Two-sample models with monotonic likelihood ratios for ordinal regression

Signal detection experiments with human observers frequently generate ordinal data which are evaluated using receiver operating characteristic (ROC) methods. These methods may include regressing a continuous two-distribution model to the data set. Because we assume that human observers will not systematically select observations absent the signal, this model should have a monotonic likelihood ratio between the distributions. This paper gives a general method for constructing pairs of distributions that have monotonic likelihood ratios and a possibly large number flexible parameters. It suggests two specific simple parametric forms of monotonic likelihood ratios, constructs new distributions with those likelihood ratios using other standard distributions, and performs ordinal regression with those new distributions to model some example data from the literature.     

Predictors of driving outcomes in advancing age

This study aimed to develop predictive models for real-life driving outcomes in older drivers. Demographics, driving history, on-road driving errors, and performance on visual, motor, and neuropsychological test scores at baseline were assessed in 100 older drivers (ages 65-89 years [72.7]). These variables were used to predict time to driving cessation, first moving violation, or crash. Using Cox proportional hazards regression models, significant individual predictors for driving cessation were greater age and poorer scores on Near Visual Acuity, Contrast Sensitivity, Useful Field of View, Judgment of Line Orientation, Trail Making Test-Part A, Benton Visual Retention Test, Grooved Pegboard, and a composite index of overall cognitive ability. Greater weekly mileage, higher education, and "serious" on-road errors predicted moving violations. Poorer scores from Trail Making Test-Part B or Trail Making Test (B-A) and serious on-road errors predicted crashes. Multivariate models using "off-road" predictors revealed (a) age and Contrast Sensitivity as best predictors for driving cessation; (b) education, weekly mileage, and Auditory Verbal Learning Task-Recall for moving violations; and (c) education, number of crashes over the past year, Auditory Verbal Learning Task-Recall, and Trail Making Test (B-A) for crashes. Diminished visual, motor, and cognitive abilities in older drivers can be easily and noninvasively monitored with standardized off-road tests, and performances on these measures predict involvement in motor vehicle crashes and driving cessation, even in the absence of a neurological disorder. (PsycINFO Database Record (c) 2012 APA, all rights reserved).     

Gay adolescents and suicide: understanding the association

Suicide among gay adolescents is a major public health concern, but it is still under-emphasized within mainstream healthcare. This article brings together information garnered from past studies, articles, and experiences to create a more in-depth understanding of why gay adolescents are at a higher risk for suicide. By creating a better understanding, this article attempts to push forward the significance of this issue and encourages physicians to take a more active role in suicide prevention. The article initially explores why issues involving homosexuality are seldom discussed and provides evidence for an increased risk in suicide among gay adolescents. By addressing major psychosocial issues faced by gay adolescents, it spends the majority of time facilitating an understanding of the increased risk. In this article "gay" refers to homosexual males and females, and bisexuals unless otherwise indicated.     

Modelling monotonic effects of ordinal predictors in Bayesian regression models

Ordinal predictors are commonly used in regression models. They are often incorrectly treated as either nominal or metric, thus under- or overestimating the information contained. Such practices may lead to worse inference and predictions compared to methods which are specifically designed for this purpose. We propose a new method for modelling ordinal predictors that applies in situations in which it is reasonable to assume their effects to be monotonic. The parameterization of such monotonic effects is realized in terms of a scale parameter b representing the direction and size of the effect and a simplex parameter modelling the normalized differences between categories. This ensures that predictions increase or decrease monotonically, while changes between adjacent categories may vary across categories. This formulation generalizes to interaction terms as well as multilevel structures. Monotonic effects may be applied not only to ordinal predictors, but also to other discrete variables for which a monotonic relationship is plausible. In simulation studies we show that the model is well calibrated and, if there is monotonicity present, exhibits predictive performance similar to or even better than other approaches designed to handle ordinal predictors. Using Stan, we developed a Bayesian estimation method for monotonic effects which allows us to incorporate prior information and to check the assumption of monotonicity. We have implemented this method in the R package brms, so that fitting monotonic effects in a fully Bayesian framework is now straightforward.     

Evaluation of dominance‐based ordinal multiple regression for variables with few categories

Dominance‐based ordinal multiple regression (DOR) is designed to answer ordinal questions about relationships among ordinal variables. Only one parameter per predictor is estimated, and the number of parameters is constant for any number of outcome levels. The majority of existing simulation evaluations of DOR use predictors that are continuous or ordinal with many categories, so the performance of the method is not well understood for ordinal variables with few categories. This research evaluates DOR in simulations using three‐category ordinal variables for the outcome and predictors, with a comparison to the cumulative logits proportional odds model (POC). Although ordinary least squares (OLS) regression is inapplicable for theoretical reasons, it was also included in the simulations because of its popularity in the social sciences. Most simulation outcomes indicated that DOR performs well for variables with few categories, and is preferable to the POC for smaller samples and when the proportional odds assumption is violated. Nevertheless, confidence interval coverage for DOR was not flawless and possibilities for improvement are suggested.     

Proximate outcomes of gatekeeper training for suicide prevention in the workplace

In this pilot study we report on proximate outcomes of a 1-hour community gatekeeper training in-service for 76 nonclinical employees in a university hospital workplace setting. Pre-post analyses resulted in positive changes in participants' knowledge about suicide and attitudes (self-efficacy) about intervening with suicidal individuals. A subset of participants engaged in role play practice of gatekeeper skills following training and rated the experience positively. Fifty-five observations were rated using an observational measure developed for this study and approximately half of these demonstrated satisfactory skills post training. Participants in this workplace gatekeeper training reported sharing new knowledge and skills with family, friends, and coworkers.     

On the use of heterogeneous thresholds ordinal regression models to account for individual differences in response style

This paper proposes a general approach to accounting for individual differences in the extreme response style in statistical models for ordered response categories. This approach uses a hierarchical ordinal regression modeling framework with heterogeneous thresholds structures to account for individual differences in the response style. Markov chain Monte Carlo algorithms for Bayesian inference for models with heterogeneous thresholds structures are discussed in detail. A simulation and two examples based on ordinal probit models are given to illustrate the proposed methodology. The simulation and examples also demonstrate that failing to account for individual differences in the extreme response style can have adverse consequences for statistical inferences.The author is grateful to Ulf Böckenholt, an associate editor, and three anonymous reviewers for helpful comments, and Kristine Kuhn and Kshiti Joshi for providing the data.     

Beyond quantity: Individual differences in working memory and the ordinal understanding of numerical symbols

In two different contexts, we examined the hypothesis that individual differences in working memory (WM) capacity are related to the tendency to infer complex, ordinal relationships between numerical symbols. In Experiment 1, we assessed whether this tendency arises in a learning context that involves mapping novel symbols to quantities by training adult participants to associate dot-quantities with novel symbols, the overall relative order of which had to be inferred. Performance was best for participants who were higher in WM capacity (HWMs). HWMs also learned ordinal information about the symbols that lower WM individuals (LWMs) did not. In Experiment 2, we examined whether WM relates to performance when participants are explicitly instructed to make numerical order judgments about highly enculturated numerical symbols by having participants indicate whether sets of three Arabic numerals were in increasing order. All participants responded faster when sequential sets (3-4-5) were in order than when they were not. However, only HWMs responded faster when non-sequential, patterned sets (1-3-5) were in order, suggesting they were accessing ordinal associations that LWMs were not. Taken together, these experiments indicate that WM capacity plays a key role in extending symbolic number representations beyond their quantity referents to include symbol-symbol ordinal associations, both in a learning context and in terms of explicitly accessing ordinal relationships in highly enculturated stimuli.     

CDAscal: An algorithm for assessing the correspondence of one or more vectors to a symmetric matrix using ordinal regression

An algorithm for assessing the correspondence of one or more attribute rating variables to a symmetric matrix of dissimilarities is presented. The algorithm is useful as an alternative to fitting property variables into a multidimensional scaling space. Rather than requiring a two step process of first deriving a multidimensional space and then fitting variables individually into the space, the algorithm directly assesses the correspondence of each variable to the symmetric matrix and permits a regression extension such that a set of variables can be considered simultaneously. The relation between the matrix and the variables is determined by evaluating pairs of pairs relations, . Though the algorithm requires only ordinal assumptions, the correspondence may be computed intervally also. Multiple ordinal regression is performed with the values derived from the matrix serving as the dependent variable and those derived from the attribute ratings serving as the independent variables. Standard multiple regression statistics forR-square,F, andt are calculated as well as measures of ordinal association between the vectors and the matrix.     

Fitting multilevel models with ordinal outcomes: performance of alternative specifications and methods of estimation

Previous research has compared methods of estimation for fitting multilevel models to binary data, but there are reasons to believe that the results will not always generalize to the ordinal case. This article thus evaluates (a) whether and when fitting multilevel linear models to ordinal outcome data is justified and (b) which estimator to employ when instead fitting multilevel cumulative logit models to ordinal data, maximum likelihood (ML), or penalized quasi-likelihood (PQL). ML and PQL are compared across variations in sample size, magnitude of variance components, number of outcome categories, and distribution shape. Fitting a multilevel linear model to ordinal outcomes is shown to be inferior in virtually all circumstances. PQL performance improves markedly with the number of ordinal categories, regardless of distribution shape. In contrast to binary data, PQL often performs as well as ML when used with ordinal data. Further, the performance of PQL is typically superior to ML when the data include a small to moderate number of clusters (i.e., ≤ 50 clusters).     

Coefficients of feasibility for ordinal response scales

Cronbach's alpha is systematically used in the social sciences to estimate internal consistency. However, this coefficient assumes the continuity of the variables and this assumption is not met by ordinal response items or Likert scales. This work shows two alternatives to the alpha coefficient, the ordinal alpha and beta coefficients. Both of them take into account the ordinal nature of the data. The new coefficients were applied to several scales and compared with the traditional estimation of alpha. This paper also shows how to estimate those coefficients.     

A primer for advancing psychology in the public-sector

Psychologists working in the public sector should, in addition to providing direct services, advocate for systems change. Although many consumers treated in the public sector face a constellation of severe life problems, working to improve the system of care is more difficult than providing treatment. Improving the quality of life of consumers of publicly funded mental health services requires that psychologists become advocates. Four prerequisites to systems change, plus coalition building, legislative advocacy, the work of state psychological associations, and forming alliances between psychologists and nonpsychological community organizations such as Rotary International, are described. In conclusion, 12 orienting ideas are listed for psychologists who want to advocate for social, institutional, and political change.     

Confidence intervals for gamma-family measures of ordinal association

This research focused on confidence intervals (CIs) for 10 measures of monotonic association between ordinal variables. Standard errors (SEs) were also reviewed because more than 1 formula was available per index. For 5 indices, an element of the formula used to compute an SE is given that is apparently new. CIs computed with different SEs were compared in simulations with small samples (N = 25, 50, 75, or 100) for variables with 4 or 5 categories. With N > 25, many CIs performed well. Performance was best for consistent CIs due to N. Cliff and colleagues (N. Cliff, 1996; N. Cliff & V. Charlin, 1991; J. D. Long & N. Cliff, 1997). CIs for Spearman's rank correlation were also examined: Parameter coverage was erratic and sometimes egregiously underestimated.     

An information theoretic measure for the evaluation of ordinal scale data

This article describes a new measure of dispersion as an indication of consensus and dissention. Building on the generally accepted Shannon entropy, this measure utilizes a probability distribution and the ordered ranking of categories in an ordinal scale distribution to yield a value confined to the unit interval. Unlike other measures that need to be normalized, this measure is always in the interval 0 to 1. The measure is typically applied to the Likert scale to determine degrees of agreement among ordinal-ranked categories when one is dealing with data collection and analysis, although other scales are possible. Using this measure, investigators can easily determine the proximity of ordinal data to consensus (agreement) or dissention. Consensus and dissention are defined relative to the degree of proximity of values constituting a frequency distribution on the ordinal scale measure. The authors identify a set of criteria that a measure must satisfy in order to be an acceptable indicator of consensus and show how the consensus measure satisfies all the criteria.