# 11.3.2 - Quasi-independence Model

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, andE= 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

G^{2},X^{2}as before.df= (usualdf) - # of cells fitted perfectly= (I-1)(I-1) - IFor our example, G

^{2}= 0.0061,df= 1,p-value = 0.938Thus, 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

aandbequal 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

model=glm(count~siskel+ebert+icon+imixed+ipro,family=poisson(link=log))

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? |