Factor retention in ordered categorical variables: Benefits and costs of polychoric correlations in eigenvalue-based testing.

An essential step in exploratory factor analysis is to determine the optimal number of factors. The Next Eigenvalue Sufficiency Test (NEST; Achim, 2017) is a recent proposal to determine the number of factors based on significance tests of the statistical contributions of candidate factors indicated by eigenvalues of sample correlation matrices. Previous simulation studies have shown NEST to recover the optimal number of factors in simulated datasets with high accuracy. However, these studies have focused on continuous variables. The present work addresses the performance of NEST for ordinal data. It has been debated whether factor models - and thus also the optimal number of factors - for ordinal variables should be computed for Pearson correlation matrices, which are known to underestimate correlations for ordinal datasets, or for polychoric correlation matrices, which are known to be instable. The central research question is to what extent the problems associated with Pearson correlations and polychoric correlations deteriorate NEST for ordinal datasets. Implementations of NEST tailored to ordinal datasets by utilizing polychoric correlations are proposed. In a simulation, the proposed implementations were compared to the original implementation of NEST which computes Pearson correlations even for ordinal datasets. The simulation shows that substituting polychoric correlations for Pearson correlations improves the accuracy of NEST for binary variables and large sample sizes (N = 500). However, the simulation also shows that the original implementation using Pearson correlations was the most accurate implementation for Likert-type variables with four response categories when item difficulties were homogeneous.

Ordered Regression Models: a Tutorial.

Ordinal outcomes are common in the social, behavioral, and health sciences, but there is no commonly accepted approach to analyzing them. Researchers make a number of different seemingly arbitrary recoding decisions implying different levels of measurement and theoretical assumptions. As a result, a wide array of models are used to analyze ordinal outcomes, including the linear regression model, binary response model, ordered models, and count models. In this tutorial, we present a diverse set of ordered models (most of which are under-utilized in applied research) and argue that researchers should approach the analysis of ordinal outcomes in a more systematic fashion by taking into consideration both theoretical and empirical concerns, and prioritizing ordered models given the flexibility they provide. Additionally, we consider the challenges that ordinal independent variables pose for analysts that often go unnoticed in the literature and offer simple ways to decide how to include ordinal independent variables in ordered regression models in ways that are easier to justify on conceptual and empirical grounds. We illustrate several ordered regression models with an empirical example, general self-rated health, and conclude with recommendations for building a sounder approach to ordinal data analysis.

Response to Fluvoxamine in the Obsessive-Compulsive Disorder Patients: Bayesian Ordinal Quantile Regression.

BACKGROUND: Obsessive-Compulsive Disorder (OCD) is a chronic neuropsychiatric disorder associated with unpleasant thoughts or mental images, making the patient repeat physical or mental behaviors to relieve discomfort. 40-60% of patients do not respond to Serotonin Reuptake Inhibitors, including fluvoxamine therapy. INTRODUCTION: The aim of the study is to identify the predictors of fluvoxamine therapy in OCD patients by Bayesian Ordinal Quantile Regression Model. METHODS: This study was performed on 109 patients with OCD. Three methods, including Bayesian ordinal quantile, probit, and logistic regression models, were applied to identify predictors of response to fluvoxamine. The accuracy and weighted kappa were used to evaluate these models. RESULTS: Our result showed that rs3780413 (mean=-0.69, sd=0.39) and cleaning dimension (mean=-0.61, sd=0.20) had reverse effects on response to fluvoxamine therapy in Bayesian ordinal probit and logistic regression models. In the 75th quantile regression model, marital status (mean=1.62, sd=0.47) and family history (mean=1.33, sd=0.61) had a direct effect, and cleaning (mean=-1.10, sd=0.37) and somatic (mean=-0.58, sd=0.27) dimensions had reverse effects on response to fluvoxamine therapy. CONCLUSION: Response to fluvoxamine is a multifactorial problem and can be different in the levels of socio-demographic, genetic, and clinical predictors. Marital status, familial history, cleaning, and somatic dimensions are associated with response to fluvoxamine therapy.

A unified model-implied instrumental variable approach for structural equation modeling with mixed variables.

The model-implied instrumental variable (MIIV) estimator is an equation-by-equation estimator of structural equation models that is more robust to structural misspecifications than full information estimators. Previous studies have concentrated on endogenous variables that are all continuous (MIIV-2SLS) or all ordinal . We develop a unified MIIV approach that applies to a mixture of binary, ordinal, censored, or continuous endogenous observed variables. We include estimates of factor loadings, regression coefficients, variances, and covariances along with their asymptotic standard errors. In addition, we create new goodness of fit tests of the model and overidentification tests of single equations. Our simulation study shows that the proposed MIIV approach is more robust to structural misspecifications than diagonally weighted least squares (DWLS) and that both the goodness of fit model tests and the overidentification equations tests can detect structural misspecifications. We also find that the bias in asymptotic standard errors for the MIIV estimators of factor loadings and regression coefficients are often lower than the DWLS ones, though the differences are small in large samples. Our analysis shows that scaling indicators with low reliability can adversely affect the MIIV estimators. Also, using a small subset of MIIVs reduces small sample bias of coefficient estimates, but can lower the power of overidentification tests of equations.

How to use χ2 test correctly——CMH χ2 tests for the data collected from the three kinds of R×C contingency tables / 四川精神卫生

The purpose of this paper was to introduce the CMH χ2 test and SAS software implementation of the three kinds of R×C contingency table data. The first type was called “two-way unordered R×C contingency table data”. The CMH χ2 test corresponding to this type of data was essentially the Pearson’s χ2 test. The second type was called “R×C contingency table data with an ordinal outcome variable”. The CMH χ2 test corresponding to this kind of data was essentially a rank sum test. The third type was called “R×C contingency table data which was of two ordinal variables with different attributes”. The CMH χ2 test corresponding to the data was essentially Pearson’s correlation analysis or Spearman’s rank correlation analysis. When there were 1 or 2 “ordinal variables” in the R×C contingency table data， it was necessary to “assign or score” the ordinal variables before performing statistical analysis. In the FREQ procedure of SAS/STAT， there were four scoring methods. With different scoring approach， both the expression form and the calculation results of CMH χ2 test statistics could change accordingly.

An Instrumental Variable Estimator for Mixed Indicators: Analytic Derivatives and Alternative Parameterizations.

Methodological development of the model-implied instrumental variable (MIIV) estimation framework has proved fruitful over the last three decades. Major milestones include Bollen's (Psychometrika 61(1):109-121, 1996) original development of the MIIV estimator and its robustness properties for continuous endogenous variable SEMs, the extension of the MIIV estimator to ordered categorical endogenous variables (Bollen and Maydeu-Olivares in Psychometrika 72(3):309, 2007), and the introduction of a generalized method of moments estimator (Bollen et al., in Psychometrika 79(1):20-50, 2014). This paper furthers these developments by making several unique contributions not present in the prior literature: (1) we use matrix calculus to derive the analytic derivatives of the PIV estimator, (2) we extend the PIV estimator to apply to any mixture of binary, ordinal, and continuous variables, (3) we generalize the PIV model to include intercepts and means, (4) we devise a method to input known threshold values for ordinal observed variables, and (5) we enable a general parameterization that permits the estimation of means, variances, and covariances of the underlying variables to use as input into a SEM analysis with PIV. An empirical example illustrates a mixture of continuous variables and ordinal variables with fixed thresholds. We also include a simulation study to compare the performance of this novel estimator to WLSMV.

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 ς 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.

A permutation method for detecting trend correlations in rare variant association studies.

In recent years, there has been an increasing interest in detecting disease-related rare variants in sequencing studies. Numerous studies have shown that common variants can only explain a small proportion of the phenotypic variance for complex diseases. More and more evidence suggests that some of this missing heritability can be explained by rare variants. Considering the importance of rare variants, researchers have proposed a considerable number of methods for identifying the rare variants associated with complex diseases. Extensive research has been carried out on testing the association between rare variants and dichotomous, continuous or ordinal traits. So far, however, there has been little discussion about the case in which both genotypes and phenotypes are ordinal variables. This paper introduces a method based on the γ-statistic, called OV-RV, for examining disease-related rare variants when both genotypes and phenotypes are ordinal. At present, little is known about the asymptotic distribution of the γ-statistic when conducting association analyses for rare variants. One advantage of OV-RV is that it provides a robust estimation of the distribution of the γ-statistic by employing the permutation approach proposed by Fisher. We also perform extensive simulations to investigate the numerical performance of OV-RV under various model settings. The simulation results reveal that OV-RV is valid and efficient; namely, it controls the type I error approximately at the pre-specified significance level and achieves greater power at the same significance level. We also apply OV-RV for rare variant association studies of diastolic blood pressure.

Stochastic approximation EM for large-scale exploratory IRT factor analysis.

A stochastic approximation EM algorithm (SAEM) is described for exploratory factor analysis of dichotomous or ordinal variables. The factor structure is obtained from sufficient statistics that are updated during iterations with the Robbins-Monro procedure. Two large-scale simulations are reported that compare accuracy and CPU time of the proposed SAEM algorithm to the Metropolis-Hasting Robbins-Monro procedure and to a generalized least squares analysis of the polychoric correlation matrix. A smaller-scale application to real data is also reported, including a method for obtaining standard errors of rotated factor loadings. A simulation study based on the real data analysis is conducted to study bias and error estimates. The SAEM factor algorithm requires minimal lines of code, no derivatives, and no large-matrix inversion. It is programmed entirely in R.

Specific agreement on ordinal and multiple nominal outcomes can be calculated for more than two raters.

OBJECTIVE: The concept of specific agreement (SA) has been proposed for dichotomous outcomes for two or more raters. We aim to extend this concept for variables with more than two ordinal or nominal categories and more than two raters. STUDY DESIGN AND SETTING: We used two data sets: four plastic surgeons classifying photographs after breast reconstruction on a 5-point ordinal scale and six raters classifying psychiatric patients into five diagnostic categories. For m raters, all (i.e., m(m-1)/2) pairwise agreement tables were summed to calculate the observed agreement (OA), SA and conditional probabilities. The 95% confidence intervals were obtained by bootstrapping. RESULTS: SA was calculated for each ordinal or nominal category to examine when one of the raters scored in a specific category, what is the probability that the other raters scored in that same category. And suppose one of the raters scored X1, what is the probability that the other raters scored X1 or any of the other categories (conditional probability). It appeared, for example, that among the psychiatric disorders, depression and personality disorders were often mixed up, whereas neurosis was rarely mixed up with schizophrenia. CONCLUSION: The concept of SA for variables with ordinal and multiple nominal categories provides relevant clinical information. The extension to conditional probabilities of alternative categories broadens the clinical application with examining which categories are most often mixed up.

Parametric Dose Standardization for Optimizing Two-Agent Combinations in a Phase I-II Trial with Ordinal Outcomes.

A Bayesian model and design are described for a phase I-II trial to jointly optimise the doses of a targeted agent and a chemotherapy agent for solid tumors. A challenge in designing the trial was that both the efficacy and toxicity outcomes were defined as four-level ordinal variables. To reflect possibly complex joint effects of the two doses on each of the two outcomes, for each marginal distribution a generalised continuation ratio model was assumed, with each agent's dose parametrically standardised in the linear term. A copula was assumed to obtain a bivariate distribution. Elicited outcome probabilities were used to construct a prior, with variances calibrated to obtain small prior effective sample size. Elicited numerical utilities of the 16 elementary outcomes were used to compute posterior mean utilities as criteria for selecting dose pairs, with adaptive randomisation to reduce the risk of getting stuck at a suboptimal pair. A simulation study showed that parametric dose standardisation with additive dose effects provides a robust, reliable model for dose pair optimisation in this setting, and it compares favourably with designs based on alternative models that include dose-dose interaction terms. The proposed model and method are applicable generally to other clinical trial settings with similar dose and outcome structures.

Assessing the inter-rater agreement for ordinal data through weighted indexes.

Assessing the inter-rater agreement between observers, in the case of ordinal variables, is an important issue in both the statistical theory and biomedical applications. Typically, this problem has been dealt with the use of Cohen's weighted kappa, which is a modification of the original kappa statistic, proposed for nominal variables in the case of two observers. Fleiss (1971) put forth a generalization of kappa in the case of multiple observers, but both Cohen's and Fleiss' kappa could have a paradoxical behavior, which may lead to a difficult interpretation of their magnitude. In this paper, a modification of Fleiss' kappa, not affected by paradoxes, is proposed, and subsequently generalized to the case of ordinal variables. Monte Carlo simulations are used both to testing statistical hypotheses and to calculating percentile and bootstrap-t confidence intervals based on this statistic. The normal asymptotic distribution of the proposed statistic is demonstrated. Our results are applied to the classical Holmquist et al.'s (1967) dataset on the classification, by multiple observers, of carcinoma in situ of the uterine cervix. Finally, we generalize the use of s* to a bivariate case.

Simulation-based study comparing multiple imputation methods for non-monotone missing ordinal data in longitudinal settings.

The application of multiple imputation (MI) techniques as a preliminary step to handle missing values in data analysis is well established. The MI method can be classified into two broad classes, the joint modeling and the fully conditional specification approaches. Their relative performance for the longitudinal ordinal data setting under the missing at random (MAR) assumption is not well documented. This article intends to fill this gap by conducting a large simulation study on the estimation of the parameters of a longitudinal proportional odds model. The two MI methods are also illustrated in quality of life data from a cancer clinical trial.

A variety of variables.

In designing studies and developing plans for analyses, we must consider which tests are appropriate for the types of variables we are using. Here I describe the types of variables available to us, and I briefly consider the appropriate tools to use in their analysis.