Survival Analysis for Randomized Clinical Trials

1. Survival Analysis for Randomized Clinical Trials The log-rank test The proportional hazard model (Cox regression) Ziad Taib March 1, 2010

2. I The log-rank test • INTRODUCTION TO SURVIVAL TIME DATA • ESTIMATING THE SURVIVAL FUNCTION • THE LOG RANK TEST

3. Survival Analysis

4. Survival Analysis in RCT • For survival analysis, the best observation plan is prospective. In clinical investigation, that is a randomized clinical trial (RCT). • Random treatment assignments. • Well-defined starting points. • Substantial follow-up time. • Exact time records of the interesting events.

5. Survival Analysis in Observational Studies • Survival analysis can be used in observational studies (cohort, case control etc) as long as you recognize its limitations. • Lack of causal interpretation. • Unbalanced subject characteristics. • Determination of the starting points. • Lost of follow-up. • Ascertainment of event times.

6. Elements of Survival Experiments • Event Definition (death, adverse events, …) • Starting time • Length of follow-up (equal length of follow-up, common stop time) • Failure time (observed time of event since start of trial) • Unobserved event time (censoring, no event recorded in the follow-up, early termination, etc) End of follow up time start Early termination event time

7. When to use survival analysis • Examples • Time to death or clinical endpoint • Time in remission after treatment of CA • Recidivism rate after alcohol treatment • When one believes that 1+ explanatory variable(s) explains the differences in time to an event • Especially when follow-up is incomplete or variable

8. Standard Notation for Survival Data • Ti -- Survival (failure) time • Ci -- Censoring time • Xi =min (Ti ,Ci) -- Observed time • Δi =I(Ti ≤Ci) -- Failure indicator: If the ith subjecthad an event before been censored, Δi=1, otherwise Δi=0. • Zi(t) – covariate vector at time t. • Data: {Xi , Δi , Zi(·) }, where i=1,2,…n.

9. Describing Survival Experiments • Central idea: the event times are realizations of an unobserved stochastic process, that can be described by a probability distribution. • Description of a probability distribution: • Cumulative distribution function • Survival function • Probability density function • Hazard function • Cumulative hazard function

10. Relationships Among Different Representations • Given any one, we can recover the others.

11. Descriptive statistics • Average survival • Can we calculate this with censored data? • Average hazard rate • Total # of failures divided by observed survival time (units are therefore 1/t or 1/pt-yrs) • An incidence rate, with a higher value indicating lower survival probability • Provides an overall statistic only

12. Estimating the survival function • There are two slightly different methods to create a survival curve. • With the actuarial method, the x axis is divided up into regular intervals, perhaps months or years, and survival is calculated for each interval. • With the Kaplan-Meier method, survival is recalculated every time a patient dies. This method is preferred, unless the number of patients is huge. • The term life-table analysis is used inconsistently, but usually includes both methods.

13. Life Tables (no censoring) In survival analysis, the object of primary interest is the survival function S(t).Therefore we need to develop methods for estimating it in a good manner. The most obvious estimate is the empirical survival function: time 0 1 2 3 4 5 6 7 8 9 10

14. Example: A RAT SURVIVAL STUDY In an experiment, 20 rats exposed to a particular type of radiation, were followed over time. The start time of follow-up was the same for each rat. This is an important difference from clinical studies where patients are recruited into the study over time and at the date of the analysis had been followed for different lengths of time. In this simple experiment all individuals have the same potential follow-up time. The potential follow-up time for each of the 20 rats is 5 days.

15. Survival Function for Ratsa

16. Proportion of rats dying on each of 5 days Survival Curve for Rat Study

17. Confidence Intervals for Survival Probabilities From above we see that the "cumulative" probability of surviving three days in the rat study is 0.25. We may want to report this probability along with its standard error. This sample proportion of 0.25 is based on 20 rats that started the study. If we assume that (i) each rat has the same unknown probability of surviving three days, S(3), and (ii) assume that the probability of one rat dying is not influenced by whether or not another rat dies, then we can use results associated with the binomial probability distribution to obtain the variance of this proportion. The variance is given by

18. This can be used to test hypotheses about the theoretical probability of surviving three days as well as to construct confidence intervals. For example, the 95% confidence interval for is given by 0.25 +/- 1.96 x 0.0968 or ( 0.060,0.440) We are 95% confident that the probability of surviving 3 days, meaning THREE OR MORE DAYS, lies between 0.060 and 0.440.

19. In general This approach has many drawbacks • Patients are recruited at different time periods • Some observations are censored • Patients can differ wrt many covariates • Continuous data is dicretised

20. Kaplan-Meier survival curves • Also known as product-limit formula • Accounts for censoring • Generates the characteristic “stair step” survival curves • Does not account for confounding or effect modification by other covariates • Is that a problem?

21. In general The same as before! Similariliy

22. Censored Observations (Kaplan-Meier) We proceed as in the case without censoring stands for the proportion of patients who survive day i among those who survive day i-1. Therefore it can be estimated according to

23. K-M Estimate: General Formula • Rank the survival times as t(1)≤t(2)≤…≤t(n). • Formula • ni patients at risk • di failures PROC LIFETEST

24. Confidence Intervals

25. Using SAS

28. Comparing Survival Functions • Question: Did the treatment make a difference in the survival experience of the two groups? • Hypothesis: H0: S1(t)=S2(t) for all t ≥ 0. • Two tests often used : • Log-rank test (Mantel-Haenszel Test); • Cox regression

29. A numerical Example

31. The Log-rank test During the th interval, let • Nt be the number of patients at risk in the drug group at the beginning of the interval. • Mt be the number of patients at risk in the placebo group at the beginning of the interval. • At the number of events during the interval in the drug group • Ct the number of events during the interval in the placebo group • Tt = Nt + Mt

32. The Log-rank test • The above contingency table is a way of summarising the data at hand. Notice though that the margins Nt and Mt are fixed. • The log-rank test is a significance test. It does not say anything about the size of the difference between the two groups.

33. This situation is similar to one where we have a total of Tt balls in an urn. These balls are of two different kinds D and K. Dt balls are drawn at random (those who experience events). • We denote by At the number of balls of type D among the Dt drawn. Dt Tt = Nt + Mt

34. Wecan thus calculate probabilities of events like {At = a} using the hypergeometric distribution. The corresponding mean and variance are.

35. The above can be used to derive the following test statistic where Nt and Mt are supposed to be large, which is often the case in RCT. and Assume now that we want to combine data from k successive such intervals. We can then define U according to And use the statistic

36. Variants of mantel- Haensel Instead of QM-H we can formulate variants with weights under Ho. • wt =1 M-H • wt = TtGehan 1965 • wt = Tarone and Ware 1977

37. A numerical example

38. Using SAS

39. Limitation of Kaplan-Meier curves • What happens when you have several covariates that you believe contribute to survival? • Example • Smoking, hyperlipidemia, diabetes, hypertension, contribute to time to myocardial infarct • Can use stratified K-M curves – but the combinatorial complexity of more than two or three covariates prevents practical use • Need another approach – multivariate Cox proportional hazards model is most commonly used • (think multivariate regression or logistic regression)

40. II Cox Regression • Introduction to the proportional hazard model (PH) • Partial likelihood • Comparing two groups • A numerical example • Comparison with the log-rank test

41. Cox Regression Model • Proportional hazard. • No specific distributional assumptions (but includes several important parametric models as special cases). • Partial likelihood estimation (Semi-parametric in nature). • Easy implementation (SAS procedure PHREG). • Parametric approaches are an alternative, but they require stronger assumptions about the hazard rate.

42. Cox Regression • In 1972 Cox suggested a model for survival data that would make it possible to take covariates into account. Up to then it was customary to discretise continuos variables and build subgroups. • Cox idea was to model the hazard rate function where h(t) is to be understood as an intensity i.e. a probability by time unit. Multiplied by time we get a probability. Think of the analogy with speed as distance by time unit. Multiplied by time we get distance.

43. Cox’ suggestion is to model h using where each parameter is a measure of the importance of the corresponding variable. A consequence of this is that two individuals with different covariate values will have hazard rate functions which differ by a multiplicative term which is the same for all values of t.

44. The covariates can be discrete, continuous or even categorical. It is also possible to generalise the model to allow time varying covariates.

45. Cox proportional hazards model, continued • Can handle both continuous and categorical predictor variables (or covariates, think: logistic, linear regression) • Without knowing baseline hazard ho(t), can still calculate coefficients for each covariate, and therefore hazard ratio • Assumes multiplicative risk—this is the proportional hazard assumption • Can be compensated in part with interaction terms

46. Example Assume we have a situation with one covariate that takes two different values 0 and 1. This is the case when we wish to compare two treatments h2(t) h1(t) t

47. Comments • One of the advantages of the PH model is that we can estimate the parameters and make comparisons between treatments without having to estimate the baseline. • The baseline can be interpreted as the case corresponding to an individual with covariate values zero. • The word semi-parametric is used to show that we do not model h0 explicitly. The usual likelihood theory does not apply. From the definition we see immediately that

48. Partial likelihood The key to understanding partial likelihood is the following conditional probability: P(patient j dies under Dt | one patient dies during Dt) =P(patient j dies under Dt)/P(one patient dies during Dt) =P(patient j dies under Dt) / Si P( one patient dies under Dt | patient i dies during Dt) P(patient i dies during Dt) =P(patient j dies under Dt) / Si P(patient i dies during Dt)

49. P(patient i dies under Dt) = • Let R(t)= # of patients at risk at time t (i.e alive and under observation at t). • Let further Rj=R(tj). Therefore P(patient j dies under Dt | one patient dies duringDt)

50. Then P(patient j dies under Dt | one patient dies during Dt) From this it follows that we can base our inference on the likelihood function