MCA and Other Statistical Techniques

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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 gt1.0 can be described as large
• A deviation lt0.5 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-multiplicativ
  e/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 gt5

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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, 92,
  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.