The formulae are easier to read from either the Vignette or the Reference
Manual (both available
here).
The assumed marginal continuation-ratio model is $$Pr(Y_{it}=j |Y_{it}
\ge j,x_{it})=F(\beta_{tj0} +\beta^{'}_{t} x_{it})$$ where \(F\) is the
cumulative distribution function determined by link. For subject
\(i\), \(Y_{it}\) is the \(t\)-th multinomial response and
\(x_{it}\) is the associated covariates vector. Finally, \(\beta_{tj0}\)
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 ordinal response \(Y_{it}\) is determined by extending the latent
variable threshold approach of Tutz (1991) as suggested in
Touloumis (2016).
When \(\beta_{tj0}=\beta_{j0}\) for all \(t\), then intercepts
should be provided as a numerical vector. Otherwise, intercepts must
be a numerical matrix such that row \(t\) contains the category-specific
intercepts at the \(t\)-th measurement occasion.
betas should be provided as a numeric vector only when
\(\beta_{t}=\beta\) 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 \(\beta_{t}\). 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 \(e^{O2}_{itj}\) in Touloumis (2016). In this
case, the algorithm forces cor.matrix to respect the local
independence assumption. 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 \(e^{O2}_{it1},...,e^{O2}_{itJ}\), respectively, for
\(t=1,\ldots,\code{clsize}\).