Ordinal outcomes: a cumulative probability model with the identity link and constrained optimization
Bibliographic record
Abstract
We present here a study of ordinal outcomes with a cumulative probability model but we consider the identity link and a 'proportional' assumption. The logit link is often used with ordinal outcomes in health research literature. With the logit link, one obtains the proportional odds model where regression coefficients are functions of cumulative odds. Recent work shows that when the log link and appropriate constraints are used, one obtains the proportional probability model where regression coefficients are functions of cumulative probabilities. However, when the identity link and appropriate parameters constraints are used, one obtains regression coefficients that are cumulative probabilities. We refer to this model as the additive probability model (APM). In using the identity link, cumulative risk differences can be estimated, and the number needed to treat. For the APM, we present the model and its constraints. The Adaptive barrier method and Augmented Lagrangian are candidate constrained optimizers to determine the maximum likelihood estimates (MLEs). Simulations are conducted to compare optimizers across metrics: non-convergence rates, absolute log-likelihood difference, bias, root mean square error, and average optimization run time. Results of the simulation provide a suggested approach. We conclude with several examples and explorations including conditions for the uniqueness of the MLE and other features of the APM. This paper is primarily about introducing the APM and constrained optimization for determining its MLEs.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.009 | 0.029 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.011 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".