Ordinal Regression Models for Continuous Scales

164

160

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. CPMs are fairly robust to minor or moderate link function misspecification in our simulations. For certain purposes, the CPM 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.

Section: Discussionmentioning

confidence: 99%

Modeling continuous response variables using ordinal regression

Liu

Shepherd

et al. 2017

164

160

“…For example, it may be worthwhile to develop CPMs that permit different relationships and different distributions for different covariate levels. Extensions of both approaches to handle correlated or longitudinal data, using a similar approach by Manuguerra and Heller, would also be beneficial.…”

Section: Discussionmentioning

confidence: 99%

An empirical comparison of two novel transformation models

Tian

Hothorn

et al. 2019

Continuous response variables are often transformed to meet modeling assumptions, but the choice of the transformation can be challenging. Two transformation models have recently been proposed: semiparametric cumulative probability models (CPMs) and parametric most likely transformation models (MLTs). Both approaches model the cumulative distribution function and require specifying a link function, which implicitly assumes that the responses follow a known distribution after some monotonic transformation. However, the two approaches estimate the transformation differently. With CPMs, an ordinal regression model is fit, which essentially treats each continuous response as a unique category and therefore nonparametrically estimates the transformation; CPMs are semiparametric linear transformation models. In contrast, with MLTs, the transformation is parameterized using flexible basis functions. Conditional expectations and quantiles are readily derived from both methods on the response variable's original scale. We compare the two methods with extensive simulations. We find that both methods generally have good performance with moderate and large sample sizes. MLTs slightly outperformed CPMs in small sample sizes under correct models. CPMs tended to be somewhat more robust to model misspecification and outcome rounding. Except in the simplest situations, both methods outperform basic transformation approaches commonly used in practice. We apply both methods to an HIV biomarker study. KEYWORDSHIV, nonparametric maximum likelihood estimation, ordinal regression model, transformation model 562

“…Before developing a new approach to modelling the cut‐point parameters γ K , it is informative to review how model has previously been modified to accommodate ‘continuous’ ordinal outcomes, albeit using a very different methodology for parameter estimation. Manuguerra and Heller discussed a cumulative logistic ordinal model for continuous response variables log( ν ∕ (1 − ν )) = g ( ν ) + x ′ β where the function g ( ν ) is a continuous analogue of the (discrete) cut‐point parameters − ∞ < γ 1 < γ 2 < ⋯ < γ K − 1 < ∞ in the conventional proportional‐odds model. The differentiable and increasing function g ( ν ) maps the continuous ordinal score ν , on the scale (0,1), to a notionally latent variable on the scale ( − ∞ , ∞ ), for instance, a generalized logistic function g ( ν ) = M + B − 1 log( Tν T ∕ (1 − ν T )), with parameters M (intercept), B (slope) and T (symmetry).…”

Section: Modelmentioning

confidence: 99%

“…Before developing a new approach to modelling the cut-point parameters K , it is informative to review how model (1) has previously been modified to accommodate 'continuous' ordinal outcomes, albeit using a very different methodology for parameter estimation. Manuguerra and Heller [24] discussed a cumulative logistic ordinal model for continuous response variables log . = .1 // D g. / C x 0ˇw here the function g. / is a continuous analogue of the (discrete) cut-point parameters 1 < 1 < 2 < < K 1 < 1 in the conventional proportional-odds model.…”

Section: Proportional Oddsmentioning

confidence: 99%

“…We can incorporate these polynomial models into existing GEE models for repeated ordinals score to allow modelling of composite scores and thereby address what appears to be the only obstacle to the widespread use of these models for scores of this type. Others have suggested modelling cut‐point parameters using generalized logistic and non‐parametric functions in proportional‐odds models for continuous ordinal scores derived from visual analogue scales in a Bayesian setting , but this is the first paper to develop models for long truly ordinal scores.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Proportional‐odds models for repeated composite and long ordinal outcome scales

Parsons

2013

Composite or long ordinal scores, that is scores that have a large number of categories and a natural ordering often resulting from the sum of a number of short ordinal scores, are widely used in many medical studies to assess function or quality of life. Typically these are analysed using unjustified assumptions of normality for the outcome measure, that are unlikely to be even approximately true. Scores of this type are better analysed using methods reserved for more conventional (short) ordinal scores, such as the proportional-odds model. The need for a large number of cut-point parameters, that define the divisions between the score categories, for long ordinal scores in the poroprtional-odds model can be avoided by the inclusion of orthogonal polynomial contrasts. The repeated measures proportional odds logistic regression model is introduced and modifications to the generalized estimating equation methodology used for parameter estimation are described for long ordinal outcomes. Data from a trial assessing two surgical interventions are introduced and briefly described and re-analysed using the new model; inferences from the new analysis are compared to previously published results for the primary outcome measure (hip function at 12 months postoperatively). A simulation study is used to illustrate how this model also has more general application for conventional short ordinal scores, to select amongst competing models of varying complexity for the cut-point parameters.