Download Statistics 262: Intermediate Biostatistics

Cox Regression II
Monday “Gut Check” Problem…

Write out the likelihood for the following data,
with weight as a time-dependent variable:
Time-to-event
(months)
Survival
(1=died/0=cen
sored)
Weight at
baseline
Weight at 3
months
Weight at 9
months
Weight at
12
months
10
0
140
145
155
.
2
1
240
.
.
.
4
0
130
130
.
.
8
1
200
210
250
.
12
0
150
145
145
140
14
0
180
180
180
175
10
1
180
190
240
.
1
0
230
.
.
.
3
0
110
110
.
.
SAS code for a time-dependent
variable…
proc phreg data=example;
model time*censor(0) = weight;
if time<3 then weight=w0;
if time>=3 and time<6 then weight=w3;
if time>=6 and time<9 then weight=w6;
if time>=9 then weight=w9;
run;
Model results


Using baseline weight: HR=2.8
Using weight as time-changing variable:
HR=9.3
1. Stratification
Violations of PH assumption can be resolved by:
•Adding time*covariate interaction
•Adding other time-dependent version of the covariate
•Stratification
Stratification
•Different stratum are allowed to have different baseline
hazard functions.
•Hazard functions do not need to be parallel between
different stratum.
•Essentially results in a “weighted” hazard ratio being
estimated: weighted over the different strata.
•Useful for “nuisance” confounders (where you do not care
to estimate the effect).
•Assumes no interaction between the stratification variable
and the main predictors.
Example: stratify on gender


L p (β) 
m

i 1
Males: 1, 3, 4, 10+, 12, 18 (subjects 1-6)
Females: 1, 4, 5, 9+ (subjects 7-10)
♂
Li  (
♀
h7 (1)
h1 (1)
) x(
)
h1 (1)  h2 (1)  h3 (1)  h4 (1)  h5 (1)  h6 (1) h7 (1)  h8 (1)  h9 (1)  h10 (1)
♀
h3 (4) ♂
h8 (4)
h2 (3)
♂
(
) x(
) x(
)
 h2 (3)  h3 (3)  h4 (3)  h5 (3)  h6 (3)  h3 (4)  h4 (4)  h5 (4)  h6 (4)  h8 (4)  h9 (4)  h10 (1)
x(
h(5)
)....
♀
h9 (5)  h10 (5)
The PL
m
L p (β)   Li 
i 1
m 0 (t  1)eβx ♂
)x
βx
βx
βx
βx
 m 0 (1)e  m 0 (1)e  m 0 (1)e  m 0 (1)
1
(
m 0 (1)eβx  m 0 (1)eβx
1
2
3
4
5
6
♀
 f 0 (1)eβx
)
  f 0 (1)eβx   f 0 (1)eβx   f 0 (1)eβx
7
(
 f 0 (1)eβx
7
8
9
10
....
m
 Lp (β)   Li  (
i 1
eβx1  eβx2
eβx1
eβx7
) x( βx7
)...
βx3
βx5
βx 6
βx8
βx9
βx10
βx 4
e e e 
e e e e
2. Using age as the time-scale
in Cox Regression




Age is a common confounder in Cox
Regression, since age is strongly related to
death and disease.
You may control for age by adding baseline
age as a covariate to the Cox model.
A better strategy for large-scale longitudinal
surveys, such as NHANES, is to use age as
your time-scale (rather than time-in-study).
You may additionally stratify on birth cohort
to control for cohort effects.
Age as time-scale



The risk set becomes everyone who was at
risk at a certain age rather than at a certain
event time.
The risk set contains everyone who was still
event-free at the age of the person who had
the event.
Requires enough people at risk at all ages
(such as in a large-scale, longitudinal survey).
The likelihood with age as time
Event times: 3, 5, 7+, 12, 13+ (years-in-study)
Baseline ages: 28, 25, 40, 29, 30 (years)
Age at event or censoring: 31, 30, 47+, 41, 43+
m
L p (β)   Li  (
i 1
(
h2 (30)
)x
h1 (30)  h2 (30)  h4 (30)  h5 (30)
h1 (31)
h4 (41)
) x(
)
h1 (31)  h4 (31)  h5 (31)  h3 (41)  h4 (41)  h5 (41)
3. Residuals


Residuals are used to investigate the
lack of fit of a model to a given subject.
For Cox regression, there’s no easy
analog to the usual “observed minus
predicted” residual of linear regression
Martingale residual

ci (1 if event, 0 if censored) minus the estimated
cumulative hazard to ti (as a function of fitted model)
for individual i:
ci-H(ti,Xi, ßi)




E.g., for a subject who was censored at 2 months, and whose predicted
cumulative hazard to 2 months was 20%

Martingale=0-.20 = -.20
E.g., for a subject who had an event at 13 months, and whose predicted
cumulative hazard to 13 months was 50%:

Martingale=1-.50 = +.50
Gives excess failures.
Martingale residuals are not symmetrically
distributed, even when the fitted model is correctly,
so transform to deviance residuals...
Deviance Residuals


The deviance residual is a normalized
transform of the martingale residual.
These residuals are much more
symmetrically distributed about zero.
Observations with large deviance
residuals are poorly predicted by the
model.
Deviance Residuals




Behave like residuals from ordinary linear
regression
Should be symmetrically distributed around 0
and have standard deviation of 1.0.
Negative for observations with longer than
expected observed survival times.
Plot deviance residuals against covariates to
look for unusual patterns.
Deviance Residuals
In SAS, option on the output statement:
Output out=outdata resdev=Varname

**Cannot get diagnostics in SAS if timedependent covariate in the model
Example: uis data
Out of 628
observations, a
few in the range
of 3-SD is not
unexpected
Pattern looks
fairly symmetric
around 0.
Example: uis data
What do you
think this
cluster
represents?
Example: censored only
Example: had event only
Schoenfeld residuals

Schoenfeld (1982) proposed the first set of
residuals for use with Cox regression
packages



Schoenfeld D. Residuals for the proportional
hazards regresssion model. Biometrika, 1982,
69(1):239-241.
Instead of a single residual for each
individual, there is a separate residual for
each individual for each covariate
Note: Schoenfeld residuals are not defined for
censored individuals.
Schoenfeld residuals


The Schoenfeld residual is defined as the covariate
value for the individual that failed minus its expected
value. (Yields residuals for each individual who failed,
for each covariate).
Expected value of the covariate at time ti = a
weighted-average of the covariate, weighted by the
likelihood of failure for each individual in the risk set at
ti.
jR( t )
residual  x ik 
x
i
jk
pj
i 1
e.g.,56 years 
jrisk set
 (age) p
j
i 1
p j  probabilit y of event now for the i th
The person who died was 56; based on the fitted
model, how likely is it that the person who died
person was 56 rather than older?
Example

5 people left in our risk set at event
time=7 months:





Female 55-year old smoker
Male 45-year old non-smoker
Female 67-year old smoker
Male 58-year old smoker
Male 70-year old non-smoker
The 55-year old female smoker is the one who
has the event…
Example
Based on our model, we can calculate a
predicted probability of death by time 7 for
each person (call it “p-hat”):





Female 55-year old smoker: p-hat=.10
Male 45-year old non-smoker : p-hat=.05
Female 67-year old smoker : p-hat=.30
Male 58-year old smoker : p-hat=.20
Male 70-year old non-smoker : p-hat=.30
Thus, the expected value for the AGE of the person who
failed is:
55(.10) + 45 (.05) + 67(.30) + 58 (.20) + 70 (.30)= 60
And, the Schoenfeld residual is: 55-60 = -5
Example
Based on our model, we can calculate a
predicted probability of death by time 7 for
each person (call it “p-hat”):





Female 55-year old smoker: p-hat=.10
Male 45-year old non-smoker : p-hat=.05
Female 67-year old smoker : p-hat=.30
Male 58-year old smoker : p-hat=.20
Male 70-year old non-smoker : p-hat=.30
The expected value for the GENDER of the person who
failed is:
0(.10) + 1(.05) + 0(.30) + 1 (.20) + 1 (.30)= .55
And, the Schoenfeld residual is: 0-.55 = -.55
Schoenfeld residuals

Since the Schoenfeld residuals are, in
principle, independent of time, a plot that
shows a non-random pattern against time is
evidence of violation of the PH assumption.


Plot Schoenfeld residuals against time to evaluate
PH assumption
Regress Schoenfeld residuals against time to test
for independence between residuals and time.
Example: no pattern with time
Example: violation of PH
Schoenfeld residuals
In SAS:
option on the output statement:
Output out=outdata ressch= Covariate1
Covariate2 Covariate3
Summary of the many ways to
evaluate PH assumption…
1. Examine log(-log(S(t)) plots
PH assumption is supported by parallel lines and refuted by lines that cross
or nearly cross
Must use categorical predictors or categories of a continuous predictor
2. Include interaction with time in the model
PH assumption is supported by non-significant interaction coefficient and
refuted by significant interaction coefficient
Retaining the interaction term in the model corrects for the violation of PH
Don’t complicate your model in this way unless it’s absolutely necessary!
3. Plot Schoenfeld residuals
PH assumption is supported by a random pattern with time and refuted by a
non-random pattern
4. Regress Schoenfeld residuals against time to test for
independence between residuals and time.
PH assumption is supported by a non-significant relationship between
residuals and time, and refuted by a significant relationship
4. Repeated events

Death (presumably) can only happen
once, but many outcomes could happen
twice…
Fractures
 Heart attacks
 Pregnancy
Etc…

Repeated events: 1


Strategy 1: run a second Cox
regression (among those who had a
first event) starting with first event time
as the origin
Repeat for third, fourth, fifth, events,
etc.

Problems: increasingly smaller and smaller
sample sizes.
Repeated events: Strategy 2

Treat each interval as a distinct
observation, such that someone who
had 3 events, for example, gives 3
observations to the dataset

Major problem: dependence between the
same individual
Strategy 3


Stratify by individual (“fixed effects partial
likelihood”)
In PROC PHREG: strata id;




Problems:
does not work well with RCT data
requires that most individuals have at least 2
events
Can only estimate coefficients for those covariates
that vary across successive spells for each
individual; this excludes constant personal
characteristics such as age, education, gender,
ethnicity, genotype
5. Competing Risks
BMT: Related vs. Unrelated
Donor
SAS Output

Patients with related donors survive longer.
37
Related/Unrelated Donor is
significant.

Can you say definitively to a patient:


38
If you find a related donor, you will have longer
survival time.
What variables could be confounders?
Survival Analysis categorizes
subjects
1
2
3
39
Event of interest was observed
Censored
Competing risk was observed
Competing Risk

an event that either precludes the event of
interest or alters its probability
Event of Interest
Competing Risk
Death from the disease
Death from other causes
Relapse
Non-relapse mortality
Relapse
Treatment complications
Local progression
Metastasis
40
BMT Example
Interested in Time to Relapse
 Competing Risks (preclude or alter probability
of relapse)
 Non-relapse mortality
 Graft-vs-host disease (GVHD)

41
Who failed from the event of
interest?
1
2
3
Event of interest was observed
Censored
Competing risk was observed

Yes
Maybe
No
Common Pitfall: treating competing risks as censoring



42
Treats nos as maybes
Puts them partially in the numerator of occurrence when they
shouldn’t be there
Thus overestimates risk (underestimates S)
What to do instead
KM estimate of event free survival (EFS)
 Cumulative Incidence Analysis

43
Event-Free Survival








In cancer, often Progression-Free Survival (PFS)
Treats competing risks as events
Can use KM
For each subject, the first event to occur
“Survival” implies death is considered an event
BMT: first of relapse, GVHD or death
Is this of interest?
May not be, e.g., Local progression and metastasis
44
Cumulative Incidence Analysis
Separates competing risks from event of
interest
 If no competing risks, equivalent to KM
 Estimates occurrence probability: F(t) = 1 –
S(t)
 Each event goes into one bin (event type)

45
GVHD
Relapse
BMT Cumulative
Incidence Curves
Death
6. Considerations when
analyzing data from an RCT…
Intention-to-Treat Analysis
Intention-to-treat analysis: compare
outcomes according to the groups to
which subjects were initially assigned,
regardless of which intervention they
actually received.
Evaluates treatment effectiveness
rather than treatment efficacy
Why intention to treat?


Non-intention-to-treat analyses lose the
benefits of randomization, as the groups may
no longer be balanced with regards to factors
that influence the outcome.
Intention-to-treat analysis simulates “real
life,” where patients often don’t adhere
perfectly to treatment or may discontinue
treatment altogether.
Drop-ins and Drop-outs:
example, WHI
Both women on
placebo and women
on active treatment
discontinued study
medications.
Women
on placebo
“dropped
Women
on treatment
in” to treatment
because
“dropped
in” to treatment
their regular
because
their doctors
doctorsput
took
them on hormones (dogma=
them off study drugs and
“hormones are good”).
put them on hormones to
insure they were on
hormones and not placebo.
Effect of Intention to treat on
the statistical analysis

Intention-to-treat analyses tend to
underestimate treatment effects;
increased variability due to switching
“waters down” results.
Example
Take the following hypothetical RCT:
Treated subjects have a 25% chance of dying during the 2-year
study vs. placebo subjects have a 50% chance of dying.
TRUE RR= 25%/50% = .50 (treated have 50% less chance of
dying)
You do a 2-yr RCT of 100 treated and 100 placebo subjects.
If nobody switched, you would see about 25 deaths in the treated
group and about 50 deaths in the placebo group (give or take a
few due to random chance).
Observed RR .50
Example, continued
BUT, if early in the study, 25 treated subjects switch
to placebo and 25 placebo subjects switch to
treatment.
You would see about
25*.25 + 75*.50 = 43-44 deaths in the placebo group
And about
25*.50 + 75*.25 = 31 deaths in the treated group
Observed RR = 31/44  .70
Diluted effect!
7. Example analysis: stress
fracture study
• Women runners may have reduced levels
of estrogen, which puts them at risk of
bone loss and stress fractures
• This was a randomized trial of hormones
(oral contraceptives) to prevent stress
fractures in women runners
• Two groups: treatment and control (no
placebo)
Baseline Description and
Comparability of Groups

Baseline descriptors are summarized as:
•
•



means and standard deviations for continuous
variables
frequencies and percentages for categorical variables
How good was the randomization?; i.e., Are the
groups indeed balanced with regards to variables
known to be prognostically related to the
outcome?
For cohort study, what factors are related to
exposure, and thus might be confounders?
Who is in the population?
Stress fracture study
Baseline characteristics by randomization assignment
control
Age (yrs)
Stress fracture (%)
Menses in past year
No. of lifetime menses
Oligo/amenorrhea (%)
Amenorrhea (%)
Oligomenorrhea (%)
Elevated EDI score (%)
Whole body BMD (g/cm2)
Total hip BMD (g/cm2)
Spine BMD (g/cm2)
Total bone mineral content (g)
Height (inches)
Weight (lbs)
BMI (kg/ m2)
Percent body fat
Calcium per day (mg)
21.9
40.0
9.5
67.4
35.8
6.2
29.6
21.0
1.10
.97
.99
2146
65.2
128.0
21.2
23.3
1412
treatment
22.4
39.1
9.4
68.9
30.0
11.4
18.6
30.0
1.11
.99
.98
2179
65.4
128.7
21.1
22.7
1401
Summary of events


Might be presented as overall incidence
rates.
If events are heterogeneous (as with
stress fractures), tabulate results.
Stress Fracture 1
Diagnostic
test
right tibial bone
right tibial bone
right tibial bone
right tibial bone
right tibial bone
right tibial bone
left tibial bone
left tibial bone
left tibial bone
left tibial bone
right foot
right foot
left third metatarsal
right 4th metatarsal
left cuboid
navicular bone
upper right femur
right femoral neck
bone scan
x-ray
bone scan
bone scan
bone scan
bone scan
bone scan
bone scan
bone scan
bone scan
bone scan
x-ray
x-ray
x-ray
MRI
bone scan
MRI
MRI
18
Stress fracture 2 Study Area
right tibial bone
right tibial bone
right femur
left foot
4
Boston
Boston
Boston
Boston
Stanford
Michigan
Boston
Michigan
Los Angeles
Michigan
Los Angeles
New York
Boston
Stanford
Stanford
Stanford
Los Angeles
Stanford
Evaluation of primary
hypothesis


Intention-to-treat analysis for RCT
Primary exposure-event hypothesis for
cohort study, adjusted for confounding
Corresponding Kaplan-Meier curve
Treatment (n=52)
6 fractures
Control (n=70)
12 fractures
Corresponding HR
Hazard Ratio (95% CI)
Randomized to treatment
.82 (0.30, 2.27)
Secondary analyses


For RCT: any non-intention to treat
analyses
For RCT and cohort: evaluate other
predictors; effect modification;
subgroups
Hazard ratios for treatment
variables
Hazard Ratio (95% CI)
Randomized to treatment
Randomized to treatment, on-protocol only (n=82)
Actually took OCs at least 1-month
Per month on OCs
Time-dependent treatment variable, when on treatment
.82 (0.30, 2.27)
.63 (0.21, 1.92)
.41 (0.15,1.08)
.92 (0.85, 0.98)
.50 (0.18,1.40)
**All analyses are stratified on site and menstrual status at baseline (amenorrheic, oligomenorrheic, or eumenorrheic),
and adjusted for age and spine Z-score at baseline using Cox Regression.
Kaplan-Meier estimates of stress fracture-free survivorship by
BMC at baseline
≥2200 g (n=52)
1800-2199 g (n=55)
<1800 g (n=15)
Kaplan-Meier estimates of stress fracture-free survivorship
by levels of daily calcium intake at baseline
1500+mg/day (n=36)
800-1499 mg/day (n=63)
<800 mg/day (n=22)
Kaplan-Meier estimates of stress fracture-free survivorship by
previous stress fracture
No previous fracture
(n=83)
Previous fracture
(n=39)
Middle two quartiles
Highest quartile of lean mass
Lowest quartile of lean mass
Risk Factors
Hazard Ratio (95% CI)
History of menstrual irregularity prior to baseline
BMC<1800g
Low calcium (<800 mg/d)
Stress fracture prior to baseline
Fat mass (per kg)
2.91 (0.81,10.43)
3.70 (1.31, 10.46)
3.60 (1.12,11.59)
5.45 (1.48,20.08)
1.05 (0.91, 1.21)
**All analyses are stratified on site and menstrual status at baseline, and adjusted for age and spine Z-score at
baseline using Cox Regression.
Other protective factors
Hazard Ratio (95% CI)
Spine BMD (per 1-standard deviation increase)
Every 100-mg/d calcium (continuous)
Lean mass (per kg), time-dependent
Change in lean mass (per kg)
Menarche (per 1-year older)
.54 (0.30, 0.96)
.90 (0.81, 0.99)
.91 (0.81, 1.02)
.83 (0.56, 1.24)
.55 (0.34,0.90)
**All analyses are stratified on site and menstrual status at baseline, and adjusted for age and spine Z-score at
baseline (except spine Z score) using Cox Regression.
References
Paul Allison. Survival Analysis Using SAS. SAS Institute Inc., Cary, NC: 2003.