Cumulative logit modelling for ordinal response variables: applications to biomedical research

Incorrect statistical methods are often used for the analysisof ordinal response data. Such data are frequently summarizedinto mean scores for comparisons, a fallacious practice becauseordinal data are inherently not equidistant. The ubiquitousPearson chi-square test is invalid because it ignores the rankingof ordinal data. Although some of the non-parametric statisticalmethods take into account the ordering of ordinal data, thesemethods do not accommodate statistical adjustment of confoundingor assessment of effect modification, two overriding analyticgoals in virtually all etiologic inference in biology and medicine.The cumulative logit model is eminently suitable for the anlaysisof ordinal response data. This multivariate method not onlyconsiders the ranked order inherent in ordinal response data,but it also allows adjustment of confounding and assessmentof effect modification based on modest sample size. A non-technicalaccount of the cumulative logit model is given and its applicationsare illustrated by two research examples. The SAS programs forthe data analysis of the research examples are available fromthe author.     

Power analysis for trend in ordinal cover classes: implications for long‐term vegetation monitoring

Question: We provide a method to calculate the power of ordinal regression models for detecting temporal trends in plant abundance measured as ordinal cover classes. Does power depend on the shape of the unobserved (latent) distribution of percentage cover? How do cover class schemes that differ in the number of categories affect power? Methods: We simulated cover class data by “cutting‐up” a continuous logit‐beta distributed variable using 7‐point and 15‐point cover classification schemes. We used Monte Carlo simulation to estimate power for detecting trends with two ordinal models, proportional odds logistic regression (POM) and logistic regression with cover classes re‐binned into two categories, a model we term an assessment point model (APM). We include a model fit to the logit‐transformed percentage cover data for comparison, which is a latent model. Results: The POM had equal or higher power compared to the APM and latent model, but power varied in complex ways as a function of the assumed latent beta distribution. We discovered that if the latent distribution is skewed, a cover class scheme with more categories might yield higher power to detect trend. Conclusions: Our power analysis method maintains the connection between the observed ordinal cover classes and the unmeasured (latent) percentage cover variable, allowing for a biologically meaningful trend to be defined on the percentage cover scale. Both the shape of the latent beta distribution and the alternative hypothesis should be considered carefully when determining sample size requirements for long‐term vegetation monitoring using cover class measurements.     

Sample size recalculation in multicenter randomized controlled clinical trials based on noncomparative data

Many late-phase clinical trials recruit subjects at multiple study sites. This introduces a hierarchical structure into the data that can result in a power-loss compared to a more homogeneous single-center trial. Building on a recently proposed approach to sample size determination, we suggest a sample size recalculation procedure for multicenter trials with continuous endpoints. The procedure estimates nuisance parameters at interim from noncomparative data and recalculates the sample size required based on these estimates. In contrast to other sample size calculation methods for multicenter trials, our approach assumes a mixed effects model and does not rely on balanced data within centers. It is therefore advantageous, especially for sample size recalculation at interim. We illustrate the proposed methodology by a study evaluating a diabetes management system. Monte Carlo simulations are carried out to evaluate operation characteristics of the sample size recalculation procedure using comparative as well as noncomparative data, assessing their dependence on parameters such as between-center heterogeneity, residual variance of observations, treatment effect size and number of centers. We compare two different estimators for between-center heterogeneity, an unadjusted and a bias-adjusted estimator, both based on quadratic forms. The type 1 error probability as well as statistical power are close to their nominal levels for all parameter combinations considered in our simulation study for the proposed unadjusted estimator, whereas the adjusted estimator exhibits some type 1 error rate inflation. Overall, the sample size recalculation procedure can be recommended to mitigate risks arising from misspecified nuisance parameters at the planning stage.     

Use of ordinal categorical variables in skeletal assessment of sex from the cranium

In anthropological studies, visual indicators of sex are traditionally scored on an ordinal categorical scale. Logistic and probit regression models are commonly used statistical tools for the analysis of ordinal categorical data. These models provide unbiased estimates of the posterior probabilities of sex conditional on observed indicators, but they do so only under certain conditions. We suggest a more general method for sexing using a multivariate cumulative probit model and examine both single indicator and multivariate indicator models on a sample of 138 crania from a Late Mississippian site in middle Tennessee. The crania were scored for five common sex indicators: superciliary arch form, chin form, size of mastoid process, shape of the supraorbital margin, and nuchal cresting. Independent assessment of sex for each individual is based on pubic indicators. The traditional logistic regressions are cumbersome because of limitations imposed by missing data. The logistic regression correctly classified 66/74 males and 46/64 females, with an overall correct classification of 81%. The cumulative probit model classified 64/74 males correctly and 51/64 females correctly for an overall correct classification rate of 83%. Finally, we apply parameters estimated from the logit and probit models to find posterior probabilities of sex assignment for 296 additional crania for which pubic indicators were absent or ambiguous. Am J Phys Anthropol 107:97–112, 1998. © 1998 Wiley-Liss, Inc.     

Deep transformation models for functional outcome prediction after acute ischemic stroke

In many medical applications, interpretable models with high prediction performance are sought. Often, those models are required to handle semistructured data like tabular and image data. We show how to apply deep transformation models (DTMs) for distributional regression that fulfill these requirements. DTMs allow the data analyst to specify (deep) neural networks for different input modalities making them applicable to various research questions. Like statistical models, DTMs can provide interpretable effect estimates while achieving the state-of-the-art prediction performance of deep neural networks. In addition, the construction of ensembles of DTMs that retain model structure and interpretability allows quantifying epistemic and aleatoric uncertainty. In this study, we compare several DTMs, including baseline-adjusted models, trained on a semistructured data set of 407 stroke patients with the aim to predict ordinal functional outcome three months after stroke. We follow statistical principles of model-building to achieve an adequate trade-off between interpretability and flexibility while assessing the relative importance of the involved data modalities. We evaluate the models for an ordinal and dichotomized version of the outcome as used in clinical practice. We show that both tabular clinical and brain imaging data are useful for functional outcome prediction, whereas models based on tabular data only outperform those based on imaging data only. There is no substantial evidence for improved prediction when combining both data modalities. Overall, we highlight that DTMs provide a powerful, interpretable approach to analyzing semistructured data and that they have the potential to support clinical decision-making.     

Applications of proportional odds ordinal logistic regression models and continuation ratio models in examining the association of physical inactivity with erectile dysfunction among type 2 diabetic patients

[Purpose]Many studies have observed a high prevalence of erectile dysfunction among individuals performing physical activity in less leisure-time. However, this relationship in patients with type 2 diabetic patients is not well studied. In exposure outcome studies with ordinal outcome variables, investigators often try to make the outcome variable dichotomous and lose information by collapsing categories. Several statistical models have been developed to make full use of all information in ordinal response data, but they have not been widely used in public health research. In this paper, we discuss the application of two statistical models to determine the association of physical inactivity with erectile dysfunction among patients with type 2 diabetes.[Methods]A total of 204 married men aged 20-60 years with a diagnosis of type 2 diabetes at the outpatient unit of the Department of Endocrinology at PSG hospitals during the months of May and June 2019 were studied. We examined the association between physical inactivity and erectile dysfunction using proportional odds ordinal logistic regression models and continuation ratio models.[Results]The proportional odds model revealed that patients with diabetes who perform leisure time physical activity for over 40 minutes per day have reduced odds of erectile dysfunction (odds ratio=0.38) across the severity categories of erectile dysfunction after adjusting for age and duration of diabetes.[Conclusion]The present study suggests that physical inactivity has a negative impact on erectile function. We observed that the simple logistic regression model had only 75% efficiency compared to the proportional odds model used here; hence, more valid estimates were obtained here.     

Sequential ordinal modeling with applications to survival data

This paper considers the class of sequential ordinal models in relation to other models for ordinal response data. Markov chain Monte Carlo (MCMC) algorithms, based on the approach of Albert and Chib (1993, Journal of the American Statistical Association 88, 669-679), are developed for the fitting of these models. The ideas and methods are illustrated in detail with a real data example on the length of hospital stay for patients undergoing heart surgery. A notable aspect of this analysis is the comparison, based on marginal likelihoods and training sample priors, of several nonnested models, such as the sequential model, the cumulative ordinal model, and Weibull and log-logistic models.     

Sample Size Requirements for Evaluation of Bridging Evidence

This paper addresses issues concerning methodologies on the sample size required for statistical evaluation of bridging evidence for a registration of pharmaceutical products in a new region. The bridging data can be either in the Complete Clinical Data Package (CCDP) generated during clinical drug development for submission to the original region or from a bridging study conducted in the new region after the pharmaceutical product was approved in the original region. When the data are in the CCDP, the randomized parallel dose‐response design stratified to the ethnic factors and region will generate internally valid data for evaluating similarity concurrently between the regions for assessment of the ability of extrapolation to the new region. Formula for sample size under this design is derived. The required sample size for evaluation of similarity between the regions can be at least four times as large as that needed for evaluation of treatment effects only. For a bridging study conducted in the new region in which the data of the foreign and new regions are not generated concurrently, a hierarchical model approach to incorporating the foreign bridging information into the data generated by the bridging study is suggested. The sample size required is evaluated. In general, the required sample size for the bridging trials in the new region is inversely proportional to equivalence limits, variability of primary endpoints, and the number of patients of the trials conducted in the original region.     

Practical and theoretical considerations in study design for detecting gene-gene interactions using MDR and GMDR approaches

Detection of interacting risk factors for complex traits is challenging. The choice of an appropriate method, sample size, and allocation of cases and controls are serious concerns. To provide empirical guidelines for planning such studies and data analyses, we investigated the performance of the multifactor dimensionality reduction (MDR) and generalized MDR (GMDR) methods under various experimental scenarios. We developed the mathematical expectation of accuracy and used it as an indicator parameter to perform a gene-gene interaction study. We then examined the statistical power of GMDR and MDR within the plausible range of accuracy (0.50～0.65) reported in the literature. The GMDR with covariate adjustment had a power of >80% in a case-control design with a sample size of ≥2000, with theoretical accuracy ranging from 0.56 to 0.62. However, when the accuracy was <0.56, a sample size of ≥4000 was required to have sufficient power. In our simulations, the GMDR outperformed the MDR under all models with accuracy ranging from 0.56～0.62 for a sample size of 1000-2000. However, the two methods performed similarly when the accuracy was outside this range or the sample was significantly larger. We conclude that with adjustment of a covariate, GMDR performs better than MDR and a sample size of 1000～2000 is reasonably large for detecting gene-gene interactions in the range of effect size reported by the current literature; whereas larger sample size is required for more subtle interactions with accuracy <0.56.     

Optimal sampling in retrospective logistic regression via two-stage method

Case-control sampling is popular in epidemiological research because of its cost and time saving. In a logistic regression model, with limited knowledge on the covariance matrix of the point estimator of the regression coefficients a priori, there exists no fixed sample size analysis. In this study, we propose a two-stage sequential analysis, in which the optimal sample fraction and the required sample size to achieve a predetermined volume of a joint confidence set are estimated in an interim analysis. Additionally required observations are collected in the second stage according to the estimated optimal sample fraction. At the end of the experiment, data from these two stages are combined and analyzed for statistical inference. Simulation studies are conducted to justify the proposed two-stage procedure and an example is presented for illustration. It is found that the proposed two-stage procedure performs adequately in the sense that the resultant joint confidence set has a well-controlled volume and achieves the required coverage probability. Furthermore, the optimal sample fractions among all the selected scenarios are close to one. Hence, the proposed procedure can be simplified by always considering a balance design.     

Validation of the Modes of Transmission Model as a Tool to Prioritize HIV Prevention Targets: A Comparative Modelling Analysis

Background

The static Modes of Transmission (MOT) model predicts the annual fraction of new HIV infections acquired across subgroups (MOT metric), and is used to focus HIV prevention. Using synthetic epidemics via a dynamical model, we assessed the validity of the MOT metric for identifying epidemic drivers (behaviours or subgroups that are sufficient and necessary for HIV to establish and persist), and the potential consequence of MOT-guided policies.

Methods and Findings

To generate benchmark MOT metrics for comparison, we simulated three synthetic epidemics (concentrated, mixed, and generalized) with different epidemic drivers using a dynamical model of heterosexual HIV transmission. MOT metrics from generic and complex MOT models were compared against the benchmark, and to the contribution of epidemic drivers to overall HIV transmission (cumulative population attributable fraction over t years, PAF_t). The complex MOT metric was similar to the benchmark, but the generic MOT underestimated the fraction of infections in epidemic drivers. The benchmark MOT metric identified epidemic drivers early in the epidemics. Over time, the MOT metric did not identify epidemic drivers. This was not due to simplified MOT models or biased parameters but occurred because the MOT metric (irrespective of the model used to generate it) underestimates the contribution of epidemic drivers to HIV transmission over time (PAF_5–30). MOT-directed policies that fail to reach epidemic drivers could undermine long-term impact on HIV incidence, and achieve a similar impact as random allocation of additional resources.

Conclusions

Irrespective of how it is obtained, the MOT metric is not a valid stand-alone tool to identify epidemic drivers, and has limited additional value in guiding the prioritization of HIV prevention targets. Policy-makers should use the MOT model judiciously, in combination with other approaches, to identify epidemic drivers.     

整合分析中两种假设模型的介绍及实例分析*

整合分析(meta-analysis)是对同一主题下多个独立实验结果进行综合的统计学方法,被认为是到目前为止最好的数量综合方法。在进行整合分析时,首选应提出统计假设,根据假设的不同可将整合分析分为固定效应模型(fixedeffect model)和随机效应模型(random effect model),前者假定有相似的多个研究在同一分组里有一个共同的真实效应值,由于取样误差,导致在实际效应值的测定中各研究间存在差别;在后者中,假定各研究间有随机变量,因此,不共享一个真实效应值。介绍了两种假设模型下整合分析的计算方法,并进行了实例分析。     

Hierarchical commensurate and power prior models for adaptive incorporation of historical information in clinical trials

Bayesian clinical trial designs offer the possibility of a substantially reduced sample size, increased statistical power, and reductions in cost and ethical hazard. However when prior and current information conflict, Bayesian methods can lead to higher than expected type I error, as well as the possibility of a costlier and lengthier trial. This motivates an investigation of the feasibility of hierarchical Bayesian methods for incorporating historical data that are adaptively robust to prior information that reveals itself to be inconsistent with the accumulating experimental data. In this article, we present several models that allow for the commensurability of the information in the historical and current data to determine how much historical information is used. A primary tool is elaborating the traditional power prior approach based upon a measure of commensurability for Gaussian data. We compare the frequentist performance of several methods using simulations, and close with an example of a colon cancer trial that illustrates a linear models extension of our adaptive borrowing approach. Our proposed methods produce more precise estimates of the model parameters, in particular, conferring statistical significance to the observed reduction in tumor size for the experimental regimen as compared to the control regimen.     

Development of experimental designs for atherosclerosis studies in mice

The mouse has become the de facto model for the majority of atherosclerosis studies. Studies involving the quantification of lesions in mouse models of the disease represent the basis of our evolving concepts on the biochemical and cellular mechanisms underlying the atherogenic process. Many issues of experimental design, including specific model, strain, gender, atherogenic stimulus, duration of study, group size, and statistical analysis may influence the outcome and interpretation of atherosclerosis studies. The selection of vascular bed in which to quantify atherosclerotic lesion size could also impact the interpretation of results. Early studies quantified atherosclerotic lesion size in either specific regions or all of the aortic sinus. Measurement of atherosclerosis throughout the aortic intimal surface has become a common mode for defining lesion size. It is likely that other vascular regions will be increasingly used. In addition to size, there is an increased emphasis on identifying and quantifying the cellular and chemical composition of atherosclerotic lesions.     

How much effort is required for proper monitoring? Assessing the effects of different survey scenarios in a dry acidic grassland

Questions: The quality of any inferences derived from field studies or monitoring programmes depends on expenditure of time and effort to make the underlying observations. Here, we used a long‐term data set from a succession‐monitoring scheme to assess the effect of different survey scenarios. We asked: (1) how well does a survey reflect successional processes if sampling effort varies (a) in space (b) in length of total observation period, (c) in observation frequency and (d) with a combination of these factors? (2) What are the practical implications for devising monitoring programmes? Location: Lignite mining region of Central Germany, post‐mining landscape of Goitzsche (Saxony‐Anhalt). Methods: Based on our full data set, we constructed subsamples. For the full data set and all subsets, we constructed Markov models and compared them based on the predictions made. We assessed effects of survey intensity on model performance using generalized linear models and multiple logistic regressions. Results: Exploring the effects of different survey scenarios revealed significant effects of all three main features of survey intensity (sample size, length, frequency). The most important sampling feature was study length. However, we found interactive effects of sample size with study length and observation interval on model predictions. This indicates that for long‐term observations with multiple recording intervals a lower sample size in space is required to reveal the same amount of information as required in a shorter study or one with fewer intervals. Conversely, a high sample size may, to some degree, compensate for relatively short study periods. Conclusions: Monitoring activities should not be restricted to intensive sampling over only a few years. With clearly limited resources, a decrease of sampling intensity in space, and stretching these resources over a longer period would probably pay off much better than totally abandoning monitoring activities after an intensive, but short, campaign.     

Mapping genes for polygenic disorders: considerations for study design in the complex trait of inflammatory bowel disease

While the methodology for the mapping of Mendelian disorders is well established, the practical and theoretical steps required for successful gene identification in a complex trait are still difficult to predict. A number of analytical models and simulations based on repetitive drawings from predefined statistical distributions are available. To supplement these analytical models, we developed an integrated simulation approach by directly simulating entire populations under a disease model based on epidemiological data. Random mating, nonoverlapping populations and the absence of differential fitness were assumed. Samples were drawn from these homogeneous and heterogeneous populations and analyzed with established analysis tools. We investigated the properties of linkage and association studies in inflammatory bowel disease - modeled as a six-locus polygenic disorder - as an example of this approach. In nonparametric linkage studies, lod scores varied widely, with the median required sample size depending on the locus-specific relative sibling risk. A fine mapping resolution <4 cM was found to require nonparametric lod scores >10. Family-based association studies (TDT test) and case-control studies showed a similar sensitivity and can identify risk loci in populations with moderate levels of linkage disequilibrium in sample sizes of 500-800 triplets. Case-control association studies were prone to false-positive results if applied in heterogeneous populations, with the false-positive rate increasing with sample size because population heterogeneity is detected with increasing power.     

Minimum required number of specimen records to develop accurate species distribution models

Species distribution models (SDMs) are widely used to predict the occurrence of species. Because SDMs generally use presence‐only data, validation of the predicted distribution and assessing model accuracy is challenging. Model performance depends on both sample size and species’ prevalence, being the fraction of the study area occupied by the species. Here, we present a novel method using simulated species to identify the minimum number of records required to generate accurate SDMs for taxa of different pre‐defined prevalence classes. We quantified model performance as a function of sample size and prevalence and found model performance to increase with increasing sample size under constant prevalence, and to decrease with increasing prevalence under constant sample size. The area under the curve (AUC) is commonly used as a measure of model performance. However, when applied to presence‐only data it is prevalence‐dependent and hence not an accurate performance index. Testing the AUC of an SDM for significant deviation from random performance provides a good alternative. We assessed the minimum number of records required to obtain good model performance for species of different prevalence classes in a virtual study area and in a real African study area. The lower limit depends on the species’ prevalence with absolute minimum sample sizes as low as 3 for narrow‐ranged and 13 for widespread species for our virtual study area which represents an ideal, balanced, orthogonal world. The lower limit of 3, however, is flawed by statistical artefacts related to modelling species with a prevalence below 0.1. In our African study area lower limits are higher, ranging from 14 for narrow‐ranged to 25 for widespread species. We advocate identifying the minimum sample size for any species distribution modelling by applying the novel method presented here, which is applicable to any taxonomic clade or group, study area or climate scenario.     

A general method for dealing with misclassification in regression: the misclassification SIMEX

We have developed a new general approach for handling misclassification in discrete covariates or responses in regression models. The simulation and extrapolation (SIMEX) method, which was originally designed for handling additive covariate measurement error, is applied to the case of misclassification. The statistical model for characterizing misclassification is given by the transition matrix Pi from the true to the observed variable. We exploit the relationship between the size of misclassification and bias in estimating the parameters of interest. Assuming that Pi is known or can be estimated from validation data, we simulate data with higher misclassification and extrapolate back to the case of no misclassification. We show that our method is quite general and applicable to models with misclassified response and/or misclassified discrete regressors. In the case of a binary response with misclassification, we compare our method to the approach of Neuhaus, and to the matrix method of Morrissey and Spiegelman in the case of a misclassified binary regressor. We apply our method to a study on caries with a misclassified longitudinal response.     

Costs and Effectiveness of Treatment Alternatives for Proximal Caries Lesions

Objectives

Invasive therapy of proximal caries lesions initiates a cascade of re-treatment cycles with increasing loss of dental hard tissue. Non- and micro-invasive treatment aim at delaying this cascade and may thus reduce both the health and economic burden of such lesions. This study compared the costs and effectiveness of alternative treatments of proximal caries lesions.

Methods

A Markov-process model was used to simulate the events following the treatment of a proximal posterior lesion (E2/D1) in a 20-year-old patient in Germany. We compared three interventions (non-invasive; micro-invasive using resin infiltration; invasive using composite restoration). We calculated the risk of complications of initial and possible follow-up treatments and modelled time-dependent non-linear transition probabilities. Costs were calculated based on item-fee catalogues in Germany. Monte-Carlo-microsimulations were performed to compare cost-effectiveness of non- versus micro-invasive treatment and to analyse lifetime costs of all three treatments.

Results

Micro-invasive treatment was both more costly and more effective than non-invasive therapy, with ceiling-value-thresholds for willingness-to-pay between 16.73 € for E2 and 1.57 € for D1 lesions. Invasive treatment was the most costly strategy. Calculated costs and effectiveness were sensitive to lesion stage, patient’s age, discounting rate and assumed initial treatment costs.

Conclusions

Non- and micro-invasive treatments have lower long-term costs than invasive therapy of proximal lesions. Micro-invasive therapy had the highest cost-effectiveness for treating D1 lesions in young patients. Decision makers with a willingness-to-pay over 16.73 € and 1.57 € for E2 and D1 lesions, respectively, will find micro-invasive treatment more cost-effective than non-invasive therapy.     

Estimating the diets of animals using stable isotopes and a comprehensive Bayesian mixing model

Using stable isotope mixing models (SIMMs) as a tool to investigate the foraging ecology of animals is gaining popularity among researchers. As a result, statistical methods are rapidly evolving and numerous models have been produced to estimate the diets of animals--each with their benefits and their limitations. Deciding which SIMM to use is contingent on factors such as the consumer of interest, its food sources, sample size, the familiarity a user has with a particular framework for statistical analysis, or the level of inference the researcher desires to make (e.g., population- or individual-level). In this paper, we provide a review of commonly used SIMM models and describe a comprehensive SIMM that includes all features commonly used in SIMM analysis and two new features. We used data collected in Yosemite National Park to demonstrate IsotopeR's ability to estimate dietary parameters. We then examined the importance of each feature in the model and compared our results to inferences from commonly used SIMMs. IsotopeR's user interface (in R) will provide researchers a user-friendly tool for SIMM analysis. The model is also applicable for use in paleontology, archaeology, and forensic studies as well as estimating pollution inputs.