In this article the authors develop goodness-of-fit tests for fuzzy-set analyses to formally assess the fit between empirical information and various causal hypotheses while accounting for measurement error in membership scores. These goodness-of-fit tests, and the accompanying logic, provide a sound inferential foundation for fuzzy-set methodology. The authors also develop descriptive measures to complement these tests. Examples from Stryker and Eliason (2003) and Mahoney (2003) show how goodness-of-fit tests and descriptive measures may be used to assess individual causal factors as well as conjunctions of factors. The authors show how these tools provide more information in a fuzzy-set analysis than do tests currently in use. In providing this inferential foundation, the authors also show that fuzzy-set methods (a) are no less amenable to falsificationist methods of the Neyman-Pearson type than are standard statistical techniques and (b) may be usefully applied in either an exploratory/inductive or a confirmatory/deductive research design.
Almond, Russell G.1995. ‘‘Fuzzy Logic: Better Science? Or Better Engineering?’’Technometrics37:267-70.
2.
Arfi, Baderine.2005. ‘‘Fuzzy Decision Making in Politics: A Linguistic Fuzzy-Set Approach (LFSA).’’ Political Analysis13:23-56.
3.
Arfi, Baderine.2006. ‘‘Linguistic Fuzzy-Logic Game Theory. ’’ Journal of Conflict Resolution50:28-57.
4.
Black, Max.1937. ‘‘Vagueness: An Exercise in Logical Analysis .’’ Philosophy of Science4:427-55.
5.
Braumoeller, Bear F. and Gary Goertz.2000. ‘‘The Methodology of Necessary Conditions.’’American Journal of Political Science44:844-58.
6.
Braumoeller, Bear F. and Gary Goertz.2003. ‘‘The Statistical Methodology of Necessary Conditions.’’ Pp. 197-223 in Necessary Conditions: Theory, Methodology, and Applications, edited by Gary Goertz and Harvey Starr.Oxford, UK: Rowan & Littlefield.
7.
Degroot, Morris.1986. Probability and Statistics, 2nd ed. Reading, MA: Addison-Wesley .
8.
Eliason, Scott R.1993. Maximum Likelihood Estimation: Logic and Practice (Sage Quantitative Applications in the Social Sciences, No. 96). Newbury Park, CA: Sage.
9.
Eliason, Scott R., Robin Stryker, and Eric Tranby.2008. ‘‘The Welfare State, Family Policies and Women’s Labor Force Participation: Combining Fuzzy-Set and Statistical Methods to Assess Causal Relations and Estimate Causal Effects.’’ Pp. 135-95 in Method and Substance in Macrocomparative Analysis, edited by Lane Kenworthy and Alex Hicks.New York: Palgrave Macmillan .
10.
Gauthier, Anne.1996. The State and the Family: A Comparative Analysis of Family Policies in Industrialized Countries. Oxford, UK: Clarendon.
11.
Goertz, Gary.2003. ‘‘The Substantive Importance of Necessary Condition Hypotheses.’’ Pp. 65-94 in Necessary Conditions: Theory, Methodology, and Applications, edited by Gary Goertz and Harvey Starr.Oxford, UK: Rowan & Littlefield.
12.
Goertz, Gary.2006. ‘‘Assessing the Trivialness, Relevance, and Relative Importance of Necessary or Sufficient Conditions in Social Science .’’ Studies in Comparative International Development41:88-109.
13.
Goertz, Gary and James Mahoney.2005. ‘‘Two-Level Theories and Fuzzy Sets. ’’ Sociological Methods & Research33:497-538.
14.
Goertz, Gary and Harvey Starr , eds. 2003. Necessary Conditions: Theory, Methodology, and Applications. Oxford, UK: Rowan & Littlefield.
15.
Heckman, James J., Robert J. LaLonde, and Jeffrey A. Smith1999. ‘‘The Economics and Econometrics of Active Labor Market Programs.’’ pp. 1865-2097 in Handbook of Labor Economics, Volume 111, edited by Orley Ashenfelter and David Card.New York: Elsevier.
16.
Holland, Paul W.1986. ‘‘Statistics and Causal Inference.’’Journal of the American Statistical Association81:945-60.
17.
Imbens, Guido W. and Donald B. Rubin.1997. ‘‘Bayesian Inference for Causal Effects in Randomized Experiments with Noncompliance.’’The Annals of Statistics25:305-327.
18.
Kandel, Abraham, Alejandro Martins, and Roberto Pacheco .1995. ‘‘On the Very Real Distinction Between Fuzzy and Statistical Methods.’’ Technometrics37:276-81.
19.
Klement, E.P., M.L. Puri, and D.A. Ralescu.1986. ‘‘Limit Theorems for Fuzzy Random Variables.’’Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Predictability in Science and Society 407:171-82.
20.
Laviolette, Michael, John W. Seaman Jr. , J. Douglas Barrett, and William H. Woodall.1995. ‘‘A Probabilistic and Statistical View of Fuzzy Methods.’’Technometrics37:249-61.
21.
Lieberson, Stanley.1991. ‘‘Small N’s and Big Conclusions: An Examination of the Reasoning in Comparative Studies Based on a Small Number of Cases.’’ Social Forces70:307-20.
22.
Lieberson, Stanley.1994. ‘‘More on the Uneasy Case for Using Mill-Type Methods in Small-N Comparative Studies.’’ Social Forces72:1225-37.
23.
Loginov, V.J.1966. ‘‘Probability Treatment of Zadeh Membership Functions and Their Use in Pattern Recognition.’’Engineering Cybernetics: 68-69.
24.
Mahoney, James.2003. ‘‘Long-Run Development and the Legacy of Colonialism in Spanish America.’’ American Journal of Sociology109:50-106.
25.
Manton, Kenneth G., Eric Stallard, Max A. Woodbury, H. Dennis Tolley, and Anatoli I. Yashin .1987. ‘‘Grade-of-Membership Techniques for Studying Complex Event History Processes With Unobserved Covariates.’’Sociological Methodology17:309-46.
26.
Manton, Kenneth G., Max A. Woodbury, Eric Stallard, and Larry S. Corder .1992. ‘‘The Use of Grade-of-Membership Techniques to Estimate Regression Relationships.’’Sociological Methodology22:321-81.
27.
Marini, Margaret Mooney and Burton Singer.1988. ‘‘Causality in the Social Sciences.’’Sociological Methodology18:347-409.
28.
McNeill, Daniel and Paul Freiberger.1993. Fuzzy Logic. New York: Simon & Schuster.
29.
Mill, John Stuart. [ 1843] 1967. A System of Logic: Ratiocinative and Inductive. Reprint, Toronto, Canada: University of Toronto.
30.
Montgomery, James D.1998. ‘‘Toward a Role-Theoretic Conception of Embeddedness.’’American Journal of Sociology104:92-125.
31.
Montgomery, James D.2000. ‘‘The Self as a Fuzzy Set of Roles, Role Theory as a Fuzzy System.’’Sociological Methodology30:261-314.
32.
Morgan, Stephen L. and Christopher Winship.2007. Counterfactuals and Causal Inference: Methods and Principles for Social Research. Cambridge, UK: Cambridge University Press.
33.
Puri, Madan L. and Dan A. Ralescu.1985. ‘‘The Concept of Normality for Fuzzy Random Variables.’’Annals of Probability13:1373-79.
34.
Ragin, Charles C.1987. The Comparative Method. Berkeley: University of California.
35.
Ragin, Charles C.2000. Fuzzy-Set Social Science. Chicago: University of Chicago.
36.
Ragin, Charles C.2003. ‘‘Fuzzy-Set Analysis of Necessary Conditions.’’ Pp. 179-96 in Necessary Conditions: Theory, Methodology, and Applications, edited by Gary Goertz and Harvey Starr.Oxford, UK: Rowan & Littlefield.
37.
Ragin, Charles C.2006. ‘‘Set Relations in Social Research: Evaluating Their Consistency and Coverage.’’Political Analysis14:291-310.
38.
Ragin, Charles C. and Paul Pennings.2005. ‘‘Fuzzy Sets’’ [special issue] . Sociological Methods & Research33.
39.
Rindfuss, Ronald R. and Futing Liao.1988. ‘‘Medical and Contraceptive Reasons for Sterilization in the United States.’’Studies in Family Planning19:370-80.
40.
Sobel, Michael E.1995. ‘‘Causal Inference in the Social and Behavioral Sciences.’’ Pp. 1- 38 in Handbook of Statistical Modeling for the Social and Behavioral Sciences, edited by G. Arminger, C. C. Clogg, and M. E. Sobel.New York : Plenum.
41.
Sobel, Michael E.1996. ‘‘An Introduction to Causal Inference.’’Sociological Methods & Research24:353-79.
42.
Stojakovic, Mila and Zoran Stojakovic.1996. ‘‘Support Functions for Fuzzy Sets.’’ Proceedings: Mathematical, Physical and Engineering Sciences452:421-38.
43.
Stryker, Robin and Scott R. Eliason.2003. ‘‘The Welfare State, Gendered Labor Markets and Aggregate Political Orientations in France, Belgium, Germany, Italy, Denmark and Britain, 1977-1994.’’Robert Schuman Centre Advanced Studies Working Paper RSC2003/ 20, European University Institute, Florence, Italy.
44.
Stuart, Alan and Keith Ord.1987. Kendall’s Advanced Theory of Statistics. Vol. 1, Distribution Theory. 5th ed. New York: Oxford University Press.
45.
Stuart, Alan, Keith Ord, and Steven Arnold.1999. Kendall’s Advanced Theory of Statistics. Vol. 2A, Classical Inference & the Linear Model. 6th ed. New York: Oxford University Press.
46.
Walker, Henry A. and Bernard P. Cohen.1985. ‘‘Scope Statements: Imperatives for Evaluating Theory.’’American Sociological Review50:288-301.
47.
Weisberg, Sanford.1985. Applied Linear Regression. 2d ed. New York: John Wiley.
48.
Western, Bruce.1998. ‘‘Causal Heterogeneity in Comparative Research: A Bayesian Hierarchical Modelling Approach.’’ American Journal of Political Science42:1233-59.
49.
Western, Bruce.2001. ‘‘Bayesian Thinking About Macrosociology. ’’ American Journal of Sociology107:353-78.
50.
Western, Bruce and Simon Jackman.1994. ‘‘Bayesian Inference for Comparative Research .’’ American Political Science Review88:412-23.
51.
Winship, Christopher and Stephen L. Morgan.1999. ‘‘The Estimation of Causal Effects from Observational Data’’Annual Review of Sociology25:659-706.
52.
Zadeh, Lotfi.1965. ‘‘Fuzzy Sets.’’Information and Control8:338-53.
53.
Zadeh, Lotfi.1995. ‘‘Probability Theory and Fuzzy Logic Are Complimentary Rather Than Competitive.’’ Technometrics37:271-76.
