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MCA and Other Statistical Techniques Johs. Hjellbrekke Department of sociology, University of Bergen, Norway.

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Brief outline of key points The standard approach and two of Benzécris principles Exploratory, confirmatory and explanatory analysis and GDA Standard causal analysis (SCA) and multiple correspondence analysis (MCA) Quantitative and geometric approaches Statistical inference in GDA Methodological Challenges.

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The Standard Approach Data are confronted with a mathematical model, assumed to underlie the observed data. Statistical analysis often a question of finding/fitting the model that best fits the data. Frequentist principles of inference far more often used than bayesian principles of inference

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Two of Benzécris principles Statistics is not probability. Under the name of mathematical statistics, authors /../ have erected a pompous discipline, rich in hypotheses which are never satisfied in practice. The model must fit the data, and not vice versa./…/ What we need is a rigorous method that extract structures from the data.

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Exploratory, confirmatory and explanatory analysis MCA often classified as an exploratory technique or statistical tool Statistical techniques are, however, per se never exploratory, explanatory or confirmatory. What they do is to provide us with a basis for these modes of reasoning Statistics does not explain anything – but provides potential elements for explanation (Lebart 1975) See also Le Roux & Rouanet 2004: chapter 1.

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Exploratory, confirmatory and explanatory analysis Basic statistics of GDA are descriptive measures But so are regression coefficients and R-squared…. The latter are often, implicitly or explicitely, interpreted causally within the classic Standard Causal Analysis (SCA)-approach In path analysis, the cold bones of correlation are turned into the warm flesh of causation with direct, total, and partial causal pathways (Holland 1993: 280) What passes for a cause in a path analysis might never get a moments notice in an experiment (Holland, ibid.)

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Standard Causal Analysis (SCA) and Multiple Correspondence Analysis (MCA) Quantitative vs Geometrical Approach: Numbers as basic ingredients and outcomes of procedures (SCA) vs. Data represented as clouds of points in geometric spaces (MCA) SCA: Primarily seeks to isolate effects of each independent variable on a dependent variable. Interaction effects often treated as secondary. Quasi- experimentation through statistical control (See Abbott 2004 for further details) MCA/GDA: relations between variables, categories/modalities and sets of variables at the center of the analysis.Not a quasi-experimental epistemological basis

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MCA and Confirmatory Analysis MCA can be used in a confirmatory and/or explanatory mode of reasoning or analysis By introducing sets of supplementary variables (Visual regression) By introducing structuring factors, i.e. the detailed study of subclouds of individuals based on the supplementary variables. Oppositions between (supplementary) categories in an MCA can also be described in standard statistical terms, similar to standardized coefficients in a regression analysis.

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Standardized Deviations in MCA Oppositions between supplementary modality points in the cloud of modalities can be described or expressed in terms of standard deviations between modality mean points in the cloud of individuals A deviation >1.0 can be described as large A deviation <0.5 can be described as small As in the case in an analysis of the Norwegian elites (analysis of the Norwegian Power and Democracy Survey 2000, Hjellbrekke & al. 2007):

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Quantitative and Geometric Approaches: The Role of the Individuals Variable centered, quantitative techniques cannot, or hardly do, examine the inviduals in the detailed way that is possible in a geometric approach Clear contrast between loglinear/log- multiplicative/latent class models and MCA/GDA

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The Cloud of Individuals – The Norwegian Elites

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MCA and Statistical Inference MCA can be combined with statistical inference Confidence intervals can be calculated for a categorys position on an axis Confidence ellipses can be calculated for a categorys position in a factorial plane

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Confidence ellipses – factorial plane 1-2,.05-level. (Analysis of the Norwegian Electoral Survey 2001, Hjellbrekke 2007)

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Confidence ellipses and confidence intervals – factorial plane 2-3,.05-levels. (Analysis of the Norwegian Electoral Survey 2001, Hjellbrekke 2007)

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Quantitative and Geometric Approaches: The number of variables Loglinear/Log-multiplicative/Latent Class Models – restricted to a small number of variables, all with few categories or modalities. GDA is not restricted in this way (the previous analysis has 31 active variables) Categories or modalities should have relative frequencies >5%

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Methodological Challenges…. We need to take a critical look the way we teach our students statistics Statistics, like social science, has a scientific history that should be integrated in our methodology courses in the same ways that we have integrated sociologys history in the introductory courses in sociology More attention should be given to the contexts of discovery of the various techniques, and to their implicit or explicit epistemological models The dominant position of the regression model has lead to unhappy orthodoxies

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References Abbott, Andrew (2004). Methods of Discovery. Heuristics for the Social Sciences. New York: W.W. Norton. Hjellbrekke, Johs. (2007). The Geometry of the Electoral Space. An analysis of the Electoral Survey 2007. In Gåsdal & al. Power, Meaning and Structure. Bergen: Fagbokforlaget (In Norwegian) Hjellbrekke & al. (2007). The Norwegian Field of Power Anno 2000. In European Society, 9:2, 245-273. Holland, Paul (1993). What Comes First, Cause or Effect?. In G. Keren & G. Lewis, A Handbook for Data Analysis in the Behavioural Sciences: Methodological Issues. Hillsdale, N.J.: Lawrence Erlbaum Ass. Publ. Le Roux, Brigitte & Rouanet, Henry (2004). Geometric Data Analysis. Dordrecht: Kluwer.

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