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STATISTICAL APPROACHES TO CAUSAL ANALYSIS Matthew McBee THE SAGE QUANTITATIVE RESEARCH KIT Statistical Approaches to Causal Analysis by Matthew McBee is the 10th volume in The SAGE Quantitative Research Kit. This book can be used together with the other titles in the Kit as a comprehensive guide to the process of doing quantitative research, but is equally valuable on its own as a practical introduction to causal inference in quantitative research. Editors of The SAGE Quantitative Research Kit: Malcolm Williams – Cardiff University, UK Richard D. Wiggins – UCL Social Research Institute, UK D. Betsy McCoach – University of Connecticut, USA Founding editor: The late W. Paul Vogt – Illinois State University, USA STATISTICAL APPROACHES TO CAUSAL ANALYSIS Matthew McBee THE SAGE QUANTITATIVE RESEARCH KIT SAGE Publications Ltd  Matthew McBee 2021 1 Oliver’s Yard 55 City Road This volume published as part of The SAGE Quantitative London EC1Y 1SP Research Kit (2021), edited by Malcolm Williams, Richard D. Wiggins and D. Betsy McCoach. SAGE Publications Inc. Apart from any fair dealing for the purposes of research, 2455 Teller Road private study, or criticism or review, as permitted under the Thousand Oaks, California 91320 Copyright, Designs and Patents Act, 1988, this publication may not be reproduced, stored or transmitted in any form, SAGE Publications India Pvt Ltd or by any means, without the prior permission in writing of B 1/I 1 Mohan Cooperative Industrial Area the publisher, or in the case of reprographic reproduction, Mathura Road in accordance with the terms of licences issued by New Delhi 110 044 the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to SAGE Publications Asia-Pacific Pte Ltd the publisher. 3 Church Street #10-04 Samsung Hub Singapore 049483 Library of Congress Control Number: 2020950514 Editor: Jai Seaman Assistant editor: Charlotte Bush British Library Cataloguing in Publication data Production editor: Manmeet Kaur Tura Copyeditor: QuADS Prepress Pvt Ltd A catalogue record for this book is available from the Proofreader: Elaine Leek British Library Indexer: Cathryn Pritchard Marketing manager: Susheel Gokarakonda Cover design: Shaun Mercier Typeset by: C&M Digitals (P) Ltd, Chennai, India Printed in the UK ISBN 978-1-5264-2473-0 At SAGE, we take sustainability seriously. Most of our products are printed in the UK using responsibly sourced papers and boards. When we print overseas, we ensure sustainable papers are used as measured by the PREPS grading system. We undertake an annual audit to monitor our sustainability. Dedication This book is dedicated to my mother, Mary Kay McBee, who passed away before she could see this book in print. Contents List of Figures and Tables xiii About the Author xxiii Acknowledgement xxv Preface xxvii 1 Introduction 1 Internal Validity 2 External Validity 2 Threats to Validity 3 Randomisation 6 Non-Experimental Research 7 A Pragmatic Definition of Causation 9 Prediction Versus Explanation 10 Causal Inference Requires External Information 11 Estimation Versus Hypothesis Testing 13 Prerequisites 13 Notation 14 The R statistical programming environment 14 Installing and Using R and RStudio 16 R Packages 16 Structure of This Book 17 2 Conditioning 19 Simulated Data Set 21 Bias and Inconsistency 21 Obtaining a Biased Estimate of the Causal Effect 25 Covariate Adjustment 26 Visualising Covariate Adjustment 27 Covariate Adjustment Depends on Strong Assumptions 30 viii STATISTICAL APPROACHES TO CAUSAL ANALYSIS Sample Selection 30 The Bias–Variance Trade-Off 32 Subclassification 33 Matching 35 Weighting 39 Computing the Weights 39 The Problem of Measurement 42 Classical Test Theory Model for Measurement Error 43 Reliability 44 Discussion 46 The ‘Curse of Dimensionality’ 47 3 Directed Acyclic Graphs 51 DAGs Are Not Path Models 53 DAG Terminology and Variable Roles 53 Exposure 54 Outcome 55 Mediator 55 Confounder 55 Proxy Confounder 56 Instrument 56 Competing Exposure 57 Collider 58 d-Separation, d-Connectedness and Statistical Independence 58 Conditioning 59 Conditioning on Colliders 61 Colliders and the Real World 62 Spurious Paths 64 Unobservables 69 Conditioning on Mediators 69 Criteria for Valid Causal Inference 72 Back-Door Criterion 72 Front-Door Criterion 73 Minimal and Sufficient Adjustment Sets 74 Simultaneous Estimation of Causal Effects 75 Measurement Error and DAGs 76 Using DAGitty 77 Practical Recommendations 81 contents ix 4 Rubin’s Causal Model and the Propensity Score 85 The Counterfactual Framework 86 Defining Causal Effects Under Rubin’s Causal Model 87 The Fundamental Problem of Causal Inference 88 Ignorability 90 Bias When Ignorability Does Not Exist 92 Baseline Bias 92 Differential Treatment Effect Bias 93 Conditional Ignorability 93 Conditional Treatment Effects 94 Example: Estimating ATT, ATU and ATE via Linear Regression 95 The Propensity Score 97 Approximating an Experiment 98 Simulated Data Set 101 Propensity Scores 101 Estimating Propensity Scores via Logistic Regression 102 Solving the Curse of Dimensionality 108 Propensity Score Estimation via Boosted Classification Trees 108 Comparing the Two Sets of Propensity Scores 115 Assumptions of Propensity Score Methods 116 Ignorability 116 Stable Unit Treatment Value Assumption 116 Positivity 117 5 Propensity Score Analysis 119 Simulated Data Set 120 Descriptive Statistics and Biased Treatment Effect Estimate 121 Obtaining a Biased Estimate of the Treatment Effect 121 Propensity Score Matching 124 Matching Algorithms 124 Estimating Treatment Effects with Matching 129 Example Analysis 129 Stratifying on the Propensity Score 131 Weighting with the propensity score 136 From Propensity Scores to Weights 139 Stabilised Weights and Truncated Weights 141 Example of an Analysis Using Propensity Score Weights 143 Doubly Robust Estimation 147