Symmetry

• Symmetry implies $\pi_{ij} = \pi_{ji}$, that the probability of changing from $i$ to $j$ is the same as that from $j$ to $i$. This can be expressed as $F_{ij} = \tau_0 \tau_{i} \tau_{j}\tau_{ij}$ where $\tau_i = \tau_j$ and $\tau_{ij} = \tau_{ij}$. One way of fitting this model is to create a new variable with a different value for each cell in one half of the table, with the other half of the table being a mirror image: that is, for each pair of cells $c_{ij}, c_{ji}$ there is a distinct value of the new variable.
• Since for the diagonal there is only one cell for each $\pi_{ii}$ it is usually weighted out.

• For a $6 \times 6$ table the symmetry variable looks like:

  -	1	2	3	4	5
  1	-	6	7	8	9
  2	6	-	10	11	12
  3	7	10	-	13	14
  4	8	11	13	-	15
  5	9	12	14	15	-

• The diagonal values are represented by `-' because they are weighted out in the analysis.

• We can use the macro tabu from http://teaching.sociology.ul.ie/~brendan/CDA/categ-macs.sps to confirm the appearance of the symmetry variable:

  tabu VAR1 VAR2 symm .
  

  This will lay it out as in the table above.

• One way of generating an design variable like this in SPSS is the following (for var1 by var2, each with 5 categories):

  compute #c=1.
  loop #i = 1 to 5.
  . loop #j = #i+1 to 5.
  .   if var1=#i and var2=#j sym = #c.
  .   if var1=#j and var2=#i sym = #c.
  .   compute #c=#c+1.
  . end loop.
  end loop.
  if var1=var2 sym=1.
  

• This will work for individual level data and for grouped data; with grouped data it is also possible to enter it by hand.

• Code to do this as a macro is also available in categ-macs.sps: the syntax is
  makesym VAR1 VAR2 n NEWVAR
  this takes the two classifying variables VAR1 and VAR2, each with n categories, and creates a variable called NEWVAR with the symmetry pattern in it.

• What does this model say about the data?
  • About the diagonal cells, nothing: since they are weighted out, they can have any value.
  • Off the diagonal $\log F_{ij} = \log F_{ji} = \lambda_0 + \sigma_{ij}$.
  • Row $i$ and Column $i$ will have the same totals: marginal homogeneity.

• This is a somewhat strange model: if we consider the symmetry term as a restricted two-way interaction, it is almost like fitting the interaction term without its first-order children, though the interaction term has consequences for the row and column totals (they match).

• It has (at least) one similarity with the mover-stayer model: it drops the diagonal entirely.

• Fitting this model takes another step, and another macro.
  • Having used makesym to create the symmetry variable, we need to make dummy variables from it.
  • This is because SPSS only recognises the classifying variables as categorical; extra variables go in to the model as covariates, which are expected to be quantitative.
  • The macro factor makes a set of dummy variables
  • factor symm s 5 creates variables s1 to s5 where s3=0 unless symm=3 when s3=1, etc. .
  • Covariates are entered in the first part of the GENLOG command:
    genlog avote evote with s2 s3 s4 s5.
  • Note the with separating classifying variables from the covariates.
  • Note that s1 is not included: this is the reference category.
  • The /design statement contains only the $n-1$ dummy variables, no `real' variable.
  • The symmetry model is fitted on a $4\times4$ table thus:

    compute offdiag = avote<>evote.
    makesym avote evote 4 symm .
    factor symm s 6.
    genlog avote evote with s2 s3 s4 s5 s6
     /cstructure = offdiag
     /print=est/plot=none
     /design= s2 s3 s4 s5 s6
     /save=adjresid.
    

  • The last subcommand /save=adjresid is not essential but it is always good to look at the pattern of residuals:

    tabu avote evote adj_1.