By Ames D.B.

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1. The ordered models can also be formulated as threshold models. The real line is divided by thresholds into C intervals, corresponding to the C ordered categories. The first threshold is γ 1 . Threshold γ 1 defines the upper bound of the interval corrsponding to observed outcome 1. Similarly, threshold γ C−1 defines the lower bound of the interval corrsponding to observed outcome C. Threshold γ c defines the boundary between the intervals corresponding to observed outcomes c − 1 and c ∗ (for c = 2, · · · , C − 1).

Tab contains the item response variable score and the explanatory variable, pupil gender (pupsex) which takes value 1 for females, and takes value 2 for males: 1. Use SabreR to fit the null model on item responses (scores), that is, the model without explanatory variables. Obtain the log likelihood, parameter estimate and standard error. Interpret the parameter estimate. 2. Add the explanatory variable, pupil gender (pupsex), to this model 20 Multivariate Generalized Linear Mixed Models Using R and re-fit the model.

If the cumulative density function of εij is F , these cumulative probabilities are denoted by: Pij(c) = F (γ c − θij ), c = 1, · · · , C − 1, where γ 0 = −∞ and γ C = +∞. Equivalently, we can write the model as a cumulative model: G Pij(c) = γ c − θij , where G = F −1 is the link function. If εij follows the logistic distribution, this results in the ordered logistic regression model, also called the ordered logit model or cumulative logit model or proportional odds model. If εij follows the standard normal distribution, this leads to the ordered probit model.