The formulae are easier to read from either the Vignette or the Reference
Manual (both available
here).
The assumed marginal baseline category logit model is $$log
\frac{Pr(Y_{it}=j |x_{it})}{Pr(Y_{it}=J |x_{it})}=(\beta_{tj0}-\beta_{tJ0})
+ (\beta^{'}_{tj}-\beta^{'}_{tJ}) x_{it}=\beta^{*}_{tj0}+ \beta^{*'}_{tj}
x_{it}$$ For subject \(i\), \(Y_{it}\) is the \(t\)-th nominal response
and \(x_{it}\) is the associated covariates vector. Also \(\beta_{tj0}\)
is the \(j\)-th category-specific intercept at the \(t\)-th measurement
occasion and \(\beta_{tj}\) is the \(j\)-th category-specific regression
parameter vector at the \(t\)-th measurement occasion.
The nominal response \(Y_{it}\) is obtained by extending the principle of
maximum random utility (McFadden, 1974) as suggested in
Touloumis (2016).
betas should be provided as a numeric vector only when
\(\beta_{tj0}=\beta_{j0}\) and \(\beta_{tj}=\beta_j\) for all \(t\).
Otherwise, betas must be provided as a numeric matrix with
clsize rows such that the \(t\)-th row contains the value of
(\(\beta_{t10},\beta_{t1},\beta_{t20},\beta_{t2},...,\beta_{tJ0},
\beta_{tJ}\)). In either case, betas should reflect the order of the
terms implied by xformula.
The appropriate use of xformula is xformula = ~ covariates,
where covariates indicate the linear predictor as in other marginal
regression models.
The optional argument xdata should be provided in ``long'' format.
The NORTA method is the default option for simulating the latent random
vectors denoted by \(e^{NO}_{itj}\) in Touloumis (2016). In this
case, the algorithm forces cor.matrix to respect the assumption of
choice independence. To import simulated values for the latent random
vectors without utilizing the NORTA method, the user can employ the
rlatent argument. In this case, row \(i\) corresponds to subject
\(i\) and columns
\((t-1)*\code{ncategories}+1,...,t*\code{ncategories}\) should contain the
realization of \(e^{NO}_{it1},...,e^{NO}_{itJ}\), respectively, for
\(t=1,\ldots,\code{clsize}\).