Modeling continuous response variables using ordinal regression
Open accessAbstract
We study the application of a widely used ordinal regression model, the cumulative probability model (CPM), for continuous outcomes. Such models are attractive for the analysis of continuous response variables because they are invariant to any monotonic transformation of the outcome and because they directly model the cumulative distribution function from which summaries such as expectations and quantiles can easily be derived. Such models can also readily handle mixed type distributions. We describe the motivation, estimation, inference, model assumptions, and diagnostics. We demonstrate that CPMs applied to continuous outcomes are semiparametric transformation models. Extensive simulations are performed to investigate the finite sample performance of these models. We find that properly specified CPMs generally have good finite sample performance with moderate sample sizes, but that bias may occur when the sample size is small. Cumulative probability models are fairly robust to minor or moderate link function misspecification in our simulations. For certain purposes, the CPMs are more efficient than other models. We illustrate their application, with model diagnostics, in a study of the treatment of HIV. CD4 cell count and viral load 6 months after the initiation of antiretroviral therapy are modeled using CPMs; both variables typically require transformations, and viral load has a large proportion of measurements below a detection limit.
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
Many models have continuous-response variables associated with achieving the desired therapeutic correction; these require ordinal regression models for predicting cumulative probability outcomes.
“Many models have continuous-response variables associated with achieving the desired therapeutic correction; these require ordinal regression models for predicting cumulative probability outcomes96.”
From How to design preclinical studies in nanomedicine and cell therapy to maximize the prospects of clinical translation (2018) · cites this paper as [96]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Cites
R: A Language and Environment for Statistical Computing
R Core Team (2000) · doi:10.32614/r.manuals · Cited by 355,706 · see citing works
Modern Applied Statistics with S
Venables, W., & Ripley, B. (2010) · Statisctics and computing/Statistics and computing · doi:10.1007/978-0-387-21706-2 · Cited by 16,188 · see citing works
An Analysis of Transformations
Box, G. E. P., & Cox, D. R. (1964) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/j.2517-6161.1964.tb00553.x · Cited by 15,059 · see citing works
Harrell (2015) · Springer Series in Statistics · doi:10.1007/978-3-319-19425-7 · Cited by 6,878 · see citing works
Regression Models for Ordinal Data
McCullagh, P. (1980) · Journal of the Royal Statistical Society Series B: Statistical Methodology · doi:10.1111/j.2517-6161.1980.tb01109.x · Cited by 4,453 · see citing works
Estimation of the Probability of an Event as a Function of Several Independent Variables
Walker, S. H., & Duncan, D. B. (1967) · Biometrika · doi:10.1093/biomet/54.1-2.167 · Cited by 1,683 · see citing works
Analysis of Ordinal Categorical Data
Agresti, A. (2010) · Wiley Series in Probability and Statistics · doi:10.1002/9780470594001 · Cited by 1,620 · see citing works
The Analysis of Cross-Classified Categorical Data.
Gokhale, D. V., & Fienberg, S. E. (1979) · Journal of the American Statistical Association · doi:10.2307/2286999 · Cited by 840 · see citing works
Regression and ordered categorical variables
Anderson, J. A. (1984) · Journal of the Royal Statistical Society Series B: Statistical Methodology · doi:10.1111/j.2517-6161.1984.tb01270.x · Cited by 574 · see citing works
Sobczyk, K., & Kirkner, D. J. (2001) · Modeling and simulation in science, engineering & technology · doi:10.1007/978-1-4612-0121-2_5 · Cited by 440 · see citing works
Maximum Likelihood Estimation in Semiparametric Regression Models with Censored Data
Zeng, D., & Lin, D. Y. (2007) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/j.1369-7412.2007.00606.x · Cited by 315 · see citing works
Sequential item response models with an ordered response
Tutz, G. (1990) · British Journal of Mathematical and Statistical Psychology · doi:10.1111/j.2044-8317.1990.tb00925.x · Cited by 228 · see citing works
McGowan, C. C., Cahn, P. E., Gotuzzo, E., Padgett, D., Pape, J. W., Wolff, M. J., Schechter, M., & Masys, D. R. (2007) · International Journal of Epidemiology · doi:10.1093/ije/dym073 · Cited by 141 · see citing works
Discussion on Professor Cox's Paper
(1972) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/j.2517-6161.1972.tb00900.x · Cited by 111 · see citing works
Alternative Analyses for the Singly-Ordered Contingency Table
Simon, G. (1974) · Journal of the American Statistical Association · doi:10.1080/01621459.1974.10480239 · Cited by 93 · see citing works
Goodman, L. A. (1983) · Biometrics · doi:10.2307/2530815 · Cited by 80 · see citing works
Ordinal Regression Models for Continuous Scales
Manuguerra, M., & Heller, G. Z. (2010) · The International Journal of Biostatistics · doi:10.2202/1557-4679.1230 · Cited by 49 · see citing works
A new residual for ordinal outcomes
Li, C., & Shepherd, B. E. (2012) · Biometrika · doi:10.1093/biomet/asr073 · Cited by 44 · see citing works
Self-citationGoodness‐of‐link testing in ordinal regression models
Genter, F. C., & Farewell, V. T. (1985) · Canadian Journal of Statistics · doi:10.2307/3315165 · Cited by 35 · see citing works
Probability‐scale residuals for continuous, discrete, and censored data
Shepherd, B. E., Li, C., & Liu, Q. (2016) · Canadian Journal of Statistics · doi:10.1002/cjs.11302 · Cited by 28 · see citing works
Self-citationThe cost of checking proportional hazards
Shepherd, B. E. (2007) · Statistics in Medicine · doi:10.1002/sim.3020 · Cited by 16 · see citing works
Self-citation
Flagged first, then by citations
Research works printed but not linked: 1
- [1]Sall J A monotone regression smoother based on ordinal cumulative logistic regression ASA Proceedings of Statistical Computing Section 1991 276 281
Cited by
Sorted by citations
Ioannidis, J. P. A., Kim, B. Y. S., & Trounson, A. (2018) · Nature Biomedical Engineering · doi:10.1038/s41551-018-0314-y · Cited by 175 · see citing works
National Variation in Congenital Heart Surgery Outcomes
Pasquali, S. K., Thibault, D., O’Brien, S. M., Jacobs, J. P., Gaynor, J. W., Romano, J. C., Gaies, M., Hill, K. D., Jacobs, M. L., Shahian, D. M., et al. (2020) · Circulation · doi:10.1161/circulationaha.120.046962 · Cited by 113 · see citing works
Health status after invasive or conservative care in coronary and advanced kidney disease
Spertus, J. A., Jones, P. G., Maron, D. J., Mark, D. B., O’Brien, S. M., Fleg, J. L., Reynolds, H. R., Stone, G. W., Sidhu, M. S., Chaitman, B. R., et al. (2020) · New England Journal of Medicine · doi:10.1056/nejmoa1916374 · Cited by 84 · see citing works
Visual-Tactile Spatial Multisensory Interaction in Adults With Autism and Schizophrenia
Noel, J.-P., Failla, M. D., Quinde-Zlibut, J. M., Williams, Z. J., Gerdes, M., Tracy, J. M., Zoltowski, A. R., Foss-Feig, J. H., Nichols, H., Armstrong, K., et al. (2020) · Frontiers in Psychiatry · doi:10.3389/fpsyt.2020.578401 · Cited by 39 · see citing works
- Authors
- Liu, Q., Shepherd, B. E., Li, C., & Harrell, F. E.
- Affiliations
- Vanderbilt University (US)
- Case Western Reserve University (US)
- Corresponding author
- Shepherd, Bryan E.
- Venue
- Statistics in Medicine
- Publisher
- Wiley
- Published
- 2017-09-05
- Volume
- 36
- Issue
- 27
- Pages
- 4316-4335
- ISSN
- 0277-6715 (ISSN-L)
- 1097-0258
- PMC
- PMC5675816
- OpenAlex
- W2752223210
- Cited by
- 208
- References
- 21
- Citations by year
- 2026: 22 · 2025: 21 · 2024: 30 · 2023: 31 · 2022: 24 · 2021: 27 · 2020: 31 · 2019: 13 · 2018: 7 · 2017: 2
- FWCI
- 10.08
- Type
- journal article
- Language
- en
- Open access
- green
- europepmc.org
- Licence
- Locations
- Statistics in Medicine (publishedVersion)
- PubMed (publishedVersion)
- Europe PMC (PubMed Central) (acceptedVersion)
- PubMed Central (acceptedVersion)
- Funders
- United States National Institutes of Health (R01AI093234 and U01AI069923)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
- Physical Sciences › Mathematics › Statistics and Probability › Probability and Statistical Research
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Distribution Estimation and Applications
- Keywords
- ordinal regression, continuous response variables, semiparametric transformation models, model diagnostics, HIV treatment, CD4 count, viral load, antiretroviral therapy, detection limit, finite sample properties
- MeSH terms
- Data Interpretation, Statistical, Humans, Probability, Regression Analysis, Models, Statistical, HIV Infections, Likelihood Functions, Treatment Outcome, Statistics, Nonparametric, CD4 Lymphocyte Count, Anti-HIV Agents, Viral Load
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
