A Correlated Probit Model for Joint Modeling of Clustered Binary and Continuous Responses

Several discrete responses, such as health status, reproduction performance and meat quality, are routinely collected for several livestock species. These traits are often of binary or discrete nature. Genetic evaluation for these traits is frequently conducted using a single-trait threshold model, or they are considered continuous responses either in univariate or in multivariate context. Implementation of threshold models in the presence of several binary responses or a mixture of binary and continuous responses is far from simple. The complexity of such implementation is primarily due to the incomplete randomness of the residual (co)variance matrix. In the current study, a multiple binary trait simulation was carried out in order to implement and validate a new procedure for dealing with the consequences of the restrictions imposed to the residual variance using threshold models. Using three and eight binary responses, the proposed method was able to estimate all unknown parameters without any noticeable bias. In fact, for simulated residual correlations ranging from −0.8 to 0.8, the resulting HPD 95% intervals included the true values in all cases. The proposed procedure involved limited additional computational cost and is straightforward to implement independent of the number of binary responses involved in the analysis. Monitoring of the convergence of the procedure must be conducted at the identifiable scale, and special care must be placed on the selection of the prior of the non-identifiable model. The latter could have serious consequences on the final results due to potential truncation of the parameter space.

In this article, we illustrate ways in which generalizability theory (G-theory) can be used with continuous latent response variables (CLRVs) to address problems of scale coarseness resulting from categorization errors caused by representing ranges of continuous variables by discrete data points and transformation errors caused by unequal interval widths between those data points. The mechanism to address these problems is applying structural equation modeling (SEM) as a tool in deriving variance components needed to estimate indices of score consistency and validity. Illustrations include quantification of multiple sources of measurement error, use of non-nested and nested designs, derivation of indices of consistency for norm- and criterion-referenced interpretation of scores, estimation of effects when changing measurement procedures and designs, and disattenuation of correlation coefficients for measurement error. These illustrations underscore the effectiveness of G-theory with continuous latent response variables in providing stable indices of reliability and validity that are reasonably independent of the number of original scale points used, unevenness of scale intervals, and average degree of item skewness. We discuss general distinctions in reliability estimation within G-theory, SEM, and classical test theory; make specific recommendations for using G-theory on raw score and CLRV metrics; and provide computer code in an online supplement for doing all key analyses demonstrated in the article using R and Mplus. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

SUMMARY Regression models containing fixed and random effects may have a response variable which is not normally distributed. The generalized mixed model includes both discrete and continuous response variables and is developed here for problems in which the regression variables enter linearly into the model. Best linear unbiased predictor methods are extended to maximum likelihood and residual maximum likelihood estimation procedures. Applications in modelling discrete response variables and in survival analysis are discussed.

Dependence in multivariate binary outcomes in longitudinal data is a challenging and an important issue to address. Numerous studies have been performed to test the dependence in binary responses either using conditional or marginal probability models. Since the con- ditional and marginal approach provide inadequate or misleading results, the joint models based on both are implemented for bivariate correlated binary responses. In the current paper, we consider a joint modeling approach and a generalized linear model (GLM) for tri-variate correlated binary responses. The link function of the GLM is used to test the dependence of response variables. The mobility index with two categories, no difficulty and difficulty, over the duration of three waves of Health and Retirement Survey (HRS) is chosen as the binary response variable. Initial analysis with Marshall-Olkin correlation coefficients and logistic regression coefficients provide moderate correlation in mobility indices implying dependence in the response variables. We also found statistically significant dependence among the response variables using the joint modeling approach. The mobility at current wave not only depends on the previous mobility status, but also depends on covariates such as age, gender, and race.

A crucial step in designing a new study is to estimate the required sample size. For a design involving cluster sampling, the appropriate sample size depends on the so-called design effect, which is a function of the average cluster size and the intracluster correlation coefficient (ICC). It is well-known that under the framework of hierarchical and generalized linear models, a reduction in residual error may be achieved by including risk factors as covariates. In this paper we show that the covariate design, indicating whether the covariates are measured at the cluster level or at the within-cluster subject level affects the estimation of the ICC, and hence the design effect. Therefore, the distinction between these two types of covariates should be made at the design stage. In this paper we use the nested-bootstrap method to assess the accuracy of the estimated ICC for continuous and binary response variables under different covariate structures. The codes of two SAS macros are made available by the authors for interested readers to facilitate the construction of confidence intervals for the ICC. Moreover, using Monte Carlo simulations we evaluate the relative efficiency of the estimators and evaluate the accuracy of the coverage probabilities of a 95% confidence interval on the population ICC. The methodology is illustrated using a published data set of blood pressure measurements taken on family members.

Analyzing angle crashes at unsignalized intersections using machine learning techniques

Generalized linear models are a general class of regressionlike models for continuous and categorical response variables. Signal detection models can be formulated as a subclass of generalized linear models, and the result is a rich class of signal detection models based on different underlying distributions. An example is a signal detection model based on the extreme value distribution. The extreme value model is shown to yield unit slope receiver operating characteristic (ROC) curves for several classic data sets that are commonly given as examples of normal or logistic ROC curves with slopes that differ from unity. The result is an additive model with a simple interpretation in terms of a shift in the location of an underlying distribution. The models can also be extended in several ways, such as to recognize response dependencies, to include random coefficients, or to allow for more general underlying probability distributions.

Optimal designs for mixed continuous and binary responses with quantitative and qualitative factors

We propose a copula-based joint modeling framework for mixed longitudinal responses. Our approach permits all model parameters to vary with time, and thus will enable researchers to reveal dynamic response-predictor relationships and response-response associations. We call the new class of models TIMECOP because we model dependence using a time-varying copula. We develop a one-step estimation procedure for the TIMECOP parameter vector, and also describe how to estimate standard errors. We investigate the finite sample performance of our procedure via three simulation studies, one of which shows that our procedure performs well under ignorable missingness. We also illustrate the applicability of our approach by analyzing binary and continuous responses from the Women's Interagency HIV Study and a smoking cessation program.

This article proposes a general parametric item response theory approach for identifying sources of misfit in response patterns that have been classified as potentially inconsistent by a global person-fit index. The approach, which is based on the weighted least squared regression of the observed responses on the model-expected responses, can be used with a variety of unidimensional and multidimensional models intended for binary, graded, and continuous responses and consists of procedures for identifying (a) general deviation trends, (b) local inconsistencies, and (c) single response inconsistencies. A free program called REG-PERFIT that implements most of the proposed techniques has been developed, described, and made available for interested researchers. Finally, the functioning and usefulness of the proposed procedures is illustrated with an empirical study based on a statistics-anxiety scale.

Quantifying ecological thresholds from response surfaces

The authors present polynomial-based Newton algorithms for maximum likelihood estimation (MLE) of the parameters of multiple exponential signals in noise. This formulation can be used in the estimation, for example, of the directions of arrival of multiple noise-corrupted narrowband plane waves using uniform linear arrays and the frequencies of multiple noise-corrupted complex sine waves. The algorithms offer rapid convergence and exhibit the computation efficiency associated with the polynomial approach. Compact, closed-form expressions are presented for the gradients and Hessians. Various model assumptions concerning the statistics of the underlying signals are considered. Numerical simulations are presented to demonstrate the algorithms' performance.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

It is known that when the maximum likelihood estimator (MLE) algorithm passes a certain point, it produces images that begin to deteriorate. We propose a quantitative criterion with a simple probabilistic interpretation that allows the user to stop the algorithm just before this effect begins. The MLE algorithm searches for the image that has the maximum probability to generate the projection data. The underlying assumption of the algorithm is a Poisson distribution of the data. Therefore, the best image, according to the MLE algorithm, is the one that results in projection means which are as close to the data as possible. It is shown that this goal conflicts with the assumption that the data are Poisson-distributed. We test a statistical hypothesis whereby the projection data could have been generated by the image produced after each iteration. The acceptance or rejection of the hypothesis is based on a parameter that decreases as the images improve and increases as they deteriorate. We show that the best MLE images, which pass the test, result in somewhat lower noise in regions of high activity than the filtered back-projection results and much improved images in low activity regions. The applicability of the proposed stopping rule to other iterative schemes is discussed.

2 - Growth Modeling with Binary Responses

Experiments and observational studies often involve gathering information on several response variables, enabling us to model their dependence on observable predictor variables. Despite associations between the response variables, they are often analysed separately using general and generalized linear models. This paper investigates applications of multivariate regression analysis to improve the accuracy of predictions and decisions, in the specific context of diagnosing arterial stenoses in human legs. Two basic models are developed for this application, using (i) four binary responses and (ii) a mixture of two binary and two normal responses. The results clearly demonstrate the potential advantages offered by this approach.