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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.