Announcements
Stat’s Seminars Update
Rest of the Fall
• Nov 4: Competing Risks, Frailty, and Violations of Proportional Hazards
Tom Greene
• Nov 11: Alternatives to Cox regression
Tom Greene & Nan Hu
• Dec 2: Chasing Tail(s): Statistical Intuition for High-Throughput Studies
Ken Rice, University of Washington
• Dec 9: Case Control Study Design
Carroll Sweeney
• Dec 16: Analyses for case-control studies
Greg Stoddard
Statistical Methods in Progress Meetings:
(also in WB – usually 238A or B)
Charlie Casper:
Nov 8, 10 AM
Mary Slattery:
Nov 22, 10 AM
Andrew Redd:
Dec 6, 10 AM
Causal Modeling Workgroup:
3rd Friday of Each Month
(3PM, Div. EPI conference room, WB)
Other Announcements
• Slides sets for seminars to be archived at
http://www.ccts.utah.edu/biostats/
• R User’s group forming
– Andrew Redd
Competing Risks, Frailty, and
Violations of Proportional Hazards
Tom Greene
References
• Aalen O, Borgan O, Gjessing H. Survival and
Event History Analysis: A Process Point of
View, Springer, 2008
• Martinussen T, Scheike T. Dynamic Regression
Models for Survival Data. Springer, 2010
Issues for Today
• Understanding what standard Kaplan-Meier
curves and competing risk cumulative incidence
curves really estimate in competing risk situations
• Same issue for standard hazard ratios and
competing risk hazard ratios for subdistributions
• When to use standard methods or competing risk
methods?
Issues for Today
• What is the difference between individual-level
hazard ratios and population hazard ratios?
• What are the implications of variation in frailty
for population hazard ratios in Cox regression ?
• What are the implications of variation in frailty
for standard survival analysis in clinical trials?
• What if there are also competing risks?
Questions for Today
• How can one check for non-proportional
hazards in Cox regression?
• How can Cox regression be modified to
account for non-proportional hazards?
Quantities in Survival Analysis
S(t) = Pr(T > t) for a survival time T
T may be subject to right censoring time C:
we observe X = min(T,C)
The hazard rate λ(t) = event rate in time interval
(t, t+ε) for the subgroup still at risk at time t
The cumulative hazard H(t) is the area under
t
the λ(u) vs. u curve up to time t: H(t )   (u )du
0
Relationship of survival and cumulative hazard:
S (t )  e
 H (t )
Estimating S(t) and H(t)
dj = # events at time tj; nj = # at risk at time tj
Kaplan-Meier estimator:
 dj
Sˆ (t )   1 

nj
t j t 




Nelson-Aalen estimator (discrete version):
ˆ
t dj

H
(t )
ˆ
Hˆ (t )  0
nj
S (t )  e
Cox Model
λ(t|Z) = λ0 (t) × exp( βZ)
Z:
λ0 (t):
β:
exp(β):
Covariate(s) of interest
Baseline hazard function when Z = 0
Cox regression coefficients
Hazard ratios under the Cox model
Two key assumptions:
1) T and C are independent given Z
2) Proportional hazards
Competing Risks Events
• An event whose occurrence precludes
the occurrence of an event of interest
• Usually a clinical event, possibly having a
direct or indirect relationship with the
event being analyzed
Competing Risk Subdistribution
• The subdistribution, or cumulative incidence
probability, corresponding to the fraction of all
patients with the event of interest, after accounting for
“true censoring”
dj ˆ
• Estimated by Fˆ sub .dist (t )  
S(t j 1)
t  tnj
j
• tj :
• d j:
• n j:
•Sˆ:
Event times (all event types)
Number of pts with event of interest at tj
Number at risk at tj
Kaplan-Meier Curve for the composite
outcome T = min(T1,T2)
Traditional Kaplan Meier and Competing Risk Cumulative Incidence
Pts > 55 Years of Age
Example:
Study of chronic
kidney disease;
Death and End
Stage Renal
Disease
(ESRD) are
competing Risks
Traditional Kaplan-Meier, ESRD censoring death
Cumulative Incidence of ESRD, Death as competing risk
Traditional Kaplan-Meier, Death censoring ESRD
Cumulative Incidence, Death as competing risk
Months Follow-up
Interpretation of Kaplan Meier curves
with competing risks
• The Kaplan Meier cumulative incidence curve
indicates the event probability for a “virtual
population” in which the competing risk does not
occur, but which is linked to the studied
population such that the target event occurs after
a competing event at the same rate as for
patients not experiencing the competing event.
• This virtual population is hard to fathom if the
competing risk and target event are associated
Competing Risk Cumulative Incidence Curves for Time to ESRD or Death prior to ESRD
Age < 55 years
ESRD or Death
ESRD
Death
Months Follow-up
Age > 55 years
ESRD or Death
ESRD
Death
Months Follow-up
Competing Risk Formulation Decomposes Cohort Into 3 or More States
Age < 55 years
Age > 55 years
Alive without ESRD
Alive without ESRD
Alive with ESRD prior to Death
Alive with ESRD prior to Death
Death prior to ESRD
Death prior to ESRD
Months Follow-up
Months Follow-up
Conventional and Competing Risk Hazard Calculation
Hazard Increment
at time tj
nj(at risk)
Competing
Risk Calculation
nj(cmp.rsk)
dj
n j (at risk )  n j (cmp .rsk )
nj(event)
nj(at risk)
Conventional
nj(cmp.rsk)
Risk Calculation
nj(event)
tj-
dj
n j (at risk)
Time
dj 

dj
n j (at risk)
 n j (cmp.rsk)
n j (at risk)  n j (cmp.rsk)
tj
Analysis of Subdistriubtion Hazard
• Typically implemented using a proportional
model (Grey’s model), analogous to
proportional Hazards Cox regression
• Implemented in R library cuminc
• Can be viewed as assigning + infinity to
survival times for those with competing risks
What to do in practice?
Example: RCT of effect of BP drug on CV composite; non-CV
death is competing risk. The trial’s focus is on the CV outcome,
but the treatment may also affect non-CV death.
• Useful to display competing risk cumulative incidence
• The standard analysis (using log rank or Cox) gives the correct
HR in the hypothetical “independent competing risk” universe
– This remains true even if the treatment affects the competing risk
• This is not true of the HR under Gray’s competing risk model
– Treatment can appear beneficial by increasing the competing risk
even if it has no effect on the target endpoint
• Hence, the “standard approach” is typically used to
compare hazard rates in RCTs
A Parable for Statisticians
A man is walking home one dark night when he sees a sage
searching for something frantically under a streetlight.
“Teacher, what have you lost?” the man asks.
“I am searching for my key,” the sage says worriedly.
“I’ll help you look,” the man says and joins the search. Soon both
are down on their knees under the streetlight, looking for the lost
key. After some time, the man asks, “Tell me, teacher, do you
remember where exactly you dropped the key?”
The sage waves his arm back toward the darkness and says,
“Over there, in my house. I lost the key inside my house…”
Shocked and exasperated, the man shouts, “Then why are you
searching for the key out here in the street?”
“Because there is more light here than inside my house,” the
teacher answers non-chalantly.
Population and Individual Level Hazards
• The hazard in the Cox model
λ(t|Z) = λ0 (t) × exp( βZ)
represents the “population hazard” given Z.
• The population hazard represents an average
of individual level hazards among patients
remaining at-risk
• The variation in individual level hazards,
conditional on Z, is captured by the concept of
frailty.
Proportional Frailty Model
• Assume the hazard function for an subject
with frailty W is α(t|W) = W × α(t).
t
Standardized
• As before: A(t )    (u )du
individual hazard,
• S(t )  E{ e
0
}  L( A(t ))
• The population hazard, λ(t) is related to the
standardized individual hazard α(t) by:
 L / ( A(t ))
(t )   (t )
L( A(t ))
WA(t )
when W = 1
Proportional Frailty Model
• Special Case: Assume W is gamma distributed
with mean 1 and variance δ
 (t )
• Then  (t ) 
1  A(t )
δ=0
δ=0.4
δ=1
δ=3
Time
Relationship
of λ(t) with
α(t) if
α(t) = t2
Relationship of Population and
Individual Hazard Ratios
• Suppose the proportional frailty model holds
in two groups, with gamma frailties with mean
1 and variance δ within each group.
• If the individual hazards in 2 groups are
W1rα(t) and W2α(t), so the individual HR is r,
then
Attenuates
2 (t )
1  A(t )
r
1(t )
1  rA(t )
towards 1 as t
increases
Attenuation in Population HR Due to
Frailty Variance
δ=0
Assumes
r = 4,
α(t) = 0.5
(i.e., ~40%
with events
δ=0.40 in 1 yr)
δ=1.00
δ=3.00
Impact of Immediate Attenuation of
Patient-Level HR to 1 Under Frailty Variation
Assumes
1) δ=1
2) r(t)=0.5 for
first year,
r(t) = 1 after
1 year
3) α(t) = 1
Competing Risks and False Protectivity
Differential Survival Bias
WB αB(t)
Alive
without
Event
WC αC(t)
Event B WB= Y0 + Y2 Y0, Y2, Y3
independent
gammas with
mean 1 and
variances δ0, δ2,
Event C WC= Y0 + Y3 and δ3


1
1

Then B (t )   B (t )

 2  4 2 AB (t ) 2  4 0 ( AB (t )  AC (t ) 
A treatment which has no effect on αB(t) but increases αC(t)
will have lower λB, and thus appear protective for event B.
Practical Implications
• Variation in population hazards ratio over time
depends both on variation in individual
hazards ratio and to frailty selection
• It is often found that HRs attenuate towards 1
or reverse over time, or among “survivors”
who reach an advance stage of a chronic
disease (e.g., those with end stage renal
disease)
Relative Mortality for Norwegian Men
1901-1905 compared to 1991
Horizontal line indicates relative risk of 1
From “Statistics Norway” as reproduced in Aalen O, Survival and Event History Analysis
Practical Implications
• Prevailing practice is to avoid covariate
adjustment for survival outcomes in RCTs.
• But adjustment for strong prognostic factors
can:
– Reduce conservative bias in estimated treatment
effect
– Increase power
– Reduce differential survival bias
Practical Implications
• Three approaches to estimating effects on
individual hazards:
– Analysis of repeat event data
– Joint analysis of longitudinal and time-to-event
outcomes
– Use background knowledge to posit assumptions for
frailty distribution and/or on the mathematical form
of α(t)
Evaluation of Proportional Hazards
• Parametric models for change in hazard ratios
over time
• Non-parametric smooths of Schoenfeld
residuals
• Non-parametric models for multiplicative
hazards
Parametric models for change in
hazard ratios over time
Example:
• Cox Regression of Effects of Dose Group
(Ktv_grp) and baseline serum albumin (Balb)
in the HEMO Study
• RCT with 871 deaths in 1871 patients; planned
follow-up 1.5 to 7 years.
Parametric models for change in
hazard ratios over time
1) Test for linear interactions of predictors with follow-up time
proc phreg data=demsum01 ;
model fu_yr * ev_d(0) = ktv_grp balb ktv_grpt balbt;
ktv_grpt = ktv_grp*fu_yr;
balbt = balb*fu_yr;
Parameter
DF
Parameter
Estimate
KTV_GRP
BALB
Ktv_grpt
Balbt
1
1
1
1
-0.06933
-1.50469
0.00686
0.16874
Standard
Error
Chi-Square
Pr > ChiSq
0.12112
0.17279
0.04406
0.06378
0.3277
75.8320
0.0243
7.0001
0.5670
<.0001
0.8762
0.0082
HR for baseline albumin (in g/dL) is 0.22 at time 0, but attenuates by a factor of
exp(0.1687) = 1.184 per year
Parametric models for change in
hazard ratios over time
2) Test for interactions of predictors with time period (> 1 yr vs. < 1yr)
proc phreg data=demsum01 ;
model fu_yr * ev_d(0) = Ktv_grp1 Balb1 Ktv_grp2 Balb2;
if fu_yr > 1 then period =1;
if . < fu_yr <= 1 then period = 0;
Ktv_grp1 = ktv_grp*(1-period);
Balb1 = balb*(1-period);
Ktv_grp2 = ktv_grp*period;
Balb2 = balb*period;
PropHazKtv: test Ktv_grp1=Ktv_grp2;
PropHazBalb: test Balb1 = Balb2;
Parametric models for change in
hazard ratios over time
2) Test for interactions of predictors with time period (> 1 yr vs. < 1yr)
Parameter
DF
Parameter
Estimate
Ktv_grp1
Balb1
Ktv_grp2
Balb2
1
1
1
1
-0.07569
-1.58168
-0.04680
-0.95959
Label
PropHazKtv
PropHazBalb
Standard
Error
Chi-Square
Pr > ChiSq
0.13413
0.18855
0.07861
0.11553
0.3185
70.3661
0.3545
68.9860
0.5725
<.0001
0.5516
<.0001
Wald
Chi-Square
DF
Pr > ChiSq
0.0345
7.9139
1
1
0.8526
0.0049
Plots of Schoenfeld Residuals
R Code:
require(survival)
HEMOCox<-coxph(Surv(fu_yr,EV_D) ~ KTV_GRP+BALB,data=hemodat)
HEMOPropchk<-cox.zph(HEMOCox)
HEMOPropchk
rho
KTV_GRP 0.0049
BALB
0.0968
GLOBAL
NA
chisq
p
0.0209 0.88510
8.2343 0.00411
8.2570 0.01611
plot(HEMOPropchk,var="KTV_GRP")
plot(HEMOPropchk,var="BALB")
plot(HEMOPropchk,var=“KTV_GRP",resid=FALSE)
plot(HEMOPropchk,var="BALB",resid=FALSE)
Schoenfeld Residual Plots with Cubic Spline Smooths: R Output
For KTV_GRP
For Baseline Albumin
Schoenfeld Residual Plots with Cubic Spline Smooths: R Output
(Omitting the residuals)
For KTV_GRP
For Baseline Albumin
Multiplicative Hazards with Time
Varying Coefficients
• Standard Cox Proportional Hazards Model
λ(t|Z) = λ0 (t) × exp(β Z)
• Cox Proportional Hazards Model with Time-Dependent
Covariates
λ(t|Z) = λ0 (t) × exp(β Z(t))
• Multiplicative Hazards Model with Fixed Covariates and
Time-Varying Coefficients
λ(t|Z) = λ0 (t) × exp(β(t) Z)
• Multiplicative Hazards Model with Time-Dependent
Covariates and Time-Varying Coefficients
λ(t|Z) = λ0 (t) × exp(β(t) Z(t))
Multiplicative Hazards Model with Fixed
Covariates and Time-Varying Coefficients
• Model
λ(t|Z) = λ0 (t) × exp(β(t) Z)
• Useful if proportional hazards assumption in doubt,
and you don’t want to assume a particular
parametric model for change in HR over time
• Estimands are cumulative Cox regression coefficients
t
B j (t )    j (u )du
0
• Can use timereg package in R (if you are careful to
center predictor variables)
Multiplicative Hazards Model with Fixed
Covariates and Time-Varying Coefficients
summary(fit)
Multiplicative Hazard Model
Test for nonparametric terms
Test for non-significant effects
Supremum-test of significance p-value H_0: B(t)=0
cKTV_GRP
1.38
0.931
cBALB
11.30
0.000
Test for time invariant effects
Kolmogorov-Smirnov test p-value H_0:constant effect
cKTV_GRP
0.163
0.986
cBALB
0.833
0.038
Cramer von Mises test p-value H_0:constant effect
cKTV_GRP
0.015
0.993
cBALB
1.040
0.026
Multiplicative Hazards Model with Fixed
Covariates and Time-Varying Coefficients
R Code:
>
>
>
>
>
>
>
cBALB<-BALB – mean(BALB)
cKTV_GRP <- KTV_GRP – mean(KTV_GRP)
require(timereg)
fit<-timecox(Surv(fu_yr,EV_D)~KTV_GRP+cBALB,max.time=5)
summary(fit)
par(mfrow(1,2)
plot(fit,c(2,3))
Multiplicative Hazards Model with Fixed
Covariates and Time-Varying Coefficients
Next Week
Alternatives to the Cox Model
• Aalen’s Additive Hazards Model
• Accelerated Failure Time Models
• Proportional Odds Models for Survival Data