11.3.2 - Quasi-symmetry Model

11.3.2 - Quasi-symmetry Model

Less 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 /
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

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

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