Presentations in this series 1.Overview and Randomization 2.Self-matching 3.Proxies 4.Intermediates 5.Instruments 6.Equipoise Avoiding Bias Due to Unmeasured Covariates Alec Walker

TD X

TD X Randomization

TD X Self-matching

TD X Proxies Randomization

6 A textbook definition from econometrics.

7 Let Obe an outcome (either T treatment or D disease) P be a proxy X be an unmeasured covariate P is a proxy for X with respect to O if the distribution 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 from econometrics.

8 Let Obe an outcome (either T treatment or D disease) P be a proxy X be an unmeasured covariate P is a proxy for X with respect to O if the distribution 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 from econometrics.

Proxy variables are Correlates 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 9

10 Interview responses may be proxies for – Historical measurements (diet, smoking, alcohol …) – Internal states – Genetic traits Biological markers are proxies for biological processes Age, sex, SES are stand-ins for their many correlates. Examples of proxies

11 Interview responses may be proxies for – Historical measurements (diet, smoking, alcohol …) – Internal states – Genetic traits Biological markers are proxies for biological processes Age, 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.

web/anonymous.portal?_nfpb=true&_pageLabel=commonConditions&contentPage =va_health_library/diabetic_retinopathy_advanced_info.html Early diabetic retinopathy Source: US Department of Veterans Affairs

web/anonymous.portal?_nfpb=true&_pageLabel=commonConditions&contentPage =va_health_library/diabetic_retinopathy_advanced_info.html microaneurysms Early diabetic retinopathy Source: US Department of Veterans Affairs

web/anonymous.portal?_nfpb=true&_pageLabel=commonConditions&contentPage =va_health_library/diabetic_retinopathy_advanced_info.html Advanced diabetic retinopathy Source: US Department of Veterans Affairs

D X 15 T

D X P 16 T

D X P 17 T D X

D X P UDUD 18 T D X UTUT

D XPUDUD 19 T D X UTUT

P UDUD UXUX 20 T D X UTUT

D XPUDUD 21 T D X UTUT UXUX

D X P UDUD 22 T D X UTUT UXUX

Thialozinediones for diabetes Acute myocardial infarction Coronary artery disease UTUT (Unmeasured) Severity of Diabetes Retinal vascular disease UDUD 23

Without mechanistic information, for each of these situations, ( covariate causes proxy proxy causes covariate both caused by a third factor ) … the proxy looks like a transformation of the predictor, with added error. Proxy value = f(Predictor value) + error 24

An accurate proxy 25

Treated Untreated The true value of the unmeasured covariate is a predictor of treatment An accurate proxy 26

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 27

The proxy almost perfectly represents the value of the unmeasured covariate. Treated Untreated An accurate proxy 28

Treated Untreated An accurate proxy 29

Treated Untreated An accurate proxy 30

An accurate proxy 31

The proportion of treated among subjects in a particular small range of proxy values An accurate proxy 32

The proportion of treated among subjects in a particular small range of proxy values An accurate proxy 33

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 34

The true value does not provide further information, if you know the proxy. An accurate proxy 35

Treated Untreated Two accurate proxies 36

Two good proxies are highly correlated with one another. Treated Untreated Two accurate proxies 37

Either proxy provides good prediction of treatment. Treated Untreated Two accurate proxies 38

Untreated Proxies with substantial random error Untreated Treated 39

Untreated The proxy is still correlated with the unknown measure. Proxies with substantial random error Untreated Treated 40

Treatment is still associated with higher values of the proxy, but the discrimination is much worse. Proxies with substantial random error 41

Proxies with substantial random error Treated Untreated 42

Proxies with substantial random error Treated Untreated The correlation between the two proxy measures is still evident. 43

Both proxies show poor discrimination between treated and untreated. Proxies with substantial random error Treated Untreated 44

The two proxies can be combined into a function that discriminates better than either proxy alone. Proxies with substantial random error Treated Untreated 45

46 A textbook definition from econometrics.

47 Let Obe an outcome (either T treatment or D disease) P be a proxy X be an unmeasured covariate P is a proxy for X with respect to O if the distribution of O given P is identical to the distribution of O given P and X. A textbook definition from econometrics.

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49

50

51 Let Obe an outcome (either T treatment or D disease) P be a proxy X be an unmeasured covariate P is a proxy for X with respect to O if the distribution 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 from econometrics.

52 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

53 Proxy variables can correspond to different components of a composite predictor. For example, “Severity of Diabetes.” Hemoglobin A1 C = f(Glucose control last 90 days) + error A Retinal vascular disease = f(Vascular damage) + error B

Thialozinediones for diabetes Acute myocardial infarction Coronary artery disease UTUT Retinal vascular disease UDUD 54 UXUX

Thialozinediones for diabetes Acute myocardial infarction Coronary artery disease UTUT Retinal vascular disease UDUD Hb A 1C UYUY Diabetes Mellitus 55 UXUX

Thialozinediones for diabetes Acute myocardial infarction Coronary artery disease UTUT Retinal vascular disease UDUD Hb A 1C UYUY Diabetes Mellitus 56 UXUX

57 Treated Untreated Treated Untreated Proxies for components of a composite variable

58 The proxy measures are uncorrelated with one another. Treated Untreated Treated Untreated Proxies for components of a composite variable

59 Proxy A captures more of the distinction. Proxy B captures none of the distinction between treatments. Treated Untreated Treated Untreated 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. 60

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. 61

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 62

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 Proxies for components of composite unmeasured covariate  Uncorrelated proxies  Keeps the correct predictor. 63

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 Proxies 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. 64

65 Multidimensional proxy variables created through the use of propensity scores

66 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. Celecoxib versus Naproxen PUB Hospital Admission MD-perceived risk of peptic ulcer & bleeding (PUB) True risk of PUB

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68 After extensive propensity matching 68

69 After control for correlates that completely capture perceived PUB diathesis, there is no further confounding. Celecoxib versus Naproxen PUB Hospital Admission

70 After control for correlates that completely capture perceived PUB diathesis, there is no further confounding. Celecoxib versus Naproxen PUB Hospital Admission

71 After control for correlates that completely capture perceived PUB diathesis, there is no further confounding. Celecoxib versus Naproxen PUB Hospital Admission

Primary Discharge Diagnosis N % N % RR With control for many, many proxies a strong effect emerges. 72

73 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 series 1.Overview and Randomization 2.Self-matching 3.Proxies 4.Intermediates 5.Instruments Avoiding Bias Due to Unmeasured Covariates Alec Walker 74