# 11.3.2 - Quasi-independence Model

Printer-friendly version

Generalization of an independence model.

Objective:

Fit the loglinear model of independence only to off-diagonal cells.

Assumptions:

Independence model holds for the off diagonal cells.

Odds ratios for off-diagonal cells equal 1.

$\theta(ij,i'j')=\dfrac{\mu_{ij}\mu_{i'j'}}{\mu_{ij'}\mu_{i'j}} \text{ for } i\neq j\text{ and } i'\neq j'$

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

Model structure:

\begin{align}
\text{log}(\mu_{ij}) &= \lambda+\lambda_i^S+\lambda_j^E & \text{ for }i\neq j\\
&= n_{ij} & \text{ for }i=j\\
\end{align}

For a single equation, specify a numerical indicator variable for each of the diagonal cells:

\begin{align}
I(i=j) &= 1 & \text{ for }i\neq j\\
&=0 & \text{elsewhere}\\
\end{align}
$\text{log}(\mu_{ij}) = \lambda+\lambda_i^S+\lambda_j^E+\delta_iI(i=j)$

Model fit:

Use G2, X2 as before. df = (usual df) - # of cells fitted perfectly= (I-1)(I-1) - I

For our example, G2 = 0.0061, df = 1, p-value = 0.938

Thus, the quasi-independence model fits well, i.e., given a change of category, Ebert's rating is independent of Siskel's (and the other way around).

Parameter estimation and interpretation:

λ′s are interpreted as before.

Odds ratios involving only off-diagonal cells are 1 by the model assumption.

For the quasi-independence model δ parameter are linked to the odds summarizing agreement for categories. The odds summarizing agreement for categories a and b equal to

$\tau_{ab}=\dfrac{\mu_{aa}\mu_{bb}}{\mu_{ab}\mu_{ba}}=\text{exp}(\delta_a+\delta_b)$

For example, the estimated odds that Siskel's rating is category 'con' rather than 'mixed' are exp(0.96+0.62) = 4.71 times as high when the Ebert's rating is 'con' than when it is 'mixed'.

In general you need to create a separate indicator (dummy) variable for each diagonal cell. The indicator is treated as a numerical variable in the model.

Let's see how we can do this in SAS and R, see.

Take a look at the SAS code, (movies.sas, movies.lst) for this example:

Part of the output:

You can find the R code for this example in movies.R.

### Quasi-Independence Model
summary(model)

And, here is a part of the output that we are interested in:

Another way to fit this model is to create a variable that takes on a unique value for each of the diagonal cells and a common value for all of the off diagonal cells. For example,

\begin{align}
qi &=1 & i=j=1\\
&=2 & i=j=2\\
&=3 & i=j=3\\
&=4 & i\neq j\\
\end{align}

This new variable is treated as a nominal variable in fitting the model.  Here is what this might look like if your were to do this in SAS with PROC GENMOD:

 Do not forget you first need to declare and create this new variable labeled as "qi" in the SAS code in order for this to run. Can you modify movies.sas and/or movies.R to fit the quasi-independence model in this way?