The impact of omitting the interaction between crossed factors in cross‐classified random effects modelling

Cross‐classified random effects modelling (CCREM) is a special case of multi‐level modelling where the units of one level are nested within two cross‐classified factors. Typically, CCREM analyses omit the random interaction effect of the cross‐classified factors. We investigate the impact of the omission of the interaction effect on parameter estimates and standard errors. Results from a Monte Carlo simulation study indicate that, for fixed effects, both coefficients estimates and accompanied standard error estimates are not biased. For random effects, results are affected at level 2 but not at level 1 by the presence of an interaction variance and/or a correlation between the residual of level two factors. Results from the analysis of the Early Childhood Longitudinal Study and the National Educational Longitudinal Study agree with those obtained from simulated data. We recommend that researchers attempt to include interaction effects of cross‐classified factors in their models.     

Bias in longitudinal data analysis with missing data using typical linear mixed‐effects modelling and pattern‐mixture approach: An analytical illustration

We analytically derive the fixed‐effects estimates in unconditional linear growth curve models by typical linear mixed‐effects modelling (TLME) and by a pattern‐mixture (PM) approach with random‐slope‐dependent two‐missing‐pattern missing not at random (MNAR) longitudinal data. Results showed that when the missingness mechanism is random‐slope‐dependent MNAR, TLME estimates of both the mean intercept and mean slope are biased because of incorrect weights used in the estimation. More specifically, the estimate of the mean slope is biased towards the mean slope for completers, whereas the estimate of the mean intercept is biased towards the opposite direction as compared to the estimate of the mean slope. We also discuss why the PM approach can provide unbiased fixed‐effects estimates for random‐coefficients‐dependent MNAR data but does not work well for missing at random or outcome‐dependent MNAR data. A small simulation study was conducted to illustrate the results and to compare results from TLME and PM. Results from an empirical data analysis showed that the conceptual finding can be generalized to other real conditions even when some assumptions for the analytical derivation cannot be met. Implications from the analytical and empirical results were discussed and sensitivity analysis was suggested for longitudinal data analysis with missing data.     

Developing an Output‐Oriented Super Slacks‐Based Measure Model with an Application to Third‐Party Reverse Logistics Providers

Outsourcing is an increasingly significant topic pursued via corporations seeking enhanced efficiency. Third‐party reverse logistics involves the employ of external firms to carry out some or all of the firm's logistics activities. Output‐oriented super slacks‐based measure (SBM) model is one of the models in data envelopment analysis (DEA). In many real‐world applications, data are often stochastic. A successful approach to address uncertainty in data is to replace deterministic data via random variables, leading to chance‐constrained DEA. In this paper, a chance‐constrained output‐oriented super SBM model is developed and also its deterministic equivalent, which is a nonlinear program, is derived. Furthermore, it is shown that the deterministic equivalent of the stochastic output‐oriented super SBM model can be converted into a quadratic program. In addition, sensitivity analysis of the stochastic output‐oriented super SBM model is discussed with respect to changes on parameters. Finally, a numerical example demonstrates the application of the proposed model. Copyright © 2011 John Wiley & Sons, Ltd.     

Multilevel Dynamic Generalized Structured Component Analysis for Brain Connectivity Analysis in Functional Neuroimaging Data

We extend dynamic generalized structured component analysis (GSCA) to enhance its data-analytic capability in structural equation modeling of multi-subject time series data. Time series data of multiple subjects are typically hierarchically structured, where time points are nested within subjects who are in turn nested within a group. The proposed approach, named multilevel dynamic GSCA, accommodates the nested structure in time series data. Explicitly taking the nested structure into account, the proposed method allows investigating subject-wise variability of the loadings and path coefficients by looking at the variance estimates of the corresponding random effects, as well as fixed loadings between observed and latent variables and fixed path coefficients between latent variables. We demonstrate the effectiveness of the proposed approach by applying the method to the multi-subject functional neuroimaging data for brain connectivity analysis, where time series data-level measurements are nested within subjects.     

Generalized linear models with ordinally‐observed covariates

An ordinally‐observed variable is a variable that is only partially observed through an ordinal surrogate. Although statistical models for ordinally‐observed response variables are well known, relatively little attention has been given to the problem of ordinally‐observed regressors. In this paper I show that if surrogates to ordinally‐observed covariates are used as regressors in a generalized linear model then the resulting measurement error in the covariates can compromise the consistency of point estimators and standard errors for the effects of fully‐observed regressors. To properly account for this measurement error when making inferences concerning the fully‐observed regressors, I propose a general modelling framework for generalized linear models with ordinally‐observed covariates. I discuss issues of model specification, identification, and estimation, and illustrate these with examples.     

Generalized and specific components of prejudice: The decomposition of intergroup context effects

Although different types of prejudice tend to be highly correlated, target‐specific and more generalized components can nevertheless be distinguished. Here, we analyze whether indicators of the intergroup context—threat, contact, and neighborhood composition—predict the target‐specific and/or generalized components of prejudice. Using data from the New Zealand Attitudes and Values Study (N = 4629), we build a multilevel model that captures the relationship between social dominance orientation, general levels of neighborhood heterogeneity, symbolic and realistic threat and cross‐group friendship (averaged across target groups), and generalized prejudice. Our model simultaneously estimates the relationship between target‐specific levels of these intergroup context indicators and target‐specific prejudice. Results indicated that social dominance orientation remained the strongest predictor of generalized prejudice when adjusting for other variables and that indicators of the intergroup context primarily explain differences between target group ratings. Aggregate levels of cross‐group friendship also had a small effect on generalized prejudice.     

Examining the role of positive and negative metacognitive beliefs in depression

Many psychological models have been developed to explain the development and maintenance of depression. The most widely evaluated model is the cognitive model of depression, and it is against this model that emerging models should be compared. Accordingly, this cross‐sectional study examined whether metacognitive beliefs, as specified in the metacognitive model of depression, would explain additional variance in depressive symptoms over dysfunctional attitudes; the core feature of the cognitive model. Moreover, mediational relationships between metacognitive beliefs, rumination, and depressive symptoms, predicted by the metacognitive model were also explored, whilst controlling for dysfunctional attitudes. A sample of 715 students completed self‐report questionnaires measuring depressive symptoms, rumination, dysfunctional attitudes, and metacognitive beliefs. Regression analyses showed that metacognitive beliefs made a significant statistical contribution to depressive symptoms, after controlling for age, gender, rumination and dysfunctional attitudes. Furthermore, as predicted by the metacognitive model, the relationship between positive metacognitive beliefs and depressive symptoms was fully mediated by rumination, whilst the relationship between negative metacognitive beliefs about uncontrollability and danger and depressive symptoms was partially mediated by rumination. The results provide further empirical support for the metacognitive model of depression and indicate that positive and negative metacognitive beliefs play an integral role in the maintenance of depressive symptoms.     

Do People Order Cues by Retrieval Fluency when Making Probabilistic Inferences?

In multiple‐cue probabilistic inferences, people infer alternatives' unknown values on decision criteria, using alternatives' attributes as cues. Some inferential strategies, like take‐the‐best, assume that people consider relevant cues sequentially in order of decreasing validity. This assumption has been deemed cognitively implausible by some, who suggest memory retrieval principles to guide cue order. We test whether memory‐based inferences are better described by a model considering cues in order of validity or in order of memory retrieval. In an experiment, we manipulated the frequency with which cues appeared in a learning phase, increasing retrieval fluency of cue values related to the more frequently appearing cue. In a subsequent decision phase, participants made a series of two‐alternative decisions based on the learned cue values. We compared two sequential sampling models, which differed in whether cues are sampled in order of subjective cue validity or in order of retrieval fluency. To model retrieval order of cues in the fluency sampling model, we used the declarative memory theory embedded in the ACT‐R cognitive architecture. Most participants' decisions were best described by the model sampling cues in order of memory retrieval. Only a minority of participants were classified as sampling cues by validity. Our result suggests that retrieval fluency is the primary driver of cue order in inferences from memory, irrespective of the cues' validities. Copyright © 2017 John Wiley & Sons, Ltd.     

Influence analysis of ranking data

This paper develops diagnostic measures to identify those observations in Thurstonian models for ranking data which unduly influence parameter estimates that are obtained by the partition maximum likelihood approach of Chan and Bentler (1998). Diagnostic measures are constructed by employing the local influence approach that uses geometric techniques to assess the effect of small perturbations on a postulated statistical model. Very little additional effort is required to compute the proposed diagnostic measures, because all of the necessary building blocks are readily available after a usual fit of the model. The work described in this paper was partially supported by the grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (RGC Ref. No. CUHK4186/98P and RGC Direct Grant ID2060178). The authors are grateful to the Editor and four anonymous referees for their helpful comments.     

Multiple layers of information processing in deductive reasoning: combining dual strategy and dual-source approaches to reasoning

The idea that inferential performance cannot be analyzed within a single model has been suggested within two theoretical contexts. The dual strategy model suggests that people reason using different approaches to processing statistical information. The dual-source model suggests that people reason probabilistically using both statistical information and some intuition about logical form. Each model suggests that people have different approaches to processing information while making inferences. The following studies examined whether these different forms of information processing were equally present during reasoning. Participants were given a series of problems designed to distinguish counterexample from statistical reasoners. They were then given a series of MP or AC inferences for which identical statistical information was provided. Results show that MP inferences were considered to be deductively valid more often than equivalent AC inferences. The effect of logical form was independent of reasoning strategy, and of relatively equivalent size for both counterexample and statistical reasoners. The second study examined explicitly probabilistic inferences, and showed smaller effects of logical form and of reasoning strategy, although with a complex set of interactions. These results show that understanding the way that people use information when making inferences requires a multidimensional approach.     

Structural Equation Modeling Approaches for Analyzing Partially Nested Data

Study designs involving clustering in some study arms, but not all study arms, are common in clinical treatment-outcome and educational settings. For instance, in a treatment arm, persons may be nested in therapy groups, whereas in a control arm there are no groups. Methodological approaches for handling such partially nested designs have recently been developed in a multilevel modeling framework (MLM-PN) and have proved very useful. We introduce two alternative structural equation modeling (SEM) approaches for analyzing partially nested data: a multivariate single-level SEM (SSEM-PN) and a multiple-arm multilevel SEM (MSEM-PN). We show how SSEM-PN and MSEM-PN can produce results equivalent to existing MLM-PNs and can be extended to flexibly accommodate several modeling features that are difficult or impossible to handle in MLM-PNs. For instance, using an SSEM-PN or MSEM-PN, it is possible to specify complex structural models involving cluster-level outcomes, obtain absolute model fit, decompose person-level predictor effects in the treatment arm using latent cluster means, and include traditional factors as predictors/outcomes. Importantly, implementation of such features for partially nested designs differs from that for fully nested designs. An empirical example involving a partially nested depression intervention combines several of these features in an analysis of interest for treatment-outcome studies.     

Dual imputation model for incomplete longitudinal data

Missing values are a practical issue in the analysis of longitudinal data. Multiple imputation (MI) is a well‐known likelihood‐based method that has optimal properties in terms of efficiency and consistency if the imputation model is correctly specified. Doubly robust (DR) weighing‐based methods protect against misspecification bias if one of the models, but not necessarily both, for the data or the mechanism leading to missing data is correct. We propose a new imputation method that captures the simplicity of MI and protection from the DR method. This method integrates MI and DR to protect against misspecification of the imputation model under a missing at random assumption. Our method avoids analytical complications of missing data particularly in multivariate settings, and is easy to implement in standard statistical packages. Moreover, the proposed method works very well with an intermittent pattern of missingness when other DR methods can not be used. Simulation experiments show that the proposed approach achieves improved performance when one of the models is correct. The method is applied to data from the fireworks disaster study, a randomized clinical trial comparing therapies in disaster‐exposed children. We conclude that the new method increases the robustness of imputations.     

Conditional mixed models with crossed random effects

The analysis of continuous hierarchical data such as repeated measures or data from meta‐analyses can be carried out by means of the linear mixed‐effects model. However, in some situations this model, in its standard form, does pose computational problems. For example, when dealing with crossed random‐effects models, the estimation of the variance components becomes a non‐trivial task if only one observation is available for each cross‐classified level. Pseudolikelihood ideas have been used in the context of binary data with standard generalized linear multilevel models. However, even in this case the problem of the estimation of the variance remains non‐trivial. In this paper, we first propose a method to fit a crossed random‐effects model with two levels and continuous outcomes, borrowing ideas from conditional linear mixed‐effects model theory. We also propose a crossed random‐effects model for binary data combining ideas of conditional logistic regression with pseudolikelihood estimation. We apply this method to a case study with data coming from the field of psychometrics and study a series of items (responses) crossed with participants. A simulation study assesses the operational characteristics of the method.     

Bayesian Analysis of Nonlinear Structural Equation Models with Nonignorable Missing Data

A Bayesian approach is developed for analyzing nonlinear structural equation models with nonignorable missing data. The nonignorable missingness mechanism is specified by a logistic regression model. A hybrid algorithm that combines the Gibbs sampler and the Metropolis–Hastings algorithm is used to produce the joint Bayesian estimates of structural parameters, latent variables, parameters in the nonignorable missing model, as well as their standard errors estimates. A goodness-of-fit statistic for assessing the plausibility of the posited nonlinear structural equation model is introduced, and a procedure for computing the Bayes factor for model comparison is developed via path sampling. Results obtained with respect to different missing data models, and different prior inputs are compared via simulation studies. In particular, it is shown that in the presence of nonignorable missing data, results obtained by the proposed method with a nonignorable missing data model are significantly better than those that are obtained under the missing at random assumption. A real example is presented to illustrate the newly developed Bayesian methodologies. This research is fully supported by a grant (CUHK 4243/03H) from the Research Grant Council of the Hong Kong Special Administration Region. The authors are thankful to the editor and reviewers for valuable comments for improving the paper, and also to ICPSR and the relevant funding agency for allowing the use of the data. Requests for reprints should be sent to Professor S.Y. Lee, Department of Statistics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong.     

Applying mixed‐effects modeling to single‐subject designs: An introduction

Behavior analysis and statistical inference have shared a conflicted relationship for over fifty years. However, a significant portion of this conflict is directed toward statistical tests (e.g., t‐tests, ANOVA) that aggregate group and/or temporal variability into means and standard deviations and as a result remove much of the data important to behavior analysts. Mixed‐effects modeling, a more recently developed statistical test, addresses many of the limitations of more basic tests by incorporating random effects. Random effects quantify individual subject variability without eliminating it from the model, hence producing a model that can predict both group and individual behavior. We present the results of a generalized linear mixed‐effects model applied to single‐subject data taken from Ackerlund Brandt, Dozier, Juanico, Laudont, & Mick, 2015, in which children chose from one of three reinforcers for completing a task. Results of the mixed‐effects modeling are consistent with visual analyses and importantly provide a statistical framework to predict individual behavior without requiring aggregation. We conclude by discussing the implications of these results and provide recommendations for further integration of mixed‐effects models in the analyses of single‐subject designs.     

Residual Normality Assumption and the Estimation of Multiple Membership Random Effects Models

While conventional hierarchical linear modeling is applicable to purely hierarchical data, a multiple membership random effects model (MMrem) is appropriate for nonpurely nested data wherein some lower-level units manifest mobility across higher-level units. Although a few recent studies have investigated the influence of cluster-level residual nonnormality on hierarchical linear modeling estimation for purely hierarchical data, no research has examined the statistical performance of an MMrem given residual non-normality. The purpose of the present study was to extend prior research on the influence of residual non-normality from purely nested data structures to multiple membership data structures. Employing a Monte Carlo simulation study, this research inquiry examined two-level MMrem parameter estimate biases and inferential errors. Simulation factors included the level-two residual distribution, sample sizes, intracluster correlation coefficient, and mobility rate. Results showed that estimates of fixed effect parameters and the level-one variance component were robust to level-two residual non-normality. The level-two variance component, however, was sensitive to level-two residual non-normality and sample size. Coverage rates of the 95% credible intervals deviated from the nominal value assumed when level-two residuals were non-normal. These findings can be useful in the application of an MMrem to account for the contextual effects of multiple higher-level units.     

Burnout among Finnish and Chinese university students

In this study the levels of experienced burnout of Finnish and Chinese university students are compared using School Burnout Inventory (SBI). This study is motivated by earlier studies, which suggest that the level of student burnout is different in the culturally distinct Finnish and Chinese university systems, but which are based on different research instruments for the two groups. The sample studied consisted of 3,035 Finnish students and 2,309 Chinese students. Because of the cross‐cultural nature of this study the level of structural equivalence of SBI between the cultural groups was examined and the effect of different response styles on the results was taken into account. Both standard and robust statistical methods were used for the analyses. The results showed that SBI with two extracted components is suitable for cross‐cultural analysis between Finnish and Chinese university students. Virtually no difference was found in experienced overall burnout between the Finnish and Chinese students, which means that both university systems contain factors causing similar levels of student burnout. This study also verified that controlling for the response styles is important in cross‐cultural studies as it was found to have a distinct effect on the results obtained from mean‐level comparisons.     

MCMC estimation for the p2 network regression model with crossed random effects

The p₂ model is a statistical model for the analysis of binary relational data with covariates, as occur in social network studies. It can be characterized as a multinomial regression model with crossed random effects that reflect actor heterogeneity and dependence between the ties from and to the same actor in the network. Three Markov chain Monte Carlo (MCMC) estimation methods for the p₂ model are presented to improve iterative generalized least squares (IGLS) estimation developed earlier, two of which use random walk proposals. The third method, an independence chain sampler, and one of the random walk algorithms use normal approximations of the binary network data to generate proposals in the MCMC algorithms. A large‐scale simulation study compares MCMC estimates with IGLS estimates for networks with 20 and 40 actors. It was found that the IGLS estimates have a smaller variance but are severely biased, while the MCMC estimates have a larger variance with a small bias. For networks with 20 actors, mean squared errors are generally comparable or smaller for the IGLS estimates. For networks with 40 actors, mean squared errors are the smallest for the MCMC estimates. Coverage rates of confidence intervals are good for the MCMC estimates but not for the IGLS estimates.