Hi Gregg,
I am sincerely thankful for this workout.
Could you please suggest any text book on how to create log-likelihood
for an ordinal model like this? Most of my online search point me
directly to some R function etc, but a theoretical discussion on this
subject would be really helpful to construct the same.
Thanks and regards,
On Mon, Apr 21, 2025 at 9:55 PM Gregg Powell <g.a.pow...@protonmail.com> wrote:
>
> Christofer,
>
> Given the constraints you mentioned—bounded parameters, no intercept, and a
> sum constraint—you're outside the capabilities of most off-the-shelf ordinal
> logistic regression functions in R like vglm or polr.
>
> The most flexible recommendation at this point is to implement custom
> likelihood optimization using constrOptim() or nloptr::nloptr() with a
> manually coded log-likelihood function for the cumulative logit model.
>
> Since you need:
>
> Coefficient bounds (e.g., lb ≤ β ≤ ub),
>
> No intercept,
>
> And a constraint like sum(β) = C,
>
> …you'll need to code your own objective function. Here's something of a
> high-level outline of the approach:
>
> A. Model Setup
> Let your design matrix X be n x p, and let Y take ordered values 1, 2, ..., K.
>
> B. Parameters
> Assume the thresholds (θ_k) are fixed (or removed entirely), and you’re
> estimating only the coefficient vector β. Your constraints are:
>
> Box constraints: lb ≤ β ≤ ub
>
> Equality constraint: sum(β) = C
>
> C. Likelihood
> The probability of category k is given by:
>
> P(Y = k) = logistic(θ_k - Xβ) - logistic(θ_{k-1} - Xβ)
>
> Without intercepts, this becomes more like:
>
> P(Y ≤ k) = 1 / (1 + exp(-(c_k - Xβ)))
>
> …where c_k are fixed thresholds.
>
> Implementation using nloptr
> This example shows a rough sketch using nloptr, which allows both equality
> and bound constraints:
>
> >library(nloptr)
> >
> ># Custom negative log-likelihood function
> >negLL <- function(beta, X, y, K, cutpoints) {
> > eta <- X %*% beta
> > loglik <- 0
> > for (k in 1:K) {
> > pk <- plogis(cutpoints[k] - eta) - plogis(cutpoints[k - 1] - eta)
> > loglik <- loglik + sum(log(pk[y == k]))
> > }
> > return(-loglik)
> >}
> >
> ># Constraint: sum(beta) = C
> >sum_constraint <- function(beta, C) {
> > return(sum(beta) - C)
> >}
> >
> ># Define objective and constraint wrapper
> >objective <- function(beta) negLL(beta, X, y, K, cutpoints)
> >eq_constraint <- function(beta) sum_constraint(beta, C = 2) # example >C
> >
> ># Run optimization
> >res <- nloptr(
> > x0 = rep(0, ncol(X)),
> > eval_f = objective,
> > lb = lower_bounds,
> > ub = upper_bounds,
> > eval_g_eq = eq_constraint,
> > opts = list(algorithm = "NLOPT_LD_SLSQP", xtol_rel = 1e-8)
> >)
>
>
>
> The next step would be writing the actual log-likelihood for your specific
> problem or verifying constraint implementation.
>
> Manually coding the log-likelihood for an ordinal model is nontrivial... so a
> bit of a challenge :)
>
>
>
> All the best,
> gregg powell
> Sierra Vista, AZ
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