Hello again Christofer,
This clarification changes the model structure somewhat significantly -it
shifts us from a standard cumulative logit model with proportional odds to a
non-parallel cumulative logit model, where each threshold has its own set of β
coefficients. At least, that is now my understanding.
So, instead of a single β vector shared across all class boundaries, you're now
specifying:
• One set of coefficients β(1)β^{(1)}β(1) for the logit of P(Y≤1)P(Y ≤ 1)P(Y≤1),
• A second, distinct set β(2)β^{(2)}β(2) for P(Y≤2)P(Y ≤ 2)P(Y≤2),
• And no intercepts, meaning the threshold-specific slope vectors must carry
all the signal.
So, we can adjust the log-likelihood accordingly:
P(Y=1)=logistic(Xβ(1))P(Y=2)=logistic(Xβ(2))−logistic(Xβ(1))P(Y=3)=1−logistic(Xβ(2))P(Y
= 1) = logistic(Xβ^{(1)}) P(Y = 2) = logistic(Xβ^{(2)}) - logistic(Xβ^{(1)})
P(Y = 3) = 1 - logistic(Xβ^{(2)})
P(Y=1)=logistic(Xβ(1))P(Y=2)=logistic(Xβ(2))−logistic(Xβ(1))P(Y=3)=1−logistic(Xβ(2))
Before I attempt a revised script, can you confirm:
1. Should the sum constraint (e.g., sum(β) = 1.60) apply to:
• Only β¹?
• Only β²?
• Or the sum of all 6 coefficients (β¹ and β² combined)?
2. Do you want to apply separate lower/upper bounds to each of the six β
coefficients (and if so, what are they for each)?
Once I understand this last part better, I’ll see about working on a version
that fits this updated structure and constraint logic.
As always – no promises.
r/
Gregg Powell
Sierra Vista, AZ
On Tuesday, April 29th, 2025 at 1:51 PM, Christofer Bogaso
<bogaso.christo...@gmail.com> wrote:
>
>
> Hi Gregg,
>
> I am just wondering if you get any time to think about this problem.
>
> As I understand, typically for this type of problem, we have different
> intercepts for different classes, while slope (beta) coefficients
> remain the same across classes.
>
> But in my case, since we do not have any intercept term, the slope
> coefficients need to be estimated separately for different classes.
> Therefore, since we have 3 classes in the response variable (i.e.
> 'apply'), there will be 3 different sets of beta-coefficients for the
> independent variables.
>
> Under this situation, I wonder how I can create the likelihood
> function subject to all applicable constraints.
>
> Any insight would be highly appreciated.
>
> Thanks and regards,
>
> On Fri, Apr 25, 2025 at 12:31 AM Gregg Powell g.a.pow...@protonmail.com wrote:
>
> > Christofer,
> > That was a detailed follow-up — you clarified the requirements precisely
> > enough providing a position to proceed from...
> >
> > To implement this, a constrained optimization procedure to estimate an
> > ordinal logistic regression model is needed (cumulative logit), based on:
> >
> > 1. Estimated Cutpoints
> > – No intercept in the linear predictor
> > – Cutpoints (thresholds) will be estimated directly from the data
> >
> > 2. Coefficient Constraints
> > – Box constraints on each coefficient
> > – For now:
> > lower = c(1, -1, 0)
> > upper = c(2, 1, 1)
> > – These apply to: pared, public, gpa (in that order)
> >
> > 3. Sum Constraint
> > – The sum of coefficients must equal 1.60
> >
> > 4. Data to use...
> > – Use the IDRE ologit.dta dataset from UCLA (for now)
> >
> > Technical Approach
> >
> > • Attempt to write a custom negative log-likelihood function using the
> > cumulative logit formulation:
> >
> > P(Y≤k∣X)=11+exp[−(θk−Xβ)]P(Y \leq k \mid X) = \frac{1}{1 + \exp[-(\theta_k
> > - X\beta)]}
> >
> > and derive P(Y=k)P(Y = k) from adjacent differences of these.
> >
> > • Cutpoints θk\theta_k will be estimated as separate parameters, with
> > constraints to ensure they’re strictly increasing for identifiability.
> >
> > • The optimization will use nloptr::nloptr(), which supports:
> > - Lower/upper bounds (box constraints)
> > - Equality constraints (for sum of β)
> > - Nonlinear inequality constraints (to keep cutpoints ordered)
> >
> > I’ll have some time later - in the next day or two to attempt a script with:
> > - Custom negative log-likelihood
> > - Parameter vector split into β and cutpoints
> > - Constraint functions: sum(β) = 1.60 and increasing cutpoints
> > - Optimization via nloptr()
> >
> > Hopefully, you’ll be able to run it locally with only the VGAM, foreign,
> > and nloptr packages.
> >
> > I’ll send the .R file along with the next email. A best attempt, anyway.
> >
> > r/
> > Gregg
> >
> > “Oh, what fun it is!”
> > —Alice, Alice’s Adventures in Wonderland by Lewis Carroll
> >
> > On Thursday, April 24th, 2025 at 1:56 AM, Christofer Bogaso
> > bogaso.christo...@gmail.com wrote:
> >
> > > Hi Gregg,
> >
> > > Many thanks for your detail feedback, those are really super helpful.
> >
> > > I have ordered a copy of Agresti from our local library, however it
> > > appears that I would need to wait for a few days.
> >
> > > Regrading my queries, it would be super helpful if we can build a
> > > optimization algo based on below criterias:
> >
> > > 1. Whether you want the cutpoints fixed (and to what values), or if
> > > you want them estimated separately (with identifiability managed some
> > > other way); I would like to have cut-points directly estimated from
> > > the data
> > > 2. What your bounds on the coefficients are (lower/upper vectors), For
> > > this question, I am having different bounds for each of the
> > > coefficients. Therefore I would have a vector of lower points and
> > > other vector of upper points. However to start with let consider lower
> > > and upper bounds as lower = c(1, -1, 0) and upper = c(2, 1, 1) for the
> > > variables pared, public, and gpa. In my model, there would not be any
> > > Intercept, hence no question on its bounds
> > > 3. What value the sum of coefficients should equal (e.g., sum(β) = 1,
> > > or something else); Let the sum be 1.60
> > > 4. And whether you're working with the IDRE example data, or something
> > > else. My original data is actually residing in a computer without any
> > > access to the internet (for data security.) However we can start with
> > > any suitable data for this experiment, so one such data may be
> > > https://stats.idre.ucla.edu/stat/data/ologit.dta. However I am still
> > > exploring if there is any possibility to extract a randomized copy of
> > > that actual data, if I can - I will share once available
> >
> > > Again, many thanks for your time and insights.
> >
> > > Thanks and regards,
> >
> > > On Wed, Apr 23, 2025 at 9:54 PM Gregg Powell g.a.pow...@protonmail.com
> > > wrote:
> >
> > > > Hello again Christofer,
> > > > Thanks for your thoughtful note — I’m glad the outline was helpful.
> > > > Apologies for the long delay getting back to you. Had to do a small bit
> > > > of research…
> >
> > > > Recommended Text on Ordinal Log-Likelihoods:
> > > > You're right that most online sources jump straight to code or canned
> > > > functions. For a solid theoretical treatment of ordinal models and
> > > > their likelihoods, consider:
> > > > "Categorical Data Analysis" by Alan Agresti
> > > > – Especially Chapters 7 and 8 on ordinal logistic regression.
> > > > – Covers proportional odds models, cumulative logits, adjacent-category
> > > > logits, and the derivation of likelihood functions.
> > > > – Provides not only equations but also intuition behind the model
> > > > structure.
> > > > It’s a standard reference in the field and explains how to build
> > > > log-likelihoods from first principles — including how the cumulative
> > > > probabilities arise and why the difference of CDFs represents a
> > > > category-specific probability.
> > > > Also helpful:
> > > > "An Introduction to Categorical Data Analysis" by Agresti (2nd ed) – A
> > > > bit more accessible, and still covers the basics of ordinal models.
> > > > ________________________________________
> >
> > > > If You Want to Proceed Practically:
> > > > In parallel with theory, if you'd like a working R example coded from
> > > > scratch — with:
> > > > • a custom likelihood for an ordinal (cumulative logit) model,
> > > > • fixed thresholds / no intercept,
> > > > • coefficient bounds,
> > > > • and a sum constraint on β
> >
> > > > I’d be happy to attempt that using nloptr() or constrOptim(). You’d be
> > > > able to walk through it and tweak it as necessary (no guarantee that I
> > > > will get it right 😊)
> >
> > > > Just let me know:
> > > > 1. Whether you want the cutpoints fixed (and to what values), or if you
> > > > want them estimated separately (with identifiability managed some other
> > > > way);
> > > > 2. What your bounds on the coefficients are (lower/upper vectors),
> > > > 3. What value the sum of coefficients should equal (e.g., sum(β) = 1,
> > > > or something else);
> > > > 4. And whether you're working with the IDRE example data, or something
> > > > else.
> >
> > > > I can use the UCLA ologit.dta dataset as a basis if that's easiest to
> > > > demo on, or if you have another dataset you’d prefer – again, let me
> > > > know.
> >
> > > > All the best!
> >
> > > > gregg
> >
> > > > On Monday, April 21st, 2025 at 11:25 AM, Christofer Bogaso
> > > > bogaso.christo...@gmail.com wrote:
> >
> > > > > 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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