1. MUSC Summer Institute May 24-25, 2018 Dr. Bethany Wolf
Survival Analysis
MUSC Summer Institute
May 24-25, 2018
Dr. Bethany Wolf

2. Outline for Today Time-to-event outcomes
Estimating survival distributions
Hypothesis testing (log-rank tests)
A brief introduction to R
Survival analysis in R, Part 1

3. Time to Event Outcomes

4. Introduction to Survival Analysis
What makes it different?
Three main variable types
Continuous
Categorical
Time-to-event
Examples

5. Time to Event Outcomes Outcomes in which we have an start Examples
Time to death
Time to relapse
Time to first opiate rescue dose after hospitalization
Time to complete recovery from illness/disease

6. Bone Marrow Transplant for Leukemia
137 patient undergoing bone marrow transplant (BMT) for acute leukemia
Some of the variables include
Disease type (ALL, AML low risk, AML high risk)
Gender
Patient and donor age
Treatment with methotrexate
Time to death or relapse
Examine survival (death or relapse) experience of these patients

7. Gender- how might we look at this?
What about patient and donor ages at time of transplant….
These are familiar types of variables and there are “standard” things we look at?

8. Patient age at transplant:
Patient Gender:
Females: n = 57, 41.6%
Males: n = 80, 57.4%
Patient age at transplant:
Mean: years
Standard deviation: years
Range: years

9. But what about time to death?
Gender and age are familiar types of variables and there are “standard” statistics we use to summarize them.
But what about time to death?
What happens if we consider standard statistical summaries?

10. Time to Death/Relapse? Number of people who died or relapsed
Mean time to “death/relpase” (all subjects)
days
Mean time to death (subjects who died)
days
Range
1 to vs. 1 to 2204 days

11. Does our summary of time to death/relapse make sense?
83 of the 137 patients relapsed but the rest were alive at the end of the study…
Does it make sense to estimate a mean (or median)
On all patients?
Only using patients who died?

12. Censoring Different types: Each leads to a different likelihood
Right
Left
Interval
Each leads to a different likelihood
Most common is right censored

13. Right Censored Data
Event is observed if it occurs before some pre-specified time
Consider an animal study
Clock starts on the first day of treatment
Clock ends at death
Always thinking about the clock

14. Treatment of Transitional Cell Carcinoma of the Bladder in Mice
Consider a study investigating how a new cancer therapeutic effects survival in mice transfected with transitional cell bladder carcinoma.
In this case, all 8 animals are treated at the same time and we follow them forward in time to observe how long it takes them to die.

15. Simple Example of Right Censoring

16. Treatment of TCC of the Bladder in Mice
What if the researcher is only interested in survival up to 12 weeks post-treatment?
animals that live longer are sacrificed at 12 weeks
What do we the information on the animals that are alive at 12 weeks?
Let’s look at our plot again…

17. Simple Example of Right Censoring

18. Simple Example of Right Censoring

19. Bone Marrow Transplant for Leukemia
Consider the bone marrow transplant study…
An investigator conducts a prospective study to investigate the impact of different factors on time to relapse or death
Disease classification
Patient and donor age
Impact of treating with methotrexate
Whether or not they exhibited platelet recovery

20. Bone Marrow Transplant for Leukemia
These data are right censored since many of the patients did not relapse or die before the study ended
However, unlike the mouse study, patients do not all enter the study at the same time.
So what might a line plot of patient survival look like?

21. More Realistic: Clinical Trial

22. More Realistic: Clinical Trials

23. Additional Issues
Patient drop-out
Loss to follow-up

24. Drop-out or LTFU

25. How do we treat these data?
Shift everything
so each patient
time represents
time in the study
Time of
Enrollment

26. Interval Censoring
Due to discrete observation times, actual time of event not observed
Examples:
Progression in cancer determined by tumor size. Measures at 3-6 month intervals and if increase occurs, known only to have happened in the last interval
Time to death in nematodes. Nematodes are grown on a plate and checked periodically for death. Due to short life span, exact time of death for each worm is not observed.
Times are biased to longer values
Challenging issue when intervals are long

27. Left Censoring The event has occurred prior to the start of the study.
OR the true survival time is less than the person’s observed survival time.
We know the event occurred, but are unsure when prior to observation.
In this kind of study, exact time would be known if it occurred after the study started.

28. Example of Left Censoring
191 California high school boys were asked “When did you first smoke marijuana?”
Possible answers include:
The exact age
“never used it”
“Used it but can not recall age”
“Never used” is a right-censored observation
“Used it but can not recall age” is a left-censored observation (we only know that the event occurred prior to the boys current age)

29. Truncation Only a subset of individuals are able to be observed
“truncation event”
Only individuals with the event are included in the study
Examples:
Left Truncation: Time to relapse (requires a complete response to cancer treatment)
Right Truncation: Patients with AIDS from transfusion (only patients who developed AIDS prior to a cutoff time are included)

30. Truncation Example: Time to AIDS
296 patients infected with HIV via blood transfusion in 1978 were sampled in 1987after they had converted to AIDS, in order to determine the time to development of AIDS among those acquiring HIV from transfusion.
According to the sampling scheme used in 1987, only those who had developed AIDS were included in the study.
This represents right-truncation since individuals who had yet to develop AIDS were not included in the sample.

31. Truncation Example
Patient must experience a response before duration of the response can be assessed
Censored
for response
response
Censored
at relapse
relapse

32. Censoring Assumption:
Potential censoring time is unrelated to potential event time
Reasonable?
Estimation approaches are biased when this is violated
Violation example:
Sick patients tend to miss clinical visits more often
High school drop-out: Kids who move may be more likely to drop-out

33. Structure of Time To Event Data
In order to evaluate survival distributions
Estimate survival rates
Test associations
Etc.
The data must have the appropriate format
Event times
Censoring times
Indicator of event occurence

34. Key Components
Event: Must have a clear definition of what constitutes an ‘event’
Death
Disease
Recurrence
Response to treatment
Need to know when the clock starts:
Age at event?
Time from study initiation?
Time from randomization?
Time since response?

35. Notation Many possible versions For our purposes, we will use
ti as the observed time to event or censoring time for the ith subject
di to indicate censoring

36. Data Structure For each subject data must include Censoring indicator:
One (or more) event/censoring time
Indicator of whether or not the subject had the event
Censoring indicator:
A binary (0/1) variable indicating whether or not a subject had the event at the specified time
For survival analysis in R
1 = subject had the event a time t
0 = subject did not have the event at time t (i.e. they were censored).

37. Data Structure Time can be specified in one of two ways:
Each subject has a single entry which shows
Their last observed time, t
Whether or not they experienced the event at that time (censoring indicator)
Or a subject can be listed on multiple rows
Each row has a start and stop time
Start and stop intervals are non-overlapping
And each entry has an indicator of whether or not they experienced the event at the end of that time interval

38. Simple Example Data on 10 subject with a time to event outcome
The event/censoring times are
10, 20+, 35, 40+, 50+, 55, 70+, 80, 90+
Here a + indicates that the subject did not experience the event

39. Simple Example: First Approach
Each subject has a single entry
Data:
10, 20+, 35, 40+, 50+, 55, 70+, 80, 90+ ID
Event Indicator
Time 1 2 3 4 5 6 7 8 9 10

40. Simple Example: Second ApproachID
Event Indicator
Start Time
Stop Time 1 10 2 20 3 35 4 40 5 …
Each subject can have more than on entry where each entry has a start/stop time
Data:
10, 20+, 35, 40+, 50+, 55, 70+, 80, 90+

41. Estimating Survival Probability and Survival Curves

42. Time to Event Outcomes Modeled using “survival analysis”
Define t = time to event
Total time they are followed after study entry
(experience the event)-(study entry)
(study end or LTFU)-(study entry)
Can be in any unit of time desired
Define the censoring indicator
1 if experienced the event in the observed time
0 if did not experience event in the observed time or where LTFU

43. Time to Event Outcomes
Key functions for characterizing time to event outcome
Survival function
Cumulative hazard
Hazard rate (or function)

44. Survival Function
S(t) = the probability of an individual surviving to time t
Basic properties
Monotonic non-increasing
S(0) = 1
S(∞) = 0*
*debatable: cure-rate distribution allow plateau at some other value

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

46. 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

47. 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

48. Cumulative Hazard Function
Often used instead of the hazard function
There is an easy mathematical relationship between H(t) and S(t)

49. More Terminology D distinct event or censoring times
t1 < t2 < t3 < … < tD
Ties allowed
At time ti, there are di events
Yi is the number of individuals at risk at ti
Yi is all the people who have event times > ti
di/Yi is an estimate of the conditional probability of an event at ti, given survival to ti

50. Kaplan-Meier Estimation
Also known as the ‘product-limit’ estimator
Step-function
Size of the steps depends of
Number of events at time t
Pattern of censoring before t

51. Simple Example Event = time to relapse Data:
10, 20+, 35, 40+, 50+, 55, 70+, 71+, 80, 90+

52. Times: 10, 20+, 35, 40+, 50+, 55, 70+, 71+, 80, 90+ 10 20 35 40 50 55 70 71 80 90

53. Times: 10, 20+, 35, 40+, 50+, 55, 70+, 71+, 80, 90+ 10
1-(0/10)=1 1 20 35 40 50 55 70 71 80 90

54. Times: 10, 20+, 35, 40+, 50+, 55, 70+, 71+, 80, 90+ 10
1-(0/10) 1
1-(1/10)=0.9
1*0.9 = 0.9 20 35 40 50 55 70 71 80 90

55. Times: 10, 20+, 35, 40+, 50+, 55, 70+, 71+, 80, 90+ 10
1-(0/10)=1 1
1-(1/10)=0.9
1*0.9 = 0.9 20 9
1-(0/9)=0.9
0.9*1 = 0.9 35 40 50 55 70 71 80 90

56. Times: 10, 20+, 35, 40+, 50+, 55, 70+, 71+, 80, 90+ 10
1-(0/10)=1 1
1-(1/10)=0.9
1*0.9 = 0.9 20 9
1-(0/9)=0.9
0.9*1 = 0.9 35 8
1-(1/8)-0.875
0.9*0.875 = 0.79 40 50 55 70 71 80 90

57. 10
1-(0/10) 1
1-(1/10)
1*0.9 = 0.9 20 9
1-(0/9)
0.9*1 = 0.9 35 8
1-(1/8)
0.9*0.875 = 0.79 40 7
1-(0/7)
0.79*1=0.79 50 6
1-(0/6)
.79*1=.79 55 5
1-(1/5)
.79*.8=.63 70 4
1-(0/4)
.63*1=.63 71 3
1-(0/3) 80 2
1-(1/2)
.63*.5=.32 90
1-(0/1)
.32*1=.32

58. Plot it…
Time di 1 10
0.9 20 35
0.79 40 50 55
0.63 70 71 80
0.32 90

59. Recall Cumulative Hazard
We said the relationship between the hazard and survival functions is:
So to look at the hazard for the Kaplan-Meier estimate use

60. 10 1
-ln(1) = 0
0.9
-ln(0.9) = 0.105 20 9 35 8
0.79
-ln(0.79) = 0.235 40 7 50 6 55 5
0.63
-ln(0.63) = 0.462 70 4 71 3 80 2
0.32
-ln(0.32) = 1.14 90

61. Cumulative Hazard Plot
Time di
H(t) 10 1
0.105 20 35
0.235 40 50 55
0.462 70 71 80
1.14 90

62. An Alternative to KM: Nelson-Aalen Estimator
Better small sample properties than KM
Variance of NA estimator

63. Uses of NA Model Identification Estimates of the hazard rate, h(t)
Determine if parametric exponential model appropriate
Estimates of the hazard rate, h(t)
Slope of H(t)
Survival Function
S(t) = exp(-H(t))
S(t) using NA for H(t) is called the Fleming-Harrington/Breslow method

64. Comparison of KM and NA Estimates10 1
0.9
0.1
0.905 20 9 35 8
0.788
0.225
0.799 40 7 50 6 55 5
0.63
0.425
0.654 70 4 71 3 80 2
0.315
0.925
0.397 90

65. Comparison of KM and NA Estimates

66. Bone Marrow Transplant
What about a more realistic example?
Let’s look at the Kaplan-Meier and Nelson Aalen estimates for survival and cumulative hazard in our bone marrow transplant patients

67. KM and NA: Time to Death/Relapse in BMT

68. Nelson Aalen vs. Kaplan Meier
Approximately the same when risk sets (Yi) large relative to number of subjects who have the event
Larger difference when ties present
Different standard errors when ties present
Smaller MSE than KM for S(t) > 0.20 and larger otherwise
NA biased upward when survival estimated close to 0

69. Why consider NA over KM
NA is the basis for all log-rank type hypothesis tests.
Additionally, NA can provide an estimate of hazard rate
Hazard rate can be estimated as slope of the cumulative hazard

70. Estimates from a Survival Curve

71. Estimates from a Survival Curve
Now we have a survival curve
We would like to be able to say something about survival in our population
Median survival time (or other quantiles)
X-Time survival rate
6 month rejection free survival rate in organ transplant
5 year survival rate in specific cancer

72. Median Survival Time
Common way to express the ‘center’ of the distribution
Find the time t where the estimated survival probability is 0.5
We don’t often have S(t) = 0.5 so we choose the first survival probability where

73. Time to Event in Toy Example
# at Risk
# Events
S(t) 10 1
0.90 35 8
0.79 55 5
0.63 80 2
0.315

74. Median Time to Death in BMT
Again, what about a more realistic example?
Time
# at Risk
# Events
S(t) …
422 72 1
0.524
456 71
0.517
466 70
0.509
467 69
0.502
481 68
0.494
486 67
0.487
487 66
0.480
526 65
0.472

75. Median Survival in BMT Data
If the median is not observed
Estimating a median can be done but it is an extrapolation
Often just state ‘median not reached’
Note we can also estimate other quantiles in the same way

76. Alternatives: X-Time Survival
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 the survival function for S(t) given time t

77. Time to Event in Toy Example
Let’s look at survival to time = 60 in the toy example
Time
# at Risk
# Events
S(t) 10 1
0.90 35 8
0.79 55 5
0.63 80 2
0.315

78. X-Time Survival in BMT
Perhaps we are interested in the probability a patient will survive (disease free) to 6 month or 2 years post-transplant
Similar to estimating the median but instead we look for the probability for that specific time
That is find
S(182.5 days = 6 months)
S(730 days = 2 years)

79. 6 month Survival in BMT Time # at Risk # Events S(t) … 164 99 1 0.715
168 98
0.708
172 97
0.701
183 96
0.693
192 95
0.686
194 94
0.679
211 93
0.672
219 92
0.664

80. 2 Year Survival in BMT Time # at Risk # Events S(t) … 625 61 1 0.450
641 60
0.444
666 59
0.435
677 58
0.427
704 57
0.420
748 56
0.412
1063 47
0.404
1074 46
0.395

81. X-Time Survival in BMT
For the median, we selected the first time where the estimate survival probability was S(t) < 0.5
For X-time survival, the probability is based on the latest time that is less than or equal to the time of interest.
So for BMT
The probability of surviving 6 months is 70.1%
The probability of surviving 2 years is 42.0%

82. Point-wise Confidence Intervals
Constructed so that the true value of survival probability, S(t) at a particular t, falls in the interval with 100 x (1 - a)% confidence
Based on the estimated variance
For Kaplan-Meier the estimated variance is :
where is the probability of survival at time t

83. Types of Point-Wise CIs
There are several point-wise confidence intervals for survival probability
Linear
Log
Log-log
Note, there are other types but these are the most common
The default in R is log confidence intervals

84. Types of Point-wise CIs
Linear
Log
Log-log

85. Types of Confidence Intervals
Linear are the easiest to compute
However the transformations, while more complicated, have better properties
All easy enough if there is software to calculate them

86. Comparison in our Toy Example

87. Comparison in BMT Data

88. Which to Use When? Linear Log Log-log
Very anti-conservative, particularly for small N
Too narrow
Log
Conservative for upper limit
Anti-conservative on lower limit
Log-log
For small N, slightly anti-conservative
A little too narrow

89. Which to Use When? For N > 25 and < 50% censoring
Log and log-log are good
Both given ~ nominal coverage for 95% CI
Exception: extreme right tail where there is little data
Linear approach requires much larger N

90. Confidence Intervals
What is most commonly produced by software packages
Valid ONLY for point-wise intervals
Problem is they are often mis-intepreted:
Plot a set of point-wise 95% CIs
Interpret as confidence “band”
These “bands” are too narrow!

91. Confidence Bands
A band for which we are 100*(1-a)% confident that the survival function fails within the band for all t in some interval
Two common approaches:
Nair: equal probability bands
Hall-Wellner bands

92. Comparison in BMT Data

93. Confidence Band Performance
The two approaches give very similar results
Both generally have accurate coverage probabilities (even for n = 20)
Confidence intervals or confidence bands?
confidence bands for overall confidence about an estimated survival curve
Confidence intervals for specific estimates of interest
Median survival time
1-year survival rate

94. Hypothesis Testing: Log-Rank Type Tests

95. Comparing two or more samples
Anova type approach
t is the largest time for which all groups have at least one subject at risk
Data can be right-censored (and left truncated) for the tests we will discuss

96. More on the Hypothesis Test
Want to compare the distribution of event times between groups
The hypothesis (for 2 groups) can be written as

97. Test for 2 Groups
If and follow some parametric distribution (e.g. normal), this is easy
Distribution of the event times across and within groups is rarely known
Need a test whose validity doesn’t depend on parametric assumptions

98. Notation
Let t1 < t2 < … < tT be distinct death times in all samples being compared
At time ti , let dij be the number of events in group j out of Yij individuals at risk (j = 1,2,…,K)
Those at risk and in the risk sets across all groups are

99. Developing a Hypothesis Test
Weighted comparison of estimated hazards between the jth groups under the null/alternative hypotheses
Based on Nelson-Aalen estimator
Remember:
If the null is true, the estimated hazard across all groups should be similar to the estimate hazard for each of the j groups

100. Basic Form of the Test
For unique times, find the weighted difference in hazard between group j and the overall hazard
If all Dj(t)’s are close to zero, there is little evidence to reject the null
Weight at ti
# events jth group
# at risk in jth grp
# events all groups/
# at risk in all groups

101. General Test Statistic
The general test statistic is
Can use Z-score or c 2
Corrects for ties

102. Log-Rank Test for 2 Groups
For log-rank W(ti)=1, the test statistic is

103. BMT Example Example categorical variables of interest
Use of Methotrexate (Yes/No)
Disease Type
ALL
AML Low Risk
AML High Risk
Compare groups via log-rank test

104. Log-Rank Tests (BMT)

105. Beyond Log-Rank
Log-rank has optimum power to detect differences when the hazard rates of our K groups are proportional
So what is meant by proportional hazard rates?

106. Graphical Depiction
Proportional hazards = hazard rates between groups is the same over time
Group 1
Group 2

107. How to Check Proportional Hazards
We can estimate cumulative hazards (H(t)) and survival probabilities (S(t)) from log-rank results
But to check proportional hazards we need the hazard rate (h(t))
The estimated slope of H(t) provides crude estimate of h(t)
Smooth the estimated h(t) for each group and compare
Estimate assuming a time x variable interaction
Estimate assuming no time x variable interaction
Let’s look at our different variables for BMT

108. Methotrexate Use

109. Disease Group

110. Another Example: Kidney Infection
Data on 119 kidney dialysis patients
Comparing time to kidney infection between two groups
Catheters placed percutaneously (n = 76)
Catheters placed surgically (n = 43)

111. Kidney Infection: Log-Rank Results

112. Kidney Infection: Hazard Rates?

113. Beyond Log-Rank So what if our hazard rates are not proportional…
We’ve mentioned using other weight functions
Depending on the choice of weight functions, we can place emphasis on different regions of the survival curve.

114. More on Weight Functions
Recall W(t) = 1
Log-rank test
As we’ve said, this has optimal power for detecting differences when hazards are proportional

115. Choices for Weight Functions
Fleming-Harrington
General case
Special cases
Log-rank: p=0 and q = 0
Mann-Whitney-Wilcoxon: p = 1, q = 0
q = 0, p > 0: gives greater weight to early departures
p = 0, q > 0: gives greater weight to late departures
Allows specific choice of influence (for better or worse!)

116. Kidney Data with Different Weights?
Log rank (p = q = 0): p-value = 0.112
Flemming-Harrington
p = 0, q = 1: p-value =
p = 1, q = 0: p-value =
p = 1, q = 1: p-value =
p = 0.5, q = 0.5: p-value =
p = 0.5, q = 2: p-value =
p = 4, q = 0: p-value =

118. Others? Many Not all available in all software
Note FH weights provide a lot of flexibility to mimic other weight functions
Not all available in all software
Worth trying a few in each situation to compare inferences

119. Take-home
Most often, we are interested in the average difference (which log-rank addresses)
If hazards are not proportional, difference may exist that log-rank misses
Difference exist at early or late times
Varying the weights allows exploration of this
But, choice of weight function can be critical
Think carefully about the distribution of weights

120. Brief Introduction to R

121. What is R
R is a comprehensive statistical and graphical programming language
Most functions are written in R
But can interface with other langauges (C, C++, Fortran,…)
R is used for data manipulation, statistics, and graphics

122. What R Provides R is open-source It provides
Data storage, analysis, and data visualization/graphing
Contributed packages for statistical analysis
Programming environment
Provides access to algorithms and their implementation
Gives you the ability to fix bugs and extend software
Promotes reproducible research by providing open and accessible tools

123. R Overview R is highly functional
Data manipulation, statistical analysis, and graphics are done using functions
Functions use strict arguments
Functions in R from contributed packages/libraries
About 25 package included with R installation
CRAN repository includes >12,500 contributed packages
We will use
survival

124. R Overview
R is an object oriented language, meaning R stores information in the R interface/ workspace as objects
Everything in R is an object
Assignment of objects is done using either
“=”
“<-”
Example: If we want a variable x that has a numeric value of 5, the “code” would be
x <- 5 or x = 5

125. R Objects Objects can be used in other calculations
There are some restrictions when giving an object a name
can’t contain `strange' symbols like !, +, -, #
dot (.) and underscore (_) are allowed
can contain numbers but can’t start with a number
case sensitive, X and x are two different objects

126. Types of Objects Numbers, characters, logical (TRUE/FALSE)
Vectors and matrices
Think of Excel spreadsheet
Data frames
Similar to matrices but allows for mixed data types
Arrays
Lists
Allows for single object to include
Functions

127. R Workspace Objects created during an R session are held in memory
The collection of objects in a current R session is called the workspace.
Stored in your working directory
The location of your working directory accessed with
getwd()
Change the working directory using
setwd(“location”)
Example location- “S:\\wolfb\\SummerInstitute2018”

128. R Workspace
The collection of objects in a current R session is called the workspace.
workspace is not saved when you close your session unless you tell R to save
This means that R objects are lost when you close R if you do not save workspace

129. R Functions Functions are available through packages
Recall CRAN repository includes >12,500 contributed packages
It can be helpful to write your own functions
Functions follow the form
Function.name(argument1, argument2, …)
Function is used by stating the function name and providing the necessary arguments

130. R Help All R functions have help documentation
Accessing help files is easy
?FunctionINeedHelpWith
help(“FunctionINeedHelpWith”)
help.search(“FunctionINeedHelpWith”)
You can also view the function and examples
getAnywhere(“FunctionINeedHelpWith”)
example(“FunctionINeedHelpWith”)

131. Let’s Try R Out…

132. Cox Regression: Overview

133. Switching Gears
The log-rank type tests provide a method for comparing survival distributions between groups
However what if we want to control for other variables
Stratified log-rank tests can be used
Unable to detect interactions
Assumes effect of the stratified variable is the same across strata
Can’t be used for continuous variables
This brings us to regression approaches

134. Cox Proportional Hazards Model
Names
Cox regression
Semi-parametric proportional hazards
Proportional hazards model
Multiplicative hazards model
Why?
Allows for adjustment of covariates (continuous and categorical) in a survival setting
Allows prediction of survival based on a set of covariates
Analogous to linear and logistic regression

135. Cox PHM Notation Data on n individuals: More complicated: Zj(t)
Tj : time on study for individual j
dj : event time indicator for individual j
Zj : vector of covariates for individual j
More complicated: Zj(t)
Covariates are time dependent
May change with time/age

136. Basic Cox Regression Model
The outcome (i.e. dependent variable) is the hazard of the event at time t given variables X
This outcome is regressed on the p (independent) variables of interest, X
Model = baseline hazard x the sum of the p regression coefficients, b, times the observed values of X

137. Comments on the Basic Model
h0(t):
Arbitrary baseline hazard
Notice that it varies by t b:
Regression coefficient vector
Interpretation is a log hazard ratio
Semi-parametric form
Non-parametric baseline hazard
Parametric form assumed for covariate effects only

138. Cox Proportional Hazards Model
Linear form
Proportional hazards because

139. BMT Example: Methotrexate Use
Evaluate the relationship between time to death/relapse and methotrexate use (MTX) as graft vs. host prophylactic
Form of the Cox Model:
h(t| I(MTX) = h0(t)exp{bprI(MTX)}
I(MTX) is an indicator of whether or not a patient received methotrexate
The actual model determined from the data is
h(t| I(platelet recovery) = h0(t)exp{0.398*I(MTX)}

140. Interpretation: Hazards Ratio
For any fixed time, individuals with platelet recovery have exp{bpr} times the risk of death compared to those who don’t.

141. Hazard Ratio So how do we interpret this model? Hazard ratio:
Interpretation:
The hazard of the death or relapse for subjects who were given MTX is 1.5 times the hazard in those that did not.
OR… There was an 50% increase in the hazard of death or relapse in patients who were given MTX.

142. Hazard Ratios Hazard function=P(event at time t|survived to time t)
Assumption: “proportional hazards”
The risk does not depend on time
That is “the risk is constant over time”
But that is still vague…
BMT example found a hazard ratio = 1.5
Patient given MTX had 50% greater risk of death relative to those not given MTX, at any given point in time
Hazard function=P(event at time t|survived to time t)

143. Hazard Ratios Hazard ratio
The hazard ratio is assumed to be constant over time

144. Categorical Variables with > 2 Levels?
What if we have a categorical variable that has more than 2 levels?
How is the hazard ratio estimated in this case?
What about the proportional hazards assumption?

145. Disease Type and Survival in BMT
Recall we had three classifications of disease type
ALL
AML Low Risk
AML High Risk
Our model looks like

146. A Slightly More Complicated Example
This model provides 3 possible hazard rates

147. A Slightly More Complicated Example
We can “compare” the different hazards between the groups by taking the ratio

148. A Slightly More Complicated Example
Interpretation:
Patients with low risk AML have 0.56 times the hazard of death or relapse relative to patients with ALL.
Patients with high risk AML have 1.47 times the hazard of death or relapse relative to patients with ALL.
Patients with high risk AML have 2.61 times the hazard of death or relapse relative to patients with low risk AML.

149. Continuous Covariates?
So far we’ve only examined categorical covariates (Methotrexate use and patient gender)
What if we wanted to look at something like patient age at time of transplant?
How is the hazard ratio estimated in this case?
What about the proportional hazards assumption?

150. Patient Age and Risk of Death
Consider a model that includes
Patient age at time of transplant (in years)
Our model looks like

151. Hazard Ratio for Patient Age
In this case we estimate the hazard ratio for some specific amount of increase (or decrease) in patient age.
The default increase for the estimated regression coefficient is 1 unit (in this case years)
Interpretation: A 1 year increase in patient age at time of transplant is associated with a 1% increase in the hazard of death or relapse.

152. Hazard Ratio for Patient Age
A 1 year increase isn’t likely clinically meaningful but we can estimate the hazard for larger increases if we want
For example, consider the impact of a 10 year increase in age
Interpretation: A 10 year increase in patient age at time of transplant is associated with a 12% increase in the hazard of death or relapse.

153. Proportional Hazards? HR for a 25 year increase in Age
HR is still assumed to be constant over time for a specific difference in Age

154. More than 1 Covariate What if we had 2 binary covariates?
How is the hazard ratio estimated in this case?
What about the proportional hazards assumption?

155. More than 1 Covariate Consider a model that includes
Methotrexate use (yes/no)
Sex (male/female)
Our model looks like

156. A Slightly More Complicated Example
This model provides 4 possible hazard rates

157. A Slightly More Complicated Example
And if we “compare” the different hazards by taking the ratio we get
Interpretation: Patients given methotrexate had 1.44 times the hazard of death relative to those not given methotrexate, controlling for sex.

158. A Slightly More Complicated Example
And if we “compare” the different hazards by taking the ratio we get
Interpretation: Male patients had 0.84 times the hazard of death relative to females, controlling for methotrexate use.

159. A Slightly More Complicated Example
But what does this mean in terms of proportional hazards?
MTX, Female
MTX, Male b1 b2 b1
No MTX, Female
No MTX, Male b2

160. What About Interactions
We might also be interested in interactions between covariates
In the BMT data, we have information on patients and donors and may want to consider interactions between these two
Patient Sex by Donor Sex
Patient Age by Donor Age

161. Models with Interactions
For these interactions our model looks like
So how do we interpret these models
We have to consider both patient and donor characteristics to estimate a hazard ratio

162. Patient and Donor Sex Interaction
Most interested in differences by patient sex so fix donor sex to interpret the hazard in male vs. female patients given male/female donor
HR for male vs. female patients given a male donor
HR for male vs. female patients given a female donor

163. Patient and Donor Age Interaction
Interactions between continuous variables are harder to understand/interpret
As in the case of gender, we are most interested in the impact of increasing patient age but how do we “fix” donor age
Consider plotting the hazard ratio for a specific increase in patient age (e.g. a 5 year increase) by donor age

164. HR for a 5 year increase in patient age with increasing donor age

165. Time-Varying Covariates
So far we’ve only considered “fixed” time covariates
However, the BMT data include several time varying covariates
Acute graft vs. host disease (AGvHD)
Chronic graft vs. host disease (CGvHD)
Platelet recovery (PR)
These all occur after BMT or not at all

166. CPHM with Time Varying Covariates
The model looks like what we’ve been working with.
The model looks similar but now we allow some of our covariates in X to be a function of t:

167. Data Structure: Time-Varying Covariates
Expand data to describe all scenarios
Need to consider the possible combinations of events
Think about the long format data we discussed in the beginning

168. Data Structure: Time-Varying Covariates
Example: Time of Platelet Recovery and Death
Possible scenarios at any point in time during the study for subject 1
No PR: Death/Relapse?
PR: Death/Relapse?
For all patients with Time PR < Time Death, need two rows in dataset to describe variation
For all patients with Time PR > Time Death, need only one row in the dataset

169. Timeline Examples: Observed Event
t0 to ta: no Platelet Recovery until ta, no event
ta to te: Platelet Recovery, event
t0 to te: no Platelet Recovery , event t0 ta te t0 te

170. Timeline Examples: Censored Event
t0 to ta: no Platelet Recovery , no event
ta to tc: Platelet Recovery, no event (censored)
t0 to tc: no Platelet Recovery, no event t0 ta tc te t0 tc te

171. Time-Varying Covariates
Estimation and inference are the same as with fixed time covariates
Difference
Data structure

172. Data Set-up
Consider the BMT data with information on death/relapse, platelet recovery, and acute and chronic graft vs. host disease ID
TTE
DFS
Time PR PR
TAGvHD
AGvHD 1
2081 13 67 2
1602 18 3
1496 12 4
1462 70 5
1433 6
1377 7
1330 17 8
996 72 …

173. Expansion Consider subject 1 (row 1) Consider row 2 Now, three rows
Row 1: start = 0, stop = 13, PR = 0, AGvHD = 0,DFS = 0
Row 2: start = 13, stop = 67, PR = 1, AGvHD = 0,DFS = 0
Row 3: start = 67, stop = 2081, PR = 1, AGvHD = 1,DFS = 0
Consider row 2
Now 2 rows
Row 1: start = 0, stop = 18, PR = 0, AGvHD = 0,DFS = 0
Row 2: start = 18, stop = 1602, PR = 1, AGvHD = 0,DFS = 0

174. Expanded Data ID Start Time Stop Time Event PR AGvHD 1 13 67 2081 2 1813 67
2081 2 18
1602 3 12
1496 4 70
1462 …

175. What About Dependence?
You might be asking whether we need to worry about correlated data?
In this case we do not need to worry about it.
There two exceptions:
When subjects have multiple events
When a subject appears in overlapping intervals
The 2nd case is almost always a data error

176. Cox Model with Time Varying Covariate
Consider a model that includes
Time to platelet recovery
Note the model looks like

177. Hazard Ratio for Platelet Recovery
We still estimate a hazard ratio in the same way
Interpretation: Patients who experience platelet recovery at a given time have 0.28 times the hazard of death or relapse relative to those who have not experienced platelet recovery at that time.

178. What About Continuous Covariates
Continuous variables can change over time also
Given the times measurements are taken, we can expand the data in the same way.
However, this assumes the value is unchanging during the intervals between which it was measured
A little unrealistic BUT…
No different from treating a single measure (e.g. blood pressure) as a fixed time covariate

179. Inference in Cox Regression

180. Global Tests of the Model
Testing that bk = 0 for all k = 1, 2, …, p
Three main tests
Chi-square/ Wald test
Likelihood ratio test
Score test
All three are chi-square distributed with p degrees of freedom
All likelihood based
Note the likelihood is a partial likelihood as it does not estimate baseline hazard

181. BMT Example: Disease Group
The model including disease group is shown below
The global p-values for this models are
Likelihood ratio test: p =
Wald Test: p =
Score Test: p =

182. Local Tests of Covariates
The global test assesses whether at least one covariates is associated with outcome
Local tests can be used to
Test associations of individual covariates while accounting for other covariates
Compare nested model to determine if a group of covariates is significant (chunk test)

183. Local Tests of Covariates
Results of local tests of associations of individual covariates are provided with model fit
Continuous covariate: significance suggest a 1 unit increase gives a HR ≠ 1, controlling for other covariates
Categorical covariate: significance suggests the HR ≠ 1 compared to the reference group, controlling for other covariates

184. BMT Example: Disease Group
Recall the global test indicated disease group as significant (so at there is a difference in at least two of the groups)
The model and corresponding local tests for the Cox model including disease group with ALL as the reference are shown here
Disease
Beta
SE(Beta)
Z score
P-value
AML low risk (vs. ALL)
-0.574
0.287
-2.00
0.046
AML high risk (vs. ALL)
0.383
0.267
1.43
0.152

185. Local Tests of Covariates
Local tests of associations of a group of covariates compares a model will all covariates to a model without the subset of covariates being evaluated
Comparison between models is based on the difference in the log-likelihood for each model
Means we are using likelihood ratio test
2(log-likelihood big model – log-likelihood smaller model)
Degrees of freedom is the number of covariates being tested

186. BMT Example Compare the following models
Model including disease group, patient age, donor age, and the interaction between patient and donor age
Model including only disease group.
Model
Log-Likelihood
Dx group, Pt age, Dn age, Pt x Dn age interaction
-361.5
Dx group
-366.6
Likelihood ratio test statistc (df=3)
10.1
P = 0.017

187. Linear Contrasts Local tests are estimated for regression coefficients
For categorical variables with >2 levels, local tests (from the model) only compare to the reference group
BMT: if ALL is the reference, local tests tell us about
AML low risk vs. ALL
AML high risk vs. ALL
We also want to compare AML low risk vs. high risk
Can be done using contrasts

188. Linear Contrasts
In the disease group example, we want to compare AML high and low risk patients
To do this, we want to test the hypothesis
H0: bAML.low = bAML.hi vs. HA: bAML.low ≠ bAML.hi
This can be done using a linear contrast
Take the difference: bAML.low - bAML.hi
We reject the null of this difference (accounting for variability) is greater than or less than 0.

189. BMT Example: Disease Group
Our local tests found
Hazard for AML low risk < hazard ALL
But found no difference between AML high risk and ALL
Comparing AML low and high risk groups using linear contrast finds
bAML.low - bAML.hi = (aka HR = 0.38)
SE(bAML.low - bAML.hi ) = 0.27
p =

190. Confidence Intervals for Hazard Ratios
We’ve learned how to estimate a hazard ratio from a Cox model
Confidence intervals for HRs are also often reported
Could also think of this as another form of hypothesis test
If the interval doesn’t include 1, no significant difference
Easy for individual covariates in a model

191. Confidence Intervals for Hazard Ratios
Examples:
95% CI for HR for Methotrexate
95% CI for 1 unit increase in Age

192. Confidence Intervals for Hazard Ratios
If we wanted to consider more than a 1 unit increase in a continuous covariate
Example: 95% CI for 5 unit increase in Age

193. Complex Hazard Ratios
What if we want to estimate hazard ratios that involve more than one covariate
AML high risk vs. low risk (includes 2 coefficients)
Interactions require us to include >2 coefficients

194. Complex Hazard Ratios
The confidence interval in these cases is calculated the same way
But now
b = combination of covariates in our HR expression
SE(b ) = standard error for the combination of covariates
If we have these two things, we an calculate the confidence interval as above

195. Complex Hazard Ratios Patient Age x Donor Age example
Interactions require us to include >2 coefficients
Our combination of b is:
Our SE(b ) is
So our final HR (95% CI) is

196. Building a Cox Regression Model

197. Model Building using PHM
Regression uses
Adjusting for potential confounders when a specific hypothesis is to be tested
Predict the distribution of the time to some event from a list of explanatory variables (“prediction”)

198. Adjustment Approach
First, perform “unadjusted” analysis of factor of interest with time to event outcome
Include factor of interest and significant confounders
One at a time
“full” model
Use some criterion (e.g. p-values) to determine if confounders should be included/retained or not

199. Prediction Approach
Interested in identifying set of variables to model survival (or use as “base” model for future testing)
Look at univariate association with time to event for individual predictors
Select a subset of predictors for consideration
Note in both approaches, we might want to consider potential interactions

200. Most Common Approach: Subset Selection
Forward entry
Start with most significant
Continue until criterion is met (e.g. p > 0.05)
Entered variables not reevaluated
Backward elimination
Put all variables in model that meet some criteria (e.g. p <0.20)
Remove least significant variables and repeat until a criterion is met (e.g. p > 0.05)
These will not be reconsidered again
Stepwise selection

201. Problems with Subset Selection
Methods can be systematic but still may not provide a clear best subset
Post-hoc so generally can’t be replicated easily
Subset selection is also discontinuous
As a result subset selection is often unstable and highly variable, particularly when p is large
small changes in the data can results in very different estimates

202. Other Approaches Information criteria
AIC = Akaike information criteria (1973), where p = number of parameters in the model
BIC = Bayesian information criteria (Schwartz, 1978)
Choose the model with the smallest AIC/BIC

203. Identifying “Best” Predictive Model
P-values aren’t useful for determining if model is best predictor
Why? Covariates with significant p-values can add more “noise” than signal to performance
Other approaches
R-squared type statistic
“Absolute prediction error”
C-index/Somer’s Dxy

204. R2 Approach Sensitive to the proportion of censored observations
Due to censoring, measure is not as straightforward as in linear regression
Compares fraction of the likelihood explained by the model relative to the likelihood for a perfect model
Sensitive to the proportion of censored observations
Can decrease substantially as % censored increases
R2 values can decrease by up to 20% for > 50% censoring

205. Absolute Prediction Error
Expected value of the absolute value between observed and predicted responses
More interpretable than others (e.g. those that use squared differences vs. absolute value)
Makes the critical difference between association and prediction

206. C-Index
Proportion of all pairs of subjects whose survival times can be ordered such that the subject with the higher predicted survival also the one whose survival is larger
Think of as the probability of concordance/agreement between observed and predicted
Relationship to Somer’s D

207. Example of Model Building: BMT
Suppose we want to find the best predictive model for time to death or relapse….

208. ALL/AML example Possible Variables:
Disease group
Methotrexate use
Patient/Donor age and sex
Platelet recovery
Also considering the between Patient age x Donor Age

209. Global Tests for Each Variable in “Univariate” Models
Global p
Disease group
0.0012
Platelet recovery
0.0005
Dx Group, Platelet Recovery (no intx)
<0.0001
Dx Group, Platelet Recovery, Intx
<0.0001, 0.042
Methotrexate
0.094
Patient Age (no interaction)
0.340
Donor Age (no interaction)
0.249
Patient Age, Donor Age, Intx
0.0037

210. Final Model via Backward Selection (based on p-values)
Variable b
SE(b ) p
AML low risk (vs. ALL)
-0.622
0.304
0.041
AML high risk (vs. ALL)
0.148
1.610
0.614
Platelet recovery
-1.06
0.346
0.002
Patient Age
-0.077
0.034
0.024
Donor Age
-0.072
0.030
0.017
Patient Age x Donor Age
0.003
0.0009
Global P
<0.0001

211. Final Model via Forward Selection (based on p-values)
Variable b
SE(b ) p
AML low risk (vs. ALL)
-0.622
0.304
0.041
AML high risk (vs. ALL)
0.148
1.610
0.614
Platelet recovery
-1.06
0.346
0.002
Patient Age
-0.077
0.034
0.024
Donor Age
-0.072
0.030
0.017
Patient Age x Donor Age
0.003
0.0009
Global P
<0.0001

212. Model Comparisons Based on AIC
Models
AIC
Disease group
737.1
Dx group + MTX
737.2
Dx group + Platelet Recovery
729.2 …
Dx group + PR + Pt Age + Dn Age
732.7
Dx group + PR + Pt Age + Dn Age + Dx x PR
730.9
Dx group + PR + Pt Age + Dn Age + Pt x Dn Age
726.6
Dx group + PR + Pt Age + Dn Age + Dx x Pr + Pt x Dn Age
725.8
Dx group + PR + MTX +Pt Age + Dn Age + Dx x Pr + Pt x Dn Age
726.3

213. R2 and C-Index for Select Models

214. Approach Depends on the Goal
Stepwise selection most common approach
But as we mentioned there are numerous issues
For model parsimony AIC (and BIC) are good choices
Nice in that they allow for comparisons of non-nested modeld
BIC more conservative (but not readily available in R)
If prediction is the goal, consider examining both R-squared and C-index

215. Model Diagnostics

216. Regression Diagnostics
Most interested in testing proportional hazards assumption
Also looking for functional form of covariates
Two types of methods
Graphical approaches
Regression approaches (sometimes combined with graphical approaches)

217. Graphical Approaches
Pretty pictures are nice and can be intuitive but…
We generally prefer a statistical means of determining if an assumption is true
This leads us to regression approaches

218. Residuals A number of residuals are available for Cox models
Schoenfeld residuals are useful to assess:
Model fit
Model assumptions
Grambsch and Therneau developed a statistical test that provides a p-value for these residuals
Assesses multivariable Cox models for
Global proportional hazards
Proportional hazards for individual covariates

219. Assessment of Proportional Hazards/Functional Form
Variable c2 p
AML low risk (vs. ALL)
0.008
0.927
AML high risk (vs. ALL)
1.112
0.292
Platelet recovery
1.655
0.684
Patient Age
0.974
0.324
Donor Age
4.882
0.027
Patient Age x Donor Age
2.943
0.086
Global P
0.362

220. Schoenfeld Residuals
We can plot the scaled Schoenfeld residuals by time to assess where any issues may be
So focus on those variables that were identified by the Grambsch-Therneau test
When the proportional hazards assumption holds, the residuals should follow an approximate straight line

221. Scaled Schoenfeld Residual Plots

222. MTX and Time to Death/Relapse
We dropped MTX from our Cox model
But we showed graphically that proportional hazards might be an issue…
We might reexamine this in our model…

223. Proportional Hazards with MTX
Variable c2 p
AML low risk (vs. ALL)
0.435
0.509
AML high risk (vs. ALL)
2.817
0.093
Platelet recovery
0.010
0.919
MTX
0.654
0.419
Patient Age
5.467
0.019
Donor Age
5.707
0.017
Patient Age x Donor Age
3.507
0.061
Global P
0.091

224. Schoenfeld Plots Again

225. How Can We Address the Problem?
Impose a time-dependent covariate into the model
General idea:
If PHM is valid, time-dependent covariate will not be significant
If time-dependent covariate is significant, then there is “something” going on in terms of the HRs that varies over time

226. Introduce Time Dependent Covariate
Create an new variable Z2(t) = Z1×g(t), where g(t) is a function of time
We don’t know the functional form of g(t)
Two good possibilities are

227. Change Point Model Choosing gives us with a change point at t
Fit proportional hazards model with Z1 and Z2(t)
Including a change point allows the HR to change after a specified time

228. Change Point Model
We could also code our change point model with the time dependent covariate
This coding yields an equivalent model but interpretation is simpler and so this may be preferred

229. Determining t
How do we select a change point for the relative risk? Where is the best change point?
Calculate log likelihood at each event time where t represents specific event times
Choose t that yields the largest log-likelihood

230. New Time Dependent Covariate
Fit proportional hazards model with Z1, Z2(t), and estimate b1, b2
Test local hypothesis:
H0: b2 = 0 vs. HA: b2 ≠ 0
Rejecting H0, reinforces that proportional hazards is doesn’t hold
Do this for each covariate in question

231. Time Varying Covariates for MTX
Start with the simplest version:
Interaction of MTX with stop time (g(t) = t)
The proportional hazards check looks good
Variable c2 p
AML low risk (vs. ALL)
0.0004
0.984
AML high risk (vs. ALL)
1.287
0.257
Platelet recovery
0.009
0.923
MTX x Time
0.015
0.901
Patient Age
0.242
0.622
Donor Age
3.917
0.048
Patient Age x Donor Age
1.551
0.213
Global P
0.501

232. Change Point for MTX Consider a change-point at 466 days
The proportional hazards check looks even better
Variable c2 p
AML low risk (vs. ALL)
0.300
0.584
AML high risk (vs. ALL)
2.538
0.111
Platelet recovery
0.786
0.375
Z1 = MTX if Time > 466
0.175
0.676
Z2 = MTX if Time < 446
0.540
0.462
Patient Age
0.072
0.788
Donor Age
2.572
0.109
Patient Age x Donor Age
0.542
Global P
0.389

233. Model with Change-Point for MTX
Variable b
SE(b ) p
AML low risk (vs. ALL)
-0.465
0.319
0.144
AML high risk (vs. ALL)
0.264
0.309
0.394
Platelet recovery
-1.240
0.362
<0.001
Z1 = MTX if Time > 466
-2.544
1.019
0.013
Z2 = MTX if Time < 446
1.219
0.302
Patient Age
-0.038
0.034
0.277
Donor Age
-0.049
0.033
0.134
Patient Age x Donor Age
0.0015
0.0009
0.107
Global P

234. Interpretation of MTX?
Up to 466 days, individuals treated with methotrexate have 3.4 times the hazard of death or relapse relative to those not treated, controlling for other factors.
However, after 466 days, those treated with methotrexate are at significantly reduced risk of death or relapse, controlling for other factors.
If you go back and look at the hazard rate plots
Hazards crossed at about 460 days

235. Other Residuals for Diagnostics
There are several other types residuals
Martingale residuals
Can be used to assess function form (i.e. identify if variable transformations are needed/help)
Continuous or ordinal variables only
Deviance residuals
Used to investigate outlier

236. Final Thoughts
All methods we discussed are for right censored, left truncated data
Parametric methods can be used for interval and left censored
There are alternative (complex) semi-parametric approaches that have been developed also

237. Final Thoughts We also assumed censoring was independent of the event
Consider if we had wanted to look at time relapse instead of relapse or death
In this case, it seems unlikely that death and relapse are independent
Competing risk analysis is an alternative that can be used in such cases