Propensity Score Adjustment
in Survival Models
Carolyn Rutter
Group Health Cooperative
rutter.c@ghc.org
June 25, 2006
AcademyHealth, Seattle WA
Outline
• Propensity Scores: General Ideas
• Background: depression & mortality among
type 2 diabetics
• Propensity Scores applied to depression & mortality
June 25, 2006
Example: Is depression associated with increased
mortality in type 2 diabetics?
Underlying question:
Does depression increase the risk of death ?
Estimate the causal effect of treatment on response
exposure
outcome
Z
Y
June 25, 2006
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Propensity Scores
Propensity score: the probability that a person receives treatment,
or is exposed, given a set of observed covariates, X.
Randomized Study: P(Tx)=0.5, the propensity score is
independent of patient characteristics and the distribution of
P(Tx) is the same across treatment groups.
Observational Study: P(Tx|X) depends on patient
characteristics and differs between treatment groups (because
Tx is associated with covariates), so that the treated group has
a higher propensity for treatment than the untreated group.
June 25, 2006
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Basic Ideas behind
Propensity Score Methods
Reduce bias by comparing treated and untreated individuals
who have the same propensity for treatment/exposure
Key assumption: Strongly Ignorable Treatment Assignment
The outcome is conditionally independent of treatment
assignment given observed covariates
Y  P(Z|X)
After adjusting for observed covariates, treatment assignment
doesn’t inform the response.
No unmeasured confounders.
June 25, 2006
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Depression & Mortality among Type 2 Diabetics
Depression is common in patients with type 2 diabetes
11% to 15% meet criteria for major depression
Depressed diabetic patients tend to have
– poorer self-management (diet, exercise, blood glucose checks)
– more lapses in refilling prescribed medications
(oral hypoglycemics, lipid lowering, anti-hypertensive)
– have cardiac risk factors (smoking, obesity, sedentary lifestyle)
Studies have linked depression to increased mortality among
diabetics, but these used a small number of patients, with
medical diagnoses based on self report
June 25, 2006
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Depression & Mortality among Type 2 Diabetics
The Pathways Study: a population-based epidemiologic
study of over 4000 patients with diabetes enrolled in an
HMO.
4262* included in following analyses
513 with major depression
3749 without major depression
Katon, Rutter, Simon et al “The association of comorbid
depression with mortality in patients with type 2 diabetes.”
Diabetes Care. 2005 Nov; 28(11):2668-72.
June 25, 2006
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3 year Mortality Outcome
All-cause mortality: May 2001(start recruitment) – May 2004
5/1/2001 – 12/31/2003 (first 31 months):
GHC automated health care records + Washington State mortality data
90% of deaths in the State mortality data were in GHC records
1/1/2004 – 4/30/2004 (last 5 months):
GHC data alone.
Censoring at the end of the study or disenrollment
Deaths over a 3-year period:
336 ( 9.0%) in 3749 patients without major depression
60 (11.7%) in 497 patients with major depression
June 25, 2006
Proportional Hazards Model
Survivor function: S(t) = Pr(T*>t)=1-F(t)
T* event time
Hazard function: instantaneous event rate
 S (t ) / t
f (t )
 (t ) 

S (t )
S (t )
Cox proportional  (t )   (t ) exp(Z )
0
hazards model
Unspecified
Baseline
hazard
June 25, 2006
AcademyHealth, Seattle WA
PH Model Results
Method
Estimate
Se(estimate) HR
P-value
Unadjusted 0.34
0.14
1.40
<0.02
Minimum
0.77
Adjustment*
0.14
2.16
<0.001
Full
0.26
Adjustment†
0.16
1.30
0.09
* Known
confounders: gender, age, race/ethnicity, education
† Potential behavioral and disease severity confounders &/or mediators:
BMI, current smoker, sedentary lifestyle, HbA1c,
use of oral hypoglycemics, use of insulin, complications of diabetes,
(pharmacy-based) comorbidity measure (excluding depression meds)
June 25, 2006
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mediator
Z
Depression
X
Self Care
Disease Severity
Education
Age, Sex
common cause
June 25, 2006
Y
Death
Z
X
mediator
common cause
June 25, 2006
Y
Propensity Score Adjustment:
3-Step Process
1. Estimate propensity score
2. Evaluate covariate balance given propensity scores
3. Incorporate propensity score in analyses to
‘synthetically balance’ the sample
•
•
•
•
June 25, 2006
Stratification
Regression
Matching
Weighting
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Step 1: Estimate propensity scores
Use logistic regression (or other method, e.g., CART) to estimate
P(Z=1|X) = i, propensity score
logit(Z) =X
Focus is on prediction rather than estimation.
– Include all potential confounders, but leave out factors related only to
the exposure or outcome (Brookhart et al, 2006, AJE)
– Include interaction effects as needed
– ROC curve can be used to evaluate fit, but doesn’t provide insight
about appropriate covariates
ˆ i the estimated propensity score for the ith individual
June 25, 2006
AcademyHealth, Seattle WA
Step 1: Estimate propensity for depression
proc logistic descending;
model major=age male smoke obese somecoll
sedentary cardio outofcontrol treatint rxrisk2
/outroc=roc;
run;
Estimated
AUC=0.72
Propensity
score missing for
6.6%
June 25, 2006
AcademyHealth, Seattle WA
Step 1: Estimate propensity for depression
proc logistic descending;
model major=age male smoke obese somecoll
sedentary cardio outofcontrol treatint rxrisk2
+ missing value indicators /outroc=roc;
run;
Estimated
AUC=0.72
None missing
propensity score
June 25, 2006
AcademyHealth, Seattle WA
Propensity Strata
Strata
1
Strata
2
Strata
3
Strata
4
Not
Depressed
826
22%
794
21%
775
21%
736
20%
339
9%
146
4%
133
4%
3749
88%
Depressed
27
5%
58
11%
78
15%
116
23%
87
17%
67
13%
80
16%
513
12%
853
20%
852
20%
853
20%
852
20%
426
10%
213
5%
213
5%
4262
Total
June 25, 2006
Strata Strata Strata
5
6
7
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Step 2: Check covariate balance
Percent Sedentary
Strata
all
1
2
3
4
5
6
7
June 25, 2006
Not depressed Depressed N
27.1
4.8
16.2
23.1
39.4
51.6
66.4
78.2
44.3
7.4
12.1
26.9
41.4
51.7
67.2
73.5
4262
853
852
853
852
426
213
213
Step 3: Incorporate Propensity Scores into
Proportional Hazards Model
 (t )  0 (t ) exp( Z )
1. Regression: Proportional hazards across different
levels of the propensity score
2. Stratification: Allow different baseline hazards
across propensity strata
3. Matching: Allow different baseline hazards for
each matched pair
4. Weighting: Assume a common baseline hazard,
June 25, 2006
AcademyHealth, Seattle WA
Regression-adjustment in the PH model
 (t )  0 (t ) exp( Z  ˆ )
Assume proportionality: check this assumption using
Shoenfeld residuals.
ˆ
rˆiZ  Z i  Z i
Zi 
Z
jRi
j
 exp( Z   ˆ  )
j
jRi
rˆi  ˆ i  ˆ i
ˆ i 
 ˆ
jRi
j
j
exp( Z j   ˆ j  )
 exp( Z   ˆ  )
jRi
June 25, 2006
exp( Z j    j  )
j
j
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Schoenfeld Residuals
Little evidence for non-proportional hazards in propensity scores.
Correlation between Schoenfeld-residual and rank-time
Depression: 0.02
June 25, 2006
Propensity: -0.06
PH Model Results
Method
Estimate
Se(estimate) HR
P-value
Min Adj
0.77
0.14
2.16
<0.001
Full Adj
0.26
Regression 0.25
0.16
0.14
1.30
1.28
0.09
0.08
June 25, 2006
AcademyHealth, Seattle WA
Stratification-adjustment in the PH model
 (t )  0m (t ) exp(Z )
mth strata
Stratified likelihood
  exp( z  ) 
j
j


L(  )   Lm ( )    

exp(
z

)
m 1
m 1 jS m kR
k
j


M
M
j : censoring indicator (1 if death obs)
June 25, 2006
AcademyHealth, Seattle WA
PH Model Results
Method
Estimate
Se(estimate) HR
P-value
Min Adj
0.77
0.14
2.16
<0.001
Fully Adj
0.26
0.16
1.30
0.09
Regression 0.25
0.14
1.28
0.08
Stratified
0.14
1.27
0.10
June 25, 2006
0.24
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Matched Propensity Score Analysis
1. Use the full sample to estimate propensity scores
2. Identify matched pairs based on linear predictor
from the propensity model. Matching within
±0.25*SE(X) is recommended by Rosenbaum &
Rubin (1983, 1985)
3. Assess matching: differences between matched and
unmatched individuals; balance within matched
sample.
4. Analyze data, accounting for matching.
June 25, 2006
AcademyHealth, Seattle WA
Matching-adjustment in the PH model
 (t )  0m (t ) exp(Z )
mth pair
 only 2/513 depressed excluded
  exp( z  ) 
j
j


L(  )   Lm ( )    

exp(
z

)
m 1
m 1 jS m kR
k
j


M
M
Within each matched pair, only the first death contributes to the
likelihood leading to additional loss of information.
June 25, 2006
AcademyHealth, Seattle WA
PH model results
Method
Estimate
Se(estimate) HR
P-value
Min Adj
0.77
0.14
2.16
<0.001
Full Adj
0.26
0.16
1.30
0.09
Regression 0.25
0.14
1.28
0.08
Stratified
0.24
0.14
1.27
0.10
Matching
0.26
0.21
1.30
0.21
June 25, 2006
AcademyHealth, Seattle WA
Weighting-adjustment in the PH model (IPW)
 (t )  0 (t ) exp(Z )
Weighted partial Likelihood Function
  w exp( z  ) 
i i
i

L(  )   
  w j exp( z j  ) 
i 1
 jRi

N
Limits options
for handling ties
up-weight individuals with ‘unexpected’ exposure
wi  ziˆ i  (1  zi )(1  ˆ i )
1
Performs best when weights are estimated
(Qi, Wang, Prentice, JASA ,2005)
June 25, 2006
AcademyHealth, Seattle WA
PH model results
Method
Se(estimate) HR
P-value
Unadjusted 0.34
0.14
1.40
<0.02
Min Adj
0.77
0.14
2.16
<0.001
Full Adj
0.26
0.16
1.30
0.09
Regression 0.25
0.14
1.28
0.08
Stratified
0.24
0.14
1.27
0.10
Matching
0.26
0.21
1.30
0.21
IPW
0.36
0.09
1.43
<0.005
June 25, 2006
Estimate
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Z
Covariate
models
X
Estimate the effect of Z on Y conditional on X
June 25, 2006
Y
Covariate
models
Z
X
Propensity
Syntheticall
y
balances
X across Z
Y
Propensity models: P(Z|X)
IPW does not depend on estimating effects of Y | (Z and X)
June 25, 2006
Combined Adjustments
 (t )  0 (t ) exp( Z  ˆ )
Regression adjust and weight.
June 25, 2006
AcademyHealth, Seattle WA
PH Model Results
Method
Estimate
Se(estimate) HR
P-value
Min Adj
0.77
0.14
2.16
<0.001
Full Adj
0.26
0.16
1.30
0.09
Regression 0.25
0.14
1.28
0.08
Stratified
0.24
0.14
1.27
0.10
Matching
0.26
0.21
1.30
0.21
IPW
0.36
0.09
1.43
<0.005
IPW+Reg
0.36
0.09
1.43
<0.005
June 25, 2006
AcademyHealth, Seattle WA
Doubly Robust
Propensity Model True
No
Regression
Model True
Yes
No
Yes
An approach that is robust to misspecification of the
regression model OR the propensity model.
June 25, 2006
AcademyHealth, Seattle WA
Doubly Robust Estimators
Idea: weighted estimators use only observed outcomes.
DR estimators incorporate unobserved outcomes
through their expected values.
 Increase efficiency, increase robustness
Adjusted Score Function:

   i  w i1 

j R w j z j exp(z j   x j  )

i  0
U A (  )    i w i  i  z i 

1


i 1 Z  0
 j R w j exp(z j   x j  )   w i


n
1
i
i
weighted score
 i indicates observing the ‘assigned’
(patient selected) treatment
June 25, 2006
Score Adjustment, i

   i  w i1 

j R w j z j exp(z j   x j  )

i  0
U A (  )    i w i  i  z i 
1



i 1 Z  0
 j R w j exp(z j   x j  )   w i


n
1
i
i
i is an augmentation term that is a function of the
regression model, M(Y|X, ,) where Y=(, T):
 


z
exp(
z


x

)

j
j
j
jRi
 M (Y X ,  ,  )
i  E  i  zi 

 

exp(
z


x

)

j
j
j

R
i

 

June 25, 2006
Doubly Robust Estimator

   i  w i1 

j R w j z j exp(z j   x j  )

i  0
U A (  )    i w i  i  z i 
1



i 1 Z  0
 j R w j exp(z j  x j )   w i


n
1
i
i
Expected value is 0
if propensity model is true
U A (  )   w i  i z i  z i (w i )  E (z i  z i )    i  0
observed
all
Expected value is 0
if regression model is
true
June 25, 2006
Doubly Robust Estimates

   i  w i1 

j R w j z j exp(z j   x j  )

i  0
U A (  )    i w i  i  z i 
1



i 1 Z  0
 j R w j exp(z j  x j )   w i


n
1
i
i
Can calculate DR estimates iteratively:
1. Calculate starting values using PH
2. Estimate i via simulation given M(Y|X, ,) and current parameter
estimates, including baseline hazard (e.g., Nelson-Aalen estimators)

e   Rik* zk exp( x jˆ )
1 m * 

ˆi    k zi  
*
*
m k 1 
e  Rik zk exp( x jˆ )   Rik (1  zk ) exp( x jˆ ) 
e  Aik* 
1 m *

ˆi    k  zi   *
* 
m k 1 
e Aik  Bik 
1.
+ TS approx
Use Newton-Raphson to solve the adjusted score for  & 
June 25, 2006
PH model results
Method
Estimate
Se(estimate) HR
P-value
Min Adj
0.77
0.14
2.16
<0.001
Full Adj
0.26
0.16
1.30
0.09
Regression 0.26
0.16
1.30
0.09
Stratified
0.24
0.14
1.27
0.10
Matching
0.26
0.21
1.30
0.21
IPW
0.36
0.09
1.43
<0.005
DR
June 25, 2006
AcademyHealth, Seattle WA
Propensity Adjustment Compared to
Inclusion of Covariates
• Separate models for treatment assignment and
outcome. Focus on synthetic balance of sample.
• Maintain power while adjusting for many covariates
– Need about 10-15 events per independent variable examined
• Multiple ways to adjust, allowing different
assumptions about proportionality of hazards
• Can no longer make inference about individual
covariates
June 25, 2006
Propensity Adjustment
for Survival Models
• Omitting covariates from PH models may result in
attenuation of estimates for included covariates
(Mitra & Heitjen, Stat in Med, 2006).
• Covariate adjustment in PH model may reduce bias
in estimates of covariate effects (Lagakos & Shoenfeld,
Biometrics, 1984) but has little effect on the variance
of estimates. (Anderson & Flemming, Biometrika, 1995)
June 25, 2006
Propensity Adjustment
for Survival Models: Recent Work
• Sturmer et al. AJE, 2005, Develop a regressioncalibration approach to adjust for error in estimated
propensity scores.
• Mitra & Heitjen, Stat in Med, 2006, develop a
method for determining the effect an umeasured
confounder would need to have to explain observed
differences.
June 25, 2006
Propensity Models
Additional research:
• More than two treatment/exposure groups
Leon AC, Mueller TI, Solomon DA, Keller MB.
2001, Stat Med.
Luellen JK, Shadish WR, & Clark MH. 2005, Evaluation
Review, & references therein
Imbens G. Biometrika, 2000.
• Continuous treatment/exposure measures
June 25, 2006
AcademyHealth, Seattle WA