Causality E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: Wiley Series in Probability and Statistics
• Sprache: Englisch

Seitenzahl: 890

Veröffentlichungsjahr: 2012

Beschreibung

A state of the art volume on statistical causality Causality: Statistical Perspectives and Applications presents a wide-ranging collection of seminal contributions by renowned experts in the field, providing a thorough treatment of all aspects of statistical causality. It covers the various formalisms in current use, methods for applying them to specific problems, and the special requirements of a range of examples from medicine, biology and economics to political science. This book: * Provides a clear account and comparison of formal languages, concepts and models for statistical causality. * Addresses examples from medicine, biology, economics and political science to aid the reader's understanding. * Is authored by leading experts in their field. * Is written in an accessible style. Postgraduates, professional statisticians and researchers in academia and industry will benefit from this book.

Leseprobe

Contents

Cover

Series

Title Page

Copyright

List of contributors

An overview of statistical causality

Chapter 1: Statistical causality: Some historical remarks

1.1 Introduction

1.2 Key issues

1.3 Rothamsted view

1.4 An earlier controversy and its implications

1.5 Three versions of causality

1.6 Conclusion

Chapter 2: The language of potential outcomes

2.1 Introduction

2.2 Definition of causal effects through potential outcomes

2.3 Identification of population causal effects

2.4 Discussion

Chapter 3: Structural equations, graphs and interventions

3.1 Introduction

3.2 Structural equations, graphs, and interventions

Chapter 4: The decision-theoretic approach to causal inference

4.1 Introduction

4.2 Decision theory and causality

4.3 No confounding

4.4 Confounding

4.5 Propensity analysis

4.6 Instrumental variable

4.7 Effect of treatment of the treated

4.8 Connections and contrasts

4.9 Postscript

4.10 Acknowledgements

Chapter 5: Causal inference as a prediction problem: Assumptions, identification and evidence synthesis

5.1 Introduction

5.2 A brief commentary on developments since 1970

5.3 Ambiguities of observational extensions

5.4 Causal diagrams and structural equations

5.5 Compelling versus plausible assumptions, models and inferences

5.6 Nonidentification and the curse of dimensionality

5.7 Identification in practice

5.8 Identification and bounded rationality

5.9 Conclusion

Acknowledgments

Chapter 6: Graph-based criteria of identifiability of causal questions

6.1 Introduction

6.2 Interventions from observations

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

6.4 The front-door criterion

6.5 Do-calculus

6.6 General identification

6.7 Dormant independences and post-truncation constraints

Chapter 7: Causal inference from observational data: A Bayesian predictive approach

7.1 Background

7.2 A model prototype

7.3 Extension to sequential regimes

7.4 Providing a causal interpretation: Predictive inference from data

7.5 Discussion

Acknowledgement

Chapter 8: Assessing dynamic treatment strategies

8.1 Introduction

8.2 Motivating example

8.3 Descriptive versus causal inference

8.4 Notation and problem definition

8.5 HIV example continued

8.6 Latent variables

8.7 Conditions for sequential plan identifiability

8.8 Graphical representations of dynamic plans

8.9 Abdominal aortic aneurysm surveillance

8.10 Statistical inference and computation

8.11 Transparent actions

8.12 Refinements

8.13 Discussion

Acknowledgements

Chapter 9: Causal effects and natural laws: Towards a conceptualization of causal counterfactuals for nonmanipulable exposures, with application to the effects of race and sex

9.1 Introduction

9.2 Laws of nature and contrary to fact statements

9.3 Association and causation in the social and biomedical sciences

9.4 Manipulation and counterfactuals

9.5 Natural laws and causal effects

9.6 Consequences of randomization

9.7 On the causal effects of sex and race

9.8 Discussion

Acknowledgements

Chapter 10: Cross-classifications by joint potential outcomes

10.1 Introduction

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

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

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

10.5 Discussion

Chapter 11: Estimation of direct and indirect effects

11.1 Introduction

11.2 Identification of the direct and indirect effect

11.3 Estimation of controlled direct effects

11.4 Estimation of natural direct and indirect effects

11.5 Discussion

Acknowledgements

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

12.1 Mediation: Direct and indirect effects

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

12.3 Relation to other methods

12.4 Conclusions

Acknowledgments

Chapter 13: The sufficient cause framework in statistics, philosophy and the biomedical and social sciences

13.1 Introduction

13.2 The sufficient cause framework in philosophy

13.3 The sufficient cause framework in epidemiology and biomedicine

13.4 The sufficient cause framework in statistics

13.5 The sufficient cause framework in the social sciences

13.6 Other notions of sufficiency and necessity in causal inference

13.7 Conclusion

Acknowledgements

Chapter 14: Analysis of interaction for identifying causal mechanisms

14.1 Introduction

14.2 What is a mechanism?

14.3 Statistical versus mechanistic interaction

14.4 Illustrative example

14.5 Mechanistic interaction defined

14.6 Epistasis

14.7 Excess risk and superadditivity

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

14.9 Collapsibility

14.10 Back to the illustrative study

14.11 Alternative approaches

14.12 Discussion

Ethics statement

Financial disclosure

Chapter 15: Ion channels as a possible mechanism of neurodegeneration in multiple sclerosis

15.1 Introduction

15.2 Background

15.3 The scientific hypothesis

15.4 Data

15.5 A simple preliminary analysis

15.6 Testing for qualitative interaction

15.7 Discussion

Acknowledgments

Chapter 16: Supplementary variables for causal estimation

16.1 Introduction

16.2 Multiple expressions for causal effect

16.3 Asymptotic variance of causal estimators

16.4 Comparison of causal estimators

16.5 Discussion

Acknowledgements

A Appendices

Chapter 17: Time-varying confounding: Some practical considerations in a likelihood framework

17.1 Introduction

17.2 General setting

17.3 Identifying assumptions

17.4 G-computation formula

17.5 Implementation by Monte Carlo simulation

17.6 Analyses of simulated data

17.7 Further considerations

17.8 Summary

Chapter 18: ‘Natural experiments’ as a means of testing causal inferences

18.1 Introduction

18.2 Noncausal interpretations of an association

18.3 Dealing with confounders

18.4 ‘Natural experiments’

18.5 Overall conclusion on ‘natural experiments’

Acknowledgement

Chapter 19: Nonreactive and purely reactive doses in observational studies

19.1 Introduction: Background, example

19.2 Various concepts of dose

19.3 Design sensitivity

19.4 Summary

Chapter 20: Evaluation of potential mediators in randomised trials of complex interventions (psychotherapies)

20.1 Introduction

20.2 Potential mediators in psychological treatment trials

20.3 Methods for mediation in psychological treatment trials

20.4 Causal mediation analysis using instrumental variables estimation

20.5 Causal mediation analysis using principal stratification

20.6 Our motivating example: The SoCRATES trial

20.7 Conclusions

Acknowledgements

Chapter 21: Causal inference in clinical trials

21.1 Introduction

21.2 Causal effect of treatment in randomized trials

21.3 Estimation for a linear structural mean model

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

21.5 Discussion

Chapter 22: Causal inference in time series analysis

22.1 Introduction

22.2 Causality for time series

22.3 Graphical representations for time series

22.4 Representation of systems with latent variables

22.5 Identification of causal effects

22.6 Learning causal structures

22.7 A new parametric model

22.8 Concluding remarks

Chapter 23: Dynamic molecular networks and mechanisms in the biosciences: A statistical framework

23.1 Introduction

23.2 SKMs and biochemical reaction networks

23.3 Local independence properties of SKMs

23.4 Modularisation of SKMs

23.5 Illustrative example – MAPK cell signalling

23.6 Conclusion

23.7 Appendix: SKM regularity conditions

Acknowledgements

Index

Series

WILEY SERIES IN PROBABILITY AND STATISTICS

ESTABLISHED BY WALTER A. SHEWHART AND SAMUEL S. WILKS

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Library of Congress Cataloging-in-Publication Data

Berzuini, Carlo. Causality : statistical perspectives and applications / Carlo Berzuini, Philip Dawid, Luisa Bernardinelli. p. cm. – (Wiley series in probability and statistics) Includes bibliographical references and index. ISBN 978-0-470-66556-5 (hardback) 1. Estimation theory. 2. Causation. 3. Causality (Physics) I. Dawid, Philip. II. Bernardinelli, Luisa. III. Title. QA276.8.B475 2012 519.5′44–dc23 2011049795

A catalogue record for this book is available from the British Library.

ISBN: 978-0-470-66556-5

List of contributors

Elja ArjasDepartment of Mathematics and Statistics University of Helsinki Helsinki, Finland

Luisa BernardinelliStatistical Laboratory Centre for Mathematical Sciences University of Cambridge Cambridge, UK

Carlo BerzuiniStatistical Laboratory Centre for Mathematical Sciences University of Cambridge Cambridge, UK

Clive G. BowsherSchool of Mathematics University of Bristol Bristol, UK

Simon CousensCentre for Statistical Methodology London School of Hygiene and Tropical Medicine, London, UK

D.R. CoxNuffield College University of Oxford Oxford, UK

Rhian DanielCentre for Statistical Methodology London School of Hygiene and Tropical Medicine, London, UK

Philip DawidStatistical Laboratory Centre for Mathematical Sciences University of Cambridge Cambridge, UK

Bianca De StavolaCentre for Statistical Methodology London School of Hygiene and Tropical Medicine, London, UK

Vanessa DidelezDepartment of Mathematics University of Bristol Bristol, UK

Graham DunnHealth Sciences Research Group School of Community Based Medicine The University of Manchester Manchester, UK

Michael EichlerDepartment of Quantitative Economics Maastricht University Maastricht, The Netherlands

Richard EmsleyHealth Sciences Research Group School of Community Based Medicine The University of Manchester Manchester, UK

Krista FischerMRC Biostatistics Unit, Cambridge, UK

and Estonian Genome Center University of Tartu, Estonia

Luisa FocoDepartment of Applied Health Sciences University of Pavia Pavia, Italy

Sander GreenlandDepartment of Epidemiology and Department of Statistics University of California Los Angeles, California, USA

Miguel A. HernánDepartment of Epidemiology Harvard School of Public Health Boston, Massachusetts, USA

Miles ParkesDepartment of Gastroenterology Addenbrookes Hospital Cambridge, UK

Roberta PastorinoDepartment of Applied Health Sciences University of Pavia Pavia, Italy

Judea PearlComputer Science Department University of California Los Angeles, California, USA

Roland R. RamsahaiStatistical Laboratory Centre for Mathematical Sciences University of Cambridge Cambridge, UK

Paul R. RosenbaumWharton School University of Pennsylvania Philadelphia, Pennsylvania, USA

Michael RutterMRC SGDP Centre Institute of Psychiatry London, UK

Ilya ShpitserDepartment of Epidemiology Harvard School of Public Health Boston, Massachusetts, USA

Arvid SjölanderDepartment of Medical Epidemiology

and Biostatistics Karolinska Institutet Stockholm, Sweden

Tyler J. VanderWeeleDepartments of Epidemiology and

Biostatistics Harvard School of Public Health Boston, Massachusetts, USA

Stijn VansteelandtGhent University, Ghent, Belgium and London School of Hygiene and Tropical

Medicine London, UK

Ian R. WhiteMRC Biostatistics Unit Cambridge, UK

Hu ZhangDepartment of Gastroenterology Addenbrookes Hospital Cambridge, UK

An overview of statistical causality

Many statistical analyses aim at a causal explanation of the data. The early observational studies on the risks of smoking (Cornfield et al., 1959), for example, aimed at something deeper than to show the poorer prognosis of a smoker. The hoped-for interpretation was causal: those who smoked would, on average, have had a better health had they not done so and, consequently, any future intervention against smoking will, at least in a similar population, have a positive impact on health. Causal interpretations and questions are the focus of this present book. They underpin many statistical studies in a variety of empirical disciplines, including natural and social sciences, psychology, and economics. The case of epidemiology and biostatistics is noted for a traditionally cautious attitude towards causality. Early researchers in these areas did not feel the need to use the word ‘causal’. Emphasis was on the requirement that the study be ‘secure’: that its conclusions should not rely on special assumptions about the nature of uncontrolled variation, something that is ideally only achieved in experimental studies. In the work of Fisher (1935), security was achieved largely by using randomization within an experimental context. This ensures that, when we form contrasts between the treatment groups, we are comparing ‘like with like’, and thus there are no systematic pre-existing differences between the treatment groups that might be alternative explanations of the observed difference in response.

Another idea originated by Fisher (1932) and later developed by Cochran (1957), Cox (1960), and Cox and McCullagh (1982) is the use of supplementary variables to improve the efficiency of estimators and of instrumental variables to make a causal effect of interest identifiable. The use of supplementary and instrumental variables in causal inference is discussed in Chapter 16 of this book, ‘Supplementary variables for causal estimation’ by Roland Ramsahai.

Early advances in the theory of experimental design, largely contributed by Rothamsted researchers, are discussed in Chapter 1, ‘Statistical causality: some historical remarks’, by David Cox. Also discussed in this chapter are some implications of the ‘Rothamsted view’ (and of the controversies that arose around it) for the current discussion on causal inference. A technical discussion of the problems of causal inference in randomized experiments in medicine is given in Chapter 21, ‘Causal inference in clinical trials’, by Krista Fischer and Ian White.

The 1960s witnessed the early development of a theory of causal inference in observational studies, a notable example being the work of Bradford Hill (1965). Hill proposed a set of guidelines to strengthen the case for a causal interpretation of the results of a given observational study. One of these guidelines, the presence of a dose–response relationship, is discussed in depth in Chapter 19, ‘Nonreactive and purely reactive doses in observational studies’, by Paul Rosenbaum. Hill's guidelines are informal, and they do not provide a definition of ‘causal’. During the 1990s, a wider community of researchers, gathered from such disciplines as statistics, philosophy, economics, social science, machine learning, and artificial intelligence, proposed a more aggressive approach to causality, reminiscent of the long philosophers’ struggle to reduce causality to probabilities. These researchers transformed cause–effect relationships into objects that can be manipulated mathematically (Pearl, 2000). They attempted to formalize concepts such as confounding and to set up various formal frameworks for causal inference from observational and experimental studies. In a given application, such frameworks allow us (i) to define the target causal effects, (ii) to express the causal assumptions in a clear way, and determine whether they are sufficient to allow estimation of the target effects from the available data, (iii) to identify analysis modalities and algorithms that render the estimate feasible, and (iv) to identify observations and experiments that would render the estimate feasible, or assumptions under which the conclusions of the analysis have a causal interpretation.

In retrospect, such effort came late. Many of the tools for conceptualizing causality had been available for some time, as in the case of the potential outcomes representation (Rubin, 1974), which Rubin adapted from experimental (Fisher, 1935) to observational studies in the early 1970s. Potential outcomes are discussed in Chapter 2, ‘The language of potential outcomes’, by Arvid Sjölander, and are used in several other chapters of this book. In this representation, any individual is characterized by a notional response Yk to each treatment Tk, regarded as fixed even before the treatment is applied. In Chapter 10, ‘Cross-classifications by joint potential outcomes’, Arvid Sjölander discusses the idea of a ‘principal stratification’ of the individuals, on the basis of the joint values of several potential outcomes of the same variable (so that each stratum specifies exactly how that variable would respond to a variety of different settings for some other variable). A sometimes serious limitation of such an approach is that typically there is no way of telling which individual falls into which stratum. In Chapter 10, principal stratification is used to bound nonidentifiable causal effects and to deal with problems due to imperfect observations. Principal stratification is also used in Chapter 21 to deal with problems of protocol nonadherence and of contamination between treatment arms, in the context of randomized clinical trials.

Another legacy from the past is the use of graphical representations of causality, predated by Wright's work on path diagrams (Wright, 1921, 1934) and later advocated by Cochran (1965). This area is currently dominated by the Non Parametric Structural Equations Models (NPSEMs) discussed in Chapter 3, ‘Structural equations, graphs and interventions’, by Ilya Shpitser. Shpitser emphasizes a conceptual symbiosis between NPSEMs and potential outcomes, where NPSEMs contribute a transparent language for expressing assumptions in terms of conditional independencies implied by the structure of a causal graph (Dawid, 1979; Geiger et al., 1990). Constructive criticism of NPSEMs is given in Chapter 5, ‘Causal inference as a prediction problem: assumptions, identification and evidence synthesis’, by Sander Greenland. A strong interpretation of an NPSEM regards each node in the model as associated with a fixed collection of potential responses to the various possible configurations of interventions on the set of parents of that node. This interpretation sheds light on some of the problems of nonstochasticity and nonidentifiability that Greenland mentions in relation to NPSEMs.

In the light of these problems, some researchers have set aside potential outcomes and NPSEMs in favour of approaches that fully acknowledge the stochastic nature of the world. One of these is the decision theoretic approach of Chapter 4. Focus here is on the assumptions under which an inference of interest, which we would ideally obtain from an experiment, can be drawn from a given set of observational data. In general, inferences are not transportable between an observational and an experimental regime of data collection, the reason being that the distributions of the domain variables in the two regimes may be completely different. The decision-theoretic approach considers special ‘regime indicator’ variables and uses them to formalize (in terms of conditional independence relationships) those conditions of invariance between regime-specific distributions that make cross-regime inference possible. In some problems of causal inference, the decision-theoretic approach leads to the same conclusions one reaches by using potential outcomes, but relaxing the strong assumptions of the latter. Chapters , , 15, and 16 further illustrate the use of a decision-theoretic formalism in combination with explicit graph representations of the assumed data-generating mechanism. A theme for future research is a comparison of different formulations of statistical causality, in relation to real data analysis situations and in terms of their ability to clarify the assumptions behind the validity of an inference.

As noted in Chapter 5 by Greenland, potential outcomes can be ambiguous in the absence of a well-defined physical mechanism that determines the value of the causal factors. Difficulties then arise with causal factors that one cannot conceivably (let alone technologically) manipulate. In fact, the definition of a causal effect of variables like sex, say, remains a veritable conundrum. The problem is discussed in Chapter 9, ‘Causal effects and natural laws: towards a conceptualization of causal counterfactuals for nonmanipulable exposures, with application to the effects of race and sex’, by Tyler VanderWeele and Miguel Hernán. The authors of this chapter argue that the notion of causal effect for nonmanipulable properties is defensible from a potential outcomes perspective if these can be assumed to affect the outcome through an underlying deterministic ‘natural law’, of the kind we encounter in physics. In the ‘sex and employment’ example of Chapter 12, Judea Pearl deals with the sex effect by invoking a hypothetical intervention that prevents the employer from being informed about the applicant's sex.

Much causal inference literature, in both observational and experimental contexts, conceptualizes causality in terms of assessing the consequences of a (future or hypothetical) intervention. This will typically involve estimating, from data, effects that are formally defined in terms of a relevant intervention distribution, induced by the intervention of interest. A given set of data will be sufficient to estimate the causal effects of interest if the relevant intervention distribution can be expressed as a function of the distribution from which the data have been obtained. A formal check of this property can be performed with the aid of the ‘do-calculus’ described in Chapter 6, ‘Graph-based criteria of identifiability of causal questions’, by Ilya Shpitser. The need for such formal machinery becomes acute in highly structured causal problems, notably in the presence of multiple causal paths (such as in Chapters and ), and of complexly structured confounding, and in the identification of dynamic plans.

The usual conception of ‘intervention’ involves fixing variables to a particular configuration of values. This definition may prove inadequate in the presence of temporal precedence relationships between exposures or actions. In such cases, interventions may be more appropriately described as plans, the archetypal example being a plan of treatment of a medical patient, where periodic decisions have to be made in the light of the outcomes of previous treatment actions. The evaluation will often be based on information gleaned from records of the performance of past decision makers. Problems of this kind are discussed in Chapters 7, 8, and 17 of this book. In Chapter 7, ‘Causal inference from observational data: a Bayesian predictive approach’, Elja Arjas approaches the problem from an entirely probabilistic point of view, and clarifies important links with mainstream analysis of time-dependent events. In Chapter 8, ‘Assessing dynamic treatment strategies’, Carlo Berzuini, Philip Dawid, and Vanessa Didelez tackle the problem from a decision-theoretic point of view, and in the context of examples in the treatment of an HIV virus infection and of an aortic aneurism. In Chapter 17, ‘Time-varying confounding: some practical considerations in a likelihood framework’, Rhian Daniel, Bianca De Stavola, and Simon Cousens tackle the problem from a potential outcomes point of view, and look more closely into aspects of statistical estimation and computation, with the aid of a simulation experiment.

Problems of the above kind typically involve biases introduced by post-treatment variables, which cannot be adjusted for within a standard regression analysis framework. Recent research in causal inference has developed algorithms to deal with a wide class of problems of this kind (albeit often only by introducing suitable parametric assumptions) and, by so doing, overcome the limits of standard regression. A notable example is Robins's G-computation algorithm (Robins, 1986). Use of this algorithm for the evaluation of dynamic plans, and the conditions under which this leads to valid causal inferences, are discussed in Chapters and . The same algorithm is discussed in Chapter 11 in relation to the analysis of mediation.

How does causal inference connect with science? One might argue that the randomization philosophy at the basis of many experimental designs (e.g. clinical trials) is a very poor example of ‘science’. Randomization may make the study conclusions more secure, but it does little to unravel the biological, or physical, or psychological processes operating behind an observed causal effect. Such a limitation may be negligible in those areas of application whose exclusive aim is to predict the consequences of a future intervention, or to identify the most promising treatment, such as often occurs in clinical trials. But there are study areas which are largely driven by the wish to understand an underlying mechanism. One example are studies of genetic association to elucidate a molecular disease mechanism, or, say, a studies to investigate whether anxiety plays a role in the response of an individual to a specific stimulus. In these and other application areas, there is a need for statistical concepts and tools that help us to unravel the mechanism behind a causal effect.

Different interpretations of the concept of mechanism will inspire different statistical approaches to mechanistic inference. One class of methods of inference about mechanism is inspired by the sufficient cause conceptualization discussed in Chapter 13, ‘The sufficient cause framework in statistics, philosophy and the biomedical and social sciences’, by Tyler VanderWeele. This models the outcome event in terms of a collection of triggers, each involving a set of component events, such that occurrence of all the component events of at least one trigger always produces the outcome event (Mackie, 1965). The above conceptualization provides a basis for considering two causal factors as interacting mechanistically – with respect to the outcome of interest – if they are jointly involved in at least one trigger. The importance of this concept in the study of mechanisms is illustrated by the deep interest of genetic epidemiologists in methods for assessing epistasis (mechanistic interaction between the effects of genes). Methodological development in this area owes much to the work of several researchers, including Rothman (1976) and Skrondal (2003). VanderWeele and Robins (2008, 2009) have provided a potential outcomes formalization of the framework and have established empirical tests for mechanistic interaction in an observational context. In Chapter 14, ‘Analysis of interaction for identifying causal mechanisms’, Carlo Berzuini, Philip Dawid, Hu Zhang, and Miles Parkes justify such tests from a decision-theoretic point of view, in a formulation that embraces continuous causal factors. They use the method to identify components of the autophagy pathway which may cause susceptibility to Crohn's Disease. In Chapter 15, ‘Ion channels as a possible mechanism of neurodegeneration in multiple sclerosis’, Luisa Bernardinelli, Carlo Berzuini, Luisa Foco, and Roberta Pastorino discuss the use of family-structured genetic association data to detect patterns of statistical interaction where the value of one causal factor affects the sign of the effect of another factor. These patterns can be reasonably interpreted in terms of mechanistic interaction, because they cannot be eliminated by variable or response transformation.

Other conceptualizations of mechanism exist. In a variety of empirical disciplines, the term ‘causal mechanism’ designates the process through which the treatment causally affects the outcome (Salmon, 1984). It is often helpful to decompose this into an indirect effect, which operates through an observed mediator of interest, and a direct effect, which includes all other possible mechanisms (Robins and Greenland,1992; Pearl, 2005). By allowing separate estimation of these two components, analysis of mediation enables the researcher to explore the role of a specific causal pathway in the transmission of an effect of interest. Many questions of scientific interest can be formulated in terms of direct and indirect effects. One example is discussed in Chapter 20, ‘Evaluation of potential mediators in randomised trials of complex interventions (psychotherapies)’ by Richard Emsley and Graham Dunn. The study described in this chapter is motivated by the question of whether the quality-of-life improvement induced by cognitive behaviour therapy in depressed patients is mediated by a therapy-induced beneficial change of beliefs. Any evidence of a direct effect of the therapy, unmediated by that change, would prompt scientific hypotheses concerning the possible involvement of further mediating mechanisms, and thus point to specific directions for future experimental investigation. Problems of analysis of mediation arise in different disciplines. In the natural sciences, direct effects reveal how Nature works. In the social sciences, they allow prediction of the effect of a normative intervention that blocks specific causal paths (for example the effect of sex discrimination on employment).

Thanks to the ability to formalize such notions as ‘holding the mediating variables fixed’ (as distinct from probabilistic conditionalization), researchers in causal inference have been able to provide analysis of mediation with a sound framework. Two chapters in this book contribute significant advances in this area. The first, Chapter 11, entitled ‘Estimation of direct and indirect effects’, is contributed by Stijn Vansteelandt. The second, Chapter 12, written by Judea Pearl, is entitled ‘The mediation formula: a guide to the assessment of causal pathways in nonlinear models’.

Causal inference research has produced important advances in the analysis of mediation. First, the traditional framework involving only linear relationships has been generalized to allow arbitrary, nonparametric, relationships. Second, it has been recognized that, in a nonlinear system, the concept of direct effect can be defined in different ways, not all of which support a meaningful definition of a corresponding indirect effect. Third, the conditions under which well-established mediation analysis methods produce causally meaningful estimates (Baron and Kenny, 1986) have been formalized.

Our earlier distinction between inference about the consequences of an intervention (ICI) and inference about mechanism (IAM) deserves further elaboration. In statistical ICI, pre-existing scientific/mechanistic knowledge about the studied system may be helpful, but will rarely be essential to the design, analysis and interpretation of a study. The validity of a clinical trial, for example, does not typically depend (at least not in an essential way) on the availability of a deep scientific understanding of the studied process. By contrast, in statistical IAM, the causal validity of the analysis conclusions will often require assumptions justified by a deep scientific understanding of the involved mechanisms, as illustrated by the genetic study example of Chapter 14.

An interesting illustration of the potential interplay between statistical inference about mechanism and scientific knowledge is given in Chapter 23, ‘Dynamic molecular networks and mechanisms in the biosciences: a statistical framework’ by Clive Bowsher. This chapter illustrates the general principle that causality operates in time, rather than instantaneously, in this specific case being governed by a continuous-time jump process called a stochastic kinetic model. In Bowsher's chapter, a priori biological knowledge and experimental evidence from previous studies are combined into a causal graph representation of the process. Considerations based on this graph will suggest experimental interventions that may reliably improve our knowledge of the system, and lead to further, incremental, refinements of the model.

The conceptualization of causality in terms of the consequences of an intervention (e.g. of the effect of aspirin in terms of the consequences I observe when I administer it to a patient) has not been universally accepted. In a time-series context, for example, the traditional view of causality is in terms of dependence not explained away by other appropriate explanatory variables, with no reference to intervention. This idea, found, for example, in the work of Granger (1969) and Schweder (1970), although understandable in disciplines (like economics) where experimental interventions are difficult to implement, misses the requirement that causal relationships should persist if certain aspects of the system are changed. Recent work by Eichler and Didelez (2010) remedies this. In Chapter 22 of this book, ‘Causal inference in time series analysis’, Michael Eichler reconciles the two views, by incorporating the idea of intervention into causal graph representations of time-series models.

The benefits of randomization are not exclusive to controlled experiments. They are also available in ‘natural experiments’, where certain variables that causally affect the outcome can (in certain conditions) be regarded as generated by Nature in a random way, unaffected by potential sources of confounding. In Chapter 18, ‘Natural experiments as a means of testing causal inferences’, Michael Rutter offers a discussion of this opportunity, and illustrates it with the aid of a wealth of scientific examples. Just one example is where an individual's genotype G, at a specific DNA locus, may be regarded as generated, by Nature, by a random draw. Assuming that G does not affect the outcome except via changes to an intermediate phenotype M, the method called Mendelian randomization provides a vehicle for an unconfounded assessment of the causal effect of M on the outcome. This will often be possible only after appropriate conditioning, e.g. on a (natural or adoption) family or environment.

In many situations no randomization, be it achieved via experimental control or through the benevolence of Nature, can be invoked. One of these is discussed in Chapter 19, ‘Nonreactive and purely reactive doses in observational studies’, by Paul Rosenbaum. In this chapter, observational data is used to investigate a possible beneficial effect of higher doses of chemotherapy in ovarian cancer. The question is a difficult one, since most of the variation in treatment dose is a response to variation in the health of patients. The method suggested deals with potential confounding by capitalizing on that portion of the observed variation in treatment dose introduced by doctors with different views on dose prescription. This chapter illustrates the effectiveness of matching techniques and the usefulness of rigorous causal inference reasoning in the context of a scientific study.

In a recent paper on statistical causality, David Cox (2004) reminds us that deep conclusions often require synthesis of evidence of different kinds. Perhaps it is in this sense that we should interpret the Fisher's famous aphorism that in order to make observational studies more like randomized experiments we should ‘make our theories elaborate’. This idea is illustrated in Chapter 15 by Luisa Bernardinelli, Carlo Berzuini, Luisa Foco, and Roberta Pastorino. The question whether ion channels are a possible cause of multiple sclerosis is here tackled by integrating different sources of statistical and experimental evidence into a coherent hypothesis about the role of specific molecular dysfunctions in susceptibility to multiple sclerosis.

Besides being an important topic in natural science disciplines, causality is important also in such normative areas as Forensic Law and Legal Philosophy. Although this book does not extensively cover this area, we should ask: How do the assumptions, methods, and requirements involved in the empirical evaluation of causal hypotheses in these normative areas differ from those relevant to the natural sciences?

In scientific applications we are typically interested in generic causal relationships. However, the focus in legal contexts will more often be on assigning causal responsibility in an individual case. Put otherwise, scientific causality is forward-looking and concerned with understanding the unfolding of generic causal processes, while legal causality is retrospective, concerned with unpicking the causes of specific observed effects. In other words, scientific causality is concerned with questions about the effects of causes, whereas legal causality is concerned about the causes of effects. This volume largely confines attention to the former enterprise; its contents amply attest that there is currently much debate and disagreement about how to formalize and analyse questions about the effects of causes. However, these already tricky philosophical and conceptual difficulties and disputes become greatly magnified when we turn to trying to understand the causes of observed effects, especially in a stochastic world (Dawid, 2011). Much work in this area tiptoes around the quicksands by ignoring stochasticity and assuming deterministic dependence of outcomes on sufficiently many inputs (unobserved as well as observed). Formalisms such as Mackie's INUS criterion are then applicable, and have proved popular with legal philosophers. Very close to INUS is a formalism that lawyers often refer to using the acronym NESS (Miller, 2011), which stands for ‘Necessary Element of a Sufficient Set’. However, when, more realistically, we wish to allow genuine indeterminism, it turns out that it may simply be impossible, even with the best data in the world, to estimate causes of effects at the individual level without making arbitrary and empirically untestable additional assumptions. However, we can sometimes extract empirically meaningful interval bounds for relevant causal quantity, as illustrated, using potential outcome methods, in Chapter 10.

This book arose out of the International Meeting on ‘Causal Inference : The State of the Art’, that we organized in 2009 at the University of Cambridge, UK, with support from the European MolPage Project and the Cambridge Statistics Initiative, and under the auspices of the Royal Statistical Society. The book project has graced us with an enriching and stimulating experience. Profound thanks are due to the chapter authors for their commitment and enthusiasm. We also express our gratitude to the Wiley series team, first and foremost to Ilaria Meliconi for her early support for this project, to Richard Davies for his indefatigable and competent assistance, and to Prachi Sinha Sahay and Baljinder Kaur for their competent management of the copyediting.

Carlo Berzuini, Philip Dawid and Luisa Bernardinelli

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1

Statistical causality: Some historical remarks

D.R. Cox

Nuffield College, University of Oxford, UK

1.1 Introduction

Some investigations are essentially descriptive. Others are concerned at least in part with probing the nature of dependences.

Examples of the former are studies to estimate the number of whales in a portion of ocean, to determine the distribution of particle size in a river bed and to find the mortality rates of smokers and of nonsmokers. Examples of the second type of investigation are experiments to compare the effect of different levels of fertilizer on agricultural yield and investigations aimed to understand any apparent differences between the health of smokers and nonsmokers, that is to study whether smoking is the explanation of differences found. Also much, but not all, laboratory work in the natural sciences comes in this category.

Briefly the objectives of the two types are respectively to describe the world and in some sense to understand it. Put slightly more explicitly, in the agricultural field trial the object is essentially to understand how the yield of a plot would differ if this level of fertilizer were used rather than that level or, in smoking studies, how the outcomes of subjects who smoke compare with what the outcomes would have been had they not smoked. These are in some sense studies of causality, even though that word seems to be sparingly used by natural scientists and until recently by statisticians.