11.3.5  Marginal Homogeneity
We have seen this when we discussed twoway tables:
Objective:
Are the row and column distributions (of a square table) the same?
H_{0}: μ_{i+} = μ_{+i}
There is no direct way to use loglinear models to fit/test this model.
To indirectly test using loglinear models (i.e., conditional likelihood ratio test), we need to do a contextual/comparision test .
Symmetry has two components: marginal homogeneity & quasisymmetry.
Symmetry is a special case of quasisymmetry. If the quasisymmetry relationship/model holds, then a way to test for marginal homogeneity is:
G^{2}(marginal homogeneity) = G^{2}(symmetry) − G^{2}(quasisymmetry) with df = I −1.
For our example, G^{2} = 0.587=0.59280.0061, df = 2, pvalue=0.746 homogeneity model fits moderately well. An alternative is to use generalized least squares instead of maximum likelihood estimation.
Here is how all the models discussed in this section relate to each other.
The most general (complex) model is quasisymmetry.
Symmetry is a special case of quasisymmetry.
Quasiindependence is a special case of quasisymmetry.
Symmetry is not a special case of quasiindependence.
Quasiindependence is not a special case of symmetry.
Summary of fits for our example (MovieCritiquesLoglin.sas and MovieCritiquesLoglin.lst or MoviesCritiquesLoglin.R):
Model

df

G^{2}

pvalue

Independence 
4

43.2325

0.0001

QuasiIndependence 
1

0.0061

0.938

Symmetry 
3

0.5928

0.900

Marginal homogeneity 
2

0.587

0.746

Quasisymmetry 
1

0.0061

0.938

So do Siskel and Ebert really disagree or only moderately agree?
These models indicate strong symmetric associations, and agreement.
General Comments:
Knowing a set of basic models is useful as you can combine those to address more specific situations. We can combine models for matched and ordinal data, as well sampling and structural zeros.
For example, for a quasiindependence model for incomplete tables see the SAS example link
https://support.sas.com/onlinedoc/913/getDoc/en/statug.hlp/catmod_sect44.htm
and/or monkey.sas example below or monkey.R.
In SAS we have ...
In R the code for this looks like this:
There are more special case models presented in Agresti(2013), ch. 11, and Agresti (2007), ch. 8.
Extensions to higher dimensions are possible by combining categories and/or collapsing categories and variables to get symmetric tables (if and when appropriate).
For more advance topics and models on how to deal with repeated measures data in higherdimensions via loglinear, logit and GLM models see Agresti(2013) ch. 11 and ch. 12, Agresti(2007), ch.9 and ch. 10, and Bishop, Holland and Fienberg (1975).