Differences and Similarities to Crisp sets. Systematic examination of consistent connections

Transcription

1 An Introduction to Fuzzy-set Analysis Vilmos Misangyi Pennsylvania State University 1 8/16/2010 Presentation for the QCA PDW AOM 2010; Montreal, Quebec Agenda Calibration Differences and Similarities to Crisp sets Analysis Systematic examination of consistent connections Empirical Example: Misangyi & Holehonnur (working paper) What combinations of governance mechanisms lead to high and low firm performance? Sample: 269 S&P 1500 firms in

2 Agenda Calibration Differences and Similarities to Crisp sets Analysis Systematic examination of consistent connections Calibration of Fuzzy-Sets Crisp sets capture differences in kind (i.e., 1/0) whereas fuzzy sets capture both differences in kind and degree Capture rich qualitative data and/or quantitative data (ordinal, ratio, or interval) Multiple calibrations (4, 5, 6 value, continuous) Four value fuzzy set: (1.0 = fully in; 0.67 = more in than out; 0.33 = more out than in; 0.0 = fully out) Continuous fuzzy set: (1.0 = fully in; 0.5 < X < 1.0 = more in than out; 0.5 = neither in nor out; 0.0 < X < 0.5 = more out than in; 0.0 = fully out) fsqca software aids in conversion of latter 2

3 Sets are Purposefully Calibrated (italics in original; Ragin, 2008: 30) Calibration is based upon theoretical and substantive knowledge. Translation based upon three qualitative anchors: full membership, full non membership, cross over point. External standards : body of social and scientific knowledge and empirical evidence. Fundamentally different from variables and ordinal scales NOT ranking of cases relative to each other as in ordinal scales Empirical Example: Monitoring/Governance Attributes Board Monitoring Outside Director Vigilance (Boeker, 1992; Daily, 1995; Hambrick & Jackson, 2000; Zajac & Westphal, 1996) FuzzyOr(Ownership, Independence) Inside Director Ownership (Fama, 1980; Rediker & Seth, 1995) Market for Corporate Control (e.g., Alchian & Demsetz, 1972; Agarwal & Knoeber, 1996) External Block Holder Ownership (e.g., Shleifer & Vishny, 1997; Tosi & Gomez Mejia, 1994) CEO Alignment (e.g., Demsetz, 1983; Bebchuck & Fried, 2004) FuzzyOr(Ownership, Incentive Pay) CEO Power (e.g., Finkelstein & D Aveni, 1994; Gompers et al., 2003) FuzzyOr(Duality, G Index) 3

4 Example: Outside Director Vigilance FuzzyOr(ownership, independence) Outside director ownership: Firm s membership in the set of firms that have a significant proportion of their board that are outsiders with a substantial stake. Measurement:» Substantial stake based on $ (Hambrick and Jackson; 2000) ($500,000 in 2000)» Must have greater than $860,000 for substantial stake. Calibration: Fuzzy set based upon the data» Fully in =.68 (75 %); crossover =.43 (mean); fully out =.18 (25%). Outside director independence: Firm s membership in the set of firms that have a significant proportion of their board that are unaffiliated outsiders. Measurement:» Proportion of directors appointed before CEO and independent per Risk Metrics Calibration: Fuzzy set based upon the distribution of the data» Fully in =.6 (75 %); crossover =.37 (median); fully out =.15 (25%). Example: High and Low Performance High Performance Firm s membership in the set of firms that have above average performance Measurement: Above average ROA based upon sector (Fiss, in press; Greckhamer et al., 2008). S&P 1500 companies in 8 sectors: Mining, Oil & Gas; Manufacturing; Transportation; Communication & Utilities; Wholesale Trade; Retail Trade; Finance; Services. Calibration: Fully in (1) = 75%; Crossover = median; Fully out = 25% Low Performance FuzzyNot(High ROA) (Fiss, in press) 4

5 Agenda Calibration Differences and Similarities to Crisp sets Analyses Systematic examination of consistent connections Fuzzy-set Analysis Systematic examination of explicit connections (Ragin, 2006; 2008) g best understood as uniformities or near uniformities in social phenomena and uniformities may reflect causal connections as well as other types of integral connections (e.g., constitutive relationships) (Ragin, 2006: 19). Subset analysis: Sufficiency of theoretical attributes/ causal conditions (e.g., governance mechanisms) for condition of interest/ outcome (e.g., firm performance). 5

6 Sufficiency Theoretical Attributes Condition of Interest Sufficiency = All cases exhibiting the theoretical ( causal ) attribute(s) display the condition of interest ( outcome ). Sufficiency of Governance Practices for Observing High Performance Governance Mechanisms Configuration High Performance Sufficiency = All cases exhibiting the Governance Mechanisms display High Performance. 6

7 Board Monitoring/Governance Mechanisms Sufficient for High and Low ROA High ROA Low ROA Outside Director Vigilance W W m m W m W W Inside Director Ownership m W W W m m m W Block Owner m m W W m m m W CEO Align m m W m m m W CEO Power W m m W W W m W Market for Corporate Control W m m m m W W W Note: core conditions are represented by W (present) and m (absent); peripheral conditions by W (present) and m (absent) In Closing. Fuzzy sets afford researchers a means by which they can capture both differences in kind and degree of the phenomena under study They are simultaneously qualitative and quantitative: they allow for qualitative assessments and afford systematic analyses using Boolean algebra and logic. The set theoretic method is a qualitative approach to research which offers researchers an interpretive algebra (Ragin, 2000: 4) that contributes to theory building by providing a rigorous way to combine verbal statements with logical relationships (Fiss, 2007: 1181). 7

8 Thank You!! BACKUP SLIDES 8

9 Three Pillars of Fuzzy-set Truth Tables Rows of truth table directly correspond to the corners of the vector space defined by fuzzy set causal conditions. Assessment of distribution of cases across the different logically possible combinations of causal conditions. Must set threshold for relevant cases vs. remainders (i.e., combinations lacking empirical instances). Assessment of the consistency of the evidence for each causal combination with the argument that it is a subset of the outcome. Evaluating Consistency (w.r.t.. Sufficiency) Set theoretic Consistency: the degree to which the cases exhibiting a given governance practice obtain high (low) performance; consistency indicates how closely the subset relation is approximated (Ragin, 2006: 2) Fuzzy sets: a subset relation exists when the membership scores in Xi are consistently less than membership scores in Yi: Consistency (Xi Yi) = (min(xi, Yi)) / (Xi) 9

10 Evaluating Coverage (w.r.t.. Sufficiency) Set theoretic Coverage: Proportion of the consistent cases displaying i the outcome that t have the relevant attribute(s). coverage gauges empirical relevance or importance of causal condition to outcome (Ragin, 2006: 2) Solution coverage: proportion of cases that total solution covers. Raw coverage: includes overlap among cases some cases conform multiple configurations. Unique coverage: assessment of relative empirical importance of each combination. Counterfactuals Counterfactuals = A combination that lacks empirical instances and thus have to be imagined Easy counterfactual = adding a redundant causal condition to a configuration known to produce the outcome. Difficult counterfactual = removing a contributing causal condition from a configuration displaying the outcome. 10

11 Solutions Solutions Complex: Does not permit any counterfactuals (no simplifying assumptions) Parsimonious Incorporates any counterfactuals (easy or difficult) that yield a simpler solution. Intermediate Incorporates only easy counterfactuals in simplifying (difficult counterfactuals not removed) Interpretation based upon parsimonious and intermediate. Core vs. Periphery Conditions Core = conditions present in all solutions (parsimonious) Periphery = conditions also present in the intermediate solution (takes difficult counterfactual to remove) 11