Presentations in this series1. Overview

and Randomization2. Self-matching3. Proxies4. Intermediates5. Instruments6. Equipoise

Avoiding Bias Due toUnmeasured Covariates

Alec Walker

T D

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T D

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T D

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Randomization

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A textbook definition fromeconometrics.

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Let O be an outcome (either T treatment or D disease)P be a proxyX be an unmeasured covariate

P is a proxy for X with respect to O if thedistribution of O given P is identical to the distribution of O given P and X

Which is to say that X adds no information about O, if you know P.

A textbook definition fromeconometrics.

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Let O be an outcome (either T treatment or D disease)P be a proxyX be an unmeasured covariate

P is a proxy for X with respect to O if thedistribution of O given P is identical to the distribution of O given P and X

Which is to say that X adds no information about O, if you know P.

Note that O, P and X could all be multidimensional, that is vectors of outcomes, proxies and unmeasured covariates, respectively. This definition could also be conditioned on other, measured covariates.

A textbook definition fromeconometrics.

Proxy variables areCorrelates of an unmeasured covariate

That are useful to the extent that they capture the influence of the unmeasured covariate on a third characteristic

Control for a proxy replaces control for the unmeasured covariate

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Interview responses may be proxies for – Historical measurements (diet, smoking, alcohol …)– Internal states– Genetic traits

Biological markers are proxies for biological processesAge, sex, SES are stand-ins for their many correlates

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Examples of proxies

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Interview responses may be proxies for – Historical measurements (diet, smoking, alcohol …)– Internal states– Genetic traits

Biological markers are proxies for biological processesAge, sex, SES are stand-ins for their many correlates

.

Examples of proxies

In diabetics, retinal vascular disease is a proxy for vascular disease more generally and is easily ascertained by funduscopic examination. In looking at determinants of myocardial infarction, control for retinal vascular disease could represent control for coexisting vascular pathology.

https://www.myhealth.va.gov/mhv-portal-web/anonymous.portal?_nfpb=true&_pageLabel=commonConditions&contentPage=va_health_library/diabetic_retinopathy_advanced_info.html

Early diabetic retinopathySource: US Department of Veterans Affairs

https://www.myhealth.va.gov/mhv-portal-web/anonymous.portal?_nfpb=true&_pageLabel=commonConditions&contentPage=va_health_library/diabetic_retinopathy_advanced_info.html

microaneurysms

Early diabetic retinopathySource: US Department of Veterans Affairs

https://www.myhealth.va.gov/mhv-portal-web/anonymous.portal?_nfpb=true&_pageLabel=commonConditions&contentPage=va_health_library/diabetic_retinopathy_advanced_info.html

Advanced diabetic retinopathySource: US Department of Veterans Affairs

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T D

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T D

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T D

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P UD

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T D

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T D

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T D

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Thialozinedionesfor diabetes

Acute myocardial infarction

Coronary artery

disease

UT

(Unmeasured) Severity of Diabetes

Retinal vascular disease

UD

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Without mechanistic information, for each of these situations,

( covariate causes proxyproxy causes covariateboth caused by a third factor )

… the proxy looks like a transformation of the predictor, with added error.

Proxy value = f(Predictor value) + error

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An accurate proxy

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Treated

Untreated

The true value of the unmeasured covariate is a predictor of treatment

An accurate proxy

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The proxy predicts treatment almost as well as does the true value.

Treated

Untreated

The true value of the unmeasured covariate is a predictor of treatment

An accurate proxy

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The proxy almost p

erfectly

represents

the value of th

e unmeasured covaria

te.

Treated

Untreated

An accurate proxy

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Treated

Untreated

An accurate proxy

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Treated

Untreated

An accurate proxy

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An accurate proxy

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The proportion of treated among subjects in a particular small range of proxy values

An accurate proxy

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The proportion of treated among subjects in a particular small range of proxy values

An accurate proxy

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The proportion of treated among subjects in a particular small range of proxy values

… is the same as the proportion of treated among subjects in the corresponding small range of true values.

An accurate proxy

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The true value does not provide further information, if you know the proxy.

An accurate proxy

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Treated

Untreated

Two accurate proxies

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Two good proxie

s are highly

correlated with

one another.Treated

Untreated

Two accurate proxies

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Either proxy provides good prediction of treatment.

Treated

Untreated

Two accurate proxies

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Untreated

Proxies with substantial

random errorUntreated

Treated

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UntreatedTh

e prox

y is s

till corre

lated

with th

e unkn

own mea

sure.

Proxies with substantial

random errorUntreated

Treated

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Treatment is still associated with higher values of the proxy, but thediscriminationis muchworse.

Proxies with substantial

random error

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Proxies with substantial

random errorTreated

Untreated

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Proxies with substantial

random errorTreated

Untreated

The corre

lation between th

e two

proxy measures is

still e

vident.

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Both proxies show poor discrimination between treated and untreated.

Proxies with substantial

random errorTreated

Untreated

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The two proxies can be combined into a function that discriminates better than either proxy alone.

Proxies with substantial

random errorTreated

Untreated

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A textbook definition fromeconometrics.

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Let O be an outcome (either T treatment or D disease)P be a proxyX be an unmeasured covariate

P is a proxy for X with respect to O if thedistribution of O given P is identical to the distribution of O given P and X.

A textbook definition fromeconometrics.

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Let O be an outcome (either T treatment or D disease)P be a proxyX be an unmeasured covariate

P is a proxy for X with respect to O if thedistribution of O given P is identical to the distribution of O given P and X.

None of the causal graphs or correlation patterns that we’ve looked at so far produce

this behavior, unless the proxy is perfect.

What are the economists talking about?

A textbook definition fromeconometrics.

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Proxy variables can correspond to different components of a composite predictor

Proxy A = f(Predictor Component A) + error A

Proxy B = f(Predictor Component B) + error B

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Proxy variables can correspond to different components of a composite predictor.For example, “Severity of Diabetes.”

Hemoglobin A1C

= f(Glucose control last 90 days) + error A

Retinal vascular disease = f(Vascular damage) + error B

Thialozinedionesfor diabetes

Acute myocardial infarction

Coronary artery

disease

UT

Retinal vascular disease

UD

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UX

Thialozinedionesfor diabetes

Acute myocardial infarction

Coronary artery

disease

UT

Retinal vascular disease

UDHb A1C

UY

Diabetes Mellitus

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UX

Thialozinedionesfor diabetes

Acute myocardial infarction

Coronary artery

disease

UT

Retinal vascular disease

UDHb A1C

UY

Diabetes Mellitus

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UX

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Treated

Untreated

Trea

ted

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reat

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Proxies for components of a composite variable

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The proxy measures are uncorrelated with one another.

Treated

Untreated

Trea

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Unt

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Proxies for components of a composite variable

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Proxy A captures more of the distinction.

Proxy B captures none of the distinction between treatments.

Treated

Untreated

Trea

ted

Unt

reat

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Proxies for components of a composite variable

When you have several candidate proxies for an unmeasured covariate, examine them simultaneously for prediction of the outcome (treatment, disease or both), and retain only those that do.

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When you have several candidate proxies for an unmeasured covariate, examine them simultaneously for prediction of the outcome (treatment, disease or both), and retain only those that do.

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When you have several candidate proxies for an unmeasured covariate, examine them simultaneously for prediction of the outcome (treatment, disease or both), and retain only those that do.

Measurement error Correlated proxies Keeps all relevant ones

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When you have several candidate proxies for an unmeasured covariate, examine them simultaneously for prediction of the outcome (treatment, disease or both), and retain only those that do.

Measurement error Correlated proxies Keeps all relevant onesProxies for components of composite unmeasured covariate Uncorrelated proxies Keeps the correct predictor.

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When you have several candidate proxies for an unmeasured covariate, examine them simultaneously for prediction of the outcome (treatment, disease or both), and retain only those that do.

Measurement error Correlated proxies Keeps all relevant onesProxies for components of composite unmeasured covariate Uncorrelated proxies Keeps the correct predictor.

Propensity scores (composite multi-variate treatment predictors), allow you to account for both settings.

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Multidimensional proxy variablescreated through the use of propensity scores

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The physician’s belief in the patient’s risk for peptic ulcer and bleeding cannot be measured directly. But we can look to known correlates of treatment choice as measures of the physician’s belief and treat these as proxy variables.

Celecoxibversus

Naproxen

PUBHospital

Admission

MD-perceived risk of peptic ulcer & bleeding (PUB)

True risk of PUB

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After extensive propensity matching

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After control for correlates that completely capture perceived PUB diathesis, there is no further confounding.

Celecoxibversus

Naproxen

PUBHospital

Admission

MD-perceived risk of peptic ulcer & bleeding (PUB)

True risk of PUB

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After control for correlates that completely capture perceived PUB diathesis, there is no further confounding.

Celecoxibversus

Naproxen

PUBHospital

Admission

MD-perceived risk of peptic ulcer & bleeding (PUB)

True risk of PUB

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After control for correlates that completely capture perceived PUB diathesis, there is no further confounding.

Celecoxibversus

Naproxen

PUBHospital

Admission

MD-perceived risk of peptic ulcer & bleeding (PUB)

True risk of PUB

Primary Discharge Diagnosis N % N % RR

With control for many, many proxies a strong effect emerges.

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A proxy is (1) a correlate that (2) captures the effect of an unmeasured covariate on either treatment or disease.

Whether a correlate is a proxy is defined only in respect of a third, predicted variable.

Strong correlates may be only weak proxies.Composite (multidimensional) proxies

are useful when no single candidate proxy captures the unmeasured covariate.

Propensity scoring creates multidimensional proxies.

Presentations in this series1. Overview

and Randomization2. Self-matching3. Proxies4. Intermediates5. Instruments

Avoiding Bias Due toUnmeasured Covariates

Alec Walker

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