Set-Theoretic Methods for the Social Sciences A Guide to Qualitative Comparative Analysis Qualitative Comparative Analysis (QCA) and other set-theoretic methods distinguish themselves from other approaches to the study of social phenomena by using sets and the search for set relations. In virtually all social science fields, statements about social phenomena can be framed in terms of set relations, and using set-theoretic methods to investigate these statements is therefore highly valuable. This book guides readers through the basic principles of set-theory and then on to the applied practices of QCA. It provides a thorough understanding of basic and advanced issues in set-theoretic methods together with tricks of the trade, software handling, and exercises.
Most arguments are introduced using examples from existing research. The use of QCA is increasing rapidly and the application of set theory is both fruitful and still widely mis- understood in current empirical comparative social research. This book provides the comprehensive guide to these methods for researchers across the social sciences. Schneider is Associate Professor in the Department of Political Science and Founding Director of the Center for the Study of Imperfections in Democracies at Central European University, Hungary.
Claudius Wagemann is Professor of Qualitative Social Science Methods at Goethe University, Frankfurt. Strategies for Social Inquiry Set-Theoretic Methods for the Social Sciences: A Guide to Qualitative Comparative Analysis Editors Colin Elman, Maxwell School of Syracuse University John Gerring, Boston University James Mahoney, Northwestern University Editorial Board Bear Braumoeller, David Collier, Francesco Guala, Peter Hedström, Theodore Hopf, Uskali Maki, Rose McDermott, Charles Ragin, Theda Skocpol, Peter Spiegler, David Waldner, Lisa Wedeen, Christopher Winship This new book series presents texts on a wide range of issues bearing upon the prac- tice of social inquiry. Strategies are construed broadly to embrace the full spectrum of approaches to analysis, as well as relevant issues in philosophy of social science. Published Titles John Gerring, Social Science Methodology: A Unified Framework, 2nd edition Michael Coppedge, Democratization and Research Methods Thad Dunning, Natural Experiments in the Social Sciences: A Design-Based Approach Forthcoming Titles Diana Kapiszewski, Lauren M.
MacLean and Benjamin L. Read, Field Research in Political Science Jason Seawright, Multi-Method Social Science: Combining Qualitative and Quantitative Tools Set-Theoretic Methods for the Social Sciences A Guide to Qualitative Comparative Analysis Carsten Q. Schneider and Claudius Wagemann CAMB RID GE UN IV E R SIT Y PRE SS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.org Information on this title: www.org/9781107601130 Translated and adapted from Qualitative Comparative Analysis (QCA) und Fuzzy Sets: Ein Lehrbuch für Anwender und alle, die es werden wollen published in German by Verlag Barbara Budrich 2007, © Verlag Barbara Budrich 2007. First published in English by Cambridge University Press 2012 as Set-Theoretic Methods for the Social Sciences: A Guide to Qualitative Comparative Analysis © Cambridge University Press 2012.
This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. Printed and bound in the United Kingdom by the MPG Books Group A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Schneider, Carsten Q. Set-theoretic methods for the social sciences : a guide to qualitative comparative analysis / Carsten Q.
Schneider and Claudius Wagemann. – (Strategies for social inquiry) Includes bibliographical references and index. Social sciences – Comparative method. Social sciences – Mathematical models.
Wagemann, Claudius, author.72–dc23 2012015930 ISBN 978-1-107-01352-0 Hardback ISBN 978-1-107-60113-0 Paperback Additional resources for this publication at www.org/schneider-wagemann Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. In honor of Philippe Schmitter, wise advisor, generous colleague, and good friend. Dedicated to Sheila, Giulia, and Leo, without whom this book would have been finished much sooner. Contents List of figures page xii List of tables xiv Acknowledgements xvi Introduction 1 Set-theoretic approaches in the social sciences 1 Qualitative Comparative Analysis as a set-theoretic approach and technique 8 Variants of QCA 13 Plan of the book 16 How to use this book 19 Part I Set-theoretic methods: the basics 21 1 Sets, set membership, and calibration 23 1.1 The notion of sets 24 1.1 Sets and concepts 24 1.2 The pros and cons of crisp sets 24 1.3 Properties of fuzzy sets 27 1.4 What fuzzy sets are not 30 1.2 The calibration of set membership 32 1.1 Principles of calibration 32 1.2 The use of quantitative scales for calibration 33 1.3 The “direct” and “indirect” methods of calibration 35 1.4 Does the choice of calibration strategy matter much? 38 1.5 Assessing calibration 40 2 Notions and operations in set theory 42 2.1 Conjunctions, Boolean and fuzzy multiplication, intersection, logical AND 42 vii viii Contents 2.2 Disjunctions, Boolean and fuzzy addition, union, logical OR 45 2.3 Negations, complements, logical NOT 47 2.4 Operations on complex expressions 47 2.1 Rules for combining logical operators 48 2.2 Negation, intersection, and union of complex sets 49 2.3 Calculating membership in complex sets 51 2.5 Relations between sets 52 2.6 Notational systems in set-theoretic methods 54 3 Set relations 56 3.3 Causal complexity in set-theoretic methods 76 3.1 Defining causal complexity 78 3.2 INUS and SUIN conditions 79 3.3 The notion of asymmetry 81 3.4 Set-theoretic methods and standard quantitative approaches 83 4 Truth tables 91 4.1 What is a truth table? 92 4.2 How to get from a data matrix to a truth table 93 4.3 Analyzing truth tables 104 4.1 Matching similar conjunctions 105 4.2 Logically redundant prime implicants 108 4.3 Issues related to the analysis of the non-occurrence of the outcome 112 Part II Neat formal logic meets noisy social science data 117 5 Parameters of fit 119 5.1 Defining and dealing with contradictory truth table rows 120 5.2 Consistency of sufficient conditions 123 5.3 Coverage of sufficient conditions 129 ix Contents 5.4 Consistency of necessary conditions 139 5.5 Coverage of necessary conditions 144 5.6 Issues related to consistency and coverage 148 6 Limited diversity and logical remainders 151 6.1 Limited diversity in set-theoretic methods: how to see it when it is there 152 6.2 Sources of limited diversity 153 6.3 What limited diversity is not 157 6.4 The Standard Analysis procedure: identifying logical remainders for crafting plausible solution terms 160 6.1 The dimension of set relations 161 6.2 The dimension of complexity 165 6.3 The dimension of types of counterfactuals 167 6.4 The Standard Analysis procedure in a nutshell 175 7 The Truth Table Algorithm 178 7.1 From the data matrix to truth table 179 7.2 Attributing an outcome value to each truth table row 182 7.3 Logically minimizing the truth table 186 7.4 Implications of the Truth Table Algorithm 190 Part III Potential pitfalls and suggestions for solutions 195 8 Potential pitfalls in the Standard Analysis procedure and suggestions for improvement 197 8.1 Beyond the Standard Analysis: expanding the types of counterfactuals 198 8.2 The Enhanced Standard Analysis: forms of untenable assumptions and how to avoid them 200 8.1 Incoherent counterfactuals I: contradicting the statement of necessity 201 8.2 Incoherent counterfactuals II: contradictory assumptions 203 8.3 Implausible counterfactuals: contradicting common sense 206 8.4 Putting the Enhanced Standard Analysis procedure into practice 209 x Contents 8.3 Theory-Guided Enhanced Standard Analysis: complementary strategies for dealing with logical remainders 211 8.1 Choosing entire truth table rows as good counterfactuals 212 8.2 Formulating conjunctural directional expectations 215 8.4 Comparing the different strategies for the treatment of logical remainders 217 9 Potential pitfalls in the analysis of necessity and sufficiency and suggestions for avoiding them 220 9.1 Pitfalls in inferring necessity from sufficiency solution terms 221 9.1 Hidden necessary conditions 221 9.2 The appearance of false necessary conditions 227 9.2 The analytic consequences of skewed set-membership scores 232 9.1 The coverage of necessary conditions and the problem of trivialness 233 9.2 The consistency of sufficient conditions and the problem of simultaneous subset relations 237 9.3 A general treatment of skewed set membership in fuzzy-set analyses 244 Part IV Variants of QCA as a technique meet QCA as an approach 251 10 Variants of QCA 253 10.1 The two-step approach 253 10.2 Multi-value QCA 255 10.1 Principles of mvQCA: notation and logical minimization 256 10.2 An assessment of mvQCA 258 10.3 Set-theoretic methods and time 263 10.1 Forms of causally relevant notions of time 264 10.2 Informal ways of integrating notions of time into set-theoretic methods 265 10.4 Temporal QCA 269 11 Data analysis technique meets set-theoretic approach 275 11.1 Recipe for a good QCA 275 11.1 The appropriateness of set-theoretic methods 276 11.2 The choice of the conditions and the outcome 276 xi Contents 11.3 The choice of the QCA variant 277 11.4 Calibration of set-membership scores 277 11.5 Analysis of necessary conditions 278 11.6 Analysis of sufficient conditions 278 11.7 Presentation of results 280 11.8 Interpretation of results 280 11.9 Reiteration of the research cycle 281 11.10 The use of software 282 11.2 Robustness and uncertainty in QCA 284 11.1 How do we see robustness in set-theoretic methods when it is there? 285 11.2 The effects of changing calibration 287 11.3 The effects of changing consistency levels 291 11.4 The effect of dropping or adding cases 293 11.3 The evaluation of theories in set-theoretic methods 295 11.1 Why standard hypothesis testing does not fit into set-theoretic methods 296 11.2 The basics of theory evaluation in set-theoretic methods 297 11.3 Extending theory evaluation by integrating consistency and coverage 300 11.4 Summarizing set-theoretic theory evaluation 304 11.4 Set-theoretic methods and case selection 305 11.1 Types of cases after a QCA 306 11.2 Forms and aims of (comparative) within-case studies after a QCA 308 11.3 Post-QCA case selection principles 310 12 Looking back, looking ahead 313 12.1 Looking back: the main topics of this book 313 12.2 Myths and misunderstandings 316 12.3 Looking ahead: tasks and developments in the coming years 318 Glossary 322 Bibliography 336 Index 346 Figures 0.1 Venn diagram for relation of sufficiency page 5 0.2 Set-theoretic approaches in the social sciences 10 1.1 Membership in fuzzy set of Länder with underdeveloped all-day schools plotted against percentage of pupils enrolled in all-day schools 36 3.1 Two-by-two table – sufficiency 59 3.2 Venn diagram – sufficiency 60 3.3 XY plots in crisp-set analysis – distribution of cases for sufficient conditions 66 3.4 XY plot – distribution of cases for sufficient condition X 67 3.5 XY plot – fully consistent sufficiency solution 69 3.6 Two-by-two table – necessity 71 3.7 Venn diagram – necessity 72 3.8 XY plot – distribution of cases for necessary condition X 76 3.9 XY plot – non-consistent necessary condition 77 3.10 Two-by-two table – necessity and sufficiency 84 3.11 XY plot – contrasting perfect set relation with perfect correlations 86 4.1 Venn diagram with three conditions 94 4.2 Three-dimensional property space 98 4.3 Logical minimization of primitive expressions to prime implicants 110 4.4 Venn diagram with logically redundant prime implicant 112 5.1 Venn diagrams – consistent and inconsistent sufficient conditions 124 5.2 XY plot – consistent and inconsistent sufficient conditions 125 5.3 Venn diagrams – different levels of coverage sufficiency 130 5.4 XY plot – different levels of coverage sufficiency 132 5.5 Venn diagram – equifinal solution term and types of coverage 135 5.6 XY plot – condition STOCK, outcome EXPORT 142 5.7 Venn diagrams – trivial and non-trivial necessary conditions 145 5.8 XY plot – condition MA+STOCK, outcome EXPORT 147 5.9 XY plot – the tension between consistency and coverage of sufficient conditions 149 6.1 Conservative, intermediate, and most parsimonious solution terms 172 6.2 Venn diagram – types of counterfactuals in Standard Analysis procedure 176 xii xiii List of figures 7.1 XY plot for path C~P 189 7.