MCA and Other Statistical Techniques

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

2. Brief outline of key points • The standard approach and two of Benzécri’s 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.

3. 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

4. Two of Benzécri’s 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.”

5. 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.

6. 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 moment’s notice in an experiment” (Holland, ibid.)

7. 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

8. 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.

9. 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):

14. 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

15. The Cloud of Individuals – The Norwegian Elites

18. MCA and Statistical Inference • MCA can be combined with statistical inference • Confidence intervals can be calculated for a category’s position on an axis • Confidence ellipses can be calculated for a category’s position in a factorial plane

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

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

21. 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%

24. 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 sociology’s 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

25. 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.