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) 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 X as 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 x such 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)
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 di indicates 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 = ?

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

51. Log-normal Just like it sounds If X ~ log-normal, then ln(X) ~ normal
Two parameters: m and s
Survival function
Median

52. Log-normal
Log-normal can work well in medical applications (e.g. age of disease onset)
Hazard is hump-shaped
Critics think that decreasing hazard at later times is unrealistic

53. Log-logistic If X ~ log-logistic, then ln(X) ~ logistic
Logistic is similar to normal, but the survival function is easier to work with
Hazard similar to Weibull, but more variable in shapes for hazard
Monotone decreasing
Hump-shaped

54. Log-logistic
Survival Function:
Hazard function:
Median:

55. Gamma Generalization of exponential Not easy to work with
-The gamma distribution represents another generalization of the exponential distribution
-If k =1 this follows a exponential distribution
-Here although there is a solution for S(t), it involves the gamma integral
-Note the hazard function is monotone increasing from – for k>1, decreasing from infinity for k<1 and approached lambda as t  infinity FOR BOTH CASES!

56. Cure Rate Distribution
Not in K & M
Assumption: fraction of individuals never fail
Violates assumption that S(∞) = 0
Useful for clinical trials in which
A fraction of the patients are cured
Event my never occur (e.g. cancer relapse)

57. Cure Rate Example
75% of women with early stage breast cancer are cured by treatment
Remaining 25% of women relapse
Assume exponential
l = 0.05

58. Cure Rate Distribution
Mixture model:
S(x) =
p =
S*(x) =
-S(t_i) is a mixture between a survival distribution and a point mass at 1
-In the above equation, p (our point mass at 1) represents the proportion of individuals we expect to never fail
-S(t) represents the overall survival
-S*(t) represents survival among the proportion of individuals whom we expect will fail

59. Cure Rate: Breast cancer example

60. R Code
par(mfrow=c(1,2)) t<-seq(0,1000,0.1) St<-0.25*exp(-0.05*t)+0.75 plot(t, St, xlim=c(0,60), ylim=c(0,1), type="l", lwd=2, xlab="Time(months)", ylab="Survival Fraction") plot(t, St, xlim=c(0,1000), ylim=c(0,1), type="l", lwd=2,

61. Competing Risks Used to be somewhat ignored Not so much anymore Idea:
Each subject can fail due to one of K causes (K > 1)
Occurrence of one event precludes us from observing the other event
Usually, quantity of interest is the cause specific hazard
Overall hazard equals sum of each hazard

62. Example
An investigator is looking at graft rejection in kidney transplant patients
However… patients can also experience graft failure and death
Treat graft failure and graft rejection events as censored observations
Why is this a problem?

63. Assumptions Dependence structure between the ‘potential’ failure times
Identifiability dilemma: Can only observe one time per person so not testable
We can not distinguish between independent and dependent competing risks

64. Useful Approaches Want to account for other causes
Adjust the denominator
Compare rates of events
Use measures of probabilities
Crude: probability of event k allowing for all other risks
Net: probability of event k if it is the ONLY risk
Partial: probability of event k is one of a subset of risks acting in the population
See K & M for more details