# Asymptotic Properties for Cumulative Probability Models for Continuous Outcomes

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. Cumulative Probability Models

#### 2.2. Cumulative Probability Models on Modified Data

#### 2.3. Asymptotic Results

- 1.
- $G\left(x\right)$ is thrice-continuously differentiable, ${G}^{\prime}\left(x\right)>0$ for any x,${G}^{\prime \prime}\left(x\right)\mathrm{sign}\left(x\right)<0$ for $\left|x\right|\ge M$, where $M>0$ is a constant, and$$\underset{x\to \infty}{lim\; inf}{G}^{\prime}\left(x\right)/\{1-G\left(x\right)\}>0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\underset{x\to -\infty}{lim\; inf}{G}^{\prime}\left(x\right)/G\left(x\right)>0.$$
- 2.
- The covariance matrix of Z is non-singular. In addition, Z and $\beta $ are bounded so that ${\beta}^{T}Z\in [-m,m]$ almost certainly for some large constant m.
- 3.
- $A\left(y\right)$ is continuously differentiable in $(-\infty ,\infty )$.

**Theorem**

**1.**

**Theorem**

**2.**

## 3. Simulation Study

#### 3.1. Simulation Set-Up

#### 3.2. Simulation Results

## 4. Example Data Analysis

## 5. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of Theorem 1

**Proof.**

**Proof.**

## Appendix B. Proof of Theorem 2

**Proof.**

## References

- Box, G.E.P.; Cox, D.R. An analysis of transformations (with Discussion). J. R. Stat. Soc. Ser. B
**1964**, 26, 211–252. [Google Scholar] - Doksum, K.A. An extension of partial likelihood methods for proportional hazard models to general transformation models. Ann. Statist.
**1987**, 15, 325–345. [Google Scholar] [CrossRef] - Cuzick, J. Rank regression. Ann. Statist.
**1988**, 16, 1369–1389. [Google Scholar] [CrossRef] - Pettitt, A.N. Inference for the linear model using a likelihood based on ranks. J. R. Statist. Soc. Ser. B
**1982**, 44, 234–243. [Google Scholar] [CrossRef] - Kalbfleisch, J.D.; Prentice, R.L. Marginal likelihoods based on Cox’s regression and life model. Biometrika
**1973**, 60, 267–278. [Google Scholar] [CrossRef] - Cheng, S.C.; Wei, L.J.; Ying, Z. Analysis of transformation models with censored data. Biometrika
**1995**, 82, 835–845. [Google Scholar] [CrossRef] - Chen, K.; Jin, Z.; Ying, Z. Semiparametric analysis of transformation models with censored data. Biometrika
**2002**, 89, 659–668. [Google Scholar] [CrossRef] - Zeng, D.; Lin, D.Y. Maximum likelihood estimation in semiparametric regression models with censored data (with Discussion). J. R. Statist. Soc. Ser. B
**2007**, 69, 507–564. [Google Scholar] [CrossRef] - Manuguerra, M.; Heller, G.Z. Ordinal regression models for continuous scales. Int. J. Biostat.
**2010**, 6, 14. [Google Scholar] [CrossRef] - Hothorn, T.; Möst, L.; Bühlmann, P. Most likely transformations. Scand. J. Stat.
**2018**, 45, 110–134. [Google Scholar] [CrossRef] - Liu, Q.; Shepherd, B.E.; Li, C.; Harrell, F.E. Modeling continuous response variables using ordinal regression. Stat. Med.
**2017**, 36, 4316–4335. [Google Scholar] [CrossRef] [PubMed] - Spertus, J.A.; Jones, P.G.; Maron, D.J.; O’Brien, S.M.; Reynolds, H.R.; Rosenberg, Y.; Stone, G.W.; Harrell, F.E.; Boden, W.E.; Weintraub, W.S.; et al. Health-status outcomes with invasive or conservative care in coronary disease. N. Engl. J. Med.
**2020**, 382, 1408–1419. [Google Scholar] [CrossRef] [PubMed] - Pun, B.T.; Badenes, R.; La Calle, G.H.; Orun, O.M.; Chen, W.; Raman, R.; Simpson, B.-G.K.; Wilson-Linville, S.; Olmedillo, B.H.; de la Cueva, A.V.; et al. Prevalence and risk factors for delirium in critically ill patients with COVID-19 (COVID-D): A multicentre cohort study. Lancet
**2021**, 9, 239–250. [Google Scholar] [CrossRef] [PubMed] - 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. National variation in congenital heart surgery outcomes. Circulation
**2020**, 142, 1351–1360. [Google Scholar] [CrossRef] [PubMed] - Williams, Z.J.; Failla, M.D.; Davis, S.L.; Heflin, B.H.; Okitondo, C.D.; Moore, D.J.; Cascio, C.J. Thermal perceptual thresholds are typical in autism spectrum disorder but strongly related to intra-individual response variability. Sci. Rep.
**2019**, 9, 12595. [Google Scholar] [CrossRef] [PubMed] - Hatch, L.D.; Scott, T.A.; Slaughter, J.C.; Xu, M.; Smith, A.H.; Stark, A.R.; Patrick, S.W.; Ely, E.W. Outcomes, resource use, and financial costs of unplanned extubations in preterm infants. Pediatrics
**2020**, 145, e20192819. [Google Scholar] [CrossRef] - Wang, J.-H.; Wong, R.C.B.; Liu, G.-S. Retinal transcriptome and cellular landscape in relation to the progression of diabetic retinopathy. Investig. Ophthalmol. Vis. Sci.
**2022**, 63, 26. [Google Scholar] [CrossRef] - Ioannidis, J.P.A.; Kim, B.Y.S.; Trounson, A. How to design preclinical studies in nanomedicine and cell therapy to maximize the prospects of clinical translation. Nat. Biomed. Eng.
**2018**, 2, 797–809. [Google Scholar] [CrossRef] - French, B.; Shotwell, M.S. Regression models for ordinal outcomes. JAMA
**2022**, 328, 772–773. [Google Scholar] [CrossRef] - Billingsley, P. Probability and Measure, 3rd ed.; Wiley: New York, NY, USA, 1995. [Google Scholar]
- Van der Vaart, A.W.; Wellner, J.A. Weak Convergence and Empirical Processes; Springer: New York, NY, USA, 1996. [Google Scholar]
- Murphy, S.A.; van der Vaart, A.W. On profile likelihood. J. Am. Stat. Assoc.
**2000**, 95, 449–465. [Google Scholar] [CrossRef] - Castilho, J.L.; Shepherd, B.E.; Koethe, J.R.; Turner, M.; Bebawy, S.; Logan, J.; Rogers, W.B.; Raffanti, S.; Sterling, T.R. CD4/CD8 ratio, age, and risk of serious non-communicable diseases in HIV-infected adults on antiretroviral therapy. AIDS
**2016**, 30, 899–908. [Google Scholar] [CrossRef] - Sauter, R.; Huang, R.; Ledergerber, B.; Battegay, M.; Bernasconi, E.; Cavassini, M.; Furrer, H.; Hoffman, M.; Rougemont, M.; Günthard, H.F.; et al. CD4/CD8 ratio and CD8 counts predict CD4 response in HIV-1-infected drug naive and in patients on cART. Medicine
**2016**, 95, e5094. [Google Scholar] [CrossRef] - Petoumenos, K.; Choi, J.Y.; Hoy, J.; Kiertiburanakul, S.; Ng, O.T.; Boyd, M.; Rajasuriar, R.; Law, M. CD4:CD8 ratio comparison between cohorts of HIV-positive Asians and Caucasians upon commencement of antiretroviral therapy. Antivir. Ther.
**2017**, 22, 659–668. [Google Scholar] - Serrano-Villar, S.; Sainz, T.; Lee, S.A.; Hunt, P.W.; Sinclair, E.; Shacklett, B.L.; Ferre, A.L.; Hayes, T.L.; Somsouk, M.; Hsue, P.Y.; et al. HIV-infected individuals with low CD4/CD8 ratio despite effective antiretroviral therapy exhibit altered T cell subsets, heightened CD8+ T cell activation, and increased risk of non-AIDS morbidity and mortality. PLoS Pathog.
**2014**, 10, e1004078. [Google Scholar] [CrossRef] - Silva, C.; Peder, L.; Silva, E.; Previdelli, I.; Pereira, O.; Teixeira, J.; Bertolini, D. Impact of HBV and HCV coinfection on CD4 cells among HIV-infected patients: A longitudinal retrospective study. J. Infect. Dev. Ctries.
**2018**, 12, 1009–1018. [Google Scholar] [CrossRef] - Gras, L.; May, M.; Ryder, L.P.; Trickey, A.; Helleberg, M.; Obel, N.; Thiebaut, R.; Guest, J.; Gill, J.; Crane, H.; et al. Determinants of restoration of CD4 and CD8 cell counts and their ratio in HIV-1-positive individuals with sustained virological suppression on antiretroviral therapy. J. Acquir. Immune Defic. Syndr.
**2019**, 80, 292–300. [Google Scholar] [CrossRef] - Serrano-Villar, S.; Perez-Elias, M.J.; Dronda, F.; Casado, J.L.; Moreno, A.; Royuela, A.; Perez-Molina, J.A.; Sainz, T.; Navas, E.; Hermida, J.M.; et al. Increased risk of serious non-AIDS-related events in HIV-infected subjects on antiretroviral therapy associated with a low CD4/CD8 ratio. PLoS ONE
**2014**, 9, e85798. [Google Scholar] [CrossRef] - Harrell, F.E., Jr. Regression Modeling Strategies, 2nd ed.; Springer: Cham, Switzerland; Berlin/Heidelberg, Germany; New York, NY, USA; Dordrecht, The Netherlands; London, UK, 2015. [Google Scholar]

**Figure 1.**Average estimate of $A\left(y\right)$ after fitting properly specified CPMs compared with the true transformation, $log\left(y\right)$. Gray curve: original data; black curve: modified data. Dashed lines are the diagonal. Top row: $(L,U)=({e}^{-4},{e}^{4})$; middle row: $(L,U)=({e}^{-2},{e}^{2})$; bottom row: $(L,U)=({e}^{-1/2},{e}^{1/2})$. Left to right: $n=100,1000,5000$. Based on 1000 replications.

**Figure 2.**Estimates of ${\beta}_{1}$ using data categorized outside $(L,U)$ compared with those using the original data and to the truth, ${\beta}_{1}=1$. Gray lines are mean estimates and dashed gray lines are the truth. Top row: $(L,U)=({e}^{-4},{e}^{4})$; middle row: $(L,U)=({e}^{-2},{e}^{2})$; bottom row: $(L,U)=({e}^{-1/2},{e}^{1/2})$. Left to right: $n=100,1000,5000$. Based on 1000 replications.

**Figure 3.**(

**a**) Histogram of CD4:CD8 ratio in our dataset. (

**b**–

**d**) Estimated outcome measures and 95% confidence intervals as functions of age, holding other covariates constant at their medians/modes. (

**b**) Median CD4:CD8 ratio; (

**c**) mean CD4:CD8 ratio; (

**d**) probability that CD4:CD8 $>1$.

**Table 1.**Simulation results for estimates from CPMs on original data and on data categorized outside $(L,U)$; $n=100,1000$; based on 1000 replications.

n | Estimand | Original | Data Categorized Outside $(\mathit{L},\mathit{U})$ | |||
---|---|---|---|---|---|---|

Data | $({\mathbf{e}}^{-\mathbf{4}},{\mathbf{e}}^{\mathbf{4}})$ | $({\mathbf{e}}^{-\mathbf{2}},{\mathbf{e}}^{\mathbf{2}})$ | $({\mathbf{e}}^{-\mathbf{1}/\mathbf{2}},{\mathbf{e}}^{\mathbf{1}/\mathbf{2}})$ | |||

100 | ${\beta}_{1}$ | bias | 0.043 | 0.043 | 0.042 | 0.048 |

SD | 0.228 | 0.228 | 0.229 | 0.260 | ||

mean SE | 0.217 | 0.217 | 0.219 | 0.251 | ||

MSE | 0.054 | 0.054 | 0.054 | 0.070 | ||

${\beta}_{2}$ | bias | –0.022 | –0.021 | –0.020 | –0.022 | |

SD | 0.119 | 0.119 | 0.120 | 0.143 | ||

mean SE | 0.110 | 0.110 | 0.111 | 0.133 | ||

MSE | 0.015 | 0.015 | 0.015 | 0.021 | ||

$A\left({e}^{0.5}\right)$ | bias | 0.019 | 0.019 | 0.019 | 0.020 | |

SD | 0.177 | 0.177 | 0.177 | 0.183 | ||

mean SE | 0.174 | 0.174 | 0.175 | 0.182 | ||

MSE | 0.032 | 0.032 | 0.032 | 0.034 | ||

$\mathrm{median}(Y\mid {X}_{1}=0,{X}_{2}=0)$ | bias | 0.022 | 0.022 | 0.023 | 0.021 | |

SD | 0.172 | 0.172 | 0.172 | 0.176 | ||

MSE | 0.030 | 0.030 | 0.030 | 0.031 | ||

$E(Y\mid {X}_{1}=0,{X}_{2}=0)$ | bias | –0.007 | - | - | - | |

SD | 0.266 | - | - | - | ||

mean SE | 0.262 | - | - | - | ||

MSE | 0.071 | - | - | - | ||

1000 | ${\beta}_{1}$ | bias | 0.007 | 0.007 | 0.007 | 0.008 |

SD | 0.068 | 0.068 | 0.068 | 0.076 | ||

mean SE | 0.067 | 0.067 | 0.068 | 0.077 | ||

MSE | 0.005 | 0.005 | 0.005 | 0.006 | ||

${\beta}_{2}$ | bias | –0.001 | –0.001 | –0.001 | –0.001 | |

SD | 0.033 | 0.033 | 0.034 | 0.040 | ||

mean SE | 0.034 | 0.034 | 0.034 | 0.041 | ||

MSE | 0.001 | 0.001 | 0.001 | 0.002 | ||

$A\left({e}^{0.5}\right)$ | bias | 0.003 | 0.003 | 0.003 | 0.003 | |

SD | 0.055 | 0.055 | 0.055 | 0.056 | ||

mean SE | 0.054 | 0.054 | 0.054 | 0.057 | ||

MSE | 0.003 | 0.003 | 0.003 | 0.003 | ||

$\mathrm{median}(Y\mid {X}_{1}=0,{X}_{2}=0)$ | bias | 0.003 | 0.003 | 0.002 | 0.002 | |

SD | 0.054 | 0.054 | 0.054 | 0.056 | ||

MSE | 0.003 | 0.003 | 0.003 | 0.003 | ||

$E(Y\mid {X}_{1}=0,{X}_{2}=0)$ | bias | –0.003 | - | - | - | |

SD | 0.081 | - | - | - | ||

mean SE | 0.083 | - | - | - | ||

MSE | 0.007 | - | - | - |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, C.; Tian, Y.; Zeng, D.; Shepherd, B.E.
Asymptotic Properties for Cumulative Probability Models for Continuous Outcomes. *Mathematics* **2023**, *11*, 4896.
https://doi.org/10.3390/math11244896

**AMA Style**

Li C, Tian Y, Zeng D, Shepherd BE.
Asymptotic Properties for Cumulative Probability Models for Continuous Outcomes. *Mathematics*. 2023; 11(24):4896.
https://doi.org/10.3390/math11244896

**Chicago/Turabian Style**

Li, Chun, Yuqi Tian, Donglin Zeng, and Bryan E. Shepherd.
2023. "Asymptotic Properties for Cumulative Probability Models for Continuous Outcomes" *Mathematics* 11, no. 24: 4896.
https://doi.org/10.3390/math11244896