groups of patients. Group 1 had a different pre-treatment régime to group 2 The time from pre-treatment to death is recorded. If a patient was still living at the end of the experiment or it

had died from a different cause then that time is considered censored (* below). A censored observation is given the value 0 in the death/censorship variable to indicate a "non-event".

Group 1: 143, 165, 188, 188, 190, 192, 206, 208, 212, 216, 220, 227, 230, 235, 246, 265, 303, 216*, 244*

Group 2: 142, 157, 163, 198, 205, 232, 232, 232, 233, 233, 233, 233, 239, 240, 261, 280, 280, 295, 295, 323, 204*, 344*

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Assumptions S is based upon the probability that an individual survives at

the end of a time interval, on the condition that the individual was present at the start of the time interval. S is the product (P) of these conditional probabilities.  If a subject is last followed up at time ti and then leaves the study for any reason

(e.g. lost to follow up) ti is counted as their censorship time.

Censored individuals have the same prospect of survival as those who continue to be followed. This can not be tested for and can lead to a bias that artificially reduces S.

Survival prospects are the same for early as for late recruits to the study (can be tested for).

The event studied (e.g. death) happens at the specified time. Late recording of the event studied will cause artificial inflation of S.

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Other Parameters The cumulative hazard function (H) is the risk of event2 (e.g. death) at time t S and H with their standard errors and confidence intervals need to be saved

Median and mean survival time The median survival time is calculated as the smallest survival time for which the survivor

function is less than or equal to 0.5. Some data sets may not get this far, in which case their median survival time is not calculated. A confidence interval for the median survival time can be constructed using a robust non-

parametric method or using a large sample estimate of the density function of the survival estimate

Mean survival time is estimated as the area under the survival curve. The estimator is based upon the entire range of data

Some software uses only the data up to the last observed event; this biases the estimate of the mean downwards, entire range of data should be used

Samples of survival times are frequently highly skewed, therefore, in survival analysis, the median is generally a better measure of central location than the mean

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Variances of S & H hat The variance of S

The confidence interval for the survivor function is not calculated directly using Greenwood's variance estimate as this would give impossible results (< 0 or > 1) at extremes of S. The confidence interval for S uses an asymptotic maximum likelihood solution by log transformation

The variance of H hat is estimated as:

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Cancer Trial Data Preparation Data is very complex, statistical skills & insight are necessary Time-to-event:

time a patient in a trial survived time to tumor progression or relapse in a patient

Event / censor code: 1 or 0 1 for event(s) happened 0 for no event or lost to follow up but survival   assumed

Stratification: e.g. centre code for a multi-centre trial Be careful with your choice of strata 

Predictors (covariates): which can be a number of variables that are thought to be related to the

event under study, e.g., drug treatment, disease stage

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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 Surv2 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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Analyzed K-M Survival Group 1 (Life Table)

Time At risk Dead Censored

S SE(S) H SE(H)

143 19 1 0 0.947368 0.051228 0.054067 0.054074165 18 1 0 0.894737 0.070406 0.111226 0.078689188 17 2 0 0.789474 0.093529 0.236389 0.118470 190 15 1 0 0.736842 0.101023 0.305382 0.137102192 14 1 0 0.684211 0.106639 0.37949 0.155857206 13 1 0 0.631579 0.110665 0.459532 0.175219208 12 1 0 0.578947 0.113269 0.546544 0.195646212 11 1 0 0.526316 0.114549 0.641854 0.217643216 10 1 1 0.473684 0.114549 0.747214 0.241825220 8 1 0 0.414474 0.114515 0.880746 0.276291227 7 1 0 0.355263 0.112426 1.034896 0.316459230 6 1 0 0.296053 0.108162 1.217218 0.365349235 5 1 0 0.236842 0.10145 1.440362 0.428345244 4 0 1 0.236842 0.10145 1.440362 0.428345246 3 1 0 0.157895 0.093431 1.845827 0.591732265 2 1 0 0.078947 0.072792 2.538974 0.922034303 1 1 0 0 * infinity *

 

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Descriptive Stats Group 1 Median survival time = 216 Andersen 95% CI for median survival time = 199.62 to

232.382 Brookmeyer-Crowley 95% CI for median survival time = 192

to 230   Mean survival time (95% CI) = 218.68 (200.36 to 237.00)

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Time At risk Dead Censored S SE(S) H SE(H)

142 22 1 0 0.954545 0.044409 0.04652 0.046524157 21 1 0 0.909091 0.061291 0.09531 0.06742163 20 1 0 0.863636 0.073165 0.146603 0.084717198 19 1 0 0.818182 0.08223 0.200671 0.100504204 18 0 1 0.818182 0.08223 0.200671 0.100504205 17 1 0 0.770053 0.090387 0.261295 0.117378232 16 3 0 0.625668 0.105069 0.468935 0.16793233 13 4 0 0.433155 0.108192 0.836659 0.249777239 9 1 0 0.385027 0.106338 0.954442 0.276184240 8 1 0 0.336898 0.103365 1.087974 0.306814261 7 1 0 0.28877 0.099172 1.242125 0.34343280 6 2 0 0.192513 0.086369 1.64759 0.44864295 4 2 0 0.096257 0.064663 2.340737 0.671772323 2 1 0 0.048128 0.046941 3.033884 0.975335344 1 0 1 0.048128 0.046941 3.033884 0.975335

 Analyzed K-M Survival Group 2 (Life Table)

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Descriptive Stats Group 2 Median survival time = 233 Andersen 95% CI for median survival time = 231.89 to 234.10 Brookmeyer-Crowley 95% CI for median survival time = 232

to 240

Mean survival time (95% CI) = 241.28 (219.59 to 262.98)

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Survival Plot

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LogNormal Survival Plot

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Hazard Rate Plot

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Log Hazard Plot

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Comparing Survival Functions Due to the censoring, classical tests such as t-test and Wilcoxon

test cannot be used for the comparison of the survival times Various tests have been designed for the comparison of survival

curves, when censoring is present • The most popular ones are:

Logrank (or Cox-Mantel or Mantel-Haenszel) test Wilcoxon (Gehan) test

The Logrank test has more power than Wilcoxon for detecting late differences

The Logrank test has less power than Wilcoxon for detecting early differences

The logrank statistic is distributed as χ2 with a H0 that survival functions of the two groups are the same

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Cox-Mantel Log Rank Test

Group Events observed Events expected

1 17 12.203684142 20 24.79631586

Chi-squareDegrees of Freedom P

3.124689787 1 0.077114564

The test statistic for equality of survival across the k groups (populations sampled) is approximately chi-square distributed on k-1 degrees of freedom.

Show the Excel Sheetfollowed by Case Study

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Case Study• Purpose: This, the largest randomized study in pancreatic

cancer performed to date, compares marimastat, the first of a new class of agents, with gemcitabine

• Context: The prognosis for unresectable pancreatic cancer remains dismal (1-year survival rate, < 10%; 5-year survival rate, < 5%)– Recent advances in conventional chemotherapy and novel

molecular treatment strategies warrant investigation

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Case Study: Patients• Histologically or cytologically proven adenocarcinomas of the pancreas

– unresectable on computed tomographic imaging• Tumors were staged – Using the International Union Against Cancer tumor-node-metastasis classification– Stage-grouped according to the American Joint Committee on Cancer Staging criteria for

pancreatic cancer• Inclusion Criteria

– Patients entered the study within 8 wks of initial diagnosis or within 8 weeks of recurrence after prior surgery

– >18 years– Karnofsky performance status (KPS) of at least 50%– At entry, patients had to have adequate bone marrow reserves, defined 1,000/µL of

granulocytes, platelet 100,000/µL, and Hb 9 g/dL. – Adequate baseline hepatic function (bilirubin 2 x ULN; AST, ALT, or alkaline phosphatase 5 x

ULN)– Adequate renal function (creatinine 2 x ULN)

• Exclusion Criteria– Any fprevious systemic anticancer therapy as a primary intervention for locally advanced or

metastatic disease – Prior exposure to a metalloproteinase inhibitor or gemcitabine– Patients with prior adjuvant or consolidation chemotherapy or radiotherapy and relapsed within

6 months after therapy – Pregnant or lactating patients were excluded– Patients who had other investigational agents within 4 weeks before study start39

Case Study: End Points• The primary study end point

– Overall survival, along with prospectively defined comparisons between gemcitabine and marimastat 25 mg bid and between gemcitabine and marimastat 10 mg bid

– Secondary study end points • Progression-free survival• Patient benefit s

– quality of life [QOL], weight loss, pain, analgesic consumption, surgical intervention to alleviate cancer symptoms, and KPS)

• Safety and tolerability

• Tumor response rate was also assessed

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Case Study: Randomization & Treatments Randomization

IC obtained from each patient before study entry A computer-generated random code used according to the method of minimization. This

method balanced the treatment groups on the basis of stage of disease (stage I/II, III, or IV), KPS (50% to 70% v 80% to 100%), sex, disease status (recurrent v newly diagnosed), and study center.

Patients received either 5 mg, 10 mg, or 25 mg of marimastat bid orally

or 1,000 mg/m2 of gemcitabine hydrochloride by intravenous infusion The marimastat dosage was double-blinded but allocations to marimastat or gemcitabine

were open-label due to the different modes of administration Treatment

Marimastat: 5 mg bid, 10 mg bid, or 25 mg bid with food. The dose of marimastat could be reduced if musculoskeletal or other toxicities developed.

Gemcitabine: 1,000 mg/m2 weekly for the first 7 weeks, no treatment in week 8, 1,000 mg/m2 weekly for 3 weeks next and nothing in the fourth week. Dose reduction (25%) permitted at granulocyte 0.5-0.99/µL or platelet 50,000 to 99,999/µL

If the counts were lower after the lower dose, the next dose was omitted.

Patients who could not be treated for 6 weeks as a result of toxicity were withdrawn from the study. No concomitant anticancer therapy

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Case Study: Statistical Analysis Sample size (n= 400: 100 per group); based on absolute

differences in survival rates at 18 months of 14.5%, power of 80% and using a significance level of .025 (log-rank test, Bonferoni adjusted)• Also based on 10% survival rate at study censure with gemcitabine and a

mortality rate 75.5% in 10- or 25-mg marimastat groups Data Analysis

• Intent-to-treat -- Kaplan-Meier• Cox proportional hazards model to identify prognostic factors and to explore their

influence on the comparative hazard of death between the treatment groups• Plots of Log (-Log, survivor function) versus time for each individual variable

were produced to ensure proportional hazard assumptions were met• In all survival analyses, patients who were lost to follow-up were censored at

their last known date alive.• Proportions were tested using the 2 test. Patient benefit data were tested using

the Wilcoxon rank sum test, and repeated measures analysis was applied to the QOL data.

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Bramhall, S. R. et al. J Clin Oncol; 19:3447-3455 2001

Primary mortality analysis by treatment arm

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log-rank test, P = .0001

Case Study: Results The 1-year survival rate:

was 19% for the gemcitabine group and 20% for the marimastat 25-mg group (2 test, P = .86). The 1-year survival rate for both the 10-mg and 5-mg groups was 14%

Progression-free survival: revealed a significant difference between the gemcitabine group and each of the

three marimastat treatment groups (log-rank test, P = .0001), with median progression-free survivals of 115, 57, 59, and 56 days for gemcitabine, marimastat 25-, 10-, and 5-mg groups, respectively

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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End of Part 1

Your ?s

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References1. Kaplan EL, Meier P. Nonparametric estimation from incomplete observations.

Journal of the American Statistical Association (1958); 53: 457-481

2. Peterson AV Jr.. Expressing the Kaplan-Meier estimator as a function of empirical subsurvival functions. Journal of the American Statistical Association (1977); 72: 854-858

3. Nelson W. Theory and applications of hazard plotting for censored failure data. Technometrics (1972); 14: 945-966

4. Aalen OO. Non parametric inference for a family of counting processes. Annals of Statistics 1978; 6: 701-726.

5. R. Peto et al. Design and analysis of randomized clinical trials requiring prolonged observation of each patient. British Journal of Cancer (1977); 31: 1-39.

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Thank You Very Much

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