Mixed Ordinal Logistic Regression

The fixed effects ordinal logistic model can be easily extended to a mixed ordinal logistic regression model by adding random effects terms and allowing covariance patterns in the residual matrix.

In Section 4.2.1 the ordinal mixed model will be specified. The residual matrix for mixed categorical models has a more complex form than for GLMMs and will be defined in Section 4.2.2. As in GLMMs, there can be benefits in reparameterising random effects models as covariance pattern models and this will be discussed in Section 4.2.3. A quasi-likelihood function for the model is defined in Section 4.2.4 and in Section 4.2.5 model fitting methods are discussed.

4.2.1 Definition of the mixed ordinal logistic regression model

The ordinal mixed model can now be specified as y = ^ + e, log(^[c]/(1 — ^[c])) = Xa + ZP, P - N(0, G), var(e) = R.

P is a vector containing the random effects. Thus, if a model fitting two treatments and three visits as fixed and patients as random were fitted to the example data above, we could write

Z is the design matrix for the random effects and has additional rows than previously to correspond to the extended number of observations. For our data Z would be

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