[R] Estimating regression with constraints in model coefficients

[Gregg Powell]

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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