Biostatistics Case Studies 2014
Session 5:
Survival Analysis Fundamentals
Youngju Pak
Biostatistician
http://research.labiomed.org/biostat
Question #1
Fourteen more subjects were assigned to one of the treatment
groups, compared to the other (Figure 1). In most randomized
studies there is a difference of at most a few subjects.
Why is it so far off here?
Question #1
P2670 , the second paragraph
Question #2
In Figure 2B, survival is 20% in the mab group at the end of the
study, yet Figure 1 says that only 243/347=70% (or perhaps 6
more to give 72%) in that group died, so why isn’t survival 30%?
243/347 = 70% Mortality
100%-20% = 80% Mortality
Kaplan-Meier: Cumulated Probabilities
• We want the probability of surviving for 54 months.
• If all subjects were followed for 54 months, then this prob is the
same as the proportion of subjects alive at that time.
• If some subjects were not followed for 54 months, then we
cannot use the proportion because we don’t know the
outcome for these subjects at 54 months, and hence the
numerator. Denominator?
• We can divide the 54 months into intervals using the follow-up
times as interval endpoints. Ns are different in these
intervals.
• Then, find proportions surviving in each interval and cumulate by
multiplying these proportions to get the survival probability.
Kaplan-Meier: Cumulated Probabilities
•Suppose 104, 93, and 46 (total 243) died in months 0-18, 18-36,
and 36-54. Proportion surviving=(347-243)/347=0.30.
• Of 104 survivors: suppose 11 had 18 months F/U, 51 had 36
months F/U, 35 had 54 months, and 7 had >54 months.
Time
(Mon)
Starting
N
N of
LOST
N
at Risk
N of
Death
N of
Survival
Survival
Rate
0-18
347
0
347
104
243
243/347 = .7
18-36
243
11
232
93
139
139/232 = .6
36-54
139
51
88
46
42
42/88 = .48
•So, 54-month survival is (243/347)(139/232)(42/88)=0.20.
• The real curve is made by creating a new interval whenever
someone dies or completes follow-up (“censored”).
Question #3
The main positive conclusion is that adding mab therapy doubled
median time until disease progression (Fig 2). Why are median,
rather than mean, times always reported in survival analyses?
Questions #4:
Why time to event as an outcome?
81.2%
73.4%
Question # 5
The authors state that the effect (on disease progression) of
adding mab therapy decreases with time. What should we look
at? Is this apparent in Figure 2A? Does it matter? Why?
Question #6
Relative risk (RR) can not be used? At end of study, would RR be
316/347 divided by 308/326, equals 0.96 (from Fig 1)? Use Fig 2
to show that approximately RR=50/73=0.68 at 1-year and
84/96=0.88 at 2-years.
Question #6
27
RR1Yr = (1-0.50)/(1-0.27)=0.68
RR2Yr = (1-0.16)/(1-0.04)=0.88
Question # 7
Even more basic, why bother with “hazards”,
since we have already solved the problem of
comparing groups with survival times?
Question # 7
Hazard: “Sort-term” incidence at a specified time.
E.g., events per 100,000 persons per day at 1 month.
determines
Prob of Survival
Hazard
3
e-1(time)
1
e-3(time)
Time
Time
Constant Hazard ↔ Exponential
Question # 7
Heuristic:
Often, HR for Group1 to Group2 ≈
Median Survival Time for Group 2
Median Survival Time for Group 1
Question # 8
• What does the requirement of “proportional
hazards” used in their (page 2670, paragraph
2, and in most paper’s) survival analyses
mean? Sketch a figure like Fig 2A where this
obviously does not occur. What could we do
in that case, since the standard analysis
needs it, and would therefore be wrong?
More on Question #8
• The Cox PH model assumes that the hazards in the
two groups remains constant over time. For example,
drug effect on the hardy is proportionally the same as
it is on the frail.
• This is difficult to see directly on Kaplan-Meier curves,
but if they are curved downward, as in Fig 2A, then the
ratio of the survival times at any survival probability is
close to the hazard ratio, and should be constant for
different y-values.
• This is obviously not the case in Fig 2A, so it is a good
example where proportional hazards do not hold.
More on Question # 8
• There are some plots (eg., Schonfeld Residual plots ,
Grambsch and Therneau test (1994) ) that can be
used to assess this. The solution is to either use a
more general Cox analysis that allows say one ratio
over some time and another ratio over other times.
• Alternatively, as in this paper, an “interaction” can be
used that puts a structured change in the hazard ratio
over time.
More Question # 8
For convex curves like
these, the hazard ratio
is approximately the
ratio of survival times
for any survival (y-axis).
HR = 6/12=0.50
HR = 12/18=0.67
HR = 24/30=0.80
So this figure
“obviously” violates
proportional hazards.
The authors used an interaction to resolve this violation
(bottom of p 2671)
More Question # 8
The circled p=0.02 verifies what seems clear in Fig 3 for
subjects >65.
Q #9: RR vs. OR
Case
Non-Case
mab
174
173
347
No mab
238
88
326
Case = 1-Yr
Progression
For mab:
Risk = Prob(Case) = 174/347 = 0.50
Odds = Prob(Case)/Prob(Non-Case) = 174/173 = 1.00
RR = (174/347)/(238/326) = 0.50/0.73 = 0.68
OR = (174/173)/(238/ 88) = 1.00/2.70 = 0.37
→ Effect by OR almost twice RR
When is Odds Ratio ≈ Relative Risk ?
Odds Ratio ≈ Relative Risk , if Prob(Case) is close to 0.
So, Odds Ratio ≈ Relative Risk in case-control studies of a
rare disease.
For most case-control studies, OR should be used but the
odds should not interpreted as a risk or chance of case since
odds is not a probability.
Question 10 :
Too many subgroup analyses?
• The overall type I error rate (False positive) will be
increased as multiple statistical tests are performed.
– See Lagakos SW. The challenge of subgroup analyses—reporting
without distorting. N Engl J Med 2006; 354(16): 1667–1669.
• The prevailing opinion is not to use p-values or adjust
for multiple tests.
– Bonferroni: if 10 tests, then p < 0.05/10=0.005
should be considered as statistically significant to
maintain the overall type I error rate of 0.05
– There are several other methods such as Tukey
– Most commercial statistical softwares have
options to adjust for multiple tests.
Funding & Conflict of Interests ?
http://www.cancer.gov/newscenter/qa/1980/AvastinQandA