Outline
Propensity Score Adjustment
in Survival Models
Carolyn Rutter
Group Health Cooperative
rutter.c@ghc.org
AcademyHealth, Seattle WA
June 25, 2006
• 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?
Propensity score: the probability that a person receives treatment,
or is exposed, given a set of observed covariates, X.
Underlying question:
Does depression increase the risk of death ?
Estimate the causal effect of treatment on response
exposure
outcome
Z
Y
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June 25, 2006
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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Depression & Mortality among Type 2 Diabetics
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
Propensity Scores
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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
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1
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
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
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.
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June 25, 2006
Deaths over a 3-year period:
336 ( 9.0%) in 3749 patients without major depression
60 (11.7%) in 497 patients with major depression
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Proportional Hazards Model
PH Model Results
Method
Survivor function: S(t) = Pr(T*>t)=1-F(t)
T* event time
Hazard function: instantaneous event rate
λ (t ) =
f (t )
− ∂S (t ) / ∂t
=
S (t )
S (t )
Se(estimate) HR
P-value
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)
Cox proportional λ (t ) = λ (t ) exp(Zβ )
0
hazards model
Unspecified
Baseline
hazard
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Estimate
Unadjusted 0.34
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June 25, 2006
mediator
Z
Depression
X
Self Care
Disease Severity
Y
Death
Z
X
Y
mediator
common cause
Education
Age, Sex
common cause
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2
Step 1: Estimate propensity scores
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
•
•
•
•
Stratification
Regression
Matching
Weighting
Use logistic regression (or other method, e.g., CART) to estimate
P(Z=1|X) = πi, propensity score
logit(Z) =Xα
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
πˆ i the estimated propensity score for the ith individual
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June 25, 2006
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
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
Propensity
score missing for
6.6%
None missing
propensity score
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June 25, 2006
Step 2: Check covariate balance
Percent Sedentary
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
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Propensity Strata
Total
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Strata Strata Strata
5
6
7
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all
1
2
3
4
5
6
7
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
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Step 3: Incorporate Propensity Scores into
Proportional Hazards Model
Regression-adjustment in the PH model
λ (t ) = λ0 (t ) exp( Zβ )
λ (t ) = λ0 (t ) exp(Zβ + πˆθ )
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,
Assume proportionality: check this assumption using
Shoenfeld residuals.
Z exp( Z β + πˆ β )
Zi =
rˆiZ = Zi − Zi
∑
j
j∈Ri
j
∑ exp(Z β + πˆ β )
j
j∈Ri
rˆiπ = πˆi −πˆi
πˆi =
∑ πˆ
j∈Ri
j
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j
exp(Z j β + πˆ j β )
∑ exp(Z β + πˆ β )
j∈Ri
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j
j
j
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June 25, 2006
Schoenfeld Residuals
PH Model Results
Little evidence for non-proportional hazards in propensity scores.
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
Correlation between Schoenfeld-residual and rank-time
Depression: 0.02
Propensity: -0.06
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PH Model Results
Stratification-adjustment in the PH model
λ (t ) = λ0m (t ) exp(Zβ )
mth strata
Stratified likelihood
⎛ δ exp( z β ) ⎞
j
j
⎟
L( β ) = ∏ Lm (β ) = ∏ ∏ ⎜
⎜
⎟
exp(
z
β
)
m =1
m =1 j∈S m ∑k∈R
k
j
⎝
⎠
M
M
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
0.24
δj : censoring indicator (1 if death obs)
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June 25, 2006
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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(Xα) 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.
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June 25, 2006
Matching-adjustment in the PH model
λ (t ) = λ0m (t ) exp(Zβ )
mth pair
⇒ only 2/513 depressed excluded
⎛ δ exp( z β ) ⎞
M
M
j
j
⎟
L( β ) = ∏ Lm (β ) = ∏ ∏ ⎜⎜
⎟
m =1 j∈S m ∑ k∈R exp( z k β )
m =1
j
⎝
⎠
Within each matched pair, only the first death contributes to the
likelihood leading to additional loss of information.
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June 25, 2006
Weighting-adjustment in the PH model (IPW)
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
λ (t ) = λ0 (t ) exp(Zβ )
Weighted partial Likelihood Function
N ⎛
δ i wi exp( z i β ) ⎞⎟
Limits options
L( β ) = ∏ ⎜
for handling ties
⎜
⎟
w
exp(
z
β
)
i =1 ∑ j∈R
j
j
⎝
⎠
i
up-weight individuals with ‘unexpected’ exposure
wi = [z i πˆ i + (1 − z i )(1 − πˆ i )]
−1
Performs best when weights are estimated
(Qi, Wang, Prentice, JASA ,2005)
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June 25, 2006
AcademyHealth, Seattle WA
June 25, 2006
PH model results
Se(estimate) HR
P-value
Unadjusted 0.34
Method
Estimate
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
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Z
Covariate
models
X
Y
Estimate the effect of Z on Y conditional on X
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5
Combined Adjustments
λ (t ) = λ0 (t ) exp(Zβ + πˆθ )
Covariate
models
Z
Y
X
Regression adjust and weight.
Propensity
Synthetically
balances
X across Z
Propensity models: P(Z|X)
IPW does not depend on estimating effects of Y | (Z and X)
June 25, 2006
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June 25, 2006
PH Model Results
Doubly Robust
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
0.14
1.27
0.10
0.24
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
Propensity Model True
No
Yes
No
Regression
Model True
Yes
An approach that is robust to misspecification of the
regression model OR the propensity model.
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June 25, 2006
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June 25, 2006
Score Adjustment, ϕi
Doubly Robust Estimators
Idea: weighted estimators use only observed outcomes.
DR estimators incorporate unobserved outcomes
through their expected values.
⇒ Increase efficiency, increase robustness
n
1
⎛
⎜
⎝
U A (β ) = ∑ ∑ Δiw i δ i ⎜ z i −
i =1 Z = 0
∑ j ∈R w j z j exp(z j β + x j γ ) ⎞⎟ ⎛ Δ i − w i−1 ⎞
⎟ϕ i = 0
−⎜
∑ j ∈R w j exp(z j β + x j γ ) ⎟⎠ ⎜⎝ w i−1 ⎟⎠
i
i
ϕi is an augmentation term that is a function of the
regression model, M(Y|X, β,γ) where Y=(δ, T):
Adjusted Score Function:
n
1
⎛
⎜
⎝
U A (β ) = ∑ ∑ Δiw i δ i ⎜ z i −
i =1 Z = 0
∑ j ∈R w j z j exp(z j β + x j γ ) ⎞⎟ ⎛ Δ i − w i−1 ⎞
⎟ϕ i = 0
−⎜
∑ j ∈R w j exp(z j β + x j γ ) ⎟⎠ ⎜⎝ w i−1 ⎟⎠
i
i
⎡ ⎛
ϕi = E⎢δi ⎜ zi −
⎢ ⎜
⎣ ⎝
weighted score
∑
∑
⎤
z j exp(z j β + x jγ ) ⎞
⎟ M (Y X , β , γ )⎥
⎟
⎥
exp(z j β + x jγ )
j∈Ri
⎠
⎦
j∈Ri
Δ i indicates observing the ‘assigned’
(patient selected) treatment
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6
Doubly Robust Estimator
Doubly Robust Estimates
n
1
⎛
⎜
⎝
U A ( β ) = ∑ ∑ Δ iw i δ i ⎜ z i −
⎛
∑ w z exp(z j β + x j γ ) ⎞⎟ ⎛ Δ i − w i−1 ⎞
⎟⎟ϕ i = 0
U A ( β ) = ∑ ∑ Δ i w i δ i ⎜ z i − j ∈R j j
−⎜
−1
⎜
i =1 Z = 0
∑ j ∈R w j exp(z j β + x jγ ) ⎟⎠ ⎜⎝ w i
⎠
⎝
n
i =1 Z = 0
1
∑ j ∈R w j z j exp(z j β + x jγ ) ⎞⎟ ⎛ Δ i − w i−1 ⎞
−⎜
−1
⎟⎟ϕ i = 0
∑ j ∈R w j exp(z j β + x j γ ) ⎟⎠ ⎜⎝ w i
⎠
i
i
i
i
Expected value is 0
if propensity model is true
Can calculate DR estimates iteratively:
1. Calculate starting values using PH
2. Estimate ϕi via simulation given M(Y|X, β,γ) and current parameter
estimates, including baseline hazard (e.g., Nelson-Aalen estimators)
U A (β ) = ∑{wi δi (z i − z i (wi ) − E (z i − z i ))} + ∑ϕi = 0
observed
all
Expected value is 0
if regression model is true
June 25, 2006
1.
ϕˆi =
⎞
eβ ∑ Rik* zk exp(x jγˆ)
1 m * ⎛⎜
⎟
δk ⎜ zi − β
∑
*
m k =1 ⎝
e ∑ Rik zk exp(x jγˆ) + ∑ Rik* (1− zk ) exp(x jγˆ) ⎟⎠
ϕˆi =
1 m *⎛
eβ A* ⎞
δk ⎜⎜ zi − β * ik * ⎟⎟
∑
e Aik + Bik ⎠
m k =1 ⎝
+ TS approx
Use Newton-Raphson to solve the adjusted score for β & γ
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
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June 25, 2006
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
Propensity Adjustment
for Survival Models: Recent Work
• 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)
• 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.
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7
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
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