Comparing covariate adjustment in interventional and observational - PowerPoint PPT Presentation

Comparing covariate adjustment in interventional and observational studies Markus Kalisch, Seminar für Statistik, ETH Zürich

What is the total causal effect ? Treatment 𝑌 Outcome 𝑍  If we apply treatment 𝑌 , how will outcome 𝑍 change ?  Data collection: - observational study - interventional study (RCT) 1

Outline for the rest of the talk  Total causal effect and covariate adjustment  Issues in observational studies  Issues in interventional studies  Insights from recent theoretical developments 2

Causal Model: How the real world might look like  We use directed acyclic graphs (DAG) – no feedback loops  Example: DAG 𝐻 𝑌 5 𝑌 2 𝑌 3 𝑍 𝑌 1  Terminology: 𝑌 4 Set of all variables: 𝒀 = {𝑌 1 , 𝑌 2 , … , 𝑌 5 , 𝑍} Path: 𝑌 1 , 𝑌 2 , 𝑌 3 , 𝑍 Directed p ath = “causal - path”: 𝑌 1 , 𝑌 2 , 𝑌 3 Not directed path = Non-causal path: (𝑌 4 ,𝑌 3 ,𝑍) Parents p𝑏 𝑌 3 = {𝑌 2 , 𝑌 4 } , Children cℎ 𝑌 1 = 𝑌 2 Think of family tree Ancestor 𝑏𝑜 , Descendant 𝑒𝑓 , Non-descendants 𝑜𝑒 3

More details: Structural Equation Model (SEM)  Example of SEM: 𝑌 1 = 𝑂 1 Causal 𝑌 2 = 4𝑌 1 + 𝑂 2 interpretation 𝑂 1 , 𝑂 2 ∼ 𝑂 0,1 𝑗𝑗𝑒  Visualization of causal structure : 𝑌 1 𝑌 2  Difference to arbitrary hierarchical system of equations: Due to causal interpretation, solving for a variable on the RHS is not meaningful in SEM. 4

Quantifying the total causal effect Define intervention distribution by replacing (some) structural equations  do-Operator Reference: Pearl, J. (2009). Causality: Models, Reasoning and Inference. 2nd edition. Cambridge Univ. Press. E.g. «intervention on 𝑌 »:  Old SEM: 𝑇 with equation 𝑌 = 2 + 𝑌 5 + 𝑂 𝑌  New SEM: መ 𝑇 with equation 𝑌 = 4  New SEM generates new distribution: 𝑄 መ 𝑇 (𝒀) = 𝑄 𝑇 (𝒀|𝑒𝑝 𝑌 = 4 ) and in particular P(𝑍|𝑒𝑝(𝑌 = 4))  Final goal: Estimate intervention distribution given observational data  Oftentimes: Expectation is enough – e.g. 𝐹(𝑍|𝑒𝑝 𝑌 = 4 ) 5

Covariate adjustment: Adjustment set  Idea: Identify intervention effects by only using conditional probabilities / expectations «do» No «do» 𝐶 𝑄 𝑍 = 𝑧 𝑒𝑝 𝑌 = 𝑦 = ෍ 𝑄 𝑍 = 𝑧 𝑌 = 𝑦, 𝐶 = 𝑐 𝑄(𝐶 = 𝑐) Adjustment 𝑐∈𝐶 𝑌 𝑍 set  Practice: Often interested in 𝐹 𝑍 = 𝑧 𝑒𝑝 𝑌 = 𝑦  Can show for multivariate Gaussian density: = 𝛽 + 𝛿𝑦 + 𝛾 𝑈 𝐹 𝐶 𝐹 𝑍 𝑒𝑝 𝑌 = 𝑦 d  Total Causal Effect: 𝑒𝑦 𝐹 𝑍 𝑒𝑝 𝑌 = 𝑦 = 𝛿 This is the regression coefficient of 𝑌 in the regression of 𝑍 on 𝑌 and 𝐶 6

Outline for the rest of the talk  Total causal effect and covariate adjustment  Issues in observational studies  Issues in interventional studies  Insights from recent theoretical developments 7

Causal Diagram: Example 1 - confounder Lab 1,2,… Treatment Outcome Should we add the lab information as covariate ? 8

Example 1 in numbers 𝑎 𝑌 𝑍  𝜁 𝑌 ∼ 𝑂(0,1) , 𝜁 𝑎 ∼ 𝑂(0,1) , 𝜁 𝑍 ~ 𝑂(0,1) independent  True causal system: 𝑎 = 𝜁 𝑎 𝑌 = 0.7 ∗ 𝑎 + 𝜁 𝑌 𝑍 = 1 ∗ 𝑌 + 0.5 ∗ 𝑎 + 𝜁 𝑍  True causal effect of 𝑌 on 𝑍 : 1 If we increase 𝑌 by one unit, 𝑍 will also increase by one unit  Can we estimate the true causal effect with a linear regression ? 9

Example 1 in numbers 𝑎 𝑌 𝑍  True causal effect of 𝑌 on 𝑍 : 1  Simple Regression: 𝑚𝑛(𝑍 ~ 𝑌) Incorrect  Multiple Regression: 𝑚𝑛(𝑍~𝑌 + 𝑎) Missing the confounder introduced a bias ! Correct 10

Causal Diagram: Example 2 – selection variable Treatment Outcome Follow-up test Should we add the info of the follow-up test as covariate ? 11

𝑌 𝑍 Example 2 in numbers 𝑎  𝜁 𝑌 ∼ 𝑂(0,1) , 𝜁 𝑎 ∼ 𝑂(0,1) , 𝜁 𝑍 ~ 𝑂(0,1) independent  True causal system: X = 𝜁 𝑌 Y = 0.7 ∗ 𝑌 + 𝜁 𝑍 Z = 0.8 ∗ 𝑌 + 0.5 ∗ 𝑍 + 𝜁 𝑎  True causal effect of 𝑌 on 𝑍 : 0.7 If we increase 𝑌 by one unit, 𝑍 will also increase by 0.7 units  Can we estimate the true causal effect with a linear regression ? 12

𝑌 𝑍 Example 2 in numbers 𝑎  True causal effect of 𝑌 on 𝑍 : 0.7  Simple Regression: 𝑚𝑛(𝑍 ~ 𝑌) Correct  Multiple Regression: 𝑚𝑛(𝑍~𝑌 + 𝑎) Including the selection variable Incorrect introduced a bias ! 13

“Parent Criterion” (PC)  Take parents of 𝑌 as adjustment set (special case of Pearl’s back -door criterion)  Sufficient but not complete  Example 1: 𝑎 𝑌 𝑍 PC: 𝑎 is a valid adjustment set; would {} be a valid adjustment set, too → ??? (perhaps we can not measure 𝑎 although we know it exists)  Example 2: 𝑌 𝑍 𝑎 PC: {} is a valid adjustment set; would 𝑎 be a valid adjustment set, too → ??? 14

Conclusion 1 In observational studies: Judging if an adjustment set is valid is not trivial 15

Outline for the rest of the talk  Total causal effect and covariate adjustment  Issues in observational studies  Issues in interventional studies  Insights from recent theoretical developments 16

RCT: Evaluation Treatment Treatment Control Control Treatment Control Treatment Control Treatment Control  Cage: Experimental Unit  5 cages with treatment ( 𝑌 = 1 ), 5 cages with control ( 𝑌 = 0 )  Randomize allocation: In causal diagram think of “deleting all incoming edges to 𝑌 ” 17

RCT in causal diagram 𝑎 1 𝑎 1 𝒀 𝐵 𝒁 𝒀 𝐵 𝒁 RCT 𝐷 𝐷 𝑎 2 𝑎 2 𝐶 𝐶 PC: PC: Valid adjustment set Valid adjustement set is {} is {𝑎 1 , 𝑎 2 } PC: {} is always valid adjustment set after randomization 18

RCT: Evaluation Treatment Treatment Control Control Treatment Control Treatment Control Treatment Control  Given a proper design, we can do a two-sample t-test with two groups (i.e. empty adjustment set).  What if we have more covariates (sex, age, intermediate blood test, follow-up information, …) ?  Is it always better to add covariates to the analysis ? 19

Messing up the evaluation of a randomized controlled trial (RCT)  You can bias ( “mess up” ), the analysis by adding the “wrong” covariates .  RCT : It is always safe to add no covariates to the analysis.  Adding the “right” covariates might increase precision. 20

Causal Diagram: Example 1 Treatment Intermediate Blood T est Outcome Should we add the intermediate blood test as covariate ? 21

Example 1 in numbers 𝑌 𝑎 𝑍  𝜁 𝑌 ∼ 𝑂(0,1) , 𝜁 𝑎 ∼ 𝑂(0,1) , 𝜁 𝑍 ~ 𝑂(0,1) independent  True causal system: 𝑌 = 𝜁 𝑌 𝑎 = 2 ∗ 𝑌 + 𝜁 𝑎 𝑍 = 0.5 ∗ 𝑎 + 𝜁 𝑍  True causal effect of 𝑌 on 𝑍 : 2 ∗ 0.5 = 1 If we increase 𝑌 by one unit, 𝑍 will also increase by one unit  Can we estimate the true causal effect with a linear regression ? 22

Example 1 in numbers 𝑌 𝑎 𝑍  True causal effect of 𝑌 on 𝑍 : 2 ∗ 0.5 = 1  Simple Regression: 𝑚𝑛(𝑍 ~ 𝑌) Correct  Multiple Regression: 𝑚𝑛(𝑍~𝑌 + 𝑎) Adding a covariate introduced a bias ! Incorrect 23

Causal Diagram: Example 2 Lab 1,2,… Treatment Outcome Should we add the lab information as covariate ? 24

Example 2 in numbers 𝑌 𝑍 𝑎  𝜁 𝑌 ∼ 𝑂(0,1) , 𝜁 𝑎 ∼ 𝑂(0,1) , 𝜁 𝑍 ~ 𝑂(0,1) independent  True causal system: 𝑌 = 𝜁 𝑌 𝑎 = 𝜁 𝑎 𝑍 = 1 ∗ 𝑌 + 0.5 ∗ 𝑎 + 𝜁 𝑍  True causal effect of 𝑌 on 𝑍 : 1 If we increase 𝑌 by one unit, 𝑍 will also increase by one unit  Can we estimate the true causal effect with a linear regression ? 25

Example 2 in numbers 𝑌 𝑍 𝑎  True causal effect of 𝑌 on 𝑍 : 1  Simple Regression: 𝑚𝑛(𝑍~𝑌) Correct  Multiple Regression: 𝑚𝑛(𝑍~𝑌 + 𝑎) • Adding a covariate did not introduce a bias Correct • Confidence interval with covariate is slightly smaller (0.12 vs 0.14) 26

Summary  Adding the wrong variable will introduce a bias “ Wrong variable ”: On causal path from 𝑌 to 𝑍 or «descendants» of those nodes ( post-intervention )  Adding the right variables might increase precision 𝑎 1 𝑎 2 “ Right variable ”: Parents of nodes on causal path from 𝑌 to 𝑍 ( pre-intervention ) 𝒀 𝐵 𝒁 𝐷  Problem in practice: 𝐶 Usually don’t know true causal structure! What are “right” and “wrong” variables ?  If in doubt, don’t use covariate !  Safe variables: Things that clearly “preceded” 𝑌 (e.g. gender) 27

Outline for the rest of the talk  Total causal effect and covariate adjustment  Issues in observational studies  Issues in interventional studies  Insights from recent theoretical developments 28

Adjustment Criteria Getting the “right estimate”:  given causal structure, criterion to check if a set is a valid adjustment set  assuming causal structure is a strong assumption in practice  discussion can shift to discussing reasonable causal structures  Pearl’s back -door criterion  Generalized Adjustment criterion 29