Models for Paired Data

With paired data, you generally know beforehand that the assumption of independence between the variables does not hold. In the case of candidate preference before and after a debate, this is because few respondents are likely to switch their preference on the basis of a single debate. However, the respondents who switched are of interest, while those who did not are not, so you want to use models that condition on the respondents who switched. This is accomplished by specifying Off-diagonal indicator as the cell structure variable in your analyses of the paired data.

Marginal homogeneity. In order to determine whether there is a statistically significant change in voter preference after the debate, the pollsters want to test the homogeneity of the marginal totals; that is, whether the probability a given candidate is preferred before the debate equals the probability they are preferred after the debate. If marginal homogeneity holds, then there is no "winner" of the debate because any supporters each candidate gained is offset by losses to the other candidates. You cannot construct a model in the General Loglinear Analsyis procedure to directly test marginal homogeneity; however, you can use the results of the symmetry and quasi-symmetry models to make conclusions about marginal homogeneity.

Symmetry. The symmetry model for paired data supposes that for each supporter a candidate gained from another contender, they lost a supporter to that same contender. If the symmetry model holds, this implies that marginal homogeneity holds; however, the symmetry condition is very strict and rarely holds.

Quasi-symmetry. The quasi-symmetry model implies that odds ratios that are formed from cell counts on one side of the table's main diagonal will equal odds ratios that are formed from the corresponding cells on the other side of the main diagonal. The quasi-symmetry condition is more general than the symmetry condition, and when symmetry holds this implies that quasi-symmetry holds.

In fact, the symmetry condition holds only when both quasi-symmetry and marginal homogeneity hold. Thus, the marginal homogeneity model can be tested by comparing the likelihood ratio statistics of the symmetry and quasi-symmetry models. See the topic Testing Marginal Homogeneity for more information.

Quasi-independence. If marginal homogeneity does not hold, the pollsters will be further interested in testing whether there is some relationship between the preferences before and after the debate; that is, whether the respondents who switched from a given candidate tended to move to particular oppenents. The quasi-independence model supposes that no such relationship exists.

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