Loglinear models

• A lot of the time we will be interested in two-time-point analyses
  • state at one wave versus state at another
  • state at successive pairs of waves

• If the variables of interest are categorical we can view the association in the form of cross-tabulations
• The standard $\chi^2$ test we apply to tables is not likely to be informative here, because the null hypothesis of independence between states at the two time points is not likely to be true
  • many people are likely to stay in the same state
  • thus the diagonal will be over-populated.

• Loglinear modelling offers us a means of testing more substantively interesting null hypotheses

• We might argue that there are two types of association in the table:
  1. some people remain in the same state over time,
  2. and those who move have a different distribution of outcome states depending on their original state.

• With a simple loglinear model we can exclude the former effect, and test a null hypothesis that
  • among those who move there is no association between original and final state

• This analogous to dropping the non-moving individuals from the table, and testing for independence in the rest.

• An example:
  • Take cjbstat and gjbstat in a wide-format file
  • Crosstabulation (collapsing non-labour-market codes together) gives:

                        GJBSTAT                        Page 1 of 1
                Count  |
               Exp Val |Self-emp Employed Unemploy Non empl
                       |loyed             ed       oyed       Row
                       |       1|       2|       3|       4| Total
    CJBSTAT    --------+--------+--------+--------+--------+
                    1  |   397  |   129  |     8  |    57  |   591
      Self-employed    |  47.1  | 306.2  |  17.7  | 220.0  |  7.8%
                       +--------+--------+--------+--------+
                    2  |   139  |  3150  |    98  |   487  |  3874
      Employed         | 308.9  |2006.9  | 116.3  |1441.9  | 50.8%
                       +--------+--------+--------+--------+
                    3  |    34  |   182  |    71  |   116  |   403
      Unemployed       |  32.1  | 208.8  |  12.1  | 150.0  |  5.3%
                       +--------+--------+--------+--------+
                    4  |    38  |   489  |    52  |  2178  |  2757
      Non employed     | 219.8  |1428.2  |  82.8  |1026.1  | 36.2%
                       +--------+--------+--------+--------+
                Column     608     3950      229     2838     7625
                 Total    8.0%    51.8%     3.0%    37.2%   100.0%
    

• There are many, many more cases on the main diagonal than exepected under independence
• But there is also off-diagonal association apparent, e.g., non-employed seem very unlikely be become self-employed but not as dramatically unlikely to become unemployed.
• Let's test: first, the loglinear model that is equivalent to independence

  genlog cjbstat gjbstat
    /print none /plot none
    /design cjbstat gjbstat.
  

• And now tell it to consider only the off-diagonal cases:

  compute offdiag = (cjbstat<>gjbstat).
  
  genlog cjbstat gjbstat
    /cstructure=offdiag
    /print none /plot none
    /design cjbstat gjbstat.
  

• The results (in terms of fit):

  LR Chi-Square       DF       Sig.
  
     5023.5965        9      .0000 <-- Independence
       20.6554        5      .0009 <-- Mover--stayer
  

  The $\chi^2$ falls enormously when the diagonal is excluded, but there is still very significant evidence of association in the rest of the table.

• There are many more models one can fit,
  • embodying other hypotheses about the association,
  • e.g., the variables may have
    • ranked categories
    • categories assigned numeric scores
    • categories with estimated numeric scores
  • and including more variable (more dimensions)
    • e.g., allow cjbstat/gjbstat association to differ by sex
    • or to model more complex relationships, e.g., between vote in wave 3, cjbstat, vote in wave 7 and gjbstat
• See http://wivenhoe.staff8.ul.ie/~brendan/CDA/ for an online course in loglinear modelling, http://teaching.sociology.ul.ie/SSS/lugano/ for a version with more weight on logistic regression
• Hagenaars (in the references) chapters 4 and 5
• Ch 3 in Dale and Davies (though not panel, good intro).