Latent Factor Regression Research Articles

An integrative latent class model of heterogeneous data modalities for diagnosing kidney obstruction.

Radionuclide imaging plays a critical role in the diagnosis and management of kidney obstruction. However, most practicing radiologists in US hospitals have insufficient time and resources to acquire training and experience needed to interpret radionuclide images, leading to increased diagnostic errors. To tackle this problem, Emory University embarked on a study that aims to develop a computer-assisted diagnostic (CAD) tool for kidney obstruction by mining and analyzing patient data comprised of renogram curves, ordinal expert ratings on the obstruction status, pharmacokinetic variables, and demographic information. The major challenges here are the heterogeneity in data modes and the lack of gold standard for determining kidney obstruction. In this article, we develop a statistically principled CAD tool based on an integrative latent class model that leverages heterogeneous data modalities available for each patient to provide accurate prediction of kidney obstruction. Our integrative model consists of three sub-models (multilevel functional latent factor regression model, probit scalar-on-function regression model, and Gaussian mixture model), each of which is tailored to the specific data mode and depends on the unknown obstruction status (latent class). An efficient MCMC algorithm is developed to train the model and predict kidney obstruction with associated uncertainty. Extensive simulations are conducted to evaluate the performance of the proposed method. An application to an Emory renal study demonstrates the usefulness of our model as a CAD tool for kidney obstruction.

High-Dimensional Mediation Analysis: A New Method Applied to Maternal Smoking, Placental DNA Methylation, and Birth Outcomes.

High-dimensional mediation analysis is an extension of unidimensional mediation analysis that includes multiple mediators, and increasingly it is being used to evaluate the indirect omics-layer effects of environmental exposures on health outcomes. Analyses involving high-dimensional mediators raise several statistical issues. Although many methods have recently been developed, no consensus has been reached about the optimal combination of approaches to high-dimensional mediation analyses. We developed and validated a method for high-dimensional mediation analysis (HDMAX2) and applied it to evaluate the causal role of placental DNA methylation in the pathway between exposure to maternal smoking (MS) during pregnancy and gestational age (GA) and birth weight of the baby at birth. HDMAX2 combines latent factor regression models for epigenome-wide association studies with tests for mediation and considers CpGs and aggregated mediator regions (AMRs). HDMAX2 was carefully evaluated using simulated data and compared to state-of-the-art multidimensional epigenetic mediation methods. Then, HDMAX2 was applied to data from 470 women of the Etude des Déterminants pré et postnatals du développement de la santé de l'Enfant (EDEN) cohort. HDMAX2 demonstrated increased power in comparison with state-of-the-art multidimensional mediation methods and identified several AMRs not identified in previous mediation analyses of exposure to MS on birth weight and GA. The results provided evidence for a polygenic architecture of the mediation pathway with a posterior estimate of the overall indirect effect of CpGs and AMRs equal to lower birth weight representing 32.1% of the total effect [standard deviation ]. HDMAX2 also identified AMRs having simultaneous effects both on GA and on birth weight. Among the top hits of both GA and birth weight analyses, regions located in COASY, BLCAP, and ESRP2 also mediated the relationship between GA and birth weight, suggesting reverse causality in the relationship between GA and the methylome. HDMAX2 outperformed existing approaches and revealed an unsuspected complexity of the potential causal relationships between exposure to MS and birth weight at the epigenome-wide level. HDMAX2 is applicable to a wide range of tissues and omic layers. https://doi.org/10.1289/EHP11559.

Are Latent Factor Regression and Sparse Regression Adequate?

We propose the Factor Augmented (sparse linear) Regression Model (FARM) that not only admits both the latent factor regression and sparse linear regression as special cases but also bridges dimension reduction and sparse regression together. We provide theoretical guarantees for the estimation of our model under the existence of sub-Gaussian and heavy-tailed noises (with bounded ( 1 + ϑ ) th moment, for all ϑ > 0 ), respectively. In addition, the existing works on supervised learning often assume the latent factor regression or sparse linear regression is the true underlying model without justifying its adequacy. To fill in such an important gap on high-dimensional inference, we also leverage our model as the alternative model to test the sufficiency of the latent factor regression and the sparse linear regression models. To accomplish these goals, we propose the Factor-Adjusted deBiased Test (FabTest) and a two-stage ANOVA type test, respectively. We also conduct large-scale numerical experiments including both synthetic and FRED macroeconomics data to corroborate the theoretical properties of our methods. Numerical results illustrate the robustness and effectiveness of our model against latent factor regression and sparse linear regression models. Supplementary materials for this article are available online.

A two-stage latent factor regression method to model the common and unique effects of multiple highly correlated exposure variables

In many epidemiological and environmental health studies, developing an accurate exposure assessment of multiple exposures on a health outcome is often of interest. However, the problem is challenging in the presence of multicollinearity, which can lead to biased estimates of regression coefficients and inflated variance estimators. Selecting one exposure variable as a surrogate of multiple highly correlated exposure variables is often suggested in the literature as a solution to handle the multicollinearity problem. However, this may lead to loss of information, since the exposure variables that are highly correlated tend to have not only common but also additional effects on the outcome variable. In this study, a two-stage latent factor regression method is proposed. The key idea is to regress the dependent variable not only on the common latent factor(s) of the explanatory variables, but also on the residuals terms from the factor analysis as the explanatory variables. The proposed method is compared to the traditional latent factor regression and principal component regression for their performance of handling multicollinearity. Two case studies are presented. Simulation studies are performed to assess their performances in terms of the epidemiological interpretation and stability of parameter estimates.

HDMAX2: A framework for High Dimensional Mediation Analysis with application to maternal smoking, placental DNA methylation and birth outcomes

Background High-dimensional mediation analysis is an extension of unidimensional mediation analysis that includes multiple mediators, and is increasingly used in environmental epidemiology to evaluate the indirect epigenetic effects of environmental exposures on health outcomes. However, analyses involving high-dimensional mediators raise several statistical issues. While many methods have recently been developed to tackle those issues, no consensus has been reached about the optimal combination of approaches. Methods We developed HDMAX2, a new multi-step approach to mediation that combines latent factor regression models for epigenome-wide association studies with max-squared tests for mediation, and considers CpGs and aggregated mediator regions (AMR). HDMAX2 was carefully evaluated on simulated data, and compared to state-of-the- art multi-dimensional epigenetic mediation methods. Then it was applied to assess the indirect effects of exposure to maternal smoking (MS) on term birth weight (BW) and gestational age (GA) at delivery in a study of 470 women from the EDEN cohort. Results HDMAX2 resulted in increased power compared to state-of-the-art multi-dimensional mediation methods, and identified several AMRs not identified in previous mediation analyses of exposure to MS on BW and GA. The results provided evidence for a polygenic architecture of the causal pathway with an overall indirect effect of CpGs and AMRs of 44.5g lower BW (32.1% of the total effect size). HDMAX2 also identified AMRs having simultaneous effects both on GA and BW. Among the top hits of both GA and BW analyses, regions located in COASY, BLCAP and ESRP2 also mediated the relationship between GA on BW, suggesting a reverse causality in the relationship between GA and the methylome. Discussion This study brought up several statistical improvements of high-dimensional mediation analyses, which revealed an unsuspected complexity of the causal relationships between exposure to MS and BW at the epigenome-wide level. Mediation; high dimension; causal inference; epigenetics; DNA methylation; pregnancy

High Dimensional Multiomics Reveals Unique Characteristics of Early Plasma Administration in Polytrauma Patients With TBI.

The authors sought to identify causal factors that explain the selective benefit of prehospital administration of thawed plasma (TP) in traumatic brain injury (TBI) patients using mediation analysis of a multiomic database. The Prehospital Air Medical Plasma (PAMPer) Trial showed that patients with TBI and a pronounced systemic response to injury [defined as endotype 2 (E2)], have a survival benefit from prehospital administration of TP. An interrogation of high dimensional proteomics, lipidomics and metabolomics previously demonstrated unique patterns in circulating biomarkers in patients receiving prehospital TP, suggesting that a deeper analysis could reveal causal features specific to TBI patients. A novel proteomic database (SomaLogic Inc., aptamer-based assay, 7K platform) was generated using admission blood samples from a subset of patients (n=149) from the PAMPer Trial. This proteomic dataset was combined with previously reported metabolomic and lipidomic datasets from these same patients. A 2-step analysis was performed to identify factors that promote survival in E2-TBI patients who had received early TP. First, features were selected using both linear and multivariate-latent-factor regression analyses. Then, the selected features were entered into the causal mediation analysis. Causal mediation analysis of observable features identified 16 proteins and 41 lipids with a high proportion of mediated effect (>50%) to explain the survival benefit of early TP in E2-TBI patients. The multivariate latent-factor regression analyses also uncovered 5 latent clusters of features with a proportion effect >30%, many in common with the observable features. Among the observable and latent features were protease inhibitors known to inhibit activated protein C and block fibrinolysis (SERPINA5 and CPB2), a clotting factor (factor XI), as well as proteins involved in lipid transport and metabolism (APOE3 and sPLA(2)-XIIA). These findings suggest that severely injured patients with TBI process exogenous plasma differently than those without TBI. The beneficial effects of early TP in E2-TBI patients may be the result of improved blood clotting and the effect of brain protective factors independent of coagulation.

Inference in latent factor regression with clusterable features

Regression models, in which the observed features X∈Rp and the response Y∈R depend, jointly, on a lower dimensional, unobserved, latent vector Z∈RK, with K≪p, are popular in a large array of applications, and mainly used for predicting a response from correlated features. In contrast, methodology and theory for inference on the regression coefficient β∈RK relating Y to Z are scarce, since typically the un-observable factor Z is hard to interpret. Furthermore, the determination of the asymptotic variance of an estimator of β is a long-standing problem, with solutions known only in a few particular cases. To address some of these outstanding questions, we develop inferential tools for β in a class of factor regression models in which the observed features are signed mixtures of the latent factors. The model specifications are both practically desirable, in a large array of applications, render interpretability to the components of Z, and are sufficient for parameter identifiability. Without assuming that the number of latent factors K or the structure of the mixture is known in advance, we construct computationally efficient estimators of β, along with estimators of other important model parameters. We benchmark the rate of convergence of β by first establishing its ℓ2-norm minimax lower bound, and show that our proposed estimator βˆ is minimax-rate adaptive. Our main contribution is the provision of a unified analysis of the component-wise Gaussian asymptotic distribution of βˆ and, especially, the derivation of a closed form expression of its asymptotic variance, together with consistent variance estimators. The resulting inferential tools can be used when both K and p are independent of the sample size n, and also when both, or either, p and K vary with n, while allowing for p>n. This complements the only asymptotic normality results obtained for a particular case of the model under consideration, in the regime K=O(1) and p→∞, but without a variance estimate. As an application, we provide, within our model specifications, a statistical platform for inference in regression on latent cluster centers, thereby increasing the scope of our theoretical results. We benchmark the newly developed methodology on a recently collected data set for the study of the effectiveness of a new SIV vaccine. Our analysis enables the determination of the top latent antibody-centric mechanisms associated with the vaccine response.

Sparse latent factor regression models for genome-wide and epigenome-wide association studies.

Association of phenotypes or exposures with genomic and epigenomic data faces important statistical challenges. One of these challenges is to account for variation due to unobserved confounding factors, such as individual ancestry or cell-type composition in tissues. This issue can be addressed with penalized latent factor regression models, where penalties are introduced to cope with high dimension in the data. If a relatively small proportion of genomic or epigenomic markers correlate with the variable of interest, sparsity penalties may help to capture the relevant associations, but the improvement over non-sparse approaches has not been fully evaluated yet. Here, we present least-squares algorithms that jointly estimate effect sizes and confounding factors in sparse latent factor regression models. In simulated data, sparse latent factor regression models generally achieved higher statistical performance than other sparse methods, including the least absolute shrinkage and selection operator and a Bayesian sparse linear mixed model. In generative model simulations, statistical performance was slightly lower (while being comparable) to non-sparse methods, but in simulations based on empirical data, sparse latent factor regression models were more robust to departure from the model than the non-sparse approaches. We applied sparse latent factor regression models to a genome-wide association study of a flowering trait for the plant Arabidopsis thaliana and to an epigenome-wide association study of smoking status in pregnant women. For both applications, sparse latent factor regression models facilitated the estimation of non-null effect sizes while overcoming multiple testing issues. The results were not only consistent with previous discoveries, but they also pinpointed new genes with functional annotations relevant to each application.

Relationship between high-risk behavior and mental health problems using latent factor regression for grouped outcomes model

article: Relationship between high-risk behavior and mental health problems using latent factor regression for grouped outcomes model - Minerva Psychiatry 2021 June;62(2):86-93 - Minerva Medica - Journals

Genetic Parameters and Genotype-by-Environment Interactions in Regional Progeny Tests of Pinus taeda L. in the Southern USA

Abstract Genetic parameters were estimated using a five-series multienvironment trial of Pinus taeda L. in the southern USA. There were 324 half-sib families planted in five test series across 37 locations. A set of six variance/covariance matrices for the genotype-by-environment (G × E) effect for tree height and diameter were compared on the basis of model fit. In single-series analysis, extended factor analytical models provided generally superior model fit to simpler models for both traits; however, in the combined-series analysis, diameter was optimally modeled using simpler variance/covariance structures. A three-way compound term for modeling G × E interactions among and within series yielded substantial improvements in terms of model fit and standard errors of predictions. Heritability of family means ranged between 0.63 and 0.90 for both height and diameter. Average additive genetic correlations among sites were 0.70 and 0.61 for height and diameter, respectively, suggesting the presence of some G × E interaction. Pairs of sites with the lowest additive genetic correlations were located at opposite ends of the latitude range. Latent factor regression revealed a small number of parents with large factor scores that changed ranks significantly between southern and northern environments.

Heterogeneous Large Datasets Integration Using Bayesian Factor Regression

Two key challenges in modern statistical applications are the large amount of information recorded per individual, and that such data are often not collected all at once but in batches. These batch effects can be complex, causing distortions in both mean and variance. We propose a novel sparse latent factor regression model to integrate such heterogeneous data. The model provides a tool for data exploration via dimensionality reduction and sparse low-rank covariance estimation while correcting for a range of batch effects. We study the use of several sparse priors (local and non-local) to learn the dimension of the latent factors. We provide a flexible methodology for sparse factor regression which is not limited to data with batch effects. Our model is fitted in a deterministic fashion by means of an EM algorithm for which we derive closed-form updates, contributing a novel scalable algorithm for non-local priors of interest beyond the immediate scope of this paper. We present several examples, with a focus on bioinformatics applications. Our results show an increase in the accuracy of the dimensionality reduction, with non-local priors substantially improving the reconstruction of factor cardinality. The results of our analyses illustrate how failing to properly account for batch effects can result in unreliable inference. Our model provides a novel approach to latent factor regression that balances sparsity with sensitivity in scenarios both with and without batch effects and is highly computationally efficient.

A Bayesian Latent Class Model to Predict Kidney Obstruction in the Absence of Gold Standard

Kidney obstruction, if untreated in a timely manner, can lead to irreversible loss of renal function. A widely used technology for evaluations of kidneys with suspected obstruction is diuresis renography. However, it is generally very challenging for radiologists who typically interpret renography data in practice to build high level of competency due to the low volume of renography studies and insufficient training. Another challenge is that there is currently no gold standard for detection of kidney obstruction. Seeking to develop a computer-aided diagnostic (CAD) tool that can assist practicing radiologists to reduce errors in the interpretation of kidney obstruction, a recent study collected data from diuresis renography, interpretations on the renography data from highly experienced nuclear medicine experts as well as clinical data. To achieve the objective, we develop a statistical model that can be used as a CAD tool for assisting radiologists in kidney interpretation. We use a Bayesian latent class modeling approach for predicting kidney obstruction through the integrative analysis of time-series renogram data, expert ratings, and clinical variables. A nonparametric Bayesian latent factor regression approach is adopted for modeling renogram curves in which the coefficients of the basis functions are parameterized via the factor loadings dependent on the latent disease status and the extended latent factors that can also adjust for clinical variables. A hierarchical probit model is used for expert ratings, allowing for training with rating data from multiple experts while predicting with at most one expert, which makes the proposed model operable in practice. An efficient MCMC algorithm is developed to train the model and predict kidney obstruction with associated uncertainty. We demonstrate the superiority of the proposed method over several existing methods through extensive simulations. Analysis of the renal study also lends support to the usefulness of our model as a CAD tool to assist less experienced radiologists in the field. Supplementary materials for this article are available online.

Bayesian latent factor regression for multivariate functional data with variable selection

In biomedical research, multivariate functional data are frequently encountered. Majority of the existing approaches for functional data analysis focus on univariate functional data and the methodology for multivariate functional data is far less studied. Particularly, the problem of investigating covariate effects on multivariate functional data has received little attention. In this research, we propose a fully Bayesian latent factor regression for studying covariate effects on multivariate functional data. The proposed model obtains a low-dimensional representation of multivariate functional data through basis expansions for splines and factor analysis for the basis coefficients. Then, the latent factors specific to each functional outcome are regressed onto covariates accounting for residual correlations among multiple outcomes. The assessment of covariate effects is conducted based on the marginal inclusion probability for each covariate, which is calculated a posteriori by assigning a stochastic search variable selection (SSVS) prior to the regression coefficients. To better control for the false discovery rate, we propose a multivariate SSVS prior that allows for a set of coefficients to be zero simultaneously. We illustrate the proposed method through a simulation study and an application to the air pollution data collected for 13 cities in China.

Hierarchical infinite factor models for improving the prediction of surgical complications for geriatric patients

Nearly a third of all surgeries performed in the United States occur for patients over the age of 65; these older adults experience a higher rate of postoperative morbidity and mortality. To improve the care for these patients, we aim to identify and characterize high risk geriatric patients to send to a specialized perioperative clinic while leveraging the overall surgical population to improve learning. To this end, we develop a hierarchical infinite latent factor model (HIFM) to appropriately account for the covariance structure across subpopulations in data. We propose a novel Hierarchical Dirichlet Process shrinkage prior on the loadings matrix that flexibly captures the underlying structure of our data while sharing information across subpopulations to improve inference and prediction. The stick-breaking construction of the prior assumes an infinite number of factors and allows for each subpopulation to utilize different subsets of the factor space and select the number of factors needed to best explain the variation. We develop the model into a latent factor regression method that excels at prediction and inference of regression coefficients. Simulations validate this strong performance compared to baseline methods. We apply this work to the problem of predicting surgical complications using electronic health record data for geriatric patients and all surgical patients at Duke University Health System (DUHS). The motivating application demonstrates the improved predictive performance when using HIFM in both area under the ROC curve and area under the PR Curve while providing interpretable coefficients that may lead to actionable interventions.

Corrigendum to "Association of Stressful Life Events with Psychological Problems: A Large-Scale Community-Based Study Using Grouped Outcomes Latent Factor Regression with Latent Predictors".

[This corrects the article DOI: 10.1155/2017/3457103.].

Primary Factors Statistically Associated with Diarrheal Occurrences

Abstract To successfully prevent diarrheal pathogen transmission, a variety of causal pathways should be considered. This study utilized a suite of tools to identify primary factors associated with diarrheal occurrences in a set of communities in Para, Brazil that had received a biosand filter (BSF). First, existing Demographic and Health Survey data sets from the departments of Para and Amazonas, Brazil were analyzed using three statistical techniques, namely: Mahalanobis-Taguchi Strategy, canonical correlation analysis, and latent factor regression. Second, results of statistical analyses were combined with a literature review and field observations to locally adapt a previously validated structural equation model (SEM) originally developed for Quiche, Guatemala. Third, a randomized household survey was used to collect data—including water sources, sanitation facilities, hygiene practices, socioeconomic statuses, education levels, BSF maintenance, and diarrheal occurrences—in Para, Brazil and analyzed w...

Spatial Bayesian latent factor regression modeling of coordinate-based meta-analysis data.

Now over 20 years old, functional MRI (fMRI) has a large and growing literature that is best synthesised with meta-analytic tools. As most authors do not share image data, only the peak activation coordinates (foci) reported in the article are available for Coordinate-Based Meta-Analysis (CBMA). Neuroimaging meta-analysis is used to (i) identify areas of consistent activation; and (ii) build a predictive model of task type or cognitive process for new studies (reverse inference). To simultaneously address these aims, we propose a Bayesian point process hierarchical model for CBMA. We model the foci from each study as a doubly stochastic Poisson process, where the study-specific log intensity function is characterized as a linear combination of a high-dimensional basis set. A sparse representation of the intensities is guaranteed through latent factor modeling of the basis coefficients. Within our framework, it is also possible to account for the effect of study-level covariates (meta-regression), significantly expanding the capabilities of the current neuroimaging meta-analysis methods available. We apply our methodology to synthetic data and neuroimaging meta-analysis datasets.

Association of Stressful Life Events with Psychological Problems: A Large-Scale Community-Based Study Using Grouped Outcomes Latent Factor Regression with Latent Predictors.

Objective The current study is aimed at investigating the association between stressful life events and psychological problems in a large sample of Iranian adults. Method In a cross-sectional large-scale community-based study, 4763 Iranian adults, living in Isfahan, Iran, were investigated. Grouped outcomes latent factor regression on latent predictors was used for modeling the association of psychological problems (depression, anxiety, and psychological distress), measured by Hospital Anxiety and Depression Scale (HADS) and General Health Questionnaire (GHQ-12), as the grouped outcomes, and stressful life events, measured by a self-administered stressful life events (SLEs) questionnaire, as the latent predictors. Results The results showed that the personal stressors domain has significant positive association with psychological distress (β = 0.19), anxiety (β = 0.25), depression (β = 0.15), and their collective profile score (β = 0.20), with greater associations in females (β = 0.28) than in males (β = 0.13) (all P < 0.001). In addition, in the adjusted models, the regression coefficients for the association of social stressors domain and psychological problems profile score were 0.37, 0.35, and 0.46 in total sample, males, and females, respectively (P < 0.001). Conclusion Results of our study indicated that different stressors, particularly those socioeconomic related, have an effective impact on psychological problems. It is important to consider the social and cultural background of a population for managing the stressors as an effective approach for preventing and reducing the destructive burden of psychological problems.

Generalized reduced rank latent factor regression for high dimensional tensor fields, and neuroimaging-genetic applications

We propose a generalized reduced rank latent factor regression model (GRRLF) for the analysis of tensor field responses and high dimensional covariates. The model is motivated by the need from imaging-genetic studies to identify genetic variants that are associated with brain imaging phenotypes, often in the form of high dimensional tensor fields. GRRLF identifies from the structure in the data the effective dimensionality of the data, and then jointly performs dimension reduction of the covariates, dynamic identification of latent factors, and nonparametric estimation of both covariate and latent response fields. After accounting for the latent and covariate effects, GRLLF performs a nonparametric test on the remaining factor of interest. GRRLF provides a better factorization of the signals compared with common solutions, and is less susceptible to overfitting because it exploits the effective dimensionality. The generality and the flexibility of GRRLF also allow various statistical models to be handled in a unified framework and solutions can be efficiently computed. Within the field of neuroimaging, it improves the sensitivity for weak signals and is a promising alternative to existing approaches. The operation of the framework is demonstrated with both synthetic datasets and a real-world neuroimaging example in which the effects of a set of genes on the structure of the brain at the voxel level were measured, and the results compared favorably with those from existing approaches.

Editorial: Measurement Invariance.

Multi-item surveys are frequently used to study scores on latent factors, like human values, attitudes, and behavior. Such studies often include a comparison, between specific groups of individuals or residents of different countries, either at one or multiple points in time (i.e., a cross-sectional or a longitudinal comparison or both). If latent factor means are to be meaningfully compared, the measurement structures of the latent factor and their survey items should be stable, that is “invariant.” As proposed by Mellenbergh (1989), “measurement invariance” (MI) requires that the association between the items (or test scores) and the latent factors (or latent traits) of individuals should not depend on group membership or measurement occasion (i.e., time). In other words, if item scores are (approximately) multivariate normally distributed, conditional on the latent factor scores, the expected values, the covariances between items, and the unexplained variance unrelated to the latent factors should be equal across groups. Many studies examining MI of survey scales have shown that the MI assumption is very hard to meet. In particular, strict forms of MI rarely hold. With “strict” we refer to a situation in which measurement parameters are exactly the same across groups or measurement occasions, that is an enforcement of zero tolerance with respect to deviations between groups or measurement occasions. Often, researchers just ignore MI issues and compare latent factor means across groups or measurement occasions even though the psychometric basis for such a practice does not hold. However, when a strict form of MI is not established and one must conclude that respondents attach different meanings to survey items, this makes it impossible to make valid comparisons between latent factor means. As such, the potential bias caused by measurement non-invariance obstructs the comparison of latent factor means (if strict MI does not hold) or regression coefficients (if less strict forms of MI do not hold). Traditionally, MI is tested for in a multiple group confirmatory factor analysis (MGCFA) with groups defined by unordered categorical (i.e., nominal) between-subject variables. In MGCFA, MI is tested at each constraint of the latent factor model using a series of nested (latent) factor models. This traditional way of testing for MI originated with Joreskog (1971), who was the first scholar to thoroughly discuss the invariance of latent factor (or measurement) structures. Additionally, Sorbom (1974, 1978) pioneered the specification and estimation of latent factor means using a multi-group SEM approach in LISREL (Joreskog and Sorbom, 1996). Following these contributions the multi-group specification of latent factor structures has become widespread in all major SEM software programs (e.g., AMOS Arbuckle, 2006, EQS Bender and Wu, 1995, LAVAAN Rosseel, 2012, Mplus Muthen and Muthen, 2013, STATA STATA, 2015, and OpenMx Boker et al., 2011). Shortly thereafter, Byrne et al. (1989) introduced the distinction between full and partial MI. Although their introduction was of great value, the first formal treatment of different forms of MI and their consequences for the validity of multi-group/multi-time comparisons is attributable to Meredith (1993). So far, a tremendous amount of papers dealing with MI have been published. The literature on MI published in the 20th century is nicely summarized by Vandenberg and Lance (2000). Noteworthy is also the overview of applications in cross-cultural studies provided by Davidov et al. (2014), as well as a recent book by Millsap (2011) containing a general systematic treatment of the topic of MI. The traditional MGCFA approach to MI-testing is described by, for example, Byrne (2004), Chen et al. (2005), Gregorich (2006), van de Schoot et al. (2012), Vandenberg (2002) and Wicherts and Dolan (2010). Researchers entering the field of MI are recommended to first consult Meredith (1993) and Millsap (2011) before reading other valuable academic works. Recent developments in statistics have provided new analytical tools for assessing MI. The aim of this special issue is to provide a forum for a discussion of MI, covering some crucial “themes”: (1) ways to assess and deal with measurement non-invariance; (2) Bayesian and IRT methods employing the concept of approximate MI; and (3) new or adjusted approaches for testing MI to fit increasingly complex statistical models and specific characteristics of survey data.