# glm link functions for multinomial and ordered probit regression?

Here's what I understand, could someone please tell me if I'm wrong, and how?

For a categorical variable $Y$, the expected value $\text{E}(Y)=\mu=\sum_{y}i\cdot\text{P}(Y=i)$. Using the descriptions from the Wikipedia pages for multinomial and order probit models, I derive that in the multinomial case $$\text{P}(Y=i)=\Phi\left(\frac{X^T\min_{j\neq i}(\beta_i-\beta_j)}{\sigma_{ii}+\sigma_{jj}+2\sigma_{ij}}\right)=\Phi\left(\frac{X^TB_i}{\sigma_{ii}+\sigma_{jj}+2\sigma_{ij}}\right)$$ where $X$ are linear predictors, $\beta_i$ are regression parameters for the $i$th latent variable $Y_i^\ast = X^T\beta_i+\epsilon_i$, $\epsilon_i \sim \mathcal{N}(0,\Sigma)$ and $\Sigma = \sigma_{ij}$, while for the ordered case $$\text{P}(Y=i)=\Phi(u_{i}-X^T\beta)-\Phi(l_{i}-X^T\beta)$$ where $l_i,u_i$ are the upper and lower cutoffs for the $i$th category. In the multinomial case, the regression is performed over $m$ latent variables while the ordered regression is performed over a single variable. Therefore, the link functions should be the inverses of the these linear combinations of normal CDFs. With a Bernouli variable, this reduces to $g^{-1}(\mu) = \Phi^{-1}(\mu)$ in both cases, with sufficient re-parameterization. Is this correct? Either way, is there another expression for the link function?

Second question: Given a variable that presents a continuous quantity (micromoles photons m^-2 s^-1, for instance) but is represented as an integer, and takes a limited number (~ 8) of values of the interval [0,1000]. Would you recommend multinomial or an ordered logit/probit regression? Or something else entirely (I considered Poisson, but it seems inappropriate given the bounded support)?

EDIT: I realize in fridge horror that the multinomial probit 'link' function isn't actually a function at all, since I can't think of any way to invert the expected value (though I suppose given the parameter vectors $\beta_i$, one could find the most likely values of the latent variables). Since GLMs generally call for an inverse, I suppose I'm done. I'm not sure the link function is necessary anyway; you could just estimate the parameters from the likelihood function as above, assuming all observations are independent. However, I likely still need a variance function since I'm trying to fit the multinomial logit/probit into the context of the 'multivariate covariance generalized linear models' introduced in Bonat and Jørgensen (2015), to allow for both correlations between observations and as to include categorical/ordinal data in multivariate analysis alongside other data under the MCGLM framework.