A Mixture Modeling Approach for Empirical Testing of Competing Theories

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A Mixture Modeling Approach for Empirical Testing of Competing Theories Kosuke Imai Dustin Tingley Princeton University January 8, 2010 Imai and Tingley (Princeton) Mixture Modeling Gakusyuin 2010 1 / 12 Motivation Empirical testing of competing theories lies at the heart of social science research Need to test the validity of alternative theories explaining the same phenomena theory confirmation is not possible when a theory is tested in isolation, regardless of the statistical approach (Clarke) Common statistical methods used in the discipline: 1 Garbage-can regressions: atheoretical, aparsimonious (Achen) 2 Model selection methods (e.g., AIC, BIC, Vuong test, J test): All or nothing, Independence of Irrelevant Alternatives (IIA) Key distinction between causal and predictive inference Imai and Tingley (Princeton) Mixture Modeling Gakusyuin 2010 2 / 12

The Proposed Approach Theoretical heterogeneity: No single theory can explain everything Explaining when each theory works 1 Testing the entire theory including its assumptions rather than just its implications 2 Leading to further theory development Finite mixture models 1 A well-known, very general class of statistical models 2 Can test more than two theories at the same time 3 Under-utilized in political science except a few studies (Hill and Kriesi; Clinton, Imai and Pemstein; Iaryczower and Shum) Quantities of interest: 1 population proportion of observations consistent with each theory 2 how this proportion varies as a function of observed characteristics 3 probability that a particular observation is consistent with a theory Imai and Tingley (Princeton) Mixture Modeling Gakusyuin 2010 3 / 12 Finite Mixture Models: A Review M competing theories, each of which implies a statistical model f m (y x) for m = 1,..., M The data generating process: Y i X i, Z i f Zi (Y i X i, θ Zi ) where Z i is the latent variable indicating the theory which generates observation i The observed-data likelihood function: N { L obs (Θ, Π {X i, Y i } N i=1 ) = M } π m f m (Y i X i, θ m ), i=1 m=1 where π m = Pr(Z i = m) is the population proportion of observations generated by theory m π m : a measure of overall performance of the theory Imai and Tingley (Princeton) Mixture Modeling Gakusyuin 2010 4 / 12

Explaining theoretical heterogeneity: Pr(Z i = m W i ) = π m (W i, ψ m ), Predicting which theory has generated a particular observation: Grouped observations: ζ i,m = Pr(Z i = m Θ, Π, {X i, Y i } N i=1 ) = π m f m (Y i X i, θ m ) M m =1 π m f m (Y i X i, θ m ) ζ i,m = π m Ji j=1 f m(y ij X ij, θ m ) M m =1 π m Ji j=1 f m (Y ij X ij, θ m ) Estimation: Expectation-Maximization or Markov chain Monte Carlo algorithm Implementation: flexmix package in R by Leisch and Gruen Imai and Tingley (Princeton) Mixture Modeling Gakusyuin 2010 5 / 12 Statistically Significantly Consistent with A Theory Identification of observations that are statistically significantly consistent with each theory Idea: If ζ i,m is greater than a threshold λ m, then include observation i in the list Problem of multiple testing: false positives Simple example: 10 Independent 0.05 level tests 1 0.95 10 0.4 chance of at least one false discovery Solution: choose the smallest value of λ m while making sure that the posterior expected value of false discovery rate on the resulting list does not exceed a prespecified threshold α m : { } N λ i=1 m = inf λ m : (1 ˆζ i,m )1{ˆζ i,m λ m } N i=1 1{ˆζ i,m λ m } + N i=1 1{ˆζ i,m < λ m } α m Imai and Tingley (Princeton) Mixture Modeling Gakusyuin 2010 6 / 12

An Application: Determinants of Trade Policies Hiscox (2002, APSR) analyzes US legislative voting on trade bills Stolper-Samuelson (SS) model: cleavages along factoral lines The highly skilled favor liberalization while the low-skilled oppose it Ricardo-Viner (RV) model: alternative cleavage along sectoral lines Exporters favor liberalization while importers oppose it Key contribution: the applicability of the two models depends on the level of factor mobility in the US economy If capital is highly mobile across industries, then the conditions for the SS model are satisfied If capital is highly specific, then the conditions for the RV model are satisfied Data Congressional voting data on 55 trade bills spanning over 150 years A combined measure of factor specificity for a given year State-level measures of relevant covariates for each model Imai and Tingley (Princeton) Mixture Modeling Gakusyuin 2010 7 / 12 Testing the Competing Theories of Trade Policy The original analysis used the J test in logistic regression with bill fixed effects The J test in its original form: Y i = (1 π)f (X i, β) + πg(x i, γ) + ɛ i, The null hypothesis, Y i = f (X i, β) + ɛ i The alternative hypothesis, Y i = g(x i, γ) + ɛ i Finite mixture models do not assume π is either 0 or 1 We fit two mixture models where π is modeled as a function of factor specificity 1 Assuming all votes for the same bill belong to the same model and using logistic regression to model π 2 Relaxing the above assumption and using the semiparametric logistic regression Imai and Tingley (Princeton) Mixture Modeling Gakusyuin 2010 8 / 12

Results with Grouped Observations 15 20 25 30 35 0.0 0.2 0.4 0.6 0.8 1.0 House Estimated Probability of Being Consistent with the Ricardo Vinor Model Factor Specificity 15 20 25 30 35 0.0 0.2 0.4 0.6 0.8 1.0 Senate Imai and Tingley (Princeton) Mixture Modeling Gakusyuin 2010 9 / 12 Results without Grouping and Parametric Assumption 15 20 25 30 35 0.0 0.2 0.4 0.6 0.8 1.0 House Estimated Probability of Being Consistent with the Ricardo Viner Model Factor Specificity Imai and Tingley (Princeton) Mixture Modeling Gakusyuin 2010 10 / 12

Classification of House Trade Bills Stolper-Samuelson Model Ricardo-Vinor Model Adams Compromise (1832) Tariff Act (1824) Clay Compromise (1833) Tariff Act (1828) Tariff Act (1842) Gorman Tariff (1894) Walker Act (1846) Underwood Tariff (1913) Tariff Act (1857) RTAA (1934) Morrill Act (1861) RTA Extension (1937) Tariff Act (1875) RTA Extension (1945) Morrison Bill (1984) RTA Extension (1955) Mills Bill (1988) Trade Expansion Act (1962) McKinley Tariff (1890) Mills Bill (1970) Dingley Tariff (1894) Trade Reform Act (1974) Payne-Aldrich Tariff (1909) Fast-Track (1991) Fordney-McCumber Tariff (1922) NAFTA (1993) Smoot-Hawley Tariff (1930) GATT (1994) Trade Remedies Reform (1984) Imai and Tingley (Princeton) Mixture Modeling Gakusyuin 2010 11 / 12 Concluding Remarks Finite mixture models offer an effective way to test competing theories Many advantages over the standard model selection procedures 1 Test any number of competing theories 2 Include nested and/or non-nested models 3 Conduct frequentist or Bayesian inference 4 Quantify the overall performance of each theory 5 Test the conditions under which each theory applies 6 Identify observations statistically significantly consistent with theory Imai and Tingley (Princeton) Mixture Modeling Gakusyuin 2010 12 / 12

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