Novel Approaches to Adjusting for Confounding: Propensity Scores, Instrumental Variables and MSMs...

Novel Approaches to Adjusting for Confounding:

Propensity Scores, Instrumental Variables and MSMs

Matthew Fox

Advanced Epidemiology

What are the exposures you are interested in studying?

Assuming I could guarantee you that you would not create bias,

which approach is better: randomization or adjustment for

every known variable?

What is intention to treat analysis?

Yesterday

Causal diagrams (DAGS)– Discussed rules of DAGS– Goes beyond statistical methods and

forces us to use prior causal knowledge– Teaches us adjustment can CREATE bias

Helps identify a sufficient set of confounders– Not how to adjust for them

This week

Beyond stratification and regression– New approaches to adjusting for (not

“controlling” ) confounding– Instrumental variables– Propensity scores (Confounder scores)– Marginal structural models

Time dependent confounding

Given the problems with the odds ratio, why does everyone use it?

Non-collapsibility of the OR (Excel; SAS)

Odds ratio collapsibility but confounding

  C+   C-   Total

  E+   E-   E+   E-   E+   E-

Disease+ 400   300   240   180   640   480

Disease- 100   200   360   720   460   920

Total 500   500   600   900   1100   1400

Risk 0.80   0.60   0.40   0.20   0.58   0.34

Odds 4.00   1.50   0.67   0.25   1.39   0.52

              Crude   Adj

RR 1.33   2.00   1.6968   1.5496

OR 2.67   2.67   2.67   2.67

RD 0.2 0.2 0.23896 0.2

Solution: SAS Code

title "Crude relative risk model";proc genmod data=odds descending;

model d = e/link=log dist=bin;run;

title "Adjusted relative risk model";proc genmod data=odds descending;

model d = e c/link=log dist=bin;run;

Model crude:

Exp(0.5288) = 1.6968Crude RR was 1.6968

Results

Results

Model adjusted: Exp(0.3794) = 1.461

MH RR was 1.55

STATA

glm d e, family(binomial) link(log) glm d e c, family(binomial) link(log)

What about risk differences?

Solution: SAS Code

title "Crude risk differences model";proc genmod data=odds descending;

model d = e/link=bin dist=identity;run;

title "Adjusted risk differences model";proc genmod data=odds descending;

model d = e c/link=bin dist=identity;run;

Model crude:

Exp(0.5288) = 1.6968Crude RR was 1.7

Results

Model crude: 0.239

Crude RD = 0.23896

Results

Adjusted model : 0.20

MH RD = 0.20

STATA

glm d e, family(binomial) link(identity) glm d e c, family(binomial) link(identity)

glm d e c c*e, family(binomial) link(identity)

Novel approaches to controlling confounding

Limitations of Stratification and Regression

Stratification/regression work well with point exposures with complete follow up and sufficient data to adjust– Limited data on confounders or small cells– No counterfactual for some people in our dataset

Regression often estimates parameters– Time dependent exposures and confounding

A common situationWith time dependence, DAGs gets complex

Randomization and Counterfactuals

Ideally, evidence comes from RCTs– Randomization gives expectation unexposed can

stand in for the counterfactual ideal Full exchangeability: E(p1=q1, p2=q2, p3=q3, p4=q4)

– In expectation, assuming no other bias [Pr(Ya=1=1) - Pr(Ya=0=1)] = [Pr(Y=1|A=1) - Pr(Y=1|A=0)]

Since we assign A, RRAC = 1– If we can’t randomize, what can we do to

approximate randomization?

How randomization works

Randomized Controlled Trial

Randomization strongly predicts exposure (ITT)

C2

C1

C3

A DRandomization

A typical observational study

Observational Study

C2

C1

C3

A D?

A typical observational study

Observational Study

C2

C1

C3

A D

Regression/stratification seeks to block backdoor path from A to D by averaging A-D associations within levels of Cx

Approach 1: Instrumental Variables

Intention to treat analysis

In an RCT we assign the exposure– e.g. assign people to take an aspirin a day vs. not– But not all will take aspirin when told to and others

will take it even if told not to

What to do with those who don’t “obey”?– The paradigm of intention to treat analysis says

analyze subject in the group they are assigned Maintains the benefits of randomization Biases towards the null at worst

Instrumental variables

An approach to dealing with confounding using a single variable– Works along the same lines as randomization

Commonly used approach in economics, yet rarely used in medical research– Suggests we are either behind the times or they are

hard to find– Party privileged in economics because little

adjustment data exists

Instrumental variables

An instrument (I):– A variable that satisfies 3 conditions:

Strongly associated with exposureHas no effect on outcome except through A (E)Shares no common causes with outcome

Ignore E-D relationship– Measure association between I and D

This is not confounded

– Approximates an ITT approach

Adjust the IV estimate

Can optionally adjust IV estimate to estimate the effect of A (exposure)– But differs from randomization

If an instrument can be found, has the advantage we can adjust for unknown confounders– This is the benefit we get from randomization?

Intention to Treat (IV Ex 1)

A(Exposure): Aspirin vs. Placebo Outcome: First MI Instrument: Randomized assignment

Therapy MI

Confounders

Randomization

Condition 1: Predictor of A ?

Condition 2: no direct effect on the outcome?

Condition 3: No common causes with outcome?

Confounding by indication (IV Ex 2)

A(Exposure): COX2 inhibitor vs NSAID Outcome: GI complications Instrument: Physician’s previous prescription

COX2/NSAID GI comp

Indications

Previous Px

Regression (17 confounders), no effect RD: -0.06/100; 95% CI -0.26 to 0.14

IV: Protective effect of COX-2 RD: -1.31/100; -2.42 to -0.20

Compatible with trial results RD: -0.65/100; -1.08 to -0.22

Unknown confounders (IV Ex 3)

A(Exposure): Childhood dehydration Outcome: Adult high blood pressure Instrument: 1st year summer climate

dehydration High BP

SES

1st year climate

Hypothesized hottest/driest summers in infancy would be associated with severe infant diarrhea/dehydration, and consequently higher blood pressure in adulthood.

For 3,964 women born 1919-1940, a 1 SD (1.3 ºC) > mean summer temp in 1st year life associated with 1.12-mmHg (95% CI: 0.33, 1.91) > adult systolic blood pressure, and 1 SD > mean summer rainfall (33.9 mm) associated with < systolic blood pressure (-1.65 mmHg, 95% CI: -2.44, -0.85).

Optionally we can adjust for “non-compliance”

Optionally if we want to estimate A-D relationship, not I-D, we can adjust:– RDID / RDIE

– Inflate the IV estimator to adjust for the lack of perfect correlation between I and E

– If I perfectly predicts E then RDIE = 1, so adjustment does nothing

Like per protocol analysis– But adjusted for confounders

To good to be true?

Maybe The assumptions needed for an

instrument are un-testable from the data– Can only determine if I is associated with A

Failure to meet the assumptions can cause strong bias– Particularly if we have a “weak” instrument

Approach 2: Propensity Scores

Comes out of a world of large datasets (Health insurance data)

Cases where we have a small (relative to the size of the dataset) exposed population and lots and lots of potential comparisons in the unexposed group– And lots of covariate data to adjust for

Then we have luxury of deciding who to include in study as a comparison group based on a counterfactual definition

Propensity Score

Model each subject’s propensity to receive the index condition as a function of confounders– Model is independent of outcomes, so good for

rare disease, common exposure Use the propensity score to balance

assignment to index or reference by:– Matching– Stratification– Modeling

Propensity Scores

The propensity score for subject i is:– Probability of being assigned to treatment A = 1 vs. reference

A = 0 given a vector xi of observed covariates:

In other words, the propensity score is:– Probability that the person got the exposure given anything

else we know about them

)|1(Pr iiiA xX

Why estimate the probability a subject receives a certain treatment when it is known what treatment they received?

How Propensity Scores Work

Quasi-experiment – Using probability a subject would have been treated

(propensity score) to adjust estimate of the treatment effect, we simulate a RCT

2 subjects with = propensity, one E+, one E- – We can think of these two as “randomly assigned”

to groups, since they have the same probability of being treated, given their covariates

– Assumes we have enough observed data that within levels of propensity E is truly random

Propensity Scores:Smoking and Colon Cancer

Have info on people’s covariates:– Alcohol use, sex, weight, age, exercise, etc:

Person A is a smoker, B is not– Both had 85% predicted probability of smoking

If “propensity” to smoke is same, only difference is 1 smoked and 1 didn’t

– This is essentially what randomization does– B is the counterfactual for A assuming a correct

model for predicting smoking

Obtaining Propensity Scores in SAS

Calculate propensity scoreproc logistic;

model exposure = cov_1 cov_2 … cov_n; output out = pscoredat pred = pscore;

run;

Either match subjects on propensity score or adjust for propensity score

proc logistic;model outcome = exposure pscore;

run;

Pros and Cons of PS

Pros– Adjustment for 1 confounder– Allows estimation of the exposure and fitting a final

model without ever seeing the outcome– Allows us to see parts of data we really should not

be drawing conclusions on b/c no counterfactual Cons

– Only works if have good overlap in pscores– Does not fix conditioning on a collider problem– Doesn’t deal with unmeasured confounders

Study of effect of neighborhood segregation on IMR

Approach 3: Marginal Structural Models

Time Dependent Confounding

Time dependent confounding:1) Time dependent covariate that is a risk

factor for or predictive of the outcome and also predicts subsequent exposure

Problematic if also:2) Past exposure history predicts

subsequent level of covariate

Example

Observational study of subjects infected with HIV– E = HAART therapy– D = All cause mortality– C = CD4 count

Time Dependent Confounding

A0 A1

C0 C1

D

A0 A1

C0 C1

D

1)

2)

Failure of Traditional Methods

Want to estimate causal effect of A on D– Can’t stratify on C (it’s an intermediate)– Can’t ignore C (it’s a confounder)

Solution – rather than stratify, weight– Equivalent to standardization

Create pseudo-population where RRCE = 1– Weight each person by “inverse probability of

treatment” they actually received– Weighting doesn’t cause problems pooling did– In DAG, remove arrow C to A, don’t box

Remember back to the SMRCrude C1 C0

E+ E- E+ E- E+ E-D+ 350 70 D+ 300 20 D+ 50 50D- 1650 1130 D- 1200 180 D- 450 950Total 2000 1200 Total 1500 200 Total 500 1000Risk 0.18 0.06 0.2 0.1 0.1 0.05RR 3.0 RR 2.0 RR 2.0

0.2

100050

*50020020

*1500

50050

*5001500300

*1500

*

*

*

*

ˆ

01

11

0

1

Nb

N

Na

N

WNb

W

WNa

W

RSM

The SMR asks, what if the exposed had also been unexposed?

Crude C1 C0

E+ E- E+ E- E+ E-D+ 350 70 D+ 300 20 D+ 50 50D- 1650 1130 D- 1200 180 D- 450 950Total 2000 1200 Total 1500 200 Total 500 1000Risk 0.18 0.06 0.2 0.1 0.1 0.05RR 3.0 RR 2.0 RR 2.0

Crude C1 C0

E+ E- E+ E- E+ E-D+ 350 D+ 300 D+ 50D- 1650 D- 1200 D- 450Total 2000 Total 1500 Total 500Risk 0.18 0.2 0.1RR RR RR

SM

R

The SMR asks, what if the exposed had also been unexposed?

Crude C1 C0

E+ E- E+ E- E+ E-D+ 350 70 D+ 300 20 D+ 50 50D- 1650 1130 D- 1200 180 D- 450 950Total 2000 1200 Total 1500 200 Total 500 1000Risk 0.18 0.06 0.2 0.1 0.1 0.05RR 3.0 RR 2.0 RR 2.0

Crude C1 C0

E+ E- E+ E- E+ E-D+ 350 D+ 300 D+ 50D- 1650 D- 1200 D- 450Total 2000 Total 1500 1500Total 500 500Risk 0.18 0.2 0.1RR RR RR

SM

R

The SMR asks, what if the exposed had also been unexposed?

Crude C1 C0

E+ E- E+ E- E+ E-D+ 350 70 D+ 300 20 D+ 50 50D- 1650 1130 D- 1200 180 D- 450 950Total 2000 1200 Total 1500 200 Total 500 1000Risk 0.18 0.06 0.2 0.1 0.1 0.05RR 3.0 RR 2.0 RR 2.0

Crude C1 C0

E+ E- E+ E- E+ E-D+ 350 D+ 300 D+ 50D- 1650 D- 1200 D- 450Total 2000 Total 1500 1500Total 500 500Risk 0.18 0.2 0.1 0.1 0.05RR RR RR

SM

R

The SMR asks, what if the exposed had also been unexposed?

Crude C1 C0

E+ E- E+ E- E+ E-D+ 350 70 D+ 300 20 D+ 50 50D- 1650 1130 D- 1200 180 D- 450 950Total 2000 1200 Total 1500 200 Total 500 1000Risk 0.18 0.06 0.2 0.1 0.1 0.05RR 3.0 RR 2.0 RR 2.0

Crude C1 C0

E+ E- E+ E- E+ E-D+ 350 D+ 300 150 D+ 50 25D- 1650 D- 1200 1350D- 450 475Total 2000 Total 1500 1500Total 500 500Risk 0.18 0.2 0.1 0.1 0.05RR RR 2.0   RR 2.0  

SM

R

The SMR asks, what if the exposed had also been unexposed?

Crude C1 C0

E+ E- E+ E- E+ E-D+ 350 70 D+ 300 20 D+ 50 50D- 1650 1130 D- 1200 180 D- 450 950Total 2000 1200 Total 1500 200 Total 500 1000Risk 0.18 0.06 0.2 0.1 0.1 0.05RR 3.0 RR 2.0 RR 2.0

Crude C1 C0

E+ E- E+ E- E+ E-D+ 350 175 D+ 300 150 D+ 50 25D- 1650 1825 D- 1200 1350D- 450 475Total 2000 2000 Total 1500 1500Total 500 500Risk 0.175 0.875 0.2 0.1 0.1 0.05RR 2.0 RR 2.0   RR 2.0  

Crude now equals the adjusted. No need to adjust.

SM

R

Could also ask, what if everyone was both exposed, unexposed?

Crude C1 C0

E+ E- E+ E- E+ E-D+ 350 70 D+ 300 20 D+ 50 50D- 1650 1130 D- 1200 180 D- 450 950Total 2000 1200 Total 1500 200 Total 500 1000Risk 0.18 0.06 0.2 0.1 0.1 0.05RR 3.0 RR 2.0 RR 2.0

Could also ask, what if everyone was both exposed, unexposed?

Crude C1 C0

E+ E- E+ E- E+ E-D+ D+ D+D- D- D-Total Total 1700 1700Total 1500 1500Risk 0.2 0.1 0.1 0.05RR RR 2.0 RR 2.0

Could also ask, what if everyone was both exposed, unexposed?

Crude C1 C0

E+ E- E+ E- E+ E-D+ D+ 340 170 D+ 150 75D- D- 1360 1530D- 1350 1425Total Total 1700 1700Total 1500 1500Risk 0.2 0.1 0.1 0.05RR RR 2.0 RR 2.0

Could also ask, what if everyone was both exposed, unexposed?

Crude C1 C0

E+ E- E+ E- E+ E-D+ 490 245 D+ 340 170 D+ 150 75D- 2710 2955 D- 1360 1530D- 1350 1425Total 3200 3200 Total 1700 1700Total 1500 1500Risk 0.153 0.077 0.2 0.1 0.1 0.05RR 2.0 RR 2.0 RR 2.0

What is Inverse Probability Weighting (IPW)?

Weight each subject by inverse probability of treatment received

Probability of treatment is:– p(receiving treatment received| covariates)– Adjust # of E+ and E- subjects in C strata

Weighting breaks E-C link only– Now Marginal (Crude) = Causal Effect

But that’s what we just did

Calculate the weightsCrude C1 C0

E+ E- E+ E- E+ E-D+ 350 70 D+ 300 20 D+ 50 50D- 1650 1130 D- 1200 180 D- 450 950Total 2000 1200 Total 1500 200 Total 500 1000Risk 0.18 0.06 0.2 0.1 0.1 0.05RR 3.0 RR 2.0 RR 2.0 PTIPTW Calculate p(receiving treatment received|C) For C=1, E=1

– PT = 1500/1700 = 0.88– IPTW = 1/0.88 = 1.13

Calculate the weightsCrude C1 C0

E+ E- E+ E- E+ E-D+ 350 70 D+ 300 20 D+ 50 50D- 1650 1130 D- 1200 180 D- 450 950Total 2000 1200 Total 1500 200 Total 500 1000Risk 0.18 0.06 0.2 0.1 0.1 0.05RR 3.0 RR 2.0 RR 2.0 PT 0.88IPTW 1.13 Calculate p(receiving treatment received|C) For C=1, E=1

– PT = 1500/1700 = 0.88– IPTW = 1/0.88 = 1.13

Calculate the weightsCrude C1 C0

E+ E- E+ E- E+ E-D+ 350 70 D+ 300 20 D+ 50 50D- 1650 1130 D- 1200 180 D- 450 950Total 2000 1200 Total 1500 200 Total 500 1000Risk 0.18 0.06 0.2 0.1 0.1 0.05RR 3.0 RR 2.0 RR 2.0 PT 0.88 0.12IPTW 1.13 8.50 Calculate p(receiving treatment received|C) For C=1, E=0

– PT = 200/1700 = 0.12– IPTW = 1/0.12 = 8.50

Calculate the weightsCrude C1 C0

E+ E- E+ E- E+ E-D+ 350 70 D+ 300 20 D+ 50 50D- 1650 1130 D- 1200 180 D- 450 950Total 2000 1200 Total 1500 200 Total 500 1000Risk 0.18 0.06 0.2 0.1 0.1 0.05RR 3.0 RR 2.0 RR 2.0 PT 0.88 0.12 0.33 0.67IPTW 1.13 8.50 3.00 1.50 Calculate p(receiving treatment received|C) For C=1, E=0

– PT = 200/1700 = 0.12– IPTW = 1/0.12 = 8.50

Multiply cell number by the weights

Apply the weightsCrude C1 C0

E+ E- E+ E- E+ E-D+ 350 70 D+ 300 20 D+ 50 50D- 1650 1130 D- 1200 180 D- 450 950Total 2000 1200 Total 1500 200 Total 500 1000Risk 0.18 0.06 0.2 0.1 0.1 0.05RR 3.0 RR 2.0 RR 2.0 PT 0.88 0.12 0.33 0.67IPTW 1.13 8.50 3.00 1.50Pseudo population

Crude C1 C0

E+ E- E+ E- E+ E-D+ D+ 340 170 D+ 150 75D- D- 1360 1530 D- 1350 1425Total Total 1700 1700 Total 1500 1500Risk 0.2 0.1 0.1 0.05RR RR 2.0 RR 2.0

CollapseCrude C1 C0

E+ E- E+ E- E+ E-D+ 350 70 D+ 300 20 D+ 50 50D- 1650 1130 D- 1200 180 D- 450 950Total 2000 1200 Total 1500 200 Total 500 1000Risk 0.18 0.06 0.2 0.1 0.1 0.05RR 3.0 RR 2.0 RR 2.0 PT 0.88 0.12 0.33 0.67IPTW 1.13 8.50 3.00 1.50Pseudo population

Crude C1 C0

E+ E- E+ E- E+ E-D+ 490 245 D+ 340 170 D+ 150 75D- 2710 2955 D- 1360 1530 D- 1350 1425Total 3200 3200 Total 1700 1700 Total 1500 1500Risk 0.153 0.077 0.2 0.1 0.1 0.05RR 2.0 RR 2.0 RR 2.0

Broke link between C and E without stratification, so no problem of conditioning on collider

Pseudo-population

The “pseudo-population” breaks the link between the exposure and the outcome without stratification– Note this is different from stratifying – Create a standard population without confounding

By creating multiple copies of people, standard errors will be biased– Use robust standard errors to adjust

Robins and Hernán

“The IPTW method effectively simulates the data that would be observed had, contrary to fact,

exposure been conditionally randomized”

Time Dependent Confounding

Extend method to time dependent confounders– Predict p(receiving treatment actually received at time t1|

covariates, treatment at t0)

Probability of treatment at t1 is:

p(receiving treatment received at t0) *

p(receiving treatment received at t1)

See Hernán for SAS code, not hard scwgt command, robust SE (repeated statement)

Time Dependent Confounding

E0 E1

C0 C1

D

E0 E1

C0 C1

D

1) Before IPTW

2) After IPTW

Limitations of MSMs

Very sensitive to weights Still need to be able to be able to predict

the exposure– The methods solves the structural problem, but

we still need the data to be able to accurately predict exposure

Still have to get the model right