I am trying to understand the following statement (Christensen and Brockhoff, 2013):

Cumulative link mixed models is a member of a class of models sometimes referred to as multivariate generalized nonlinear mixed models (Fahrmeir and Tutz, 2001).

I am particularly finding it difficult to understand if cumulative link models are themselves multinomial generalized nonlinear models. I am assuming that if such a non-linearity exits, it would exist because conditional expectation of response variable is assumed to be a non-linear function of predictors. I say this because, as McCullagh and Nelder (1989, Chapter 11) say:

... all generalized linear models are in a strict sense non-linear. However, their non-linearity is limited in that it enters only through the variance function and the link function, the linearity of terms contributing to the linear predictor being preserved.

For example, if we consider a cumulative logit model with proportional odds assumption from Agresti (2010, Chapter 3) for a response variable $\mathbf{Y}$ with 3 possible ordered outcomes (1, 2 and 3) with a predictor $x$ and $N$ observations of $(x, Y)$, i.e.

$$\begin{align} \operatorname {logit} [P(Y_i \le j \mid x)] = \alpha_j - x\beta, \end{align}$$

can I write the expectation of $i^{th}$ observation falling in the $j^{th}$ category, where $j = 1, 2$, as:

$$\begin{align} E[Y_i = j \mid x] = N p_{i,j} &= N \left(\frac{\exp(\alpha_{j} - x\beta)}{1 + \exp(\alpha_{j} - x\beta)} - \frac{\exp(\alpha_{j - 1} - x\beta)}{1 + \exp(\alpha_{j - 1} - x\beta)}\right) \\ & where \ \alpha_{j} > \alpha_{j - 1}, \alpha_{0} = -\infty, \alpha_{3} = \infty \end{align}$$

If the above formulation is correct, the conditional expectation of $Y_{i}$ being $j$, $j \in\{1, 2\}$ is a nonlinear function of $\beta$.

However, I can't validate if my formulation is correct, and thus, I can't understand if cumulative link models are really multinomial generalized nonlinear model. Could someone help me with this?

Christensen, R. H. B., & Brockhoff, P. B. (2013). Analysis of sensory ratings data with cumulative link models. Journal de la Société Française de Statistique, 154(3), 58-79.

McCullagh, P. & Nelder, J. (1989). Generalized Linear Models. Chapman & Hall/CRC, second edition.

Agresti, A. (2010). Categorical data analysis. John Wiley & Sons, second edition.



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