We have seen this when we discussed the two-way table for the siblings' puzzle times. Marginal homogeneity means the row and column distributions (of a square table) are the same. That is, 

\(\mu_{i+}=\mu_{+i}\)

for all \(i\). There is no log-linear model that corresponds directly to marginal homogeneity, but we can use the log-linear models of symmetry and quasi-symmetry to indirectly test for marginal homogeneity.

Recall that symmetry has two restrictions:

• \(\lambda_{ij}^{SE}=\lambda^{SE}_{ji}\)
• \(\lambda^S_i=\lambda^E_i\)

And this suggests that symmetry consists of restrictions from both quasi-symmetry and marginal homogeneity. This correspondence is actually one-to-one, meaning that both quasi-symmetry and marginal homogeneity together are necessary and sufficient for symmetry. And thus if a symmetry model lacks fit when compared with a quasi-symmetry model, it must be due to the marginal homogeneity assumption. This gives us an avenue for testing marginal homogeneity:

\(G^2\) (marginal homogeneity versus saturated) \(= G^2\) (symmetry versus quasi-symmetry) \)

For our example, we calculated \(G^2 = 0.5928\) for the test of symmetry versus saturated (with 3 df) and \(G^2=0.0061\) for the test of quasi-symmetry saturated (with 1 df), giving a likelihood ratio test of \(G^2=0.5928-0.0061=0.5867\) for the test of symmetry versus quasi-symmetry (with 2 df). Indirectly, this serves as our test for marginal homogeneity, and in this case, it would be a reasonable fit to the data (p-value 0.746).

A diagram of the models discussed in the lesson are depicted below.

The most general (complex) model is the saturated model. Each of quasi-symmetry and marginal homogeneity is a special case of the saturated model, but neither is a special case of the other. And symmetry is both quasi-symmetry and marginal homogeneity.

In terms of deviance statistics versus saturated, the table below summarizes the test results. For this example, the symmetry model would be preferred because it's the most restrictive model that fits the data reasonably well.

Thus, Siskel and Ebert tend to agree on their ratings of movies.