Statistics and Causal Inference
01 Dec 1986-Journal of the American Statistical Association (Taylor & Francis Group)-Vol. 81, Iss: 396, pp 945-960
TL;DR: In this article, the authors use a particular model for causal inference (Holland and Rubin 1983; Rubin 1974) to critique the discussions of other writers on causation and causal inference.
Abstract: Problems involving causal inference have dogged at the heels of statistics since its earliest days. Correlation does not imply causation, and yet causal conclusions drawn from a carefully designed experiment are often valid. What can a statistical model say about causation? This question is addressed by using a particular model for causal inference (Holland and Rubin 1983; Rubin 1974) to critique the discussions of other writers on causation and causal inference. These include selected philosophers, medical researchers, statisticians, econometricians, and proponents of causal modeling.
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Journal Article•
TL;DR: This book by a teacher of statistics (as well as a consultant for "experimenters") is a comprehensive study of the philosophical background for the statistical design of experiment.
Abstract: THE DESIGN AND ANALYSIS OF EXPERIMENTS. By Oscar Kempthorne. New York, John Wiley and Sons, Inc., 1952. 631 pp. $8.50. This book by a teacher of statistics (as well as a consultant for \"experimenters\") is a comprehensive study of the philosophical background for the statistical design of experiment. It is necessary to have some facility with algebraic notation and manipulation to be able to use the volume intelligently. The problems are presented from the theoretical point of view, without such practical examples as would be helpful for those not acquainted with mathematics. The mathematical justification for the techniques is given. As a somewhat advanced treatment of the design and analysis of experiments, this volume will be interesting and helpful for many who approach statistics theoretically as well as practically. With emphasis on the \"why,\" and with description given broadly, the author relates the subject matter to the general theory of statistics and to the general problem of experimental inference. MARGARET J. ROBERTSON
13,333 citations
TL;DR: For instance, King, Keohane, Verba, and Verba as mentioned in this paper have developed a unified approach to valid descriptive and causal inference in qualitative research, where numerical measurement is either impossible or undesirable.
Abstract: While heated arguments between practitioners of qualitative and quantitative research have begun to test the very integrity of the social sciences, Gary King, Robert Keohane, and Sidney Verba have produced a farsighted and timely book that promises to sharpen and strengthen a wide range of research performed in this field. These leading scholars, each representing diverse academic traditions, have developed a unified approach to valid descriptive and causal inference in qualitative research, where numerical measurement is either impossible or undesirable. Their book demonstrates that the same logic of inference underlies both good quantitative and good qualitative research designs, and their approach applies equally to each. Providing precepts intended to stimulate and discipline thought, the authors explore issues related to framing research questions, measuring the accuracy of data and uncertainty of empirical inferences, discovering causal effects, and generally improving qualitative research. Among the specific topics they address are interpretation and inference, comparative case studies, constructing causal theories, dependent and explanatory variables, the limits of random selection, selection bias, and errors in measurement. Mathematical notation is occasionally used to clarify concepts, but no prior knowledge of mathematics or statistics is assumed. The unified logic of inference that this book explicates will be enormously useful to qualitative researchers of all traditions and substantive fields.
6,233 citations
TL;DR: This update of the CONSORT statement improves the wording and clarity of the previous checklist and incorporates recommendations related to topics that have only recently received recognition, such as selective outcome reporting bias.
Abstract: Overwhelming evidence shows the quality of reporting of randomised controlled trials (RCTs) is not optimal. Without transparent reporting, readers cannot judge the reliability and validity of trial findings nor extract information for systematic reviews. Recent methodological analyses indicate that inadequate reporting and design are associated with biased estimates of treatment effects. Such systematic error is seriously damaging to RCTs, which are considered the gold standard for evaluating interventions because of their ability to minimise or avoid bias. A group of scientists and editors developed the CONSORT (Consolidated Standards of Reporting Trials) statement to improve the quality of reporting of RCTs. It was first published in 1996 and updated in 2001. The statement consists of a checklist and flow diagram that authors can use for reporting an RCT. Many leading medical journals and major international editorial groups have endorsed the CONSORT statement. The statement facilitates critical appraisal and interpretation of RCTs. During the 2001 CONSORT revision, it became clear that explanation and elaboration of the principles underlying the CONSORT statement would help investigators and others to write or appraise trial reports. A CONSORT explanation and elaboration article was published in 2001 alongside the 2001 version of the CONSORT statement. After an expert meeting in January 2007, the CONSORT statement has been further revised and is published as the CONSORT 2010 Statement. This update improves the wording and clarity of the previous checklist and incorporates recommendations related to topics that have only recently received recognition, such as selective outcome reporting bias. This explanatory and elaboration document-intended to enhance the use, understanding, and dissemination of the CONSORT statement-has also been extensively revised. It presents the meaning and rationale for each new and updated checklist item providing examples of good reporting and, where possible, references to relevant empirical studies. Several examples of flow diagrams are included. The CONSORT 2010 Statement, this revised explanatory and elaboration document, and the associated website (www.consort-statement.org) should be helpful resources to improve reporting of randomised trials.
5,957 citations
Book•
01 Jan 1987TL;DR: In this article, the authors present a general classification notation for multilevel models and a discussion of the general structure and maximum likelihood estimation for a multi-level model, as well as the adequacy of Ordinary Least Squares estimates.
Abstract: Contents Dedication Preface Acknowledgements Notation A general classification notation and diagram Glossary Chapter 1 An introduction to multilevel models 1.1 Hierarchically structured data 1.2 School effectiveness 1.3 Sample survey methods 1.4 Repeated measures data 1.5 Event history and survival models 1.6 Discrete response data 1.7 Multivariate models 1.8 Nonlinear models 1.9 Measurement errors 1.10 Cross classifications and multiple membership structures. 1.11 Factor analysis and structural equation models 1.12 Levels of aggregation and ecological fallacies 1.13 Causality 1.14 The latent normal transformation and missing data 1.15 Other texts 1.16 A caveat Chapter 2 The 2-level model 2.1 Introduction 2.2 The 2-level model 2.3 Parameter estimation 2.4 Maximum likelihood estimation using Iterative Generalised Least Squares (IGLS) 2.5 Marginal models and Generalized Estimating Equations (GEE) 2.6 Residuals 2.7 The adequacy of Ordinary Least Squares estimates. 2.8 A 2-level example using longitudinal educational achievement data 2.9 General model diagnostics 2.10 Higher level explanatory variables and compositional effects 2.11 Transforming to normality 2.12 Hypothesis testing and confidence intervals 2.13 Bayesian estimation using Markov Chain Monte Carlo (MCMC) 2.14 Data augmentation Appendix 2.1 The general structure and maximum likelihood estimation for a multilevel model Appendix 2.2 Multilevel residuals estimation Appendix 2.3 Estimation using profile and extended likelihood Appendix 2.4 The EM algorithm Appendix 2.5 MCMC sampling Chapter 3. Three level models and more complex hierarchical structures. 3.1 Complex variance structures 3.2 A 3-level complex variation model example. 3.3 Parameter Constraints 3.4 Weighting units 3.5 Robust (Sandwich) Estimators and Jacknifing 3.6 The bootstrap 3.7 Aggregate level analyses 3.8 Meta analysis 3.9 Design issues Chapter 4. Multilevel Models for discrete response data 4.1 Generalised linear models 4.2 Proportions as responses 4.3 Examples 4.4 Models for multiple response categories 4.5 Models for counts 4.6 Mixed discrete - continuous response models 4.7 A latent normal model for binary responses 4.8 Partitioning variation in discrete response models Appendix 4.1. Generalised linear model estimation Appendix 4.2 Maximum likelihood estimation for generalised linear models Appendix 4.3 MCMC estimation for generalised linear models Appendix 4.4. Bootstrap estimation for generalised linear models Chapter 5. Models for repeated measures data 5.1 Repeated measures data 5.2 A 2-level repeated measures model 5.3 A polynomial model example for adolescent growth and the prediction of adult height 5.4 Modelling an autocorrelation structure at level 1. 5.5 A growth model with autocorrelated residuals 5.6 Multivariate repeated measures models 5.7 Scaling across time 5.8 Cross-over designs 5.9 Missing data 5.10 Longitudinal discrete response data Chapter 6. Multivariate multilevel data 6.1 Introduction 6.2 The basic 2-level multivariate model 6.3 Rotation Designs 6.4 A rotation design example using Science test scores 6.5 Informative response selection: subject choice in examinations 6.6 Multivariate structures at higher levels and future predictions 6.7 Multivariate responses at several levels 6.8 Principal Components analysis Appendix 6.1 MCMC algorithm for a multivariate normal response model with constraints Chapter 7. Latent normal models for multivariate data 7.1 The normal multilevel multivariate model 7.2 Sampling binary responses 7.3 Sampling ordered categorical responses 7.4 Sampling unordered categorical responses 7.5 Sampling count data 7.6 Sampling continuous non-normal data 7.7 Sampling the level 1 and level 2 covariance matrices 7.8 Model fit 7.9 Partially ordered data 7.10 Hybrid normal/ordered variables 7.11 Discussion Chapter 8. Multilevel factor analysis, structural equation and mixture models 8.1 A 2-stage 2-level factor model 8.2 A general multilevel factor model 8.3 MCMC estimation for the factor model 8.4 Structural equation models 8.5 Discrete response multilevel structural equation models 8.6 More complex hierarchical latent variable models 8.7 Multilevel mixture models Chapter 9. Nonlinear multilevel models 9.1 Introduction 9.2 Nonlinear functions of linear components 9.3 Estimating population means 9.4 Nonlinear functions for variances and covariances 9.5 Examples of nonlinear growth and nonlinear level 1 variance Appendix 9.1 Nonlinear model estimation Chapter 10. Multilevel modelling in sample surveys 10.1 Sample survey structures 10.2 Population structures 10.3 Small area estimation Chapter 11 Multilevel event history and survival models 11.1 Introduction 11.2 Censoring 11.3 Hazard and survival funtions 11.4 Parametric proportional hazard models 11.5 The semiparametric Cox model 11.6 Tied observations 11.7 Repeated events proportional hazard models 11.8 Example using birth interval data 11.9 Log duration models 11.10 Examples with birth interval data and children s activity episodes 11.11 The grouped discrete time hazards model 11.12 Discrete time latent normal event history models Chapter 12. Cross classified data structures 12.1 Random cross classifications 12.2 A basic cross classified model 12.3 Examination results for a cross classification of schools 12.4 Interactions in cross classifications 12.5 Cross classifications with one unit per cell 12.6 Multivariate cross classified models 12.7 A general notation for cross classifications 12.8 MCMC estimation in cross classified models Appendix 12.1 IGLS Estimation for cross classified data. Chapter 13 Multiple membership models 13.1 Multiple membership structures 13.2 Notation and classifications for multiple membership structures 13.3 An example of salmonella infection 13.4 A repeated measures multiple membership model 13.5 Individuals as higher level units 13.5.1 Example of research grant awards 13.6 Spatial models 13.7 Missing identification models Appendix 13.1 MCMC estimation for multiple membership models. Chapter 14 Measurement errors in multilevel models 14.1 A basic measurement error model 14.2 Moment based estimators 14.3 A 2-level example with measurement error at both levels. 14.4 Multivariate responses 14.5 Nonlinear models 14.6 Measurement errors for discrete explanatory variables 14.7 MCMC estimation for measurement error models Appendix 14.1 Measurement error estimation 14.2 MCMC estimation for measurement error models Chapter 15. Smoothing models for multilevel data. 15.1 Introduction 15.2. Smoothing estimators 15.3 Smoothing splines 15.4 Semi parametric smoothing models 15.5 Multilevel smoothing models 15.6 General multilevel semi-parametric smoothing models 15.7 Generalised linear models 15.8 An example Fixed Random 15.9 Conclusions Chapter 16. Missing data, partially observed data and multiple imputation 16.1 Creating a completed data set 16.2 Joint modelling for missing data 16.3 A two level model with responses of different types at both levels. 16.4 Multiple imputation 16.5 A simulation example of multiple imputation for missing data 16.6 Longitudinal data with attrition 16.7 Partially known data values 16.8 Conclusions Chapter 17 Multilevel models with correlated random effects 17.1 Non-independence of level 2 residuals 17.2 MCMC estimation for non-independent level 2 residuals 17.3 Adaptive proposal distributions in MCMC estimation 17.4 MCMC estimation for non-independent level 1 residuals 17.5 Modelling the level 1 variance as a function of explanatory variables with random effects 17.6 Discrete responses with correlated random effects 17.7 Calculating the DIC statistic 17.8 A growth data set 17.9 Conclusions Chapter 18. Software for multilevel modelling References Author index Subject index
5,839 citations
TL;DR: Propensity score matching (PSM) has become a popular approach to estimate causal treatment effects as discussed by the authors, but empirical examples can be found in very diverse fields of study, and each implementation step involves a lot of decisions and different approaches can be thought of.
Abstract: Propensity score matching (PSM) has become a popular approach to estimate causal treatment effects. It is widely applied when evaluating labour market policies, but empirical examples can be found in very diverse fields of study. Once the researcher has decided to use PSM, he is confronted with a lot of questions regarding its implementation. To begin with, a first decision has to be made concerning the estimation of the propensity score. Following that one has to decide which matching algorithm to choose and determine the region of common support. Subsequently, the matching quality has to be assessed and treatment effects and their standard errors have to be estimated. Furthermore, questions like 'what to do if there is choice-based sampling?' or 'when to measure effects?' can be important in empirical studies. Finally, one might also want to test the sensitivity of estimated treatment effects with respect to unobserved heterogeneity or failure of the common support condition. Each implementation step involves a lot of decisions and different approaches can be thought of. The aim of this paper is to discuss these implementation issues and give some guidance to researchers who want to use PSM for evaluation purposes.
5,510 citations
Cites background or methods from "Statistics and Causal Inference"
...bootstrapping methods, the variance approximation proposed in Lechner (2001a) and the variance estimators proposed by Abadie and Imbens (2006a)....
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...Imbens (2004) or Wooldridge (2004), Section 18.3.2, for a formal description of weighting on propensity score estimators. 26. See Imbens (2004) for a formal proof that this weighting estimator removes the bias due to different distributions of the covariates between treated and untreated individuals....
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...To obtain standard errors the user can choose between bootstrapping and the variance approximation proposed by Lechner (2001a). Additionally the authors offer balancing tests (blocking, stratification) as discussed in Section 3.4. Leuven and Sianesi (2003) provide the programme psmatch2 for implementing different kinds of matching estimators including covariate and propensity score matching....
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...The multiple treatment case (as discussed in Imbens (2000) and Lechner (2001a)) consists of more than two alternatives, for example when an individual is faced with the choice to participate in job-creation schemes, vocational training or wage subsidy programmes or to not participate at all (we will describe this approach in more detail in Section 3.10). For that case it is well known that the multinomial logit is based on stronger assumptions than the multinomial probit model, making the latter the preferable option.8 However, since the multinomial probit is computationally more burdensome, a practical alternative is to estimate a series of binomial models as suggested by Lechner (2001a). Bryson et al....
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...One is by ‘brute force’ (Imbens, 2004), i.e. by estimating the five components of the variance σ(2)0(X ), σ 2 1(X ), E(Y (1)|X ), E(Y (0)|X ) and P(X) using kernel methods or series. Even though this is consistently possible and hence the asymptotic variance will be consistent, too, Imbens (2004) notes that this might be an additional computational burden. Hence, practical alternatives are called for and we are going to present three of them. Two of them, bootstrapping and the variance approximation by Lechner (2001a), are very common in the applied literature....
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References
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TL;DR: The authors discusses the central role of propensity scores and balancing scores in the analysis of observational studies and shows that adjustment for the scalar propensity score is sufficient to remove bias due to all observed covariates.
Abstract: : The results of observational studies are often disputed because of nonrandom treatment assignment. For example, patients at greater risk may be overrepresented in some treatment group. This paper discusses the central role of propensity scores and balancing scores in the analysis of observational studies. The propensity score is the (estimated) conditional probability of assignment to a particular treatment given a vector of observed covariates. Both large and small sample theory show that adjustment for the scalar propensity score is sufficient to remove bias due to all observed covariates. Applications include: matched sampling on the univariate propensity score which is equal percent bias reducing under more general conditions than required for discriminant matching, multivariate adjustment by subclassification on balancing scores where the same subclasses are used to estimate treatment effects for all outcome variables and in all subpopulations, and visual representation of multivariate adjustment by a two-dimensional plot. (Author)
23,744 citations
TL;DR: In this article, the cross spectrum between two variables can be decomposed into two parts, each relating to a single causal arm of a feedback situation, and measures of causal lag and causal strength can then be constructed.
Abstract: There occurs on some occasions a difficulty in deciding the direction of causality between two related variables and also whether or not feedback is occurring. Testable definitions of causality and feedback are proposed and illustrated by use of simple two-variable models. The important problem of apparent instantaneous causality is discussed and it is suggested that the problem often arises due to slowness in recording information or because a sufficiently wide class of possible causal variables has not been used. It can be shown that the cross spectrum between two variables can be decomposed into two parts, each relating to a single causal arm of a feedback situation. Measures of causal lag and causal strength can then be constructed. A generalisation of this result with the partial cross spectrum is suggested.
16,349 citations
Journal Article•
TL;DR: This book by a teacher of statistics (as well as a consultant for "experimenters") is a comprehensive study of the philosophical background for the statistical design of experiment.
Abstract: THE DESIGN AND ANALYSIS OF EXPERIMENTS. By Oscar Kempthorne. New York, John Wiley and Sons, Inc., 1952. 631 pp. $8.50. This book by a teacher of statistics (as well as a consultant for \"experimenters\") is a comprehensive study of the philosophical background for the statistical design of experiment. It is necessary to have some facility with algebraic notation and manipulation to be able to use the volume intelligently. The problems are presented from the theoretical point of view, without such practical examples as would be helpful for those not acquainted with mathematics. The mathematical justification for the techniques is given. As a somewhat advanced treatment of the design and analysis of experiments, this volume will be interesting and helpful for many who approach statistics theoretically as well as practically. With emphasis on the \"why,\" and with description given broadly, the author relates the subject matter to the general theory of statistics and to the general problem of experimental inference. MARGARET J. ROBERTSON
13,333 citations
Book•
01 Jan 1970
TL;DR: Hume's early years and education is described in a treatise of human nature as discussed by the authors. But it is not a complete account of the early years of his life and education.
Abstract: PART 1: INTRODUCTORY MATERIAL How to Use this Book List of Abbreviations Editor's Introduction Hume's Early years and Education A Treatise of Human Nature Book 1: Of the Understanding Book 1 part 1: The Elements of the Mental World Book 1 Part 2: The Ideas of Space and Time Book 1 Part 3: Knowledge, Probability, Belief, and Causation Book 1 Part 4: Forms of Scepticism Book 2: Of the passions Book 2 Part 1: The Indirect Passions of Pride and Humility Book 2 Part 2: The Indirect Passions of Love and Hatred Book 2 part 3: The Direct Passions and the Will Book 3: Of Morals Book 3 Part 1: The Source of Moral Distinctions Book 3 Part 2: The Artificial Virtues Book 3 Part 3: Natural Virtues and Natural Abilities The Abstract and the Early Reception of the Treatise Supplementary Reading A Note on the Texts of this Edition PART 2: THE TEXT Advertisement Introduction Book 1: Of the Understanding Part 1: Of ideas, their origin, composition, connexion, abstraction, etc Sect 1: Of the origin of our ideas Sect 2: Division of the subject Sect 3: Of the ideas of the memory and imagination Sect 4: Of the connexion of association of ideas Sect 5 Of relations Sect 6 Of modes and substances Sect 7: Of abstract ideas Part 2: Of ideas of space and time Sect 1: Of the infinite divisibility of our ideas of space and time Sect 2: Of the infinite divisibility of space and time Sect 3 Of the other qualities of our ideas of space and time Sect 4 Objections answered Sect 5: The same subject continued Sect 6: Of the idea of existence and of external existence Part 3: of knowledge and probability Sect 1: Of knowledge Sect 2 Of probability and of the idea of cause and effect Sect 3: Why a cause is always necessary Sect 4: Of the component parts of our reasonings concerning cause and effect Sect 5: Of the impressions of the senses and memory Section 6: Of the inference from the impression to the idea Sect 7: Of the nature of the idea or belief Sect 8: Of the causes of belief Sect 9: Of the effects of other relations and other habits Sect 10 Of the influence of belief Sect 11: Of the probability of chances Sect 12: Of the probability of causes Sect 13: Of unphilosophical probability Sect 14: Of the idea of necessary connexion Sect 15: Rules by which to judge of causes and effects Sect 16: Of the reason of animals Part 4: Of the sceptical and other systems of philosophy Sect 1: Of scepticism with regard to reason Sect 2: Of scepticism with regard to the senses Sect 3 Of the ancient philosophy Sect 4 Of the modern philosophy Sect 5: Of the immateriality of the soul Sect 6: Of personal identity Sect 7: Conclusion of this book Book 2: Of the Passions Part 1: Of pride and humility Sect 1: Division of the subject Sect 2: Of pride and humility their objects and causes Sect 3: Whence these objects and causes are derived Sect 4: Of the relations of impressions and ideas Sect 5: Of the influence of these relations on pride and humility Sect 6: Limitations of this system Sect 7: Of vice and virtue Sect 8: Of beauty and deformity Sect 9: Of external advantages and disadvantages Sect 10: Of property and riches Sect 11: Of the love of fame Sect 12: Of the pride and humility of animals Part 2: Of love and hatred Sect 1: Of the objects and causes of love and hatred Sect 2: Experiments to confirm this system Sect 3: Difficulties solved Sect 4: Of the love of relations Sect 5: Of our esteem for the rich and powerful Sect 6: Of benevolence and anger Sect 7: Of compassion Sect 8: Of malice and envy Sect 9: Of the mixture of benevolence and anger with compassion and malice Sect 10 Of respect and contempt Sect 11: Of the amorous passion, or love betwixt the sexes Sect 12: Of the love and hatred of animals Part 3: Of the will and direct passions Sect 1: Of liberty and necessity Sect 2: The same subject continued Sect 3: Of the influencing motives of the will Sect 4: Of the causes of the violent passions Sect 5: Of the effects of custom Sect Of the influence of the imagination on passions Sect 7: Of contiguity and distance in space and time Sect 8: The same subject continued Sect 9: Of the direct passions Sect 10: Of curiosity, or the love of truth Book 3: Of Morals Advertisement Part 1: Of virtue and vice in general Sect 1: Moral distinctions not derived from reason Sect 2: Moral distinctions derived from a moral sense Part 2: Of justice and injustice Sect 1: Justice, whether a natural or artificial virtue? Sect 2: Of the origin of justice and property Sect 3: Of the rules, which determine property Sect 4: Of the transference of property by consent Sect 5: Of the obligation of promises Sect 6: Some farther reflections concerning justice and injustice Sect 7: Of the origin of government Sect 8: Of the source of allegiance Sect 9: Of the measures of allegiance Sect 10: Of the objects of allegiance Sect 11: Of the laws of nations Sect 12: Of chastity and modesty Part 3: Of the other virtues and vices Sect 1: Of the origin of the natural virtues and vices Sect 2: Of greatness of mind Sect 3 Of goodness and benevolence Sect 4: Of natural abilities Sect 5: Some farther reflections concerning the natural virtues Sect 6: Conclusion of this book Appendix An Abstract of A Treatise of Human Nature PART 3 SUPPLEMENTARY MATERIAL Editors' Annotations Annotations to the Treatise Annotations to the Abstract Glossary References Index
10,342 citations
TL;DR: A discussion of matching, randomization, random sampling, and other methods of controlling extraneous variation is presented in this paper, where the objective is to specify the benefits of randomization in estimating causal effects of treatments.
Abstract: A discussion of matching, randomization, random sampling, and other methods of controlling extraneous variation is presented. The objective is to specify the benefits of randomization in estimating causal effects of treatments. The basic conclusion is that randomization should be employed whenever possible but that the use of carefully controlled nonrandomized data to estimate causal effects is a reasonable and necessary procedure in many cases. Recent psychological and educational literature has included extensive criticism of the use of nonrandomized studies to estimate causal effects of treatments (e.g., Campbell & Erlebacher, 1970). The implication in much of this literature is that only properly randomized experiments can lead to useful estimates of causal effects. If taken as applying to all fields of study, this position is untenable. Since the extensive use of randomized experiments is limited to the last half century,8 and in fact is not used in much scientific investigation today,4 one is led to the conclusion that most scientific "truths" have been established without using randomized experiments. In addition, most of us successfully determine the causal effects of many of our everyday actions, even interpersonal behaviors, without the benefit of randomization. Even if the position that causal effects of treatments can only be well established from randomized experiments is taken as applying only to the social sciences in which
8,377 citations