This document discusses analyzing data from a study where rats were exposed to different doses of a potential carcinogen. Rats were followed until a tumor developed or 104 days. The study aims to determine if exposed rats develop tumors faster than unexposed rats. It discusses challenges in analyzing time-to-event data like censoring and lack of symmetry. Key methods covered include hazard ratios, log-rank tests, and comparing tumor rates between groups while accounting for varying follow-up times.

1. Survival Analysis for Lab
Scientists
Mike LaValley
1/24/2011

2. Rats Data
 Rats treated with a potential
carcinogen
 Control rats – no exposure
 Different doses
 Low dose
 Medium dose
 High dose
 Rats followed until a tumor developed or
until day 104 (15 weeks)

3. Rats Data
 Experiment terminated at 104 days
and any tumor free rats are sacrificed
 Censoring
 If a rat escapes or is hurt during
follow-up, follow-up is terminated
 Censoring
 Question – do exposed rats develop
tumors at a faster rate than
unexposed?

4. Time-to-Event Data
 Many different types of studies collect data
on the length of time until a particular
event happens
 Time to heart attack or stroke in the
Framingham Heart Study
 Time to AIDS in treatment studies with HIV-
infected persons
 Time to return to work following a workplace
injury
 Time to disease onset in hamsters following
exposure to C. difficile

5. Time-to-Event Data
 Time-to-event data has 2
characteristics that complicate its
analysis
 Times to event are not symmetrically
distributed
 Not every person/hamster/rat/specimen
has the event under observation

6. Lack of Symmetry
4 6 8 10 12 14 16
01020304050
temp.norm
4 6 8 10 12 14 16 18
020406080100
temp.logs
0 10 20 30 40 50 60
050100150
temp.exp
0 5 10 15 20 25
010203040 temp.weib
The top 2 bar
charts show
samples from
symmetric
distributions
The bottom 2 bar
charts show
samples from
right-skewed
distributions

7. Lack of Symmetry
 When data aren’t symmetric the usual
statistics don’t work well
 The mean is no longer in the center of the
sample
 It is shifted to the right
 The standard deviation isn’t very useful
 Measuring one standard deviation to the right
from the mean gives a lower percentage of the
population than measuring one standard
deviation to the left

8. Lack of Symmetry

9. Lack of Symmetry
 Ways to cope with a lack of symmetry
 Transform the data using a log or square root
 This tends to make the distribution more
symmetric
 Use robust or non-parametric methods
 Medians instead of means
 Inter-quartile ranges instead of standard
deviations
 Rank-based Wilcoxon tests instead of t-tests

10. Lack of Symmetry
 In survival analysis, there is an
emphasis on use of non-parametric
methods because of the skewness
 However, it is also possible to use
non-symmetric distributions to cope
with the lack of symmetry
 Exponential distribution
 Weibull distribution

11. Censoring
 That not every
person/hamster/rat/specimen has the
event under observation is a much
harder problem
 This is called censoring
 Censoring is what prevents us from
being able to use standard methods
on time-to-event data

12. Censoring
 Censoring creates a difference
between
 What we get to see on everyone – the
follow-up time
 Time to the event or censoring
 What we want to see on everyone – the
time to event

13. Censoring
 The follow-up times are always less
than or equal to the time-to-event
 This is a big problem
 The follow-up times are systematically
too short
 The mean follow-up time is less than the
mean time-to-event
 The median follow-up time is less than
the median time-to-event

14. Censoring
The figure shows a
study with 7 subjects.
Circles show when
subjects had events
and squares show
when they were
censored. The solid
blue lines are the
follow-up times, and
the red dotted lines
are the times from
censoring until the
event.
median follow-up time is 2.5 years
median survival time is 4.0 years

15. Censoring
 The more censoring there is, the
worse the discrepancy between
follow-up time and time to event
 Need to remove the effect of
censoring from the follow-up times to
get a sense of time-to-event
 This is what survival analysis is all
about

16. Hazard
 In longitudinal studies, data is collected on
a sample of individuals over time
 Often we are interested in the occurrence
of an important event in our sample
 Death
 Heart attack
 Tumor

17. Risk
 If our interest is on whether subjects
ever experience the event in the
study
 Analysis of risk of the event
 Based on the proportion of subjects
who have the event
 Unadjusted analysis done using
contingency tables and chi-square tests
 Adjusted analysis using logistic
regression

18. Hazard
 If our interest is on how long until
subjects experience the event in the
study
 Analysis of the hazard of the event
 Based on the rate of occurrence of the
event
 The difference between risk and hazard
is that the hazard incorporates time

19. Hazard
 A risk analysis is based on the
number of events per person
 A hazard analysis is based on the
number of events per person-year (or
other person-time measure)
 A hazard analysis would be more
appropriate if the length of follow-up
varies

20. Hazard
 Risk: 2 out of 6
subjects have the
event (33%)
 Hazard: 2 events
per 38 person-
weeks (0.05 events
per person-week)

21. Censoring
 In the example we just saw, one subject withdrew
and did not complete the study or have the event
 Some other subjects completed the study but never
had the event
 In a hazard analysis, both types of subject are
considered censored and their follow-up time is used
in the denominator
 In a risk analysis, incorporating the first type of
subject requires some arbitrary rules
 Restricting analysis to subjects who do not
withdraw
 Assigning the event (or absence of event) to
subjects who withdraw

22. Hazard
 A person-years analysis is fine as long
as we assume that the hazard rate is
constant
 Constant hazard -> exponential
distribution
 Most modern survival analysis is done in
a way to avoid assuming that the hazard
rate is constant over time
 Makes the notation and terminology more abstract
 Analyses are more complicated than taking the ratio
of events to person-time

23. Hazard
 To move away from having a
constant hazard over time
 To calculate the hazard at a particular
time t
 Look only in subjects eligible to have the
event at t
 The hazard at t is the person-time event
rate for a very short period of time
starting at t (limit as the period of time
shrinks to 0)

24. Hazard Ratio
 If we follow two groups over time, we
may want to compare their hazard
rates
 The hazard ratio is the hazard (at
time t) in group 1 divided by the
hazard (at time t) in group 2
( )
( )
( )
1
2
Group hazard t
Hazard Ratio t
Group hazard t
=

25. Hazard Ratio
 The hazard ratio has similar
construction to the relative risk,
 Except that the hazards can change
with time
 To simplify the models, we often
assume Proportional Hazards
 i.e. the hazard ratio is constant over
time

26. Hazard Ratio
 Under proportional hazards
 Although the hazards within each group
may change over time
 The ratio of the hazards between groups
stays the same over time
( )
( )
1
2
Group hazard t
Hazard Ratio
Group hazard t
=

27. Hazard Ratio
These hazard
curves are
proportional
There is a
constant hazard
ratio of 0.5
comparing
group 1 to
group 2

28. Testing Survival Difference
 Nonparametric tests are typically
used to test for survival differences
between groups
 Logrank test – best test if the hazards
are proportional between groups
 Gives equal weight to events that happen
throughout follow-up
 Wilcoxon test – gives more weight to
events early in follow-up

29. Testing Survival Difference
 Parametric test assuming exponential
survival is also sometimes used
 Best test if the hazard is constant within
the groups being compared
 Goes along with the person-time analysis

30. Treatment
Groups
Tumors Average
Follow-Up
(days)
Tumors /
Rat-Day
Control 3/30 90 0.0011
Low Dose 2/10 89 0.0022
Medium
Dose
4/10 86 0.0047
High Dose 4/10 91 0.0043

36. Survival Comparison
 Comparing the 4 groups
 Logrank test p=0.095
 Wilcoxon test p=0.311
 Exponential test p=0.171
 So, we don’t find a difference between
groups that couldn’t be attributed to
chance
 But the logrank result is suggestive that
with a larger experiment we might attain
statistical significance

37. Analyzing Time-to-Event
Outcomes
 Becoming more and more common
 Challenge mostly due to censoring
 Most basic analysis – exponential
hazard is estimated by
 #events/total person-time
 Modern methods avoid assuming
exponential data
 Logrank test