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Part 2 Cox Regression


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Part 2 Cox Regression

  1. 1. Survival Analysis and CoxRegression 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 1
  2. 2. Part 2: Cox Regression Analysis of Cancer CTs2
  3. 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 scrutiny3
  4. 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 5
  5. 5. 6 Source: WHO
  6. 6. 7
  7. 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]). 8
  8. 8. 10
  9. 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 ? 11
  10. 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 studies12
  11. 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)13
  12. 12. Event Test Non-EventSample of Target Randomize Population Event Control Non-Event Time to Event 14
  13. 13. Illustration of Survival Data15
  14. 14. 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 16
  15. 15. Understanding the Effect of Co-variables17
  16. 16. 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 18 subject
  17. 17. 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 19 Source: John Fox
  18. 18. Cox Regression is a Proportional HazardsModel 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 20
  19. 19. Exploring Co-variables by Cox Regression21 Source: Yesilda Balavarca, Internet
  20. 20. Interpretation of Results h1 (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 22
  21. 21. 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.23
  22. 22. Proportionality of Hazards: SchoenfeldResiduals24
  23. 23. Cox Regression is a Proportional HazardsModel 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 associated 25
  24. 24. Remember the Survival Data inPart 1?26
  25. 25. Organized Input DataGroup Surv Time Surv Censor Surv Group Surv Time Surv Censor Surv 2 142 1 2 232 1 1 143 1 2 232 1 2 157 1 2 232 1 2 163 1 2 233 1 1 165 1 2 233 1 1 188 1 2 233 1 1 188 1 2 233 1 1 235 1 1 190 1 2 239 1 1 192 1 2 240 1 2 198 1 1 244 0 2 204 0 1 246 1 2 205 1 2 261 1 1 206 1 1 265 1 1 208 1 2 280 1 1 212 1 2 280 1 1 216 0 2 295 1 1 216 1 2 295 1 1 220 1 1 303 1 1 227 1 2 323 1 1 230 1 2 344 0 27
  26. 26. Hazard Rate Plot28
  27. 27. Log Hazard Plot29
  28. 28. Cox Hazard Analysis 95% Conf. Hazard = Coefficient (±) Std.Error P Exp(Coef.) Group Surv -0.5861172 0.6726008 0.343165 0.0876 0.55648The significance test for the coefficient b1 tests the null hypothesis that itequals zero and thus that its exponent equals oneThe confidence interval for b1 is therefore the confidence interval for therelative death rate or hazard ratioWhat is your conclusion of this analysis?30
  29. 29. And the Case Study Data inPart 1?31
  30. 30. 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 gemcitabine32
  31. 31. 33
  32. 32. 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 34
  33. 33. 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 35
  34. 34. 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 (χ2 ) 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 36
  35. 35. Notes 137
  36. 36. Notes 238
  37. 37. End of Part 2 Your ?s39
  38. 38. Thank You Very Much40