1. Survival Analysis and Cox
Regression for Cancer Trials
Presented at PG Department of Statistics,
Sardar Patel University January 29, 2013
Dr. Bhaswat S. Chakraborty
Sr. VP & Chair, R&D Core Committee
Cadila Pharmaceuticals Ltd., Ahmedabad
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3. 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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4. 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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7. 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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9. 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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10. 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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11. 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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12. Event
Test
Non-Event
Sample of Target Randomize
Population
Event
Control
Non-Event
Time to Event
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14. 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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15. 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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17. 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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18. 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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19. 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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20. Other Parameters
The cumulative hazard function (H) is the risk of event 2 (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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21. 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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22. 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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25. 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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27. 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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32. 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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34. Cox-Mantel Log Rank Test
Group Events observed Events expected
1 17 12.20368414
2 20 24.79631586
Degrees of
Chi-square 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.
36. 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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37. 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 start
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38. 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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39. 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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40. 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 χ2 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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42. Primary mortality analysis by treatment arm
log-rank test,
P = .0001
Bramhall, S. R. et al. J Clin Oncol; 19:3447-3455 2001
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43. Case Study: Results
The 1-year survival rate:
was 19% for the gemcitabine group and 20% for the marimastat 25-mg group (χ2
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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48. References
1. 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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