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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.

List of contributors xv

An overview of statistical causality xvii
Carlo Berzuini, Philip Dawid and Luisa Bernardinelli

1 Statistical causality: Some historical remarks 1
D.R. Cox

1.1 Introduction 1

1.2 Key issues 2

1.3 Rothamsted view 2

1.4 An earlier controversy and its implications 3

1.5 Three versions of causality 4

1.6 Conclusion 4

References 4

2 The language of potential outcomes 6
Arvid Sjölander

2.1 Introduction 6

2.2 Definition of causal effects through potential outcomes 7

2.2.1 Subject-specific causal effects 7

2.2.2 Population causal effects 8

2.2.3 Association versus causation 9

2.3 Identification of population causal effects 9

2.3.1 Randomized experiments 9

2.3.2 Observational studies 11

2.4 Discussion 11

References 13

3 Structural equations, graphs and interventions 15
Ilya Shpitser

3.1 Introduction 15

3.2 Structural equations, graphs, and interventions 16

3.2.1 Graph terminology 16

3.2.2 Markovian models 17

3.2.3 Latent projections and semi-Markovian models 19

3.2.4 Interventions in semi-Markovian models 19

3.2.5 Counterfactual distributions in NPSEMs 20

3.2.6 Causal diagrams and counterfactual independence 22

3.2.7 Relation to potential outcomes 22

References 23

4 The decision-theoretic approach to causal inference 25
Philip Dawid

4.1 Introduction 25

4.2 Decision theory and causality 26

4.2.1 A simple decision problem 26

4.2.2 Causal inference 27

4.3 No confounding 28

4.4 Confounding 29

4.4.1 Unconfounding 29

4.4.2 Nonconfounding 30

4.4.3 Back-door formula 31

4.5 Propensity analysis 33

4.6 Instrumental variable 34

4.6.1 Linear model 36

4.6.2 Binary variables 36

4.7 Effect of treatment of the treated 37

4.8 Connections and contrasts 37

4.8.1 Potential responses 37

4.8.2 Causal graphs 39

4.9 Postscript 40

Acknowledgements 40

References 40

5 Causal inference as a prediction problem: Assumptions, identification and evidence synthesis 43
Sander Greenland

5.1 Introduction 43

5.2 A brief commentary on developments since 1970 44

5.2.1 Potential outcomes and missing data 45

5.2.2 The prognostic view 45

5.3 Ambiguities of observational extensions 46

5.4 Causal diagrams and structural equations 47

5.5 Compelling versus plausible assumptions, models and inferences 47

5.6 Nonidentification and the curse of dimensionality 50

5.7 Identification in practice 51

5.8 Identification and bounded rationality 53

5.9 Conclusion 54

Acknowledgments 55

References 55

6 Graph-based criteria of identifiability of causal questions 59
Ilya Shpitser

6.1 Introduction 59

6.2 Interventions from observations 59

6.3 The back-door criterion, conditional ignorability, and covariate adjustment 61

6.4 The front-door criterion 63

6.5 Do-calculus 64

6.6 General identification 65

6.7 Dormant independences and post-truncation constraints 68

References 69

7 Causal inference from observational data: A Bayesian predictive approach 71
Elja Arjas

7.1 Background 71

7.2 A model prototype 72

7.3 Extension to sequential regimes 76

7.4 Providing a causal interpretation: Predictive inference from data 80

7.5 Discussion 82

Acknowledgement 83

References 83

8 Assessing dynamic treatment strategies 85
Carlo Berzuini, Philip Dawid, and Vanessa Didelez

8.1 Introduction 85

8.2 Motivating example 86

8.3 Descriptive versus causal inference 87

8.4 Notation and problem definition 88

8.5 HIV example continued 89

8.6 Latent variables 89

8.7 Conditions for sequential plan identifiability 90

8.7.1 Stability 90

8.7.2 Positivity 91

8.8 Graphical representations of dynamic plans 92

8.9 Abdominal aortic aneurysm surveillance 94

8.10 Statistical inference and computation 95

8.11 Transparent actions 97

8.12 Refinements 98

8.13 Discussion 99

Acknowledgements 99

References 99

9 Causal effects and natural laws: Towards a conceptualization of causal counterfactuals for nonmanipulable exposures, with application to the effects of race and sex 101
Tyler J. VanderWeele and Miguel A. Hernán

9.1 Introduction 101

9.2 Laws of nature and contrary to fact statements 102

9.3 Association and causation in the social and biomedical sciences 103

9.4 Manipulation and counterfactuals 103

9.5 Natural laws and causal effects 104

9.6 Consequences of randomization 107

9.7 On the causal effects of sex and race 108

9.8 Discussion 111

Acknowledgements 112

References 112

10 Cross-classifications by joint potential outcomes 114
Arvid Sjölander

10.1 Introduction 114

10.2 Bounds for the causal treatment effect in randomized trials with imperfect compliance 115

10.3 Identifying the complier causal effect in randomized trials with imperfect compliance 119

10.4 Defining the appropriate causal effect in studies suffering from truncation by death 121

10.5 Discussion 123

References 124

11 Estimation of direct and indirect effects 126
Stijn Vansteelandt

11.1 Introduction 126

11.2 Identification of the direct and indirect effect 127

11.2.1 Definitions 127

11.2.2 Identification 129

11.3 Estimation of controlled direct effects 132

11.3.1 G-computation 132

11.3.2 Inverse probability of treatment weighting 133

11.3.3 G-estimation for additive and multiplicative models 137

11.3.4 G-estimation for logistic models 141

11.3.5 Case-control studies 142

11.3.6 G-estimation for additive hazard models 143

11.4 Estimation of natural direct and indirect effects 146

11.5 Discussion 147

Acknowledgements 147

References 148

12 The mediation formula: A guide to the assessment of causal pathways in nonlinear models 151
Judea Pearl

12.1 Mediation: Direct and indirect effects 151

12.1.1 Direct versus total effects 151

12.1.2 Controlled direct effects 152

12.1.3 Natural direct effects 154

12.1.4 Indirect effects 156

12.1.5 Effect decomposition 157

12.2 The mediation formula: A simple solution to a thorny problem 157

12.2.1 Mediation in nonparametric models 157

12.2.2 Mediation effects in linear, logistic, and probit models 159

12.2.3 Special cases of mediation models 164

12.2.4 Numerical example 169

12.3 Relation to other methods 170

12.3.1 Methods based on differences and products 170

12.3.2 Relation to the principal-strata direct effect 171

12.4 Conclusions 173

Acknowledgments 174

References 175

13 The sufficient cause framework in statistics, philosophy and the biomedical and social sciences 180
Tyler J. VanderWeele

13.1 Introduction 180

13.2 The sufficient cause framework in philosophy 181

13.3 The sufficient cause framework in epidemiology and biomedicine 181

13.4 The sufficient cause framework in statistics 185

13.5 The sufficient cause framework in the social sciences 185

13.6 Other notions of sufficiency and necessity in causal inference 187

13.7 Conclusion 188

Acknowledgements 189

References 189

14 Analysis of interaction for identifying causal mechanisms 192
Carlo Berzuini, Philip Dawid, Hu Zhang and Miles Parkes

14.1 Introduction 192

14.2 What is a mechanism? 193

14.3 Statistical versus mechanistic interaction 193

14.4 Illustrative example 194

14.5 Mechanistic interaction defined 196

14.6 Epistasis 197

14.7 Excess risk and superadditivity 197

14.8 Conditions under which excess risk and superadditivity indicate the presence of mechanistic interaction 200

14.9 Collapsibility 201

14.10 Back to the illustrative study 202

14.11 Alternative approaches 204

14.12 Discussion 204

Ethics statement 205

Financial disclosure 205

References 206

15 Ion channels as a possible mechanism of neurodegeneration in multiple sclerosis 208
Luisa Bernardinelli, Carlo Berzuini, Luisa Foco, and Roberta Pastorino

15.1 Introduction 208

15.2 Background 209

15.3 The scientific hypothesis 209

15.4 Data 210

15.5 A simple preliminary analysis 211

15.6 Testing for qualitative interaction 213

15.7 Discussion 214

Acknowledgments 216

References 216

16 Supplementary variables for causal estimation 218
Roland R. Ramsahai

16.1 Introduction 218

16.2 Multiple expressions for causal effect 220

16.3 Asymptotic variance of causal estimators 222

16.4 Comparison of causal estimators 222

16.4.1 Supplement C with L or not 223

16.4.2 Supplement L with C or not 224

16.4.3 Replace C with L or not 225

16.5 Discussion 226

Acknowledgements 226

Appendices 227

16.A Estimator given all X’s recorded 227

16.B Derivations of asymptotic variances 227

16.C Expressions with correlation coefficients 229

16.D Derivation of I’s 230

16.E Relation between ρ2 rl|t and ρ2 rl|c 231

References 232

17 Time-varying confounding: Some practical considerations in a likelihood framework 234
Rhian Daniel, Bianca De Stavola and Simon Cousens

17.1 Introduction 234

17.2 General setting 235

17.2.1 Notation 235

17.2.2 Observed data structure 235

17.2.3 Intervention strategies 236

17.2.4 Potential outcomes 237

17.2.5 Time-to-event outcomes 237

17.2.6 Causal estimands 238

17.3 Identifying assumptions 238

17.4 G-computation formula 239

17.4.1 The formula 239

17.4.2 Plug-in regression estimation 240

17.5 Implementation by Monte Carlo simulation 242

17.5.1 Simulating an end-of-study outcome 242

17.5.2 Simulating a time-to-event outcome 242

17.5.3 Inference 242

17.5.4 Losses to follow-up 243

17.5.5 Software 243

17.6 Analyses of simulated data 243

17.6.1 The data 243

17.6.2 Regimes to be compared 244

17.6.3 Parametric modelling choices 245

17.6.4 Results 246

17.7 Further considerations 249

17.7.1 Parametric model misspecification 249

17.7.2 Competing events 249

17.7.3 Unbalanced measurement times 250

17.8 Summary 251

References 251

18 ‘Natural experiments’ as a means of testing causal inferences 253
Michael Rutter

18.1 Introduction 253

18.2 Noncausal interpretations of an association 253

18.3 Dealing with confounders 255

18.4 ‘Natural experiments’ 256

18.4.1 Genetically sensitive designs 257

18.4.2 Children of twins (CoT) design 259

18.4.3 Strategies to identify the key environmental risk feature 261

18.4.4 Designs for dealing with selection bias 263

18.4.5 Instrumental variables to rule out reverse causation 264

18.4.6 Regression discontinuity (RD) designs to deal with unmeasured confounders 265

18.5 Overall conclusion on ‘natural experiments’ 266

18.5.1 Supported causes 266

18.5.2 Disconfirmed causes 267

Acknowledgement 267

References 268

19 Nonreactive and purely reactive doses in observational studies 273
Paul R. Rosenbaum

19.1 Introduction: Background, example 273

19.1.1 Does a dose–response relationship provide information that distinguishes treatment effects from biases due to unmeasured covariates? 273

19.1.2 Is more chemotherapy for ovarian cancer more effective or more toxic? 274

19.2 Various concepts of dose 277

19.2.1 Some notation: Covariates, outcomes, and treatment assignment in matched pairs 277

19.2.2 Reactive and nonreactive doses of treatment 278

19.2.3 Three test statistics that use doses in different ways 279

19.2.4 Randomization inference in randomized experiments 280

19.2.5 Sensitivity analysis 281

19.2.6 Sensitivity analysis in the example 283

19.3 Design sensitivity 284

19.3.1 What is design sensitivity? 284

19.3.2 Comparison of design sensitivity with purely reactive doses 286

19.4 Summary 287

References 287

20 Evaluation of potential mediators in randomised trials of complex interventions (psychotherapies) 290
Richard Emsley and Graham Dunn

20.1 Introduction 290

20.2 Potential mediators in psychological treatment trials 291

20.3 Methods for mediation in psychological treatment trials 293

20.4 Causal mediation analysis using instrumental variables estimation 297

20.5 Causal mediation analysis using principal stratification 301

20.6 Our motivating example: The SoCRATES trial 302

20.6.1 What are the joint effects of sessions attended and therapeutic alliance on the PANSS score at 18 months? 303

20.6.2 What is the direct effect of random allocation on the PANSS score at 18 months and how is this influenced by the therapeutic alliance? 304

20.6.3 Is the direct effect of the number of sessions attended on the PANSS score at 18 months influenced by therapeutic alliance? 305

20.7 Conclusions 305

Acknowledgements 306

References 307

21 Causal inference in clinical trials 310
Krista Fischer and Ian R. White

21.1 Introduction 310

21.2 Causal effect of treatment in randomized trials 312

21.2.1 Observed data and notation 312

21.2.2 Defining the effects of interest via potential outcomes 312

21.2.3 Adherence-adjusted ITT analysis 314

21.3 Estimation for a linear structural mean model 316

21.3.1 A general estimation procedure 316

21.3.2 Identifiability and closed-form estimation of the parameters in a linear SMM 317

21.3.3 Analysis of the EPHT trial 319

21.4 Alternative approaches for causal inference in randomized trials comparing experimental treatment with a control 321

21.4.1 Principal stratification 321

21.4.2 SMM for the average treatment effect on the treated (ATT) 322

21.5 Discussion 324

References 325

22 Causal inference in time series analysis 327
Michael Eichler

22.1 Introduction 327

22.2 Causality for time series 328

22.2.1 Intervention causality 328

22.2.2 Structural causality 331

22.2.3 Granger causality 332

22.2.4 Sims causality 334

22.3 Graphical representations for time series 335

22.3.1 Conditional distributions and chain graphs 336

22.3.2 Path diagrams and Granger causality graphs 337

22.3.3 Markov properties for Granger causality graphs 338

22.4 Representation of systems with latent variables 339

22.4.1 Marginalization 341

22.4.2 Ancestral graphs 342

22.5 Identification of causal effects 343

22.6 Learning causal structures 346

22.7 A new parametric model 349

22.8 Concluding remarks 351

References 352

23 Dynamic molecular networks and mechanisms in the biosciences: A statistical framework 355
Clive G. Bowsher

23.1 Introduction 355

23.2 SKMs and biochemical reaction networks 356

23.3 Local independence properties of SKMs 358

23.3.1 Local independence and kinetic independence graphs 358

23.3.2 Local independence and causal influence 361

23.4 Modularisation of SKMs 362

23.4.1 Modularisations and dynamic independence 362

23.4.2 MIDIA Algorithm 363

23.5 Illustrative example – MAPK cell signalling 365

23.6 Conclusion 369

23.7 Appendix: SKM regularity conditions 369

Acknowledgements 370

References 370

Index 371