1. A. Analysis of count data
Introduction to log-linear models

2. Log-linear analysis Contingency-table analysis
Categorical data analysis
Discrete multivariate analysis (Bishop, Fienberg and Holland, 1975)
Analysis of cross-classified data
Multivariate analysis of qualitative data (Goodman, 1978)
Count data analysis

3. Contrast Coding Log-linear models for two-way tables
Saturated log-linear model:
Overall effect (level)
Main effects
(marginal freq.)
Interaction effect
In case of 2 x 2 table:
4 observations
9 parameters
Normalisation constraints

4. Survey: leaving parental home in the Netherlands

5. Descriptive statistics
Leaving home
Descriptive statistics
Counts
Percentages
Odds of leaving home early rather than late
Reference category

6. Log-linear models for two-way tables 4 models
Leaving home
Log-linear models for two-way tables 4 models
Model 1: Null model or overall effect model
All categories are equiprobable (an observation is equally likely to fall into any cell)
for all i and j
Exp(4.887) = 132.5
= 530/4
 = s.e
ij is expected count (frequency) in cell (ij): category i of variable A (row) and category j of variable B (column)

7. Leaving home
Where ij is a cell frequency generated by a Poisson process and Var[aX] = a2 Var[X] where a is a constant (e.g. Fingleton, 1984, p. 29) 

8. Log-linear models for two-way tables
Leaving home
Log-linear models for two-way tables
Model 2: B null model
Categories of variable B (sex) are equiprobable within levels of variable A (age)
for all j
GLIM
estimate s.e Parameter Exp(parameter)
Overall effect
TIME(1)
TIME(2)

9. Log-linear models for two-way tables
Leaving home
Log-linear models for two-way tables
Model 3: B null model
Categories of variable A (age) are equiprobable within levels of variable B (sex)
for all j
SPSS
estimate s.e Parameter Exp(parameter)
Overall effect
TIME(1)
TIME(2)

10. Log-linear models for two-way tables
Leaving home
Log-linear models for two-way tables
Model 4: independence model (unsaturated model)
Categories of variable B (sex) are not equiprobable but the probability is independent of levels of variable A (time)
estimate s.e Parameter Exp(parameter)
Overall effect
TIME(2)
SEX(2)
GLIM

11. LOG-LINEAR MODEL: predictions Females leaving home early: 109.62
Females leaving home late: * =
Males leaving home early: * = 99.37
Males leaving home late: * * =

12. SPSS Parameter Estimate SE 1 5.0280 .0721 Overall effect
Leaving home
SPSS
Parameter Estimate SE
Overall effect
Time(1)
Time(2)
Sex(1)
Sex (2)

13. Log-linear models for two-way tables
Leaving home
Log-linear models for two-way tables
Model 5: saturated model
The values of categories of variable B (sex) depend on levels of variable A (time)
estimate s.e parameter
Overall effect
TIME(2)
SEX(2)
TIME(2).SEX(2)
GLIM

14. Parameter Estimate SE Parameter 1 5.1846 .0748 Overall effect
Leaving home
Parameter Estimate SE Parameter
Overall effect
Time(1)
Time(2)
Sex(1)
Sex(2)
Time(1) * Sex(1)
Time(1) * Sex(2)
Time(2) * Sex(1)
Time(2) * Sex(2)
SPSS

15. LOG-LINEAR MODEL: predictions Expected frequencies
Leaving home
LOG-LINEAR MODEL: predictions
Expected frequencies
Observed Model 1 Model 2 Model 3 Model 4 Model 5
Fem_<20 F
Mal_<20 F
Fem_>20 F
Mal_>20 F
D:\s\1\liebr\2_2\2_2.wq2

16. Relation log-linear model and Poisson regression model
are dummy variables (0 if i or j is equal to 1and1 if i or j equal to 2) and interaction variable is

23. Log-linear model fit a model to a table of frequencies
Data: survey of political attitudes of British electors
Source: Payne, C. (1977) The log-linear model for contingency. In: C.O. Muircheartaigh and C. Payne eds. The analysis of survey data. Vol 2: Model fitting, Wiley, New York, pp [data p. 106].(from Butler and Stokes, ‘Political change in Britain’, Macmillan, 2nd edidition, 1974)

24. The classical approach
Geometric means (Birch, 1963)
Effect coding (mean is ref. Cat.)
Birch, M.W. (1963) ‘Maximum likelihood in three-way contingency tables’,J. Royal Stat. Soc. (B), 25:

25. The basic model Political attitudes Overall effect : 22.98/4 = 5.7456
Effect of party : Conservative : 11.49/ =
Labour : 11.49/ =
Effect of gender : Male : 11.44/ =
Female : 11.54/ =
Interaction effects: Gender-Party interaction effect
Male conservative : =
Female conservative : =
Male labour : =
Female labour : =

26. The basic model (Effect Coding: Mean)
Political attitudes
The basic model (Effect Coding: Mean)
Birch, M.W. (1963) ‘Maximum likelihood in three-way contingency tables’,J. Royal Stat. Soc. (B), 25:
Coding: effect coding
Parameters are subject to constraints: normalisation constraints
Only first-order contrasts can be estimated:

27. Political attitudes
The basic model (GLIM)
Estimate S.E.

28. Political attitudes
The basic model (SPSS)

29. The basic model (1) Political attitudes
ln 11 = =
ln 12 = =
ln 21 = =
ln 22 = =

30. The design-matrix approach

31. I. Design matrix: Effect Coding unsaturated log-linear model
Number of parameters exceeds number of equations  need for additional equations
(X’X)-1 is singular  identify linear dependencies

32. I. Design matrix unsaturated log-linear model
(additional eq.)
Coding!

33. 3 unknowns  3 equations
where
is the frequency predicted by the model

34. Political attitudes

35.  Political attitudes 314.17*1.0040*0.9772 = 308.23
314.17*[1/1.0040]* =

36. Design matrix Saturated log-linear model

37. Political attitudes
exp[ ] = exp[5.6312] = 279
exp[ ] = 335

38. Political attitudes

39. Other Ways of Restricting II. Design Matrix: Contrast Coding

40. III. Design matrix: other restrictions on parameters saturated log-linear model
(SPSS)

41. Political attitudes

42. Political attitudes

43. Political attitudes

44. Political attitudes

45. Prediction of counts or frequencies:
Political attitudes
Prediction of counts or frequencies:
A. Effect coding
279 = * * *
352 = * * *
335 = * * *
291 = * * *
B. Contrast coding: GLIM
291 = 279 * * * (females voting labour)
279 = 279 * * * (males voting conservative = ref.cat)
352 = 279 * * * (females voting conservative)
335 = 279 * * * (males voting labour)
C. Contrast coding: SPSS (SPSS adds 0.5 to observed values )
279.5 = * * *
352.5 = * * * 1
291.5 = * * * 1 (females voting labour = ref.cat)
335.5 = * * * 1