MCA and Other Statistical Techniques

Johs. Hjellbrekke

Department of sociology,

University of Bergen, Norway.

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.

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

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.”

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.

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.)

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

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.

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):

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

The Cloud of Individuals – The Norwegian Elites

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

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

Hjellbrekke 2007)

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

Survey 2001, Hjellbrekke 2007)

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%

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

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.