June 25, 2006 Propensity Score Adjustment in Survival Models Carolyn Rutter Group Health Cooperative [email protected] AcademyHealth, Seattle WA.

June 25, 2006

Propensity Score Adjustmentin Survival Models

Carolyn RutterGroup Health Cooperative

[email protected]

AcademyHealth, Seattle WA

June 25, 2006

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

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June 25, 2006

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.

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June 25, 2006

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.

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June 25, 2006

Depression & Mortality among Type 2 Diabetics

Depression is common in patients with type 2 diabetes11% 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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June 25, 2006

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 depression3749 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.

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Depression & Mortality among Type 2 Diabetics

June 25, 2006

3 year Mortality OutcomeAll-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 timeHazard function: instantaneous event rate

Cox proportionalhazards model

)exp()()( Ztt 0

UnspecifiedBaseline hazard

)()(

)(/)(

)(tStf

tSttS

t

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June 25, 2006

PH Model Results

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* 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)

Method Estimate

Se(estimate)

HR P-value

Unadjusted

0.34 0.14 1.40 <0.02

Minimum Adjustment*

0.77 0.14 2.16 <0.001

Full Adjustment†

0.26 0.16 1.30 0.09

June 25, 2006

ZDepression

XSelf Care

Disease Severity

Age, Sex

Education

YDeath

mediator

common cause

June 25, 2006

Z X Y

mediator

common cause

June 25, 2006

Propensity Score Adjustment:

3-Step Process1. Estimate propensity score 2. Evaluate covariate balance given

propensity scores3. Incorporate propensity score in analyses

to ‘synthetically balance’ the sample• Stratification• Regression• Matching• Weighting

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June 25, 2006

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

the estimated propensity score for the ith individuali

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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

Propensity score missing for 6.6%

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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 + missing value indicators /outroc=roc;

run;

Estimated AUC=0.72

None missing

propensity score

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June 25, 2006

Propensity Strata

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Strata1

Strata2

Strata3

Strata4

Strata5

Strata6

Strata7

NotDepressed

82622%

79421%

77521%

73620%

3399%

1464%

1334%

374988%

Depressed

275%

5811%

7815%

11623%

8717%

6713%

8016%

51312%

Total 85320%

85220%

85320%

85220%

42610%

2135%

2135%

4262

June 25, 2006

Step 2: Check covariate balance

Strata Not depressed

Depressed

N

all 27.1 44.3 4262

1 4.8 7.4 853

2 16.2 12.1 852

3 23.1 26.9 853

4 39.4 41.4 852

5 51.6 51.7 426

6 66.4 67.2 213

7 78.2 73.5 213

Percent Sedentary

June 25, 2006

Step 3: Incorporate Propensity Scores into Proportional Hazards

Model

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,

)exp()()( 0 Ztt

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June 25, 2006

Regression-adjustment in the PH model

Assume proportionality: check this assumption usingShoenfeld residuals.

)ˆexp()()( 0 Ztt

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iiiZ ZZr ˆ

i

i

Rjjj

Rjjjj

i Z

ZZ

Z)ˆexp(

)ˆexp(

iiir ˆˆˆ

i

i

Rjjj

Rjjjj

i Z

Z

)ˆexp(

)ˆexp(ˆ

ˆ

June 25, 2006

Schoenfeld ResidualsLittle evidence for non-proportional hazards in propensity

scores.

Correlation between Schoenfeld-residual and rank-time

Depression: 0.02 Propensity: -0.06

June 25, 2006

PH Model Results

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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

June 25, 2006

Stratification-adjustment in the PH model

Stratified likelihood

j : censoring indicator (1 if death obs)

)exp()()( Ztt m0 mth strata

M

m Sj Rk k

jjM

mm

mj

z

zLL

11 )exp(

)exp()()(

AcademyHealth, Seattle WA

June 25, 2006

PH Model Results

AcademyHealth, Seattle WA

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

0.24 0.14 1.27 0.10

June 25, 2006

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.

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June 25, 2006

Matching-adjustment in the PH model

only 2/513 depressed excluded

Within each matched pair, only the first death contributes to the likelihood leading to additional loss of information.

)exp()()( Ztt m0 mth pair

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M

m Sj Rk k

jjM

mm

mj

z

zLL

11 )exp(

)exp()()(

June 25, 2006

PH model results

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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

)exp()()( Ztt 0

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Weighting-adjustment in the PH model (IPW)

Weighted partial Likelihood Function

N

i Rj jj

iii

izw

zwL

1 )exp(

)exp()(

up-weight individuals with ‘unexpected’ exposure

1)ˆ1)(1(ˆ iiiii zzw

Limits optionsfor handling ties

Performs best when weights are estimated (Qi, Wang, Prentice, JASA ,2005)

June 25, 2006

PH model results

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Method Estimate

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

Z X Y

Covariatemodels

Estimate the effect of Z on Y conditional on X

June 25, 2006

IPW does not depend on estimating effects of Y | (Z and X)

Z X Y

Covariatemodels

Propensity

Synthetically

balancesX across

Z

Propensity models: P(Z|X)

June 25, 2006

Combined Adjustments

Regression adjust and weight.

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)ˆexp()()( 0 Ztt

June 25, 2006

PH Model Results

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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

Doubly Robust

RegressionModel True

Propensity Model True

No Yes

No

Yes

An approach that is robust to misspecification of theregression model OR the propensity model.

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June 25, 2006

Doubly Robust EstimatorsIdea: weighted estimators use only

observed outcomes. DR estimators incorporate unobserved outcomes through their expected values. Increase efficiency, increase robustness

Adjusted Score Function:

iindicates observing the ‘assigned’ (patient selected) treatment

weighted score

01

1

01

1

n

i Zi

i

ii

Rj jjj

Rj jjjjiiiiA w

wxzw

xzzwzwU

i

i

)exp(

)exp()(

June 25, 2006

Score Adjustment, i

i is an augmentation term that is a function of the regression model, M(Y|X, ) where Y=(, T):

),,()exp(

)exp(

XYM

xz

xzzzE

i

i

Rj jj

Rj jjj

iii

01

1

01

1

n

i Zi

i

ii

Rj jjj

Rj jjjjiiiiA w

wxzw

xzzwzwU

i

i

)exp(

)exp()(

June 25, 2006

Doubly Robust Estimator

01

1

01

1

n

i Zi

i

ii

Rj jjj

Rj jjjjiiiiA w

wxzw

xzzwzwU

i

i

)exp(

)exp()(

0 all

iobserved

iiiiiiiA zzEwzzwU )()()(

Expected value is 0 if regression model is true

Expected value is 0 if propensity model is true

June 25, 2006

Doubly Robust Estimates

Can calculate DR estimates iteratively: 1. Calculate starting values using PH2. Estimate i via simulation given M(Y|X, ) and current

parameter estimates, including baseline hazard (e.g., Nelson-Aalen estimators)

1. Use Newton-Raphson to solve the adjusted score for

01

1

01

1

n

i Zi

i

ii

Rj jjj

Rj jjjjiiiiA w

wxzw

xzzwzwU

i

i

)exp(

)exp()(

m

k ikik

ikiki

m

k jkikjkik

jkikiki

BAe

Aez

m

xzRxzRe

xzRez

m

1**

**

1**

**

1ˆ

)ˆexp()1()ˆexp(

)ˆexp(1ˆ

+ TS approx

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

DRAcademyHealth, Seattle WA

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

• 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

regression-calibration 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 ModelsAdditional research:

• More than two treatment/exposure groupsLeon 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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