A symmetry model is important because testing symmetry is often the preliminary analysis on symmetric tables.

It is very restrictive because it has implications for both the association between the variables and the margins of the table.

The symmetry model rarely fits data very well. Example of a perfectly symmetric table:

Objective:

Fit the loglinear model of symmetry to test if a square table exhibits symmetry.

Assumptions:

The off diagonal cells have equal expected counts.

μ_ij = μ_ij for i ≠ j
= n_ii for i = j

and row and column margins are equal, i.e., μ_i+ = μ_+i

For our example, let S = classification by Siskel, and E = classification by Ebert.

Model structure:

log(μ_ij) = λ + λ_i + λ_j + λ_ij

where

λ_ij = λ_ji and λ_i = λ_j when i = j.

The second constraint indicates that the main/marginal effect terms are the same for the rows and columns; thus there are no superscripts.

Model fit:

Use G², X² as before. df = (# cells) - (# non-redundant parameters) = # off diagonal cells - # unique parameters = I(I - 1) - I(I - 1)/2 = I(I - 1)/2

G² = 0.59, df = 3, p-value=0.90

The symmetry model seems to fit this data well. This also implies strong agreement. Recall that only strong agreement is not enough to indicate symmetry!

Parameter estimation and interpretation:

The MLE estimates are \(\hat{\mu}_{ij}=\hat{\mu}_{ji}=\dfrac{n_{ij}+n_{ji}}{2}\)

λ′s are interpreted as before.

There are many different ways to fit these models. I will only discuss one here that comes from Agresti.

You need to create a variable that takes on a unique value for each diagonal cell and a unique value of each pair of cells. For example,

\begin{align}
symm &=1 & i=j=1\\
&=2 & i=j=2\\
&=3 & i=j=3\\
&=4 & (i,j)=(1,2)=(2,1)\\
&=5 & (i,j)=(1,3)=(3,1)\\
&=6 & (i,j)=(2,3)=(3,2)\\
\end{align}

This new variable is treated as a nominal variable in fitting the model.

• Using SAS
• Using R