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This article considers the identification conditions of confirmatory factor analysis (CFA) models for ordered categorical outcomes with invariance of different types of parameters across groups. The current practice of invariance testing is to first identify a model with only configural invariance and then test the invariance of parameters based on this identified baseline model. This approach is not optimal because different identification conditions on this baseline model identify the scales of latent continuous responses in different ways. Once an invariance condition is imposed on a parameter, these identification conditions may become restrictions and define statistically non-equivalent models, leading to different conclusions. By analyzing the transformation that leaves the model-implied probabilities of response patterns unchanged, we give identification conditions for models with invariance of different types of parameters without referring to a specific parametrization of the baseline model. Tests based on this approach have the advantage that they do not depend on the specific identification condition chosen for the baseline model.

To evaluate model fit in confirmatory factor analysis, researchers compare goodness-of-fit indices (GOFs) against fixed cutoff values (e.g., CFI > .950) derived from simulation studies. Methodologists have cautioned that cutoffs for GOFs are only valid for settings similar to the simulation scenarios from which cutoffs originated. Despite these warnings, fixed cutoffs for popular GOFs (i.e., χ 2 , χ 2 / df , CFI, RMSEA, SRMR) continue to be widely used in applied research. We (1) argue that the practice of using fixed cutoffs needs to be abandoned and (2) review time-honored and emerging alternatives to fixed cutoffs. We first present the most in-depth simulation study to date on the sensitivity of GOFs to model misspecification (i.e., misspecified factor dimensionality and unmodeled cross-loadings) and their susceptibility to further data and analysis characteristics (i.e., estimator, number of indicators, number and distribution of response options, loading magnitude, sample size, and factor correlation). We included all characteristics identified as influential in previous studies. Our simulation enabled us to replicate well-known influences on GOFs and establish hitherto unknown or underappreciated ones. In particular, the magnitude of the factor correlation turned out to moderate the effects of several characteristics on GOFs. Second, to address these problems, we discuss several strategies for assessing model fit that take the dependency of GOFs on the modeling context into account. We highlight tailored (or “dynamic”) cutoffs as a way forward. We provide convenient tables with scenario-specific cutoffs as well as regression formulae to predict cutoffs tailored to the empirical setting of interest.

The high mountain regions of Asia contain more glacial ice than anywhere on the planet outside of the polar regions. Because of the large population living in the Indus watershed region who are reliant on melt from these glaciers for fresh water, understanding the factors that affect glacial melt along with the impacts of climate change on the region is important for managing these natural resources. While there are multiple climate data products (e.g., reanalysis and global climate models) available to study the impact of climate change on this region, each product will have a different amount of skill in projecting a given climate variable, such as precipitation. In this research, we develop a spatially varying mixture model to compare the distribution of precipitation in the High Mountain Asia region as produced by climate models with the corresponding distribution from in situ observations from the Asian Precipitation—Highly Resolved Observational Data Integration Towards Evaluation (APHRODITE) data product. Parameter estimation is carried out via a computationally efficient Markov chain Monte Carlo algorithm. Each of the estimated climate distributions from each climate data product is then validated against APHRODITE using a spatially varying Kullback–Leibler divergence measure. Supplementary materials accompanying this paper appear online.

Increasingly, industrial experiments use multistratum designs, such as split-plot and strip-plot designs. Often, these experiments span more than one processing stage. The challenge is to identify an appropriate multistratum design, along with an appropriate statistical model. In this article, we introduce Hasse diagrams in the response surface context as a tool to visualize the unit structure of the experimental design, the randomization and sampling approaches used, the stratum in which each experimental factor is applied, and the degrees of freedom available in each stratum to estimate main effects, interactions, and variance components. We illustrate their use on several responses measured in a large study of the adhesion properties of coatings to polypropylene. We discuss quantitative, binary, and ordered categorical responses, for designs ranging from a simple split-plot to a strip-plot that involves repeated measurements of the response. The datasets discussed in this article are available online as supplementary materials, along with sample SAS programs.

In this paper, ordered categorical variables are used to compare between linear and nonlinear interactions of fixed covariate and latent variables Bayesian structural equation models. Gibbs sampling method is applied for estimation and model comparison. Hidden continuous normal distribution (censored normal distribution) is used to handle the problem of ordered categorical data. Statistical inferences, which involve estimation of parameters and their standard deviations, and residuals analyses for testing the selected model, are discussed. The proposed procedure is illustrated by a simulation data obtained from R program. Analysis are done by using OpenBUGS program.

In this paper, we summarize some recent developments in the analysis of nonparametric models where the classical models of ANOVA are generalized in such a way that not only the assumption of normality is relaxed but also the structure of the designs is introduced in a broader framework and also the concept of treatment effects is redefined. The continuity of the distribution functions is not assumed so that not only data from continuous distributions but also data with ties are included in this general setup. In designs with independent observations as well as in repeated measures designs, the hypotheses are formulated by means of the distribution functions. The main results are given in a unified form. Some applications to special designs are considered, where in simple designs, some well known statistics (such as the Kruskal-Wallis statistic and the χ2-statistic for dichotomous data) come out as special cases. The general framework presented here enables the nonparametric analysis of data with continuous distribution functions as well as arbitrary discrete data such as count data, ordered categorical and dichotomous data.

Though ubiquitous, Likert scaling's traditional mode of analysis is often unable to uncover all of the valid information in a data set. Here, the authors discuss a solution to this problem based on methodology developed by quantum physicists: the state multipole method. The authors demonstrate the relative ease and value of this method by examining college students' endorsement of one possible cause of prejudice: segregation. Though the mean level of students' endorsement did not differ among ethnic groups, an examination of state multipoles showed that African Americans had a level of polarization in their endorsement that was not reflected by Hispanics or European Americans. This result could not have been obtained with the traditional approach and demonstrates the new method's utility for social science research.

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.

Currently, no general methods have been developed to relate pharmacologically based models, such as indirect response models, to discrete or ordered categorical data. We propose the use of an unobservable latent variable (LV), through which indirect response models can be linked with drug exposure. The resulting indirect latent variable response model (ILVRM) is demonstrated using a case study of a JAK3 inhibitor, which was administered to patients in a rheumatoid arthritis (RA) study. The clinical endpoint for signs and symptoms in RA is the American College of Rheumatology response criterion of 20%—a binary response variable. In this case study, four exposure-response models, which have different pharmacological interpretations, were constructed and fitted using the ILVRM method. Specifically, two indirect response models, an effect compartment model, and a model which assumes instantaneous (direct) drug action were assessed and compared for their ability to predict the response data. In general, different model interpretations can influence drug inference, such as time to drug effect onset, as well as affect extrapolations of responses to untested experimental conditions, and the underlying pharmacology that operates to generate key response features does not change because the response was measured discretely. Consideration of these model interpretations can impact future study designs and ultimately provide greater insight into drug development strategies.

This study developed and validated the Turkish version of the Self-Directed Online Learning Scale (SDOLS-T) for assessing students’ perceptions of their self-directed learning (SDL) ability in an online environment. Specifically, this study conducted in two stages multiple categorical confirmatory factor analyses factoring in the ordered categorical structure of the SDOLS-T data. The data in this study came from a parent study which utilized the SDOLS-T and other instruments for data collection. From among the three competing models the literature recommends examining to explain the shared variance of items in a survey, the results at stage 1 showed that the correlated, two-factor structure, originally proposed for the SDOLS, was also the best-fit model for the SDOLS-T. At stage 2, using the best-fit model from stage 1, measurement invariance analyses were conducted to examine the extent to which SDL under the SDOLS-T was understood and measured equivalently across the groups specified by four dichotomous demographic variables: gender, network connection, online learning experience, and grade. The stage 2 results indicate the SDOLS-T reached scalar invariance at least for gender and network connection, thus allowing the comparison of latent or manifest means, or any other scores (e.g., total scores, Rasch scores), across the groups by these two demographic variables. In the end, the findings support the SDOLS-T for use in facilitating educational practice (e.g., improving instructional design), advancing scholarly literature (e.g., investigating SDL measurement and content area issues), and informing policy/decision-making (e.g., increasing retention rates and reducing dropout) in online education in Turkey.