Products of Variables in Structural Equation Models.

A general method is introduced in which variables that are products of other variables in the context of a structural equation model (SEM) can be decomposed into the sources of variance due to the multiplicands. The result is a new category of SEM which we call a Products of Variables Model (PoV). Some useful and practical features of PoV models include estimation of interactions between latent variables, latent variable moderators, manifest moderators with missing values, and manifest or latent squared terms. Expected means and covariances are analytically derived for a simple product of two variables and it is shown that the method reproduces previously published results for this special case. It is shown algebraically that using centered multiplicands results in an unidentified model, but if the multiplicands have non-zero means, the result is identified. The method has been implemented in OpenMx and Ωnyx and is applied in five extensive simulations.

Associations Between Traumatic Stress, Brain Volumes and Post-traumatic Stress Disorder Symptoms in Children: Data from the ABCD Study.

Reduced volumes in brain regions of interest (ROIs), primarily from adult samples, are associated with posttraumatic stress disorder (PTSD). We extended this work to children using data from the Adolescent Brain Cognitive Development (ABCD) Study® (N = 11,848; Mage = 9.92). Structural equation modeling and an elastic-net (EN) machine-learning approach were used to identify potential effects of traumatic events (TEs) on PTSD symptoms (PTSDsx) directly, and indirectly via the volumes 300 subcortical and cortical ROIs. We then estimated the genetic and environmental variation in the phenotypes. TEs were directly associated with PTSDsx (r = 0.92) in children, but their indirect effects (r < 0.0004)-via the volumes of EN-identified subcortical and cortical ROIs-were negligible at this age. Additive genetic factors explained a modest proportion of the variance in TEs (23.4%) and PTSDsx (21.3%), and accounted for most of the variance of EN-identified volumes of four of the five subcortical (52.4-61.8%) three of the nine cortical ROIs (46.4-53.3%) and cerebral white matter in the left hemisphere (57.4%). Environmental factors explained most of the variance in TEs (C = 61.6%, E = 15.1%), PTSDsx (residual-C = 18.4%, residual-E = 21.8%), right lateral ventricle (C = 15.2%, E = 43.1%) and six of the nine EN-identified cortical ROIs (C = 4.0-13.6%, E = 56.7-74.8%). There is negligible evidence that the volumes of brain ROIs are associated with the indirect effects of TEs on PTSDsx at this age. Overall, environmental factors accounted for more of the variation in TEs and PTSDsx. Whereas additive genetic factors accounted for most of the variability in the volumes of a minority of cortical and in most of subcortical ROIs.

Using Genetic Marginal Effects to Study Gene-Environment Interactions with GWAS Data.

Gene-environment interactions (GxE) play a central role in the theoretical relationship between genetic factors and complex traits. While genome wide GxE studies of human behaviors remain underutilized, in part due to methodological limitations, existing GxE research in model organisms emphasizes the importance of interpreting genetic associations within environmental contexts. In this paper, we present a framework for conducting an analysis of GxE using raw data from genome wide association studies (GWAS) and applying the techniques to analyze gene-by-age interactions for alcohol use frequency. To illustrate the effectiveness of this procedure, we calculate genetic marginal effects from a GxE GWAS analysis for an ordinal measure of alcohol use frequency from the UK Biobank dataset, treating the respondent's age as the continuous moderating environment. The genetic marginal effects clarify the interpretation of the GxE associations and provide a direct and clear understanding of how the genetic associations vary across age (the environment). To highlight the advantages of our proposed methods for presenting GxE GWAS results, we compare the interpretation of marginal genetic effects with an interpretation that focuses narrowly on the significance of the interaction coefficients. The results imply that the genetic associations with alcohol use frequency vary considerably across ages, a conclusion that may not be obvious from the raw regression or interaction coefficients. GxE GWAS is less powerful than the standard "main effect" GWAS approach, and therefore require larger samples to detect significant moderated associations. Fortunately, the necessary sample sizes for a successful application of GxE GWAS can rely on the existing and on-going development of consortia and large-scale population-based studies.

GW-SEM 2.0: Efficient, Flexible, and Accessible Multivariate GWAS.

Most genome-wide association study (GWAS) analyses test the association between single-nucleotide polymorphisms (SNPs) and a single trait or outcome. While valuable second-step analyses of these associations (e.g., calculating genetic correlations between traits) are common, single-step multivariate analyses of GWAS data are rarely performed. This is unfortunate because multivariate analyses can reveal information which is irrevocably obscured in multi-step analysis. One simple example is the distinction between variance common to a set of measures, and variance specific to each. Neither GWAS of sum- or factor-scores, nor GWAS of the individual measures will deliver a clean picture of loci associated with each measure's specific variance. While multivariate GWAS opens up a broad new landscape of feasible and informative analyses, its adoption has been slow, likely due to the heavy computational demands and difficulties specifying models it requires. Here we describe GW-SEM 2.0, which is designed to simplify model specification and overcome the inherent computational challenges associated with multivariate GWAS. In addition, GW-SEM 2.0 allows users to accurately model ordinal items, which are common in behavioral and psychological research, within a GWAS context. This new release enhances computational efficiency, allows users to select the fit function that is appropriate for their analyses, expands compatibility with standard genomic data formats, and outputs results for seamless reading into other standard post-GWAS processing software. To demonstrate GW-SEM's utility, we conducted (1) a series of GWAS using three substance use frequency items from data in the UK Biobank, (2) a timing study for several predefined GWAS functions, and (3) a Type I Error rate study. Our multivariate GWAS analyses emphasize the utility of GW-SEM for identifying novel patterns of associations that vary considerably between genomic loci for specific substances, highlighting the importance of differentiating between substance-specific use behaviors and polysubstance use. The timing studies demonstrate that the analyses take a reasonable amount of time and show the cost of including additional items. The Type I Error rate study demonstrates that hypothesis tests for genetic associations with latent variable models follow the hypothesized uniform distribution. Taken together, we suggest that GW-SEM may provide substantially deeper insights into the underlying genomic architecture for multivariate behavioral and psychological systems than is currently possible with standard GWAS methods. The current release of GW-SEM 2.0 is available on CRAN (stable release) and GitHub (beta release), and tutorials are available on our github wiki ( https://jpritikin.github.io/gwsem/ ).

Combining Structural-Equation Modeling with Genomic-Relatedness-Matrix Restricted Maximum Likelihood in OpenMx.

There is a long history of fitting biometrical structural-equation models (SEMs) in the pregenomic behavioral-genetics literature of twin, family, and adoption studies. Recently, a method has emerged for estimating biometrical variance-covariance components based not upon the expected degree of genetic resemblance among relatives, but upon the observed degree of genetic resemblance among unrelated individuals for whom genome-wide genotypes are available-genomic-relatedness-matrix restricted maximum-likelihood (GREML). However, most existing GREML software is concerned with quickly and efficiently estimating heritability coefficients, genetic correlations, and so on, rather than with allowing the user to fit SEMs to multitrait samples of genotyped participants. We therefore introduce a feature in the OpenMx package, "mxGREML", designed to fit the biometrical SEMs from the pregenomic era in present-day genomic study designs. We explain the additional functionality this new feature has brought to OpenMx, and how the new functionality works. We provide an illustrative example of its use. We discuss the feature's current limitations, and our plans for its further development.

An exploratory factor model for ordinal paired comparison indicators.

Suppose the same contestants play in tournaments of chess, shogi, and Go. Per-tournament rankings can be estimated. We may also try to recover a latent board game skill that accounts for some proportion of the variance in per-board game rankings. To accomplish this, a factor model is introduced. Identification issues with the ordinal paired item model are discussed. Simulation studies are presented to provide some guidance about sample size requirements. Both single item and multivariate correlation and factor model are validated using simulation-based calibration. We recommend leave-one-out cross-validation to assess model fit. To ease application of the methods described, an open-source companion R extension, pcFactorStan, is published on the Comprehensive R Archive Network. Application of pcFactorStan is demonstrated by analysis of a real-world dataset.

OpenMx: A Modular Research Environment for Item Response Theory Method Development.

There are many item response theory software packages designed for users. Here, the authors introduce an environment tailored to method development and simulation. Implementations of a selection of classic algorithms are available as well as some recently developed methods. Source code is developed in public repositories on GitHub; your collaboration is welcome.

Cloud computing for voxel-wise SEM analysis of MRI data.

As data collection costs fall and vast quantities of data are collected, data analysis time can become a bottleneck. For massively parallel analyses, cloud computing offers the short-term rental of ample processing power. Recent software innovations have reduced the offort needed to take advantage of cloud computing. To demonstrate, we replicate a voxel-wise examination of the genetic contributions to cortical development by age using evidence from 1,748 MRI scans. Specifically, we employ off-the-shelf Kubernetes software that permits us to re-run our analyses using almost the same computer code as was published in the original article. Large, well funded institutions may continue to maintain their own computing clusters. However, the modest cost of renting and ease of utilizing cloud computing services makes unprecedented compute power available to all researchers, whether or not affliated with a research institution. We expect this to spur innovation in the sophisticated modeling of large datasets.

A Comprehensive Quantitative Genetic Analysis of Cerebral Surface Area in Youth.

The genetics of cortical arealization in youth is not well understood. In this study, we use a genetically informative sample of 677 typically developing children and adolescents (mean age 12.72 years), high-resolution MRI, and quantitative genetic methodology to address several fundamental questions on the genetics of cerebral surface area. We estimate that >85% of the phenotypic variance in total brain surface area in youth is attributable to additive genetic factors. We also observed pronounced regional variability in the genetic influences on surface area, with the most heritable areas seen in primary visual and visual association cortex. A shared global genetic factor strongly influenced large areas of the frontal and temporal cortex, mirroring regions that are the most evolutionarily novel in humans relative to other primates. In contrast to studies on older populations, we observed statistically significant genetic correlations between measures of surface area and cortical thickness (rG = 0.63), suggestive of overlapping genetic influences between these endophenotypes early in life. Finally, we identified strong and highly asymmetric genetically mediated associations between Full-Scale Intelligence Quotient and left perisylvian surface area, particularly receptive language centers. Our findings suggest that spatially complex and temporally dynamic genetic factors are influencing cerebral surface area in our species.SIGNIFICANCE STATEMENT Over evolution, the human cortex has undergone massive expansion. In humans, patterns of neurodevelopmental expansion mirror evolutionary changes. However, there is a sparsity of information on how genetics impacts surface area maturation. Here, we present a systematic analysis of the genetics of cerebral surface area in youth. We confirm prior research that implicates genetics as the dominant force influencing individual differences in global surface area. We also find evidence that evolutionarily novel brain regions share common genetics, that overlapping genetic factors influence both area and thickness in youth, and the presence of strong genetically mediated associations between intelligence and surface area in language centers. These findings further elucidate the complex role that genetics plays in brain development and function.

The Dynamic Associations Between Cortical Thickness and General Intelligence are Genetically Mediated.

The neural substrates of intelligence represent a fundamental but largely uncharted topic in human developmental neuroscience. Prior neuroimaging studies have identified modest but highly dynamic associations between intelligence and cortical thickness (CT) in childhood and adolescence. In a separate thread of research, quantitative genetic studies have repeatedly demonstrated that most measures of intelligence are highly heritable, as are many brain regions associated with intelligence. In the current study, we integrate these 2 streams of prior work by examining the genetic contributions to CT-intelligence relationships using a genetically informative longitudinal sample of 813 typically developing youth, imaged with high-resolution MRI and assessed with Wechsler Intelligence Scales (IQ). In addition to replicating the phenotypic association between multimodal association cortex and language centers with IQ, we find that CT-IQ covariance is nearly entirely genetically mediated. Moreover, shared genetic factors drive the rapidly evolving landscape of CT-IQ relationships in the developing brain.

Multivariate normal maximum likelihood with both ordinal and continuous variables, and data missing at random.

A novel method for the maximum likelihood estimation of structural equation models (SEM) with both ordinal and continuous indicators is introduced using a flexible multivariate probit model for the ordinal indicators. A full information approach ensures unbiased estimates for data missing at random. Exceeding the capability of prior methods, up to 13 ordinal variables can be included before integration time increases beyond 1 s per row. The method relies on the axiom of conditional probability to split apart the distribution of continuous and ordinal variables. Due to the symmetry of the axiom, two similar methods are available. A simulation study provides evidence that the two similar approaches offer equal accuracy. A further simulation is used to develop a heuristic to automatically select the most computationally efficient approach. Joint ordinal continuous SEM is implemented in OpenMx, free and open-source software.

Likelihood-based confidence intervals for a parameter with an upper or lower bound.

The precision of estimates in many statistical models can be expressed by a confidence interval (CI). CIs based on standard errors (SE) are common in practice, but likelihood-based CIs are worth consideration. In comparison to SEs, likelihood-based CIs are typically more difficult to estimate, but are more robust to model (re)parameterization. In latent variable models, some parameters may take on values outside of their interpretable range. Therefore, it is desirable to place a bound to keep the parameter interpretable. For likelihood-based CI, a correction is needed when a parameter is bounded. The correction is known (Wu & Neale, 2012), but is difficult to implement in practice. A novel automatic implementation that is simple for an applied researcher to use is introduced. A simulation study demonstrates the accuracy of the correction using a latent growth curve model and the method is illustrated with a multilevel confirmatory factor analysis.

Many-level multilevel structural equation modeling: An efficient evaluation strategy.

Structural equation models are increasingly used for clustered or multilevel data in cases where mixed regression is too inflexible. However, when there are many levels of nesting, these models can become difficult to estimate. We introduce a novel evaluation strategy, Rampart, that applies an orthogonal rotation to the parts of a model that conform to commonly met requirements. This rotation dramatically simplifies fit evaluation in a way that becomes more potent as the size of the data set increases. We validate and evaluate the implementation using a 3-level latent regression simulation study. Then we analyze data from a state-wide child behavioral health measure administered by the Oklahoma Department of Human Services. We demonstrate the efficiency of Rampart compared to other similar software using a latent factor model with a 5-level decomposition of latent variance. Rampart is implemented in OpenMx, a free and open source software.

OpenMx 2.0: Extended Structural Equation and Statistical Modeling.

The new software package OpenMx 2.0 for structural equation and other statistical modeling is introduced and its features are described. OpenMx is evolving in a modular direction and now allows a mix-and-match computational approach that separates model expectations from fit functions and optimizers. Major backend architectural improvements include a move to swappable open-source optimizers such as the newly written CSOLNP. Entire new methodologies such as item factor analysis and state space modeling have been implemented. New model expectation functions including support for the expression of models in LISREL syntax and a simplified multigroup expectation function are available. Ease-of-use improvements include helper functions to standardize model parameters and compute their Jacobian-based standard errors, access to model components through standard R $ mechanisms, and improved tab completion from within the R Graphical User Interface.

Maintained Individual Data Distributed Likelihood Estimation (MIDDLE).

Maintained Individual Data Distributed Likelihood Estimation (MIDDLE) is a novel paradigm for research in the behavioral, social, and health sciences. The MIDDLE approach is based on the seemingly impossible idea that data can be privately maintained by participants and never revealed to researchers, while still enabling statistical models to be fit and scientific hypotheses tested. MIDDLE rests on the assumption that participant data should belong to, be controlled by, and remain in the possession of the participants themselves. Distributed likelihood estimation refers to fitting statistical models by sending an objective function and vector of parameters to each participant's personal device (e.g., smartphone, tablet, computer), where the likelihood of that individual's data is calculated locally. Only the likelihood value is returned to the central optimizer. The optimizer aggregates likelihood values from responding participants and chooses new vectors of parameters until the model converges. A MIDDLE study provides significantly greater privacy for participants, automatic management of opt-in and opt-out consent, lower cost for the researcher and funding institute, and faster determination of results. Furthermore, if a participant opts into several studies simultaneously and opts into data sharing, these studies automatically have access to individual-level longitudinal data linked across all studies.

Modular Open-Source Software for Item Factor Analysis.

This paper introduces an Item Factor Analysis (IFA) module for OpenMx, a free, open-source, and modular statistical modeling package that runs within the R programming environment on GNU/Linux, Mac OS X, and Microsoft Windows. The IFA module offers a novel model specification language that is well suited to programmatic generation and manipulation of models. Modular organization of the source code facilitates the easy addition of item models, item parameter estimation algorithms, optimizers, test scoring algorithms, and fit diagnostics all within an integrated framework. Three short example scripts are presented for fitting item parameters, latent distribution parameters, and a multiple group model. The availability of both IFA and structural equation modeling in the same software is a step toward the unification of these two methodologies.