Carlo Berzuini and Philip Dawid, Statistical Labority, centre for Mathematical Sciences, University of Cambridge, UK. Luisa Bernardinelli, MRC Biostatistics Unit, Institute of Public Health, Cambridge, UK.

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• List of contributors xvAn overview of statistical causality xviiCarlo Berzuini, Philip Dawid and Luisa Bernardinelli1 Statistical causality: Some historical remarks 1D.R. Cox1.1 Introduction 11.2 Key issues 21.3 Rothamsted view 21.4 An earlier controversy and its implications 31.5 Three versions of causality 41.6 Conclusion 4References 42 The language of potential outcomes 6Arvid Sjölander2.1 Introduction 62.2 Definition of causal effects through potential outcomes 72.2.1 Subject-specific causal effects 72.2.2 Population causal effects 82.2.3 Association versus causation 92.3 Identification of population causal effects 92.3.1 Randomized experiments 92.3.2 Observational studies 112.4 Discussion 11References 133 Structural equations, graphs and interventions 15Ilya Shpitser3.1 Introduction 153.2 Structural equations, graphs, and interventions 163.2.1 Graph terminology 163.2.2 Markovian models 173.2.3 Latent projections and semi-Markovian models 193.2.4 Interventions in semi-Markovian models 193.2.5 Counterfactual distributions in NPSEMs 203.2.6 Causal diagrams and counterfactual independence 223.2.7 Relation to potential outcomes 22References 234 The decision-theoretic approach to causal inference 25Philip Dawid4.1 Introduction 254.2 Decision theory and causality 264.2.1 A simple decision problem 264.2.2 Causal inference 274.3 No confounding 284.4 Confounding 294.4.1 Unconfounding 294.4.2 Nonconfounding 304.4.3 Back-door formula 314.5 Propensity analysis 334.6 Instrumental variable 344.6.1 Linear model 364.6.2 Binary variables 364.7 Effect of treatment of the treated 374.8 Connections and contrasts 374.8.1 Potential responses 374.8.2 Causal graphs 394.9 Postscript 40Acknowledgements 40References 405 Causal inference as a prediction problem: Assumptions, identification and evidence synthesis 43Sander Greenland5.1 Introduction 435.2 A brief commentary on developments since 1970 445.2.1 Potential outcomes and missing data 455.2.2 The prognostic view 455.3 Ambiguities of observational extensions 465.4 Causal diagrams and structural equations 475.5 Compelling versus plausible assumptions, models and inferences 475.6 Nonidentification and the curse of dimensionality 505.7 Identification in practice 515.8 Identification and bounded rationality 535.9 Conclusion 54Acknowledgments 55References 556 Graph-based criteria of identifiability of causal questions 59Ilya Shpitser6.1 Introduction 596.2 Interventions from observations 596.3 The back-door criterion, conditional ignorability, and covariate adjustment 616.4 The front-door criterion 636.5 Do-calculus 646.6 General identification 656.7 Dormant independences and post-truncation constraints 68References 697 Causal inference from observational data: A Bayesian predictive approach 71Elja Arjas7.1 Background 717.2 A model prototype 727.3 Extension to sequential regimes 767.4 Providing a causal interpretation: Predictive inference from data 807.5 Discussion 82Acknowledgement 83References 838 Assessing dynamic treatment strategies 85Carlo Berzuini, Philip Dawid, and Vanessa Didelez8.1 Introduction 858.2 Motivating example 868.3 Descriptive versus causal inference 878.4 Notation and problem definition 888.5 HIV example continued 898.6 Latent variables 898.7 Conditions for sequential plan identifiability 908.7.1 Stability 908.7.2 Positivity 918.8 Graphical representations of dynamic plans 928.9 Abdominal aortic aneurysm surveillance 948.10 Statistical inference and computation 958.11 Transparent actions 978.12 Refinements 988.13 Discussion 99Acknowledgements 99References 999 Causal effects and natural laws: Towards a conceptualization of causal counterfactuals for nonmanipulable exposures, with application to the effects of race and sex 101Tyler J. VanderWeele and Miguel A. Hernán9.1 Introduction 1019.2 Laws of nature and contrary to fact statements 1029.3 Association and causation in the social and biomedical sciences 1039.4 Manipulation and counterfactuals 1039.5 Natural laws and causal effects 1049.6 Consequences of randomization 1079.7 On the causal effects of sex and race 1089.8 Discussion 111Acknowledgements 112References 11210 Cross-classifications by joint potential outcomes 114Arvid Sjölander10.1 Introduction 11410.2 Bounds for the causal treatment effect in randomized trials with imperfect compliance 11510.3 Identifying the complier causal effect in randomized trials with imperfect compliance 11910.4 Defining the appropriate causal effect in studies suffering from truncation by death 12110.5 Discussion 123References 12411 Estimation of direct and indirect effects 126Stijn Vansteelandt11.1 Introduction 12611.2 Identification of the direct and indirect effect 12711.2.1 Definitions 12711.2.2 Identification 12911.3 Estimation of controlled direct effects 13211.3.1 G-computation 13211.3.2 Inverse probability of treatment weighting 13311.3.3 G-estimation for additive and multiplicative models 13711.3.4 G-estimation for logistic models 14111.3.5 Case-control studies 14211.3.6 G-estimation for additive hazard models 14311.4 Estimation of natural direct and indirect effects 14611.5 Discussion 147Acknowledgements 147References 14812 The mediation formula: A guide to the assessment of causal pathways in nonlinear models 151Judea Pearl12.1 Mediation: Direct and indirect effects 15112.1.1 Direct versus total effects 15112.1.2 Controlled direct effects 15212.1.3 Natural direct effects 15412.1.4 Indirect effects 15612.1.5 Effect decomposition 15712.2 The mediation formula: A simple solution to a thorny problem 15712.2.1 Mediation in nonparametric models 15712.2.2 Mediation effects in linear, logistic, and probit models 15912.2.3 Special cases of mediation models 16412.2.4 Numerical example 16912.3 Relation to other methods 17012.3.1 Methods based on differences and products 17012.3.2 Relation to the principal-strata direct effect 17112.4 Conclusions 173Acknowledgments 174References 17513 The sufficient cause framework in statistics, philosophy and the biomedical and social sciences 180Tyler J. VanderWeele13.1 Introduction 18013.2 The sufficient cause framework in philosophy 18113.3 The sufficient cause framework in epidemiology and biomedicine 18113.4 The sufficient cause framework in statistics 18513.5 The sufficient cause framework in the social sciences 18513.6 Other notions of sufficiency and necessity in causal inference 18713.7 Conclusion 188Acknowledgements 189References 18914 Analysis of interaction for identifying causal mechanisms 192Carlo Berzuini, Philip Dawid, Hu Zhang and Miles Parkes14.1 Introduction 19214.2 What is a mechanism? 19314.3 Statistical versus mechanistic interaction 19314.4 Illustrative example 19414.5 Mechanistic interaction defined 19614.6 Epistasis 19714.7 Excess risk and superadditivity 19714.8 Conditions under which excess risk and superadditivity indicate the presence of mechanistic interaction 20014.9 Collapsibility 20114.10 Back to the illustrative study 20214.11 Alternative approaches 20414.12 Discussion 204Ethics statement 205Financial disclosure 205References 20615 Ion channels as a possible mechanism of neurodegeneration in multiple sclerosis 208Luisa Bernardinelli, Carlo Berzuini, Luisa Foco, and Roberta Pastorino15.1 Introduction 20815.2 Background 20915.3 The scientific hypothesis 20915.4 Data 21015.5 A simple preliminary analysis 21115.6 Testing for qualitative interaction 21315.7 Discussion 214Acknowledgments 216References 21616 Supplementary variables for causal estimation 218Roland R. Ramsahai16.1 Introduction 21816.2 Multiple expressions for causal effect 22016.3 Asymptotic variance of causal estimators 22216.4 Comparison of causal estimators 22216.4.1 Supplement C with L or not 22316.4.2 Supplement L with C or not 22416.4.3 Replace C with L or not 22516.5 Discussion 226Acknowledgements 226Appendices 22716.A Estimator given all X’s recorded 22716.B Derivations of asymptotic variances 22716.C Expressions with correlation coefficients 22916.D Derivation of I’s 23016.E Relation between ρ2 rl|t and ρ2 rl|c 231References 23217 Time-varying confounding: Some practical considerations in a likelihood framework 234Rhian Daniel, Bianca De Stavola and Simon Cousens17.1 Introduction 23417.2 General setting 23517.2.1 Notation 23517.2.2 Observed data structure 23517.2.3 Intervention strategies 23617.2.4 Potential outcomes 23717.2.5 Time-to-event outcomes 23717.2.6 Causal estimands 23817.3 Identifying assumptions 23817.4 G-computation formula 23917.4.1 The formula 23917.4.2 Plug-in regression estimation 24017.5 Implementation by Monte Carlo simulation 24217.5.1 Simulating an end-of-study outcome 24217.5.2 Simulating a time-to-event outcome 24217.5.3 Inference 24217.5.4 Losses to follow-up 24317.5.5 Software 24317.6 Analyses of simulated data 24317.6.1 The data 24317.6.2 Regimes to be compared 24417.6.3 Parametric modelling choices 24517.6.4 Results 24617.7 Further considerations 24917.7.1 Parametric model misspecification 24917.7.2 Competing events 24917.7.3 Unbalanced measurement times 25017.8 Summary 251References 25118 ‘Natural experiments’ as a means of testing causal inferences 253Michael Rutter18.1 Introduction 25318.2 Noncausal interpretations of an association 25318.3 Dealing with confounders 25518.4 ‘Natural experiments’ 25618.4.1 Genetically sensitive designs 25718.4.2 Children of twins (CoT) design 25918.4.3 Strategies to identify the key environmental risk feature 26118.4.4 Designs for dealing with selection bias 26318.4.5 Instrumental variables to rule out reverse causation 26418.4.6 Regression discontinuity (RD) designs to deal with unmeasured confounders 26518.5 Overall conclusion on ‘natural experiments’ 26618.5.1 Supported causes 26618.5.2 Disconfirmed causes 267Acknowledgement 267References 26819 Nonreactive and purely reactive doses in observational studies 273Paul R. Rosenbaum19.1 Introduction: Background, example 27319.1.1 Does a dose–response relationship provide information that distinguishes treatment effects from biases due to unmeasured covariates? 27319.1.2 Is more chemotherapy for ovarian cancer more effective or more toxic? 27419.2 Various concepts of dose 27719.2.1 Some notation: Covariates, outcomes, and treatment assignment in matched pairs 27719.2.2 Reactive and nonreactive doses of treatment 27819.2.3 Three test statistics that use doses in different ways 27919.2.4 Randomization inference in randomized experiments 28019.2.5 Sensitivity analysis 28119.2.6 Sensitivity analysis in the example 28319.3 Design sensitivity 28419.3.1 What is design sensitivity? 28419.3.2 Comparison of design sensitivity with purely reactive doses 28619.4 Summary 287References 28720 Evaluation of potential mediators in randomised trials of complex interventions (psychotherapies) 290Richard Emsley and Graham Dunn20.1 Introduction 29020.2 Potential mediators in psychological treatment trials 29120.3 Methods for mediation in psychological treatment trials 29320.4 Causal mediation analysis using instrumental variables estimation 29720.5 Causal mediation analysis using principal stratification 30120.6 Our motivating example: The SoCRATES trial 30220.6.1 What are the joint effects of sessions attended and therapeutic alliance on the PANSS score at 18 months? 30320.6.2 What is the direct effect of random allocation on the PANSS score at 18 months and how is this influenced by the therapeutic alliance? 30420.6.3 Is the direct effect of the number of sessions attended on the PANSS score at 18 months influenced by therapeutic alliance? 30520.7 Conclusions 305Acknowledgements 306References 30721 Causal inference in clinical trials 310Krista Fischer and Ian R. White21.1 Introduction 31021.2 Causal effect of treatment in randomized trials 31221.2.1 Observed data and notation 31221.2.2 Defining the effects of interest via potential outcomes 31221.2.3 Adherence-adjusted ITT analysis 31421.3 Estimation for a linear structural mean model 31621.3.1 A general estimation procedure 31621.3.2 Identifiability and closed-form estimation of the parameters in a linear SMM 31721.3.3 Analysis of the EPHT trial 31921.4 Alternative approaches for causal inference in randomized trials comparing experimental treatment with a control 32121.4.1 Principal stratification 32121.4.2 SMM for the average treatment effect on the treated (ATT) 32221.5 Discussion 324References 32522 Causal inference in time series analysis 327Michael Eichler22.1 Introduction 32722.2 Causality for time series 32822.2.1 Intervention causality 32822.2.2 Structural causality 33122.2.3 Granger causality 33222.2.4 Sims causality 33422.3 Graphical representations for time series 33522.3.1 Conditional distributions and chain graphs 33622.3.2 Path diagrams and Granger causality graphs 33722.3.3 Markov properties for Granger causality graphs 33822.4 Representation of systems with latent variables 33922.4.1 Marginalization 34122.4.2 Ancestral graphs 34222.5 Identification of causal effects 34322.6 Learning causal structures 34622.7 A new parametric model 34922.8 Concluding remarks 351References 35223 Dynamic molecular networks and mechanisms in the biosciences: A statistical framework 355Clive G. Bowsher23.1 Introduction 35523.2 SKMs and biochemical reaction networks 35623.3 Local independence properties of SKMs 35823.3.1 Local independence and kinetic independence graphs 35823.3.2 Local independence and causal influence 36123.4 Modularisation of SKMs 36223.4.1 Modularisations and dynamic independence 36223.4.2 MIDIA Algorithm 36323.5 Illustrative example – MAPK cell signalling 36523.6 Conclusion 36923.7 Appendix: SKM regularity conditions 369Acknowledgements 370References 370Index 371

Carlo Berzuini and Philip Dawid, Statistical Labority, centre for Mathematical Sciences, University of Cambridge, UK. Luisa Bernardinelli, MRC Biostatistics Unit, Institute of Public Health, Cambridge, UK.

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• List of contributors xvAn overview of statistical causality xviiCarlo Berzuini, Philip Dawid and Luisa Bernardinelli1 Statistical causality: Some historical remarks 1D.R. Cox1.1 Introduction 11.2 Key issues 21.3 Rothamsted view 21.4 An earlier controversy and its implications 31.5 Three versions of causality 41.6 Conclusion 4References 42 The language of potential outcomes 6Arvid Sjölander2.1 Introduction 62.2 Definition of causal effects through potential outcomes 72.2.1 Subject-specific causal effects 72.2.2 Population causal effects 82.2.3 Association versus causation 92.3 Identification of population causal effects 92.3.1 Randomized experiments 92.3.2 Observational studies 112.4 Discussion 11References 133 Structural equations, graphs and interventions 15Ilya Shpitser3.1 Introduction 153.2 Structural equations, graphs, and interventions 163.2.1 Graph terminology 163.2.2 Markovian models 173.2.3 Latent projections and semi-Markovian models 193.2.4 Interventions in semi-Markovian models 193.2.5 Counterfactual distributions in NPSEMs 203.2.6 Causal diagrams and counterfactual independence 223.2.7 Relation to potential outcomes 22References 234 The decision-theoretic approach to causal inference 25Philip Dawid4.1 Introduction 254.2 Decision theory and causality 264.2.1 A simple decision problem 264.2.2 Causal inference 274.3 No confounding 284.4 Confounding 294.4.1 Unconfounding 294.4.2 Nonconfounding 304.4.3 Back-door formula 314.5 Propensity analysis 334.6 Instrumental variable 344.6.1 Linear model 364.6.2 Binary variables 364.7 Effect of treatment of the treated 374.8 Connections and contrasts 374.8.1 Potential responses 374.8.2 Causal graphs 394.9 Postscript 40Acknowledgements 40References 405 Causal inference as a prediction problem: Assumptions, identification and evidence synthesis 43Sander Greenland5.1 Introduction 435.2 A brief commentary on developments since 1970 445.2.1 Potential outcomes and missing data 455.2.2 The prognostic view 455.3 Ambiguities of observational extensions 465.4 Causal diagrams and structural equations 475.5 Compelling versus plausible assumptions, models and inferences 475.6 Nonidentification and the curse of dimensionality 505.7 Identification in practice 515.8 Identification and bounded rationality 535.9 Conclusion 54Acknowledgments 55References 556 Graph-based criteria of identifiability of causal questions 59Ilya Shpitser6.1 Introduction 596.2 Interventions from observations 596.3 The back-door criterion, conditional ignorability, and covariate adjustment 616.4 The front-door criterion 636.5 Do-calculus 646.6 General identification 656.7 Dormant independences and post-truncation constraints 68References 697 Causal inference from observational data: A Bayesian predictive approach 71Elja Arjas7.1 Background 717.2 A model prototype 727.3 Extension to sequential regimes 767.4 Providing a causal interpretation: Predictive inference from data 807.5 Discussion 82Acknowledgement 83References 838 Assessing dynamic treatment strategies 85Carlo Berzuini, Philip Dawid, and Vanessa Didelez8.1 Introduction 858.2 Motivating example 868.3 Descriptive versus causal inference 878.4 Notation and problem definition 888.5 HIV example continued 898.6 Latent variables 898.7 Conditions for sequential plan identifiability 908.7.1 Stability 908.7.2 Positivity 918.8 Graphical representations of dynamic plans 928.9 Abdominal aortic aneurysm surveillance 948.10 Statistical inference and computation 958.11 Transparent actions 978.12 Refinements 988.13 Discussion 99Acknowledgements 99References 999 Causal effects and natural laws: Towards a conceptualization of causal counterfactuals for nonmanipulable exposures, with application to the effects of race and sex 101Tyler J. VanderWeele and Miguel A. Hernán9.1 Introduction 1019.2 Laws of nature and contrary to fact statements 1029.3 Association and causation in the social and biomedical sciences 1039.4 Manipulation and counterfactuals 1039.5 Natural laws and causal effects 1049.6 Consequences of randomization 1079.7 On the causal effects of sex and race 1089.8 Discussion 111Acknowledgements 112References 11210 Cross-classifications by joint potential outcomes 114Arvid Sjölander10.1 Introduction 11410.2 Bounds for the causal treatment effect in randomized trials with imperfect compliance 11510.3 Identifying the complier causal effect in randomized trials with imperfect compliance 11910.4 Defining the appropriate causal effect in studies suffering from truncation by death 12110.5 Discussion 123References 12411 Estimation of direct and indirect effects 126Stijn Vansteelandt11.1 Introduction 12611.2 Identification of the direct and indirect effect 12711.2.1 Definitions 12711.2.2 Identification 12911.3 Estimation of controlled direct effects 13211.3.1 G-computation 13211.3.2 Inverse probability of treatment weighting 13311.3.3 G-estimation for additive and multiplicative models 13711.3.4 G-estimation for logistic models 14111.3.5 Case-control studies 14211.3.6 G-estimation for additive hazard models 14311.4 Estimation of natural direct and indirect effects 14611.5 Discussion 147Acknowledgements 147References 14812 The mediation formula: A guide to the assessment of causal pathways in nonlinear models 151Judea Pearl12.1 Mediation: Direct and indirect effects 15112.1.1 Direct versus total effects 15112.1.2 Controlled direct effects 15212.1.3 Natural direct effects 15412.1.4 Indirect effects 15612.1.5 Effect decomposition 15712.2 The mediation formula: A simple solution to a thorny problem 15712.2.1 Mediation in nonparametric models 15712.2.2 Mediation effects in linear, logistic, and probit models 15912.2.3 Special cases of mediation models 16412.2.4 Numerical example 16912.3 Relation to other methods 17012.3.1 Methods based on differences and products 17012.3.2 Relation to the principal-strata direct effect 17112.4 Conclusions 173Acknowledgments 174References 17513 The sufficient cause framework in statistics, philosophy and the biomedical and social sciences 180Tyler J. VanderWeele13.1 Introduction 18013.2 The sufficient cause framework in philosophy 18113.3 The sufficient cause framework in epidemiology and biomedicine 18113.4 The sufficient cause framework in statistics 18513.5 The sufficient cause framework in the social sciences 18513.6 Other notions of sufficiency and necessity in causal inference 18713.7 Conclusion 188Acknowledgements 189References 18914 Analysis of interaction for identifying causal mechanisms 192Carlo Berzuini, Philip Dawid, Hu Zhang and Miles Parkes14.1 Introduction 19214.2 What is a mechanism? 19314.3 Statistical versus mechanistic interaction 19314.4 Illustrative example 19414.5 Mechanistic interaction defined 19614.6 Epistasis 19714.7 Excess risk and superadditivity 19714.8 Conditions under which excess risk and superadditivity indicate the presence of mechanistic interaction 20014.9 Collapsibility 20114.10 Back to the illustrative study 20214.11 Alternative approaches 20414.12 Discussion 204Ethics statement 205Financial disclosure 205References 20615 Ion channels as a possible mechanism of neurodegeneration in multiple sclerosis 208Luisa Bernardinelli, Carlo Berzuini, Luisa Foco, and Roberta Pastorino15.1 Introduction 20815.2 Background 20915.3 The scientific hypothesis 20915.4 Data 21015.5 A simple preliminary analysis 21115.6 Testing for qualitative interaction 21315.7 Discussion 214Acknowledgments 216References 21616 Supplementary variables for causal estimation 218Roland R. Ramsahai16.1 Introduction 21816.2 Multiple expressions for causal effect 22016.3 Asymptotic variance of causal estimators 22216.4 Comparison of causal estimators 22216.4.1 Supplement C with L or not 22316.4.2 Supplement L with C or not 22416.4.3 Replace C with L or not 22516.5 Discussion 226Acknowledgements 226Appendices 22716.A Estimator given all X’s recorded 22716.B Derivations of asymptotic variances 22716.C Expressions with correlation coefficients 22916.D Derivation of I’s 23016.E Relation between ρ2 rl|t and ρ2 rl|c 231References 23217 Time-varying confounding: Some practical considerations in a likelihood framework 234Rhian Daniel, Bianca De Stavola and Simon Cousens17.1 Introduction 23417.2 General setting 23517.2.1 Notation 23517.2.2 Observed data structure 23517.2.3 Intervention strategies 23617.2.4 Potential outcomes 23717.2.5 Time-to-event outcomes 23717.2.6 Causal estimands 23817.3 Identifying assumptions 23817.4 G-computation formula 23917.4.1 The formula 23917.4.2 Plug-in regression estimation 24017.5 Implementation by Monte Carlo simulation 24217.5.1 Simulating an end-of-study outcome 24217.5.2 Simulating a time-to-event outcome 24217.5.3 Inference 24217.5.4 Losses to follow-up 24317.5.5 Software 24317.6 Analyses of simulated data 24317.6.1 The data 24317.6.2 Regimes to be compared 24417.6.3 Parametric modelling choices 24517.6.4 Results 24617.7 Further considerations 24917.7.1 Parametric model misspecification 24917.7.2 Competing events 24917.7.3 Unbalanced measurement times 25017.8 Summary 251References 25118 ‘Natural experiments’ as a means of testing causal inferences 253Michael Rutter18.1 Introduction 25318.2 Noncausal interpretations of an association 25318.3 Dealing with confounders 25518.4 ‘Natural experiments’ 25618.4.1 Genetically sensitive designs 25718.4.2 Children of twins (CoT) design 25918.4.3 Strategies to identify the key environmental risk feature 26118.4.4 Designs for dealing with selection bias 26318.4.5 Instrumental variables to rule out reverse causation 26418.4.6 Regression discontinuity (RD) designs to deal with unmeasured confounders 26518.5 Overall conclusion on ‘natural experiments’ 26618.5.1 Supported causes 26618.5.2 Disconfirmed causes 267Acknowledgement 267References 26819 Nonreactive and purely reactive doses in observational studies 273Paul R. Rosenbaum19.1 Introduction: Background, example 27319.1.1 Does a dose–response relationship provide information that distinguishes treatment effects from biases due to unmeasured covariates? 27319.1.2 Is more chemotherapy for ovarian cancer more effective or more toxic? 27419.2 Various concepts of dose 27719.2.1 Some notation: Covariates, outcomes, and treatment assignment in matched pairs 27719.2.2 Reactive and nonreactive doses of treatment 27819.2.3 Three test statistics that use doses in different ways 27919.2.4 Randomization inference in randomized experiments 28019.2.5 Sensitivity analysis 28119.2.6 Sensitivity analysis in the example 28319.3 Design sensitivity 28419.3.1 What is design sensitivity? 28419.3.2 Comparison of design sensitivity with purely reactive doses 28619.4 Summary 287References 28720 Evaluation of potential mediators in randomised trials of complex interventions (psychotherapies) 290Richard Emsley and Graham Dunn20.1 Introduction 29020.2 Potential mediators in psychological treatment trials 29120.3 Methods for mediation in psychological treatment trials 29320.4 Causal mediation analysis using instrumental variables estimation 29720.5 Causal mediation analysis using principal stratification 30120.6 Our motivating example: The SoCRATES trial 30220.6.1 What are the joint effects of sessions attended and therapeutic alliance on the PANSS score at 18 months? 30320.6.2 What is the direct effect of random allocation on the PANSS score at 18 months and how is this influenced by the therapeutic alliance? 30420.6.3 Is the direct effect of the number of sessions attended on the PANSS score at 18 months influenced by therapeutic alliance? 30520.7 Conclusions 305Acknowledgements 306References 30721 Causal inference in clinical trials 310Krista Fischer and Ian R. White21.1 Introduction 31021.2 Causal effect of treatment in randomized trials 31221.2.1 Observed data and notation 31221.2.2 Defining the effects of interest via potential outcomes 31221.2.3 Adherence-adjusted ITT analysis 31421.3 Estimation for a linear structural mean model 31621.3.1 A general estimation procedure 31621.3.2 Identifiability and closed-form estimation of the parameters in a linear SMM 31721.3.3 Analysis of the EPHT trial 31921.4 Alternative approaches for causal inference in randomized trials comparing experimental treatment with a control 32121.4.1 Principal stratification 32121.4.2 SMM for the average treatment effect on the treated (ATT) 32221.5 Discussion 324References 32522 Causal inference in time series analysis 327Michael Eichler22.1 Introduction 32722.2 Causality for time series 32822.2.1 Intervention causality 32822.2.2 Structural causality 33122.2.3 Granger causality 33222.2.4 Sims causality 33422.3 Graphical representations for time series 33522.3.1 Conditional distributions and chain graphs 33622.3.2 Path diagrams and Granger causality graphs 33722.3.3 Markov properties for Granger causality graphs 33822.4 Representation of systems with latent variables 33922.4.1 Marginalization 34122.4.2 Ancestral graphs 34222.5 Identification of causal effects 34322.6 Learning causal structures 34622.7 A new parametric model 34922.8 Concluding remarks 351References 35223 Dynamic molecular networks and mechanisms in the biosciences: A statistical framework 355Clive G. Bowsher23.1 Introduction 35523.2 SKMs and biochemical reaction networks 35623.3 Local independence properties of SKMs 35823.3.1 Local independence and kinetic independence graphs 35823.3.2 Local independence and causal influence 36123.4 Modularisation of SKMs 36223.4.1 Modularisations and dynamic independence 36223.4.2 MIDIA Algorithm 36323.5 Illustrative example – MAPK cell signalling 36523.6 Conclusion 36923.7 Appendix: SKM regularity conditions 369Acknowledgements 370References 370Index 371

Carlo Berzuini and Philip Dawid, Statistical Labority, centre for Mathematical Sciences, University of Cambridge, UK. Luisa Bernardinelli, MRC Biostatistics Unit, Institute of Public Health, Cambridge, UK.

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• List of contributors xvAn overview of statistical causality xviiCarlo Berzuini, Philip Dawid and Luisa Bernardinelli1 Statistical causality: Some historical remarks 1D.R. Cox1.1 Introduction 11.2 Key issues 21.3 Rothamsted view 21.4 An earlier controversy and its implications 31.5 Three versions of causality 41.6 Conclusion 4References 42 The language of potential outcomes 6Arvid Sjölander2.1 Introduction 62.2 Definition of causal effects through potential outcomes 72.2.1 Subject-specific causal effects 72.2.2 Population causal effects 82.2.3 Association versus causation 92.3 Identification of population causal effects 92.3.1 Randomized experiments 92.3.2 Observational studies 112.4 Discussion 11References 133 Structural equations, graphs and interventions 15Ilya Shpitser3.1 Introduction 153.2 Structural equations, graphs, and interventions 163.2.1 Graph terminology 163.2.2 Markovian models 173.2.3 Latent projections and semi-Markovian models 193.2.4 Interventions in semi-Markovian models 193.2.5 Counterfactual distributions in NPSEMs 203.2.6 Causal diagrams and counterfactual independence 223.2.7 Relation to potential outcomes 22References 234 The decision-theoretic approach to causal inference 25Philip Dawid4.1 Introduction 254.2 Decision theory and causality 264.2.1 A simple decision problem 264.2.2 Causal inference 274.3 No confounding 284.4 Confounding 294.4.1 Unconfounding 294.4.2 Nonconfounding 304.4.3 Back-door formula 314.5 Propensity analysis 334.6 Instrumental variable 344.6.1 Linear model 364.6.2 Binary variables 364.7 Effect of treatment of the treated 374.8 Connections and contrasts 374.8.1 Potential responses 374.8.2 Causal graphs 394.9 Postscript 40Acknowledgements 40References 405 Causal inference as a prediction problem: Assumptions, identification and evidence synthesis 43Sander Greenland5.1 Introduction 435.2 A brief commentary on developments since 1970 445.2.1 Potential outcomes and missing data 455.2.2 The prognostic view 455.3 Ambiguities of observational extensions 465.4 Causal diagrams and structural equations 475.5 Compelling versus plausible assumptions, models and inferences 475.6 Nonidentification and the curse of dimensionality 505.7 Identification in practice 515.8 Identification and bounded rationality 535.9 Conclusion 54Acknowledgments 55References 556 Graph-based criteria of identifiability of causal questions 59Ilya Shpitser6.1 Introduction 596.2 Interventions from observations 596.3 The back-door criterion, conditional ignorability, and covariate adjustment 616.4 The front-door criterion 636.5 Do-calculus 646.6 General identification 656.7 Dormant independences and post-truncation constraints 68References 697 Causal inference from observational data: A Bayesian predictive approach 71Elja Arjas7.1 Background 717.2 A model prototype 727.3 Extension to sequential regimes 767.4 Providing a causal interpretation: Predictive inference from data 807.5 Discussion 82Acknowledgement 83References 838 Assessing dynamic treatment strategies 85Carlo Berzuini, Philip Dawid, and Vanessa Didelez8.1 Introduction 858.2 Motivating example 868.3 Descriptive versus causal inference 878.4 Notation and problem definition 888.5 HIV example continued 898.6 Latent variables 898.7 Conditions for sequential plan identifiability 908.7.1 Stability 908.7.2 Positivity 918.8 Graphical representations of dynamic plans 928.9 Abdominal aortic aneurysm surveillance 948.10 Statistical inference and computation 958.11 Transparent actions 978.12 Refinements 988.13 Discussion 99Acknowledgements 99References 999 Causal effects and natural laws: Towards a conceptualization of causal counterfactuals for nonmanipulable exposures, with application to the effects of race and sex 101Tyler J. VanderWeele and Miguel A. Hernán9.1 Introduction 1019.2 Laws of nature and contrary to fact statements 1029.3 Association and causation in the social and biomedical sciences 1039.4 Manipulation and counterfactuals 1039.5 Natural laws and causal effects 1049.6 Consequences of randomization 1079.7 On the causal effects of sex and race 1089.8 Discussion 111Acknowledgements 112References 11210 Cross-classifications by joint potential outcomes 114Arvid Sjölander10.1 Introduction 11410.2 Bounds for the causal treatment effect in randomized trials with imperfect compliance 11510.3 Identifying the complier causal effect in randomized trials with imperfect compliance 11910.4 Defining the appropriate causal effect in studies suffering from truncation by death 12110.5 Discussion 123References 12411 Estimation of direct and indirect effects 126Stijn Vansteelandt11.1 Introduction 12611.2 Identification of the direct and indirect effect 12711.2.1 Definitions 12711.2.2 Identification 12911.3 Estimation of controlled direct effects 13211.3.1 G-computation 13211.3.2 Inverse probability of treatment weighting 13311.3.3 G-estimation for additive and multiplicative models 13711.3.4 G-estimation for logistic models 14111.3.5 Case-control studies 14211.3.6 G-estimation for additive hazard models 14311.4 Estimation of natural direct and indirect effects 14611.5 Discussion 147Acknowledgements 147References 14812 The mediation formula: A guide to the assessment of causal pathways in nonlinear models 151Judea Pearl12.1 Mediation: Direct and indirect effects 15112.1.1 Direct versus total effects 15112.1.2 Controlled direct effects 15212.1.3 Natural direct effects 15412.1.4 Indirect effects 15612.1.5 Effect decomposition 15712.2 The mediation formula: A simple solution to a thorny problem 15712.2.1 Mediation in nonparametric models 15712.2.2 Mediation effects in linear, logistic, and probit models 15912.2.3 Special cases of mediation models 16412.2.4 Numerical example 16912.3 Relation to other methods 17012.3.1 Methods based on differences and products 17012.3.2 Relation to the principal-strata direct effect 17112.4 Conclusions 173Acknowledgments 174References 17513 The sufficient cause framework in statistics, philosophy and the biomedical and social sciences 180Tyler J. VanderWeele13.1 Introduction 18013.2 The sufficient cause framework in philosophy 18113.3 The sufficient cause framework in epidemiology and biomedicine 18113.4 The sufficient cause framework in statistics 18513.5 The sufficient cause framework in the social sciences 18513.6 Other notions of sufficiency and necessity in causal inference 18713.7 Conclusion 188Acknowledgements 189References 18914 Analysis of interaction for identifying causal mechanisms 192Carlo Berzuini, Philip Dawid, Hu Zhang and Miles Parkes14.1 Introduction 19214.2 What is a mechanism? 19314.3 Statistical versus mechanistic interaction 19314.4 Illustrative example 19414.5 Mechanistic interaction defined 19614.6 Epistasis 19714.7 Excess risk and superadditivity 19714.8 Conditions under which excess risk and superadditivity indicate the presence of mechanistic interaction 20014.9 Collapsibility 20114.10 Back to the illustrative study 20214.11 Alternative approaches 20414.12 Discussion 204Ethics statement 205Financial disclosure 205References 20615 Ion channels as a possible mechanism of neurodegeneration in multiple sclerosis 208Luisa Bernardinelli, Carlo Berzuini, Luisa Foco, and Roberta Pastorino15.1 Introduction 20815.2 Background 20915.3 The scientific hypothesis 20915.4 Data 21015.5 A simple preliminary analysis 21115.6 Testing for qualitative interaction 21315.7 Discussion 214Acknowledgments 216References 21616 Supplementary variables for causal estimation 218Roland R. Ramsahai16.1 Introduction 21816.2 Multiple expressions for causal effect 22016.3 Asymptotic variance of causal estimators 22216.4 Comparison of causal estimators 22216.4.1 Supplement C with L or not 22316.4.2 Supplement L with C or not 22416.4.3 Replace C with L or not 22516.5 Discussion 226Acknowledgements 226Appendices 22716.A Estimator given all X’s recorded 22716.B Derivations of asymptotic variances 22716.C Expressions with correlation coefficients 22916.D Derivation of I’s 23016.E Relation between ρ2 rl|t and ρ2 rl|c 231References 23217 Time-varying confounding: Some practical considerations in a likelihood framework 234Rhian Daniel, Bianca De Stavola and Simon Cousens17.1 Introduction 23417.2 General setting 23517.2.1 Notation 23517.2.2 Observed data structure 23517.2.3 Intervention strategies 23617.2.4 Potential outcomes 23717.2.5 Time-to-event outcomes 23717.2.6 Causal estimands 23817.3 Identifying assumptions 23817.4 G-computation formula 23917.4.1 The formula 23917.4.2 Plug-in regression estimation 24017.5 Implementation by Monte Carlo simulation 24217.5.1 Simulating an end-of-study outcome 24217.5.2 Simulating a time-to-event outcome 24217.5.3 Inference 24217.5.4 Losses to follow-up 24317.5.5 Software 24317.6 Analyses of simulated data 24317.6.1 The data 24317.6.2 Regimes to be compared 24417.6.3 Parametric modelling choices 24517.6.4 Results 24617.7 Further considerations 24917.7.1 Parametric model misspecification 24917.7.2 Competing events 24917.7.3 Unbalanced measurement times 25017.8 Summary 251References 25118 ‘Natural experiments’ as a means of testing causal inferences 253Michael Rutter18.1 Introduction 25318.2 Noncausal interpretations of an association 25318.3 Dealing with confounders 25518.4 ‘Natural experiments’ 25618.4.1 Genetically sensitive designs 25718.4.2 Children of twins (CoT) design 25918.4.3 Strategies to identify the key environmental risk feature 26118.4.4 Designs for dealing with selection bias 26318.4.5 Instrumental variables to rule out reverse causation 26418.4.6 Regression discontinuity (RD) designs to deal with unmeasured confounders 26518.5 Overall conclusion on ‘natural experiments’ 26618.5.1 Supported causes 26618.5.2 Disconfirmed causes 267Acknowledgement 267References 26819 Nonreactive and purely reactive doses in observational studies 273Paul R. Rosenbaum19.1 Introduction: Background, example 27319.1.1 Does a dose–response relationship provide information that distinguishes treatment effects from biases due to unmeasured covariates? 27319.1.2 Is more chemotherapy for ovarian cancer more effective or more toxic? 27419.2 Various concepts of dose 27719.2.1 Some notation: Covariates, outcomes, and treatment assignment in matched pairs 27719.2.2 Reactive and nonreactive doses of treatment 27819.2.3 Three test statistics that use doses in different ways 27919.2.4 Randomization inference in randomized experiments 28019.2.5 Sensitivity analysis 28119.2.6 Sensitivity analysis in the example 28319.3 Design sensitivity 28419.3.1 What is design sensitivity? 28419.3.2 Comparison of design sensitivity with purely reactive doses 28619.4 Summary 287References 28720 Evaluation of potential mediators in randomised trials of complex interventions (psychotherapies) 290Richard Emsley and Graham Dunn20.1 Introduction 29020.2 Potential mediators in psychological treatment trials 29120.3 Methods for mediation in psychological treatment trials 29320.4 Causal mediation analysis using instrumental variables estimation 29720.5 Causal mediation analysis using principal stratification 30120.6 Our motivating example: The SoCRATES trial 30220.6.1 What are the joint effects of sessions attended and therapeutic alliance on the PANSS score at 18 months? 30320.6.2 What is the direct effect of random allocation on the PANSS score at 18 months and how is this influenced by the therapeutic alliance? 30420.6.3 Is the direct effect of the number of sessions attended on the PANSS score at 18 months influenced by therapeutic alliance? 30520.7 Conclusions 305Acknowledgements 306References 30721 Causal inference in clinical trials 310Krista Fischer and Ian R. White21.1 Introduction 31021.2 Causal effect of treatment in randomized trials 31221.2.1 Observed data and notation 31221.2.2 Defining the effects of interest via potential outcomes 31221.2.3 Adherence-adjusted ITT analysis 31421.3 Estimation for a linear structural mean model 31621.3.1 A general estimation procedure 31621.3.2 Identifiability and closed-form estimation of the parameters in a linear SMM 31721.3.3 Analysis of the EPHT trial 31921.4 Alternative approaches for causal inference in randomized trials comparing experimental treatment with a control 32121.4.1 Principal stratification 32121.4.2 SMM for the average treatment effect on the treated (ATT) 32221.5 Discussion 324References 32522 Causal inference in time series analysis 327Michael Eichler22.1 Introduction 32722.2 Causality for time series 32822.2.1 Intervention causality 32822.2.2 Structural causality 33122.2.3 Granger causality 33222.2.4 Sims causality 33422.3 Graphical representations for time series 33522.3.1 Conditional distributions and chain graphs 33622.3.2 Path diagrams and Granger causality graphs 33722.3.3 Markov properties for Granger causality graphs 33822.4 Representation of systems with latent variables 33922.4.1 Marginalization 34122.4.2 Ancestral graphs 34222.5 Identification of causal effects 34322.6 Learning causal structures 34622.7 A new parametric model 34922.8 Concluding remarks 351References 35223 Dynamic molecular networks and mechanisms in the biosciences: A statistical framework 355Clive G. Bowsher23.1 Introduction 35523.2 SKMs and biochemical reaction networks 35623.3 Local independence properties of SKMs 35823.3.1 Local independence and kinetic independence graphs 35823.3.2 Local independence and causal influence 36123.4 Modularisation of SKMs 36223.4.1 Modularisations and dynamic independence 36223.4.2 MIDIA Algorithm 36323.5 Illustrative example – MAPK cell signalling 36523.6 Conclusion 36923.7 Appendix: SKM regularity conditions 369Acknowledgements 370References 370Index 371