ANALYZING CATEGORICAL DATA WITH GRAPHICAL MODELS A SOCIAL SCIENCE APPLICATION

1
ANALYZING CATEGORICAL DATA WITH GRAPHICAL
MODELS- A SOCIAL SCIENCE APPLICATION -

• Renáta Németh 1, Tamás Rudas 2, Wicher Bergsma 3
• 1 nmthrnt_at_freemail.hu. Department of Statistics,
  Faculty of Social Sciences, Eötvös Loránd
  University
• 2 rudas_at_tarki.hu. Department of Statistics,
  Faculty of Social Sciences, Eötvös Loránd
  University,
• 3 W.P.Bergsma_at_lse.ac.uk. Department of
  Statistics, London School of Economics and
  Political Science
• SMABS/EAM Conference
• 5 July 2006, Budapest

2
Overview

• 1. Marginal models
• 2. Graphical models
• 4. Application the classic Duncan model and its
  modified version (1992, Czechoslovakia, Hungary,
  USA)

3. Graphical models for categorical data
3
Marginal models

• They impose restrictions on the marginal
  distributions of the contingency table
• Their parameterization can be obtained by
  marginal loglinear parameters (Bergsma, Rudas,
  2002)
• Example the value of the parameter pertaining to
  the EI effect depend on the marginal from which
  it is computed (see the Simpson paradox)
• F fathers occupation, E educational
  background, O occupation, I income

4
Parameterizing marginal models

• There are many different ways to parameterize a
  certain table using marginal log-linear
  parameters. Two parameterizations for the EOI
  joint distribution

• Consider the model (Education ? Income
  Occupation).
• It holds if and only if ?EIEOI 0.
• General requirements for a parameterization
• Independence models should be identified by
  certain parameters being zero
• The existence of a distribution when some of the
  parameters are linearly restricted
• Parameters should be easily interpretable

5
Graphical models(see eg. Lauritzen 1996)

• Applications in statistical physics, genetics,
  artificial intelligence...
• Graph node variable, edge/arrow
  undirected/directed association. Unconnected
  nodes are assumed to be conditionally
  independent.
• Modularity by combining simpler parts
• Complex systems can be in easily visualized and
  interpreted

6
Conditional independences in graphical models I

• Directed acyclic graph (DAG) a variable V is
  conditionally independent from its nondescendants
  conditioned on its parents.
• Education of roofs
• Fathers Income
• occupation Occupation of birth of storks

7
Conditional independences in graphical models II

• Chain graph model the variables are divided into
  groups that are ordered, variables in a group are
  responses to variables in earlier groups.
  Associations within the groups are represented by
  edges, the arrows point from the explanatories to
  the responses.
• A missing edge between two variables means
  conditional independence of these given all other
  variables in the group and in all earlier groups.
• A missing arrow means conditional independence of
  the explanatory and the response, given all other
  variables in the group of the response and in all
  earlier groups.
• Example
• Determinants of health from a 2-waves panel data
• Socioeconomic (wave 1) Health behaviour
  (wave 1) Health (wave 1) Health
  (wave 2)

• Income
• Marital status

• Smoking
• Alcohol consumption
• Physical activity

• Coronary heart disease
• High blood pressure

• Coronary heart disease
• High blood pressure

8
Categorical data graphical models as marginal
models

• To obtain parameterization of DAG or chain-graph
  models, marginal log-linear parameters can be
  used. The parameterization can be chosen in a way
  that ordered decomplosability and hierarchicity
  hold (Rudas, Bergsma, 2004, Rudas, Bergsma,
  Németh, 2006 a, 2006 b), which implies the
  following (Bergsma, Rudas, 2002)
• the model can be defined by setting the values of
  some of the parameters to zero, the distributions
  within the model are parameterized by the
  remaining unrestricted parameters
• the parameters are variation independent (hence
  they are well-interpretable and any evaluation of
  them defines non-empty model),
• the model has standard asymptotic behavior (MLH
  estimates for parameters, LR statistic for
  testing models).

9
Application, Model 1

• Status Attainment Model
• (Duncan et al, 1968)
• Fe fathers education,
• Fo fathers occupation
• E education
• O occupation
• I income

Conditional independences O ? Fe EFo, I ?
FeFo EO. Zero parameters ?FeOFeFoEO,
?FeIFeFoEOI, ?FoIFeFoEOI Free parameters
?FeFo , ?EFeFoE , ?OFeFoEO , ?EOFeFoEO ,
?FoOFeFoEO , ?FoEOFeFoEO , ?IFeFoEOI ,
?EIFeFoEOI , ?OIFeFoEOI , ?EOIFeFoEOI.
10
Application, Model 2

• The modified Status Attainment Model
• a chain graph model
• (Boguszak et al, 1990)

Conditional independences I ? FeFo EO. Zero
parameters ?FeIFeFoEOI, ?FoIFeFoEOI Free
parameters ?FeFo , ?FeFoEO , ?IFeFoEOI ,
?EIFeFoEOI , ?OIFeFoEOI , ?EOIFeFoEOI. Zero
parameters of Model 1 are also zero parameters of
Model 2.
11
Application comparing Model 2 to Model 1

• Data for Czechoslovakia, Hungary, and the USA,
  from the International Social Survey Programme
  (ISSP) 1992.
• Variable categories
• E and Fe 1 - below higher education, 2 higher
  education
• I 1 - below country-specific sample median
  income 2 - above the median
• O and Fo 1 - lower class 2 - middle class 3 -
  upper class
• Results
• Model 1 Model 2 Model 2 to Model 1
• df 42 30 12
• USA L2 25.6 14.7 25.6-14.710.9
• p .978 .991 0.538
• Hungary L2 38.1 23.3 38.1-23.314.8
• p .643 .802 0.253
• Czechoslovakia L2 38.8 25.5 38.8-25.513.3
• p .614 .700 0.348

12
Application
Model 1, estimates for ? parameters From top to
bottom USA, Hungary, Czechoslovakia
13
References

• Bergsma, W., Rudas, T. (2002) Marginal models
  for categorical data. The Annals of Statistics,
  (30/1), 140-159.
• Boguszak, Marek, Gabal, Ivan, Mateju, Petr
  (1990) Ke koncepcím vývoje sociální struktury v
  CSSR. Sociologický casopis (26/3), 168- 186.
• Internation Social Survey Programme (ISSP)
  Social Inequality II, 1992. Zentralarchiv für
  Empirische Sozialforschung, Köln
• Lauritzen, S.L. (1996). Graphical Models. Oxford
  University Press.
• Rudas, Tamás, Bergsma, Wicher, Németh, Renáta
  (2006a) Parameterization and estimation of path
  models for categorical data. Proceeding of the
  IASC 17th Compstat Symposium, 2006, Rome
• Rudas, T., Bergsma, W., Németh, R. (2006b)
  Graphical and path models for categorical
  variables. (manuscript)
• Rudas, Tamás, Bergsma, Wicher (2004) On
  application of marginal models for categorical
  data. Metron, (62/1), 1-23.
• Duncan, O. D., Featherman, D. L., Duncan, B.
  (1968) Socioeconomic Background and Occupational
  Achievement Extensions of a Basic Model.
  Washington, D. C. U. S. Department of Health,
  Education, and Welfare, Office of Education,
  Bureau of Research.