# 11.3.2 - Quasi-symmetry Model

11.3.2 - Quasi-symmetry ModelLess restrictive than the symmetry model but still assuming some agreement compared with the fully saturated (unspecified) model, the quasi-symmetry model assumes that

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

Note that, compared with the symmetry model, the quasi-symmetry model does not require marginal homogeneity, which is that \(\lambda^S_i=\lambda^E_i\). In terms of the notation we have so far, the model now becomes

\(\log(\mu_{ij}) = \lambda+\lambda_i^S+\lambda_j^E+ \beta_4I_{12}+\beta_5I_{13}+\beta_6I_{23} \)

Thus, we use the separate S and E factors to allow for more general marginal distributions but apply a restriction to their joint distribution.

Generally for an \(I\times I\) table, the number of parameters breaks down as follows:

\(1+(I-1)+(I-1)+\dfrac{I(I-1)}{2}\)

The first three quantities allow one parameter for \(\lambda\) and \(I-1\) parameters for each marginal distribution separately for S and E. The last quantity \(I(I-1)/2\) corresponds to the number of off-diagonal counts and is identical to the symmetry model. But the symmetry model requires only \(I\) parameters for a common marginal distribution, whereas the quasi-symmetry model here allows for additional parameters and is hence less restrictive.

While it may seem at a glance that using the same indicators for the off-diagonal table counts would require symmetry, note that the additional parameters allowed in the marginal distributions for S and E influence the off-diagonal counts as well. For example, the expected count for the \((1,2)\) cell would be

\(\mu_{12}=\exp[\lambda+\lambda_1^S+\lambda_2^E+\beta_4]\)

while the expected count for the \((2,1)\) cell would be

\(\mu_{21}=\exp[\lambda+\lambda_2^S+\lambda_1^E+\beta_4]\)

For the \(3\times3\) table, we're currently working with, the quasi-symmetry model has 8 parameters, and so the deviance test versus the saturated model will have only a single degree of freedom. But for larger tables, it can be a more moderate appreciable reduction. As it is, we see below when fitting this model with software, the test statistic is \(G^2=0.0061\) and is very similar to the saturated model.

In SAS we have...

```
/* quasi-symmetry */
proc genmod data=critic order=data;
class siskel ebert;
model count=siskel ebert I12 I13 I23 /
link=log dist=poisson predicted;
title "Quasi-Symmetry Model";
run;
```

And from the output, we can verify the deviance statistic above as well as the non-symmetrical form of the fitted values.

Criteria For Assessing Goodness Of Fit | |||
---|---|---|---|

Criterion | DF | Value | Value/DF |

Deviance | 1 | 0.0061 | 0.0061 |

Scaled Deviance | 1 | 0.0061 | 0.0061 |

Pearson Chi-Square | 1 | 0.0061 | 0.0061 |

Scaled Pearson X2 | 1 | 0.0061 | 0.0061 |

Log Likelihood | 351.5763 | ||

Full Log Likelihood | -20.0988 | ||

AIC (smaller is better) | 56.1975 | ||

AICC (smaller is better) | . | ||

BIC (smaller is better) | 57.7753 |

Observation Statistics | |||||
---|---|---|---|---|---|

Observation | count | Predicted Value | Linear Predictor | Standard Error of the Linear Predictor | HessWgt |

1 | 24 | 24 | 3.1780538 | 0.2041241 | 24 |

2 | 8 | 8.0980871 | 2.0916279 | 0.3150296 | 8.0980871 |

3 | 13 | 12.901913 | 2.5573756 | 0.2606862 | 12.901913 |

4 | 8 | 7.9019129 | 2.0671049 | 0.3179476 | 7.9019129 |

5 | 13 | 13 | 2.5649494 | 0.2773501 | 13 |

6 | 11 | 11.098087 | 2.4067728 | 0.2778455 | 11.098087 |

7 | 10 | 10.098087 | 2.312346 | 0.2888566 | 10.098087 |

8 | 9 | 8.9019129 | 2.1862662 | 0.3037655 | 8.9019129 |

9 | 64 | 64 | 4.1588831 | 0.125 | 64 |

In R we have...

```
# quasi-symmetry model
fit.q = glm(count~S+E+I12+I13+I23,family=poisson(link='log'))
```

And from the output, we can verify the deviance statistic above as well as the non-symmetrical form of the fitted values.

```
> summary(fit.q)
Call:
glm(formula = count ~ S + E + I12 + I13 + I23, family = poisson(link = "log"))
Deviance Residuals:
1 2 3 4 5 6 7 8
0.00000 -0.03454 0.02727 0.03482 0.00000 -0.02949 -0.03092 0.03281
9
0.00000
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 3.1781 0.2041 15.569 < 2e-16 ***
Smixed -0.3188 0.2594 -1.229 0.21913
Spro 0.3679 0.2146 1.714 0.08652 .
Emixed -0.2943 0.2594 -1.134 0.25666
Epro 0.6129 0.2146 2.856 0.00430 **
I12 -0.7921 0.3036 -2.609 0.00907 **
I13 -1.2336 0.2414 -5.110 3.22e-07 ***
I23 -1.0654 0.2712 -3.928 8.55e-05 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 1.0221e+02 on 8 degrees of freedom
Residual deviance: 6.0515e-03 on 1 degrees of freedom
AIC: 56.198
> mu.q = fitted.values(fit.q)
> matrix(mu.q,nrow=3)
[,1] [,2] [,3]
[1,] 24.000000 7.901913 10.098087
[2,] 8.098087 13.000000 8.901913
[3,] 12.901913 11.098087 64.000000
```