Continuation Ratio Model Research Articles

Using ordinal logistic regression to evaluate the performance of laser-Doppler predictions of burn-healing time

BackgroundLaser-Doppler imaging (LDI) of cutaneous blood flow is beginning to be used by burn surgeons to predict the healing time of burn wounds; predicted healing time is used to determine wound treatment as either dressings or surgery. In this paper, we do a statistical analysis of the performance of the technique.MethodsWe used data from a study carried out by five burn centers: LDI was done once between days 2 to 5 post burn, and healing was assessed at both 14 days and 21 days post burn. Random-effects ordinal logistic regression and other models such as the continuation ratio model were used to model healing-time as a function of the LDI data, and of demographic and wound history variables. Statistical methods were also used to study the false-color palette, which enables the laser-Doppler imager to be used by clinicians as a decision-support tool.ResultsOverall performance is that diagnoses are over 90% correct. Related questions addressed were what was the best blood flow summary statistic and whether, given the blood flow measurements, demographic and observational variables had any additional predictive power (age, sex, race, % total body surface area burned (%TBSA), site and cause of burn, day of LDI scan, burn center). It was found that mean laser-Doppler flux over a wound area was the best statistic, and that, given the same mean flux, women recover slightly more slowly than men. Further, the likely degradation in predictive performance on moving to a patient group with larger %TBSA than those in the data sample was studied, and shown to be small.ConclusionModeling healing time is a complex statistical problem, with random effects due to multiple burn areas per individual, and censoring caused by patients missing hospital visits and undergoing surgery. This analysis applies state-of-the art statistical methods such as the bootstrap and permutation tests to a medical problem of topical interest. New medical findings are that age and %TBSA are not important predictors of healing time when the LDI results are known, whereas gender does influence recovery time, even when blood flow is controlled for.The conclusion regarding the palette is that an optimum three-color palette can be chosen 'automatically', but the optimum choice of a 5-color palette cannot be made solely by optimizing the percentage of correct diagnoses.

Creation of a community violence exposure scale: Accounting for what, who, where, and how often

Previous research has used the Rasch model, a method for obtaining a continuous scale from dichotomous survey items measuring a single latent construct, to create a scale of community violence exposure. The authors build upon previous work and describe the application of a Rasch model using the continuation ratio model to create an exposure to community violence (ETV) scale including event circumstance information previously shown to modify the impact of experienced events. They compare the Rasch ETV scale to a simpler sum ETV score, and estimate the effect of ETV on child posttraumatic stress symptoms. Incorporating detailed event circumstance information that is grounded in traumatic stress theory may reduce measurement error in the assessment of children's community violence exposure.

Three Classes of Nonparametric Differential Step Functioning Effect Estimators

The examination of measurement invariance in polytomous items is complicated by the possibility that the magnitude and sign of lack of invariance may vary across the steps underlying the set of polytomous response options, a concept referred to as differential step functioning (DSF). This article describes three classes of nonparametric DSF effect estimators that are based on the models specified by the graded response model (GRM), the continuation ratio model, and the generalized partial credit model. A simulation study was conducted to compare the statistical properties of the three DSF effect estimators under a variety of conditions. The results of the simulation study indicate that the DSF effect estimator based on the GRM yielded the highest level of stability while maintaining the lowest overall level of bias.

Ordinal logistic regression models: application in quality of life studies

Quality of life has been increasingly emphasized in public health research in recent years. Typically, the results of quality of life are measured by means of ordinal scales. In these situations, specific statistical methods are necessary because procedures such as either dichotomization or misinformation on the distribution of the outcome variable may complicate the inferential process. Ordinal logistic regression models are appropriate in many of these situations. This article presents a review of the proportional odds model, partial proportional odds model, continuation ratio model, and stereotype model. The fit, statistical inference, and comparisons between models are illustrated with data from a study on quality of life in 273 patients with schizophrenia. All tested models showed good fit, but the proportional odds or partial proportional odds models proved to be the best choice due to the nature of the data and ease of interpretation of the results. Ordinal logistic models perform differently depending on categorization of outcome, adequacy in relation to assumptions, goodness-of-fit, and parsimony.

Diabetes-Related Complications, Glycemic Control, and Falls in Older Adults

Older adults with type 2 diabetes are more likely to fall, but little is known about risk factors for falls in this population. We determined whether diabetes-related complications or treatments are associated with risk of falls in older diabetic adults. In the Health, Aging, and Body Composition cohort of well-functioning older adults, participants reported falls in the previous year at annual visits. Odds ratios (ORs) for more frequent falls among 446 diabetic participants whose mean age was 73.6 years, with an average follow-up of 4.9 years, were estimated with continuation ratio models. In the first year, 23[corrected]% reported falling; 22, 26, 30[corrected], and 31[corrected]% fell in subsequent years. In adjusted models, reduced peroneal nerve response amplitude (OR 1.50 -95% CI 1.07-2.12], worst quartile versus others); higher cystatin-C, a marker of reduced renal function (1.38 [1.11-1.71], for 1 SD increase); poorer contrast sensitivity (1.41 [0.97-2.04], worst quartile versus others); and low A1C in insulin users (4.36 [1.32-14.46], A1C <or=6 vs. >8%) were associated with risk of falls. In those using oral hypoglycemic medications but not insulin, low A1C was not associated with risk of falls (1.29 [0.65-2.54], A1C <or=6 vs. >8%). Adjustment for physical performance explained some, but not all, of these associations. In older diabetic adults, reducing diabetes-related complications may prevent falls. Achieving lower A1C levels with oral hypoglycemic medications was not associated with more frequent falls, but, among those using insulin, A1C <or=6% increased risk of falls.

Characterization of stage-classified biological processes using multinomial models: a case study of anchovy (Engraulis encrasicolus) eggs in the Bay of Biscay

This paper describes how multinomial models can be used to study developmental rates of biological processes and how these are affected by different factors such as temperature or parental effects. First, the most common multinomial models for ordinal categorical data (namely, proportional odds, continuation-ratio, and extended continuation-ratio models) are described within the particular framework of developmental rates for discrete and ordered stage classified biological processes. Then, the extraction from the fitted models of a series of biologically meaningful parameters is proposed. In particular, maximum probable age and several alternative definitions for stage duration are presented and pros and cons of each of them are discussed. As an example, the methodology is applied to analyze the temperature dependent development of anchovy (Engraulis encrasicolus) eggs in the Bay of Biscay.

Conserved oviposition preferences in alpine leaf beetle populations despite host shifts and isolation

Abstract 1. Choosing the plant on which to lay their eggs is the last act of care that most female herbivorous insects bestow upon their offspring. These decisions play a pivotal role in insect–plant interactions, placing host preference under strong selection and contributing to the diversity of phytophagous insects as one of the first traits to adapt to new hosts.2. This study presents a test of whether extreme isolation and exposure to different host plants can produce intra‐specific divergence in oviposition preference in alpine insects. Geographic variation should impose selection to fine‐tune host plant ranking and specificity to the plants normally encountered, to avoid wasting time during the very limited reproductive season experienced at high altitudes.3. Beetles from five populations of Oreina elongata differing in host availability were offered three natural hosts: Cirsium spinosissimum, Adenostyles alliariae, and Adenostyles glabra. A novel application of a continuation ratio model (logistic regression) was made to sequential no‐choice experiments, combined with quasi‐likelihood analysis of multiple‐choice experiments.4. The results show little geographic variation in host plant choice: all populations strongly preferred Cirsium in multiple‐choice trials, and in no‐choice experiments laid around 47% of their remaining eggs during each stage, almost regardless of the host present.5. Enemy‐free space seems to explain the preference for Cirsium, but isolation and exposure to different plants has clearly not caused local adaptation in host plant ranking or specificity. Reasons for this conservatism despite divergence in other characteristics are discussed.

An adaptive phase I design for identifying a biologically optimal dose for dual agent drug combinations

Historically, designs for dose seeking trials using chemotherapeutic drug combinations have been geared towards finding the maximum tolerated dose, with safety as the primary outcome. With target based agents whose dose-efficacy curves are unknown and dose-toxicity relationships may be minimal, alternative designs are needed. In this paper, we propose an extension to an adaptive single agent dose-finding design previously reported. A generalization of the continuation ratio model allowing separate toxicity and efficacy curves for each agent in a dual agent combination, generating a dose success surface for the combination, is proposed. A continual reassessment approach with a straightforward dose selection criterion using the accumulated data from all patients treated to that point is employed. Our simulation studies demonstrate favourable operating characteristics in terms of experimentation and recommendation rates, and the average sample size, under a variety of scenarios. The proposed approach allowing the incorporation of both the toxicity and efficacy of each agent into the identification of an optimal dosing region for a combination is novel and warrants further consideration.

Markov Models for Repeated Ordinal Data

To demonstrate the application of Markov models, especially for ordinal outcomes, within the context of regression models for correlated data. A brief review of regression methods for correlated data is given. A proportional odds model and a continuation ratio model is applied to repeated measurements of macular pigment density, obtained in an intervention study on the supplementation of macular carotenoids. The correlation between repeated assessments is assumed to follow a first-order Markov model. The models are implemented with standard statistical software. Both models, though not directly comparable, provide a similar conclusion. The application of these models with standard statistical software is straightforward. Markov models can be valuable alternatives to random effects modes or procedures based on generalized estimation equations.

An adaptive dose-finding design incorporating both toxicity and efficacy

Novel therapies are challenging the standards of drug development. Agents with specific biologic targets and limited toxicity require novel designs to determine doses to be taken forward into larger studies. In this paper, we describe an approach that incorporates both toxicity and efficacy data into the estimation of the biologically optimal dose of an agent in a phase I trial. The approach is based on the flexible continuation-ratio model, and uses straightforward optimal dose selection criteria. Dose selection is based on all patients treated up until that time point, using a continual reassessment method approach. Dose-outcome curves considered include monotonically increasing, monotonically decreasing, and unimodal curves. Our simulation studies demonstrate that the proposed design, which we call TriCRM, has favourable operating characteristics.

Identifying methodologies in the assessment of treatment effects on the repeated occurrences of hot flashes in postmenopausal women

Two methods that are commonly used in most clinical trials for evaluating the effect of treatment on hot flashes are frequency only and severity score methods. However, in these crude methods, composite recurrent hot flash data on a patient are reduced to a one-dimensional summary measure, an approach which may not be optimal. In addition, the duration of a hot flash event provides additional information, which is not accounted for in the two methods mentioned above. We propose two new methods in the analysis of hot flash data in this paper. The first method utilizes a generalization of the continuation ratio model to model the severity of a hot flash event. The model accounts for individual random effects and dependence of recurrent events within an individual. The other method that accounts for the duration of a hot flash event is a 3-Dimensional (3D) score method, which incorporates hot flash information on frequency, severity, and duration. Simulation studies were conducted to evaluate the properties of our proposed methods. The results indicate that our proposed generalized continuation ratio model has appreciably high power in detecting small treatment effects on severity compared with the two commonly used methods. The proposed 3D score, on the other hand, is sensitive in detecting treatment effects on both frequency and duration, but has low power in detecting treatment effects on severity alone. Our proposed generalized continuation ratio model approach provides a valuable and powerful alternative to the two commonly used methods in the analysis of treatment effects on hot flash severity. Our proposed 3D score method incorporates the additional information hot flash duration provides, and is another analysis option for the practitioners to employ.

Modelling the association between adherence and viral load in HIV-infected patients

The primary objective of this paper is to investigate the effect of adherence to prescribed antiretroviral therapy on virologic response measured repeatedly over time in HIV-infected patients. To this end observations on plasma viral load (HIV RNA) assessed in copies/ml are categorized into four clinically meaningful states, [0--50[, [50--400[, [400--2000[, [2000 and up. A time-dependent continuation ratio model is used to analyse longitudinal ordinal responses. The main challenge lies in modelling dependencies over time and using information contained in the data efficiently to establish a dynamic relation between observed patient adherence and viral load. Among the several measures of adherence investigated, two specifically account for long periods of time without intake. One is derived from the third moment of the inter-dose interval distribution, while the second reflects internal drug exposure using pharmacokinetic parameters. The approach is applied to a clinical trial involving 35 patients who were followed over 12 months. Results demonstrate a significant relation between patient adherence and virologic response.

Analysis of a Longitudinal Ordinal Response Clinical Trial Using Dynamic Models

SummaryIn many areas of pharmaceutical research, there has been increasing use of categorical data and more specifically ordinal responses. In many cases, complex models are required to account for different types of dependences among the responses. The clinical trial that is considered here involved patients who were required to remain in a particular state to enable the doctors to examine their heart. The aim of this trial was to study the relationship between the dose of the drug administered and the time that was spent by the patient in the state permitting examination. The patient’s state was measured every second by a continuous Doppler signal which was categorized by the doctors into one of four ordered categories. Hence, the response consisted of repeated ordinal series. These series were of different lengths because the drug effect wore off faster (or slower) on certain patients depending on the drug dose administered and the infusion rate, and therefore the length of drug administration. A general method for generating new ordinal distributions is presented which is sufficiently flexible to handle unbalanced ordinal repeated measurements. It consists of obtaining a cumulative mixture distribution from a Laplace transform and introducing into it the integrated intensity of a binary logistic, continuation ratio or proportional odds model. Then, a multivariate distribution is constructed by a procedure that is similar to the updating process of the Kalman filter. Several types of history dependences are proposed.

Optimal designs and limiting optimal designs for a trinomial response

Designs for the continuation-ratio model for a trinomial response will be described. For the situation where the three response categories are: “no response”, “efficacy” and “adverse reaction”, both D-optimal designs and c-optimal designs for estimating the dose with the maximum probability of efficacy are found. Optimal designs are not available in closed form but designs with closed form expression are found which are approximately optimal. Motivated by these designs, a new concept is defined, “limiting optimality”, where a sequence of designs is said to be optimal in an asymptotic sense for a sequence of prior distributions. A member of the sequence is approximately optimal for the corresponding prior distribution. Algebraic forms of limiting optimal designs are derived for a special case of the model where the slopes are equal. They are shown to be very efficient, provide insight, and also provide starting designs for numerical algorithms.

Semi- and Nonparametric Modeling of Ordinal Data

Parametric models for categorical ordinal response variables, like the proportional odds model or the continuation ratio model, assume that the predictor is given by a linear form of covariates. In this article the parametric models are extended to include smooth components in a semiparametric or partially parametric fashion. Parts of the covariates are thereby modeled linearly while other covariates are modeled as unspecified but smooth functions. Estimation is based on a combination of local likelihood and profile likelihood and asymptotic properties of the estimates are derived. In a simulation study it is demonstrated that the profile likelihood approach is to be preferred over a backfitting procedure. Two data examples demonstrate the applicability of the models.

Conditional risk models for ordinal response data: simultaneous logistic regression analysis and generalized score tests

A general class of conditional risk models is introduced for ordinal regression. Special cases include the cumulative logit models, continuation ratio models and adjacent category odds models. A simultaneous logistic regression (SLR) approach is introduced for fitting the models in a unified fashion. Inferences are obtained by adapting the theory of generalized estimating equations. SLR is fully efficient for the continuation ratio model and has high efficiency in other cases. The general approach applies to other link functions such as ordinal probit analysis as well. Rao-type generalized score tests are developed for model assessment within this framework. These tests are useful in testing for parallelism within the general class of models. Real data examples illustrate the unified modeling made possible by this approach.

On measurement properties of continuation ratio models

Three classes of polytomous IRT models are distinguished. These classes are the adjacent category models, the cumulative probability models, and the continuation ratio models. So far, the latter class has received relatively little attention. The class of continuation ratio models includes logistic models, such as the sequential model (Tutz, 1990), and nonlogistic models, such as the acceleration model (Samejima, 1995) and the nonparametric sequential model (Hemker, 1996). Four measurement properties are discussed. These are monotone likelihood ratio of the total score, stochastic ordering of the latent trait by the total score, stochastic ordering of the total score by the latent trait, and invariant item ordering. These properties have been investigated previously for the adjacent category models and the cumulative probability models, and for the continuation ratio models this is done here. It is shown that stochastic ordering of the total score by the latent trait is implied by all continuation ratio models, while monotone likelihood ratio of the total score and stochastic ordering on the latent trait by the total score are not implied by any of the continuation ratio models. Only the sequential rating scale model implies the property of invariant item ordering. Also, we present a Venn-diagram showing the relationships between all known polytomous IRT models from all three classes.

Regression models for unconstrained, partially or fully constrained continuation odds ratios.

Epidemiologists frequently encounter studies with ordered responses. Standard ordered response logit models, such as the continuation ratio model, constrain exposure to have a homogeneous effect across thresholds of the ordered response. We demonstrate a method for fitting regression models for unconstrained, partially or fully constrained continuation odds ratios using a 'person-threshold' data set. For each subject, we create a separate record for each response threshold the subject is 'at risk' of passing and then apply standard binary logistic regression to estimate the continuation-ratio model. An example demonstrates the unconstrained, partially and fully constrained continuation-ratio model, while a small simulation study examines some properties of the proposed 'person-threshold' approach. Finally, we present a brief discussion of statistical software to implement the method.

Calculating Ordinal Regression Models in SAS and S-Plus

Although a number of regression models for ordinal responses have been proposed, these models are not widely known and applied in epidemiology and biomedical research. Overviews of these models are either highly technical or consider only a small part of this class of models so that it is difficult to understand the features of the models and to recognize important relations between them. In this paper we give an overview of logistic regression models for ordinal data based upon cumulative and conditional probabilities. We show how the most popular ordinal regression models, namely the proportional odds model and the continuation ratio model, are embedded in the framework of generalized linear models. We describe the characteristics and interpretations of these models and show how the calculations can be performed by means of SAS and S-Plus. We illustrate and compare the methods by applying them to data of a study investigating the effect of several risk factors on diabetic retinopathy. A special aspect is the violation of the usual assumption of equal slopes which makes the correct application of standard models impossible. We show how to use extensions of the standard models to work adequately with this situation.

Multivariate continuation ratio models: connections and caveats.

We develop semiparametric estimation methods for a pair of regressions that characterize the first and second moments of clustered discrete survival times. In the first regression, we represent discrete survival times through univariate continuation indicators whose expectations are modeled using a generalized linear model. In the second regression, we model the marginal pairwise association of survival times using the Clayton-Oakes cross-product ratio (Clayton, 1978, Biometrika 65, 141-151; Oakes, 1989, Journal of the American Statistical Association 84, 487-493). These models have recently been proposed by Shih (1998, Biometrics 54, 1115-1128). We relate the discrete survival models to multivariate multinomial models presented in Heagerty and Zeger (1996, Journal of the American Statistical Society 91, 1024-1036) and derive a paired estimating equations procedure that is computationally feasible for moderate and large clusters. We extend the work of Guo and Lin (1994, Biometrics 50, 632-639) and Shih (1998) to allow covariance weighted estimating equations and investigate the impact of weighting in terms of asymptotic relative efficiency. We demonstrate that the multinomial structure must be acknowledged when adopting weighted estimating equations and show that a naive use of GEE methods can lead to inconsistent parameter estimates. Finally, we illustrate the proposed methodology by analyzing psychological testing data previously summarized by TenHave and Uttal (1994, Applied Statistics 43, 371-384) and Guo and Lin (1994).