The formulae are easier to read from either the Vignette or the Reference
Manual (both available
here).
The assumed marginal adjacent-category logit model is $$log
\frac{Pr(Y_{it}=j |x_{it})}{Pr(Y_{it}=j+1 |x_{it})}=\beta_{tj0}
+ \beta^{'}_{t} x_{it}$$
For subject \(i\), \(Y_{it}\) is the \(t\)-th ordinal 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 \(\beta_{t}\) is the regression
parameter vector at the \(t\)-th measurement occasion.
The ordinal response \(Y_{it}\) is obtained by utilizing the threshold
approach described in the Vignette. This approach is based on the connection
between baseline-category and adjacent-category logit models.
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 \(e^{O3}_{itj}\) in the Vignette. 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 \(e^{O3}_{it1},...,e^{O3}_{itJ}\), respectively, for
\(t=1,\ldots,\code{clsize}\).