Parametric Distributions in Survival Analysis: Functions and Median Survival

1. Lecture 2: Key Functions and Parametric Distributions Survival Function Hazard Function Median Survival Common Parametric Distributions

2. But First • Let’s think a little more about censoring and truncation using an example… • An investigator is interested in determining if treatment with amoxetine leads to recovery of cognitive function in rats with brain lesions that mimic Parkinson’s disease. • The outcome of interest is time to “complete” recovery of cognitive function • i.e. time it takes to return to baseline cognitive function after treatment with amoxetine.

3. Amoxetine and Cognitive Function • Collect baseline measure of cognitive function • Time to correctly perform water radial arm maze (WARM) task • Induce cognitive impairment • Treat 4 week old rats with N-(2-chloroethyl)-N-ethyl-bromo-benzylamine (DSP-4) • causes noradrenergic lesions in the locus coeruleus. • Treat lesioned animals with Amoxetine • daily dose for 4 weeks (ages 4 to 8 weeks) • 0, 0.3, 1.0, or 3.0 mg/kg • Measures cognitive performance post treatment • weekly for 16 weeks (ages 8 to 24 weeks) • Endpoint: time it takes reach >75% baseline cognitive function

4. Describe the type of censoring • Rat survives to 24 weeks of age but never achieves complete cognitive recovery • Rat does not achieve complete cognitive recovery at 12 weeks but does by 13 weeks • Rat that dies at 16 weeks but has not yet achieve complete cognitive recovery

5. Describe the type of censoring • Rat doesn’t develop brain lesions due to misplaced DSP-4 treatment and shows complete cognitive recovery at 8 weeks • Rat shows complete cognitive recovery 8 at weeks

6. Let’s Draw These 5 Animals

7. Time to Event Outcomes • Modeled using “survival analysis” • Define X = time to event • X is a random variable • Realizations of X are denoted x • X>0 • Key characterizing functions • Survival functions • Hazard rate (or function) • Probability density function • Mean residual life

8. PDF, survival function, hazard rate, and mean residual life • S(x) • f(x)

9. PDF, survival function, hazard rate, and mean residual life • h(x) • mrl(x)

10. Survival Function • S(x) = the probability of an individual surviving to time x • Basic properties:

11. Types of time to event data • Continuous t • Observe actual time • Discrete t • Interval censoring • Grouping into intervals Where p(xj) is the probability mass function, P(X = xj)

12. Example of Discrete Time to Event • Discrete Uniform (3 times possible)

13. Hazard Rate • A little harder to conceptualize • Instantaneous failure rate or conditional failure rate • Interpretation: probability that a person at time t experiences the event in the interval (x, x+Dx) given survival to time x.

14. Hazard Rate • Relationship between h(x), S(x) and pdf (continuous):

15. Hazard Function • Useful for conceptualizing how the chance of an event changes over time • i.e. consider hazard ‘relative’ over time • Examples: • Treatment related mortality • Early on, high risk of death • Later on, risk of death decreases • Aging • Early on, low risk of death • Later on, high risk of death

16. Shapes of Hazard Functions • Increasing • Natural aging and wear • Decreasing • Early failures due to device or transplant failures • Bathtub • Populations followed from birth • Hump Shaped • Initial risk of event, followed by decreasing chance of event

17. Examples

18. R Code for Hazard Function Shapes #Examples of hazard function shapes weibull.hazard<-function(x,alp,lam) { h<-alp*lam*x^(alp-1) return(h) } loglogistic.hazard<-function(x,alp,lam) { h<-alp*lam*x^(alp-1)/(1+lam*x^alp) return(h) } x<-seq(0, 6, 0.05) h1<-weibull.hazard(x, 1.5, 0.25) plot(x, h1, type="l", lwd=2, ylab="Hazard Function", xlab="Time", ylim=c(0,1)) h2<-loglogistic.hazard(x, 0.5, 0.25) lines(x, h2, lwd=2, col=2) h3<-loglogistic.hazard(x, 2, 1) lines(x, h3, lwd=2, col=3) h4<-0.01*(x-3)^4 lines(x, h4, lwd=2, col=4)

19. Cumulative Hazard Function • Often used instead of the hazard function • Relationship between H(x) and S(x) • More on this later or model checking…

20. What if time is discrete? • So far we’ve focused on time Xas a continuous r.v. • Discrete x • Interval censoring • Grouping into intervals • Depending on level of discreteness, use discrete data approach where p(xj) is a pmf(P(X= xj)).

21. Complications • How can we use this to define our “discrete” hazard function?

22. Complications • How can we use this to define our “discrete” hazard function?

23. Mean Residual Life • Biomedical applications • Median is very common • MRL is not common • MRL = the expected residual life • Theoretically, could be useful to predict survival times given survival to a certain point in time.

24. Mean • We do not see the mean quantified very often in biomedical applications • Why? • Recall our censoring issue • Empirical means depend on parametric model • Means can only be ‘model-based’ • Somewhat counterintuitive, especially when alternatives exist • More common: median

25. Median • Very/Most common way to express the ‘center’ of the distribution • Rarely see another quantile expressed • Find time xsuch that • Complication: in some applications, median is not reached empirically • Reported median based on model seems like an extrapolation • Often just state ‘median not reached’ and given alternative point estimates

26. X-Year Survival Rate • Many applications have ‘landmark’ times that historically used to quantify survival • Examples: • Breast cancer: 5 year relapse-free survival • Pancreatic cancer: 6 month survival • Acute myeloid leukemia (AML): 12 month relapse-free survival • Solve for S(x) given x

27. Common Parametric Distributions • Course will focus on non-parametric and semi-parametric methods • But… some parametrics can be useful • Especially for trial design • Note that power and precision are improved under parametric approaches versus others

28. Example 1: Exponential • Recall the exponential distribution • f(x) = • F(x) = • What is S(x) based on F(x) and f(x) • S(x) =

29. Example 1: Exponential • What about H(x) and h(x) • H(x) = • h(x) = • l represents the failure rate per unit of time • Large l, rapid decay • Small l, slow decay

30. Example 1: Exponential

31. R Code for the Plot time<-seq(0, 60, 0.1) S1<-exp(-0.1*time) S2<-exp(-0.05*time) S3<-exp(-0.01*time) plot(time, S1, xlab="Time", ylab="Survival Function", col=3 , lwd=2, type="l") lines(time, S2, col=2 , lwd=2) lines(time, S3, col=4 , lwd=2) labs<-c(expression(paste(lambda, " = ",0.1, sep="")), expression(paste(lambda, " = ",0.05, sep="")), expression(paste(lambda, " = ",0.01, sep=""))) legend(x=45, y=.95, labs, col=c(3,2,4), lty=c(1,1,1), lwd=(2,2,2), cex=0.9)

32. Example: Kidney Infection after Catheterization • Kidney infection after catheter insertion in patients using portable dialysis equipment • Time to event was time to catheter removal BUT should be noted that catheter can be removed for reasons other than infection (right censored) • Only 76 observations (!) • Time to infection is outcome of interest • Question: can we describe it using a parametric approach?

33. Kidney Infection Example:Survival curve and 95% confidence intervals

34. Exponential • Overly used due to simplicity • One parameter • Recall: S(x) = e-lx • So let’s revisit the hazard function:

35. Exponential • Mean = • Median =

36. Exponential • MRL = • “lack of memory” • Realistic?

37. Exponential • Recall the cumulative hazard function H(x) • For exponential: • Plot of ln(H(x)) vs. ln(x) should be a straight line with: • Slope = • Intercept = • Use to check model with non-parametric distribution of H(x)

38. Does Exponential Fit the Kidney Data?

39. R Code library(survival) surv.kid<-Surv(kidney$time, kidney$status) fit.kid<-survfit(surv.kid~1) exp.kid<-survreg(surv.kid~1, dist="exp") plot(fit.kid, xlab="Time", ylab="Survival Fraction") # summarize KM estimator to get median survival summary(fit.kid) names(fit.kid) # define log cumulative hazard and log time logHt<-log(-log(fit.kid$surv)) logt<-log(fit.kid$time) # Plot log cumulative hazard vs. log time plot(logt, logHt, lwd=2, type="l", xlab="log(t)", ylab="log(H(t))") points(logt, logHt, pch=16) # Add plot of x=y line. If exponential fits, should be parallel. abline(-exp.kid$coef, 1, lwd=2, col="red")

40. Exponential • Another alternative model check • What about plotting –ln(S(x)) versus x? • Should be a straight line with • Slope = • Intercept = • Why would the previous be preferred? • It can accommodate Weibull as we will see….

41. Another Exponential Check

42. More Model Checking • We will build likelihood later • For now, accept that the MLE of l is • Where diindicates whether the event is observed or censored for patient i, an ti is the event or censoring time • Here: • This implies a model such that S(x) =

43. Compare Fitted and Observed S(t)

44. What about specific survival time? Median survival? Mean survival? • Empirical: • 200 day survival = 21.0% • Median survival = 66 days • Mean survival = ? • Exponential Model: • 200 day survival = S(200) = ? • Median survival = ? • Mean survival = ?

45. Weibull • Generalization of the Exponential • VERY common for survival, but not always perfect • Shape and Scale parameters: a and l • Variable hazard • Increasing • Decreasing • Constant (a = 1)

46. Weibull: Generalization of Exponential • Shape Parameter: a • Scale Parameter: l • Note: There are different parameterizations for the Weibull

47. Weibull Example

48. R Code for the Weibull Plot #Weibull time<-seq(0,60, 0.1) S1<-exp(-0.05*time^.5) S2<-exp(-0.05*time^1) S3<-exp(-0.01*time^0.5) S4<-exp(-0.01*time^1) plot(time, S1, xlab="Time", ylab="Survival Function", col=2, lwd=2, type="l", ylim=c(0,1)) lines(time, S2, col=1, lwd=2) lines(time, S3, col=3, lwd=2) lines(time, S4, col=4, lwd=2) labs<-c(expression(paste(lambda, " = ",0.05, ", ", alpha, " = ",0.5, sep="")), expression(paste(lambda, " = ",0.05, ", ", alpha, " = ",1, sep="")), expression(paste(lambda, " = ",0.01, ", ", alpha, " = ",0.5, sep="")), expression(paste(lambda, " = ",0.01, ", ", alpha, " = ",1, sep=""))) legend(x=0, y=.25, labs, col=c(2,1,3,4), lty=1, lwd=2,cex=0.9)

49. Effect of Shape Parameter

50. Weibull • Mean: • Median: • Model checking: • More later when we discuss likelihoods