Causal Hazard Ratio Estimation By Instrumental Variables or Principal Stratification. Todd MacKenzie, PhD

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1 Causal Hazard Ratio Estimation By Instrumental Variables or Principal Stratification Todd MacKenzie, PhD

2 Collaborators A. James O Malley Tor Tosteson Therese Stukel 2

3 Overview 1. Instrumental variable (I.V.) 2. I.V.s for Hazard Ratio Estimation 3. Principal stratification (P.S.) 4. P.S. for Hazard Ratio Estimation 3

4 Observational Studies and Confounding Confounding is always a threat to observational studies Therefore we prefer when possible to conduct 4

5 Randomized Studies Randomized studies yield fair comparisons However Require greater resources Tend to be conducted in artificial settings Therefore, observational studies deserve more consideration 5

6 Current Arsenal of Statistical Methods for Overcoming Confounding 1. Adjustment (regression) models 2. Propensity score matching or stratification 3. Inverse propensity weighted marginal structural models Bias is removed only to the extent that confounders are included in the model 6

7 Confounding that cannot be removed Goes by different names Omitted covariates Unmeasured confounders Residual confounding 7

8 Instrumental Variables Estimation by I.V can yield unbiased estimates without observing all confounders 8

9 Instrumental Variables for the Linear Model Idea is accessible to any student of an introductory mathematical statistics class 9

10 Instrumental Variables for the Linear Model 1. Y = a + bx + ε such that Cov(I.V., ε)=0 2. Cov(Y,W) = Cov(a+bX+ε, I.V.) = bcov(x, I.V.) 3. Therefore b = Cov(Y, I.V.) / Cov(X, I.V.) 10

11 Assumptions 1. Nonzero association of X and the I.V. Usual rule of thumb: F-test of I.V. and X exceeds 10 11

12 Assumptions 2. The effect of the I.V. on the Y is strictly through the effect of the I.V on X: a. There is no direct effect of the I.V. on Y b. There is no intermediate variable for the effect of the I.V on Y except X: Exclusion Criteria or Absence of Indirect Effect c. There are no variables that effect both I.V. and Y: No I.V.- Outcome Confounders or Randomization 12

13 IV X Y 13

14 There are no paths between IV and Y except through X IV X Y 14

15 Faith based statistics: This second assumption cannot be empirically validated IV X Y 15

16 Other Assumptions Not Necessarily Worth Mentioning SUTVA: Stable Unit Treatment Value Assumption My treatment does not effect your outcome Monotonicity 16

17 Do instrumental variables exist? Yes, man-made instruments exist: Randomized Studies Randomization is an instrument for the effect of treatment on an endpoint The arm a subject is randomized to should have no effect on the endpoint through its effects on the treatment received 17

18 Do instrumental variables exist in the real world? Happenstance Be on the lookout for natural experiments Commonly employed instrumental variables: Geographic variables (regional rates, distances) Care provider propensity to prescribe a treatment Calendar time Before after a health policy change See regression discontinuity 18

19 Mendelian Randomization: Assessing the Effect of a Phenotype on a Dependent Variable Gene for Some Phenotype That Phenotype Endpoint 19

20 Instrumental Variables Theory Is Well Developed for Linear Models ˆ T 1 β = ( IV X ) ( IV T Y ) Equivalent to Two Stage Least Squares 1. regress X on IV and save predicted values as P 2. regress Y on P, use coefficient of P Equivalent to Control Function 1. regress X on IV and save residuals, R 2. regress Y on X and R, use coefficient of X 20

21 Our Aim To develop an estimator of the hazard ratio using an instrumental variable 21

22 Challenges Non-linear parameterization (e.g., Cox model) Right censoring 22

23 Prior Work in the Literature Some in econometrics literature Stukel et al (2007) JAMA 23

24 Original Motivation Stukel et al Analysis of Observational Studies in the Presence of Treatment Selection Bias: Effects of Invasive Cardiac Management on AMI Survival Using Propensity Score and Instrumental Variable Methods JAMA (2007) 24

25 Cox Proportional Hazards Model Suppose the effect of a treatment, X, on an individual s potential time-to-events, {T(x)} x, is to multiply the hazard by e β Pr[ T ( x) = t T ( x) t] = e βx λ Estimand of interest: β, the log hazard ratio ( t) 25

26 Non-Collapsibility of Cox s Model Like most non-linear models (e.g. logistic regression) Cox s model is not collapsible That is, if the conditional distribution of timeto-event T given Z 1 and Z 2 is a Cox model then the marginal conditional distribution of T given Z 1 (obtained by integrating over Z 2 ) is not a Cox model unless Z 1 is independent of T 26

27 A model that collapses to Cox The model for the joint effect of X and Z (e.g., an omitted covariate), collapses to (integrates out to) under certain conditions ) ( ] ) ( ) ( Pr[ t e t x T t x T βx λ = = z t e z Z t x T t x T x θ λ β + = = = ) ( ], ) ( ) ( Pr[ 27

28 Incorporating the I.V. Assuming that the I.V. is independent of the omitted covariate, Z, an argument using risk sets or martingales, e.g., βx E[ I. V. { dn( t) e λ( t)}] 0 can be made that results in the estimating equation on the next slide 28

29 Estimation of the Hazard Ratio Using Instrumental Variables Solve the estimating equation below for β Counting process notation In conventional notation ) ( ] ) ( ) ( [ t dn e t Y e t IV Y IV i n i n j X j n j X j j i j j = = = = τ β β ] [ 0 1 ) ( ) ( = = n i R j X R j X j i i i j i j e e IV IV β β δ 29

30 Simulations Conducted simulations to evaluate bias For the unmeasured confounder two models were assume 1. Additive effect of the confounder on the hazard 2. Multiplicative effect (Cox) of the confounder on the hazard 30

31 Simulation Results for Multiplicative Omitted Covariate 31

32 Application using data from Stukel et al (2007) JAMA National cohort of 120,000 patients on Medicare, hospitalized with acute myocardial infarction (AMI) Exposure: coronary catheterization within 30 days of hospitalization Endpoint: mortality by 4 years after hospitalization Covariates: rich set of prognostic variables 32

33 Aim of Stukel s study Estimate the effect of coronary catheterization on survival using three different approaches Regression (Cox s model) Propensity Scores (with and without matching) Instrumental variables Compare the estimates to those obtained in randomized studies 33

34 Instrumental Variables in Stukel et al (2007) Instrument: regional cardiac catheterization rate proportion of eligible patients receiving cardiac catheterization within 30 days of admission in each area 566 areas Assumption: the effect of this proportion on survival is strictly through its association with an individual s propensity to have a card. cath. Methods for linear instrument variable models applied to survival Used endpoint of survival at 1 year (4 years) Excludes subjects who were censored before that time Converted coefficient from linear model into hazard ratio 34

35 Example: Effect of coronary catheterization on survival Approach Hazard Ratio (95%CI) Cox adjusted Cox adjusted excluding.rst 31 days Prop Score matched within Stukel L.M. Instrumental Variable at 1 Yr Stukel L.M. Instrumental Variable at 4 Yrs Instrumental Variable Estimator (Overall) Based on First Year Whereas randomized studies yielded estimates of 0.80 to

36 Related Work Accelerated Failure Time model Additive Hazards Model (Jason Fine) Mitra & Small, Chan & Small Tchetgen Tchetgen 36

37 Principal Stratification

38 Randomized Trial Goal: Estimate treatment effect e.g., drug vs placebo 38

39 Compliance Not all participants comply 39

40 Common Estimators from a Randomized Trial Intention-to-treat As treated Per protocol 40

41 Intention-to-treat Analyze as randomized Unbiased estimate of treatment assignment It does not estimate the effect of treatment, were a subject to comply 41

42 As Treated Selection bias is introduced Treatment is selected as opposed to randomized 42

43 Per Protocol Excludes subjects who did not comply What is it estimating? 43

44 How to Estimate the effect of treatment? What is the effect of treatment, if someone complies with treatment Principal Stratification is a framework for defining a meaningful estimand To start with, consider the relationship of the exposure a subject receives to the arm of the study they are randomized to 44

45 Mappings from Support of the Binary Assignment to the Support of the Binary Expsosure? There are 4 mappings from the two possible values of the assignment, R, to the two possible values of the exposure, X Mapping R to X Principal Strata Name Identity 0 to 0, 1 to 1 Compliers Contant Zero Constant Unity 0 to 0, 1 to 0 Never Takers 0 to 1, 1 to 1 Always Takers Reverse 0 to 1, 1 to 0 Defiers 45

46 The Principal Strata Latent Not directly observable Names are suggestive but should not be taken at face value 46

47 No Defiers It appears reasonable to assume that there is no defiers (or at least they are very rare) Furthermore this assumption facilitates identification of some of the latent strata This assumption is called Monotonicity 47

48 Identification of Strata from the Observations Assignment Exposure Principal Strata 0 0 Complier (Co) or Never Taker (NT) 0 1 Always Taker (AT) 1 0 Never Taker (NT) 1 1 Complier (Co) or Always Taker (AT) 48

49 Calculating the Strata Probabilities Assignment Exposure Principal Strata 0 0 Pr[Co]+Pr[NT] = Pr[X=0 R=0] 0 1 Pr[AT] = Pr[X=1 R=0] 1 0 Pr[NT] = Pr[X=0 R=1] 1 1 Pr[Co]+Pr[AT] = Pr[X=1 R=1] Arithmetic shows that Pr[Co] = Pr[X=0 R=0] - Pr[X=0 R=1] or = Pr[X=1 R=1] - Pr[X=1 R=0] Thus it is possible to identify the strata probabilities 49

50 Complier Average Causal Effect for Continuous Endpoints 50

51 Extending Principal Stratification to Non-Linear Parameter Estimation The following slides show a way of arriving at estimates of the distributions of an endpoint in treated compliers and in untreated compliers These respective distributions can then be used to estimate parameters that characterize these distributions 51

52 Mixture Distributions R X Observed Mixture of Weights Y(X) 0 0 Y(X) R=0,X=0 Y(0) PS=Co and Y(0) PS=NT (p C0, p NT ) / (p C0 + p NT ) 0 1 Y(X) R=0,X=1 Y(1) PS=AT 1 0 Y(X) R=1,X=0 Y(0) PS=NT 1 1 Y(X) R=1,X=1 Y(1) PS=Co and Y(1) PS=AT (p C0, p AT ) / (p C0 + p AT ) 52

53 Arithmetic Shows The Potential Outcomes in the Compliers Can be Identified Using the Observable Distributions Potential Outcomes in the Compliers Mixture of Weights Y(0) PS=Co Y(X) R=0,X=0 and Y(X) PS=NT (1+p NT /p C0, -p NT /p Co ) Y(1) PS=Co Y(X) R=1,X=1 and Y(X) PS=AT (1+p Co /p AT, -p AT /p Co ) 53

54 Operationalizing: Weights To estimate the distribution, or parameters related to the distribution of the potential outcomes on the compliers, use the observeable samples but weight them in proportion to the weights on the previous slide Note: Some of the weights are negative! Negative weights not readily implemented with some software (e.g. R) 54

55 Principal Stratification Weights for Hazard Ratio Estimation To estimate the hazard ratio of Cox s model we apply weights to the observed follow-up and censoring indicators to create samples from treated and untreated individuals from the complier strata 55

56 Results of Simulations 56

57 Summary It is possible to estimate the causal hazard ratio with little or no bias in the setting of allor-nothing compliance I.T.T., Per Protocol and As Treated Estimators are Biased 57

58 Future Steps Instrumental Variables Time-varying exposure e.g. compliance that varies with time Weak instruments Principal Stratification Smooth weights to remove negative values Generalization of three principal strata (NT, Co, AT) to a latent variable that is continuous variable indicating propensity for obtaining treatment 58

59 59

60 60

61 Causal Inference Syntax Exposure X: 1=treatment, 0=no treatment Potential outcomes Y(0) is outcome if there is no treatment Y(1) is outcome if there is treatment We observe only one of the potential outcomes, Y(X) The others are counterfactuals 61

62 Resemblance to maximum partial likelihood Estimating Equation with I.V. ) ( ] ) ( ) ( [ t dn e t Y e t W Y W i n i n j X j n j X j j i j j = = = = τ β β Partial Likelihood Score ) ( ] ) ( ) ( [ t dn e t Y e t Y X X i n i n j X j n j X j j i j j = = = = τ β β 62

63 Simulation Results for Additive Omitted Covariate 63

64 Principal Stratification: When treatment is not accessible to controls Two classes One sided compliance 1. Compliers: do as assigned 2. Never takers: will not do treatment regardless of assignment 64

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