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We consider structural equation models in which variables can be written as a function of their parents and noise terms, which are assumed to be jointly independent. Corresponding to each structural equation model is a directed acyclic graph describing the relationships between the variables. In Gaussian structural equation models with linear functions, the graph can be identified from the joint distribution only up to Markov equivalence classes, assuming faithfulness.In this work, we prove full identifiability in the case where all noise variables have the same variance: the directed acyclic graph can be recovered from the joint Gaussian distribution.Our result has direct implications for causal inference: if the data follow a Gaussian structural equation model with equal error variances, then, assuming that all variables are observed, the causal structure can be inferred from observational data only. We propose a statistical method and an algorithm based on our theoretical findings.

We develop estimation for potentially high-dimensional additive structural equation models. A key component of our approach is to decouple order search among the variables from feature or edge selection in a directed acyclic graph encoding the causal structure. We show that the former can be done with nonregularized (restricted) maximum likelihood estimation while the latter can be efficiently addressed using sparse regression techniques. Thus, we substantially simplify the problem of structure search and estimation for an important class of causal models. We establish consistency of the (restricted) maximum likelihood estimator for low- and high-dimensional scenarios, and we also allow for misspecification of the error distribution. Furthermore, we develop an efficient computational algorithm which can deal with many variables, and the new method's accuracy and performance is illustrated on simulated and real data.

Due to the coupled or interconnected relationships among frequent climate extremes, unique geological conditions, discontinuous soil distribution, rugged geomorphology, and highly heterogeneous landscapes in different karst watersheds, few studies were conducted to decouple the relative magnitudes of the climate, lithology, soil, topography, and landscape on soil erosion in karst regions. The objective of this study was to quantify the relative importance of these influencing factors on runoff and sediment yield (SY) in 40 typical karst watersheds in southwest China. To address this issue, the Pearson correlation and random forest were first to select the dominant factors influencing runoff and SY. Subsequently, the partial least squares‐structural equation model (PLS‐SEM) was used to decouple the complex relationships among runoff, SY and their potential influencing factors. Results showed that climate, lithology, soil, topography and landscape could explain 79% of the runoff variation, and only climate factors have significant impact on runoff for heterogeneous karst watersheds (P < 0.01, path coefficient (β) = 0.589). The explanation of five factors to SY variability is 59%, and the landscape has the greatest impact on SY (P < 0.01, β = −0.458). Different from runoff, climatic factors have no significant influence on SY. By elucidating a complex coupled relationship framework, this study can provide a scientific basis for the formulation of soil and water loss program, and the optimization of land resources and ecological environment sustainable development in karst watersheds. Plain Language Summary The variations of runoff and sediment yield (SY) among heterogeneous watersheds are expected to be influenced by climate, lithology, soil, topography and landscape, which are generally interconnected and coupled. Therefore, it is important to decouple the complex relationship between runoff, SY and their potential influencing factors, especially in karst areas where ecologically fragile regions experience severe soil erosion. The results show that only climate factors play a significant influence on runoff. Unlike runoff, all factors have significant effects on SY except climate factors. Interestingly, landscape factors have the greatest influence on SY. Our findings are conducive to better understanding the hydrological and sediment transport characteristics of karst watersheds, and provide scientific basis for soil erosion control and sustainable development of ecological environment. Key Points We decoupled the effects of climate, lithology, soil, topography and landscape on runoff and sediment yield (SY) Climate and landscape exerted the largest influence on runoff and SY, respectively Lithology, soil, topography, and landscape significantly affected SY rather than runoff variability

We consider the problem of regularized maximum likelihood estimation for the structure and parameters of a high-dimensional, sparse directed acyclic graphical (DAG) model with Gaussian distribution, or equivalently, of a Gaussian structural equation model. We show that the ℓ 0 -penalized maximum likelihood estimator of a DAG has about the same number of edges as the minimal-edge I-MAP (a DAG with minimal number of edges representing the distribution), and that it converges in Frobenius norm. We allow the number of nodes p to be much larger than sample size n but assume a sparsity condition and that any representation of the true DAG has at least a fixed proportion of its nonzero edge weights above the noise level. Our results do not rely on the faithfulness assumption nor on the restrictive strong faithfulness condition which are required for methods based on conditional independence testing such as the PC-algorithm.

In this study, a one-year program was conducted to investigate the relationships between students' perceptions of mobile learning and their tendencies of peer interaction and higher-order thinking in issue-based mobile learning activities. To achieve the research objective, a survey consisting of eight scales, namely, usability, continuity, adaptive content, collaboration, communication, problem-solving, critical thinking and creativity, was developed. A total of 658 students from 38 high schools in Taiwan filled in the questionnaire after experiencing issue-based mobile learning activities. From the exploratory and confirmatory factor analyses, it was found that the questionnaire had high reliability and validity. The structural equation model further revealed that the provision of adaptive content in the mobile learning had positive impacts on the students' tendency to interact with peers (i.e., collaboration and communication), which further affected their tendency to engage in higher-order thinking (i.e., problem-solving, critical thinking, and creativity). The findings of this study provide a good reference for researchers and school teachers who intend to promote mobile learning in school settings.

A person's socioeconomic status (SES) can affect health (social causation) and health can affect SES (health selection). The findings for each of these pathways may depend on how SES is measured. We study (1) whether social causation or health selection is more important for overall health inequalities, (2) whether this differs between stages of the life course, and (3) between measures of SES. Using retrospective survey data from 10 European countries (SHARELIFE, n{\\thinspace}={\\thinspace}18,734), and structural equation models in a cross-lagged panel design, we determine the relative explanatory power of social causation and health selection through childhood, adulthood, and old age. We use three ways to measure SES: First, as a latent variable capturing different aspects of SES, second as material wealth, and third as occupational skill level. Between childhood and adulthood, social causation and health selection are equally important. In the transition from adulthood to old age, social causation becomes more important than health selection, making it the dominant mechanism in old age. The three measures of SES produce similar results. Only material wealth shows a stronger effect on health (between childhood and adulthood); it is also more affected by health (between adulthood and old age) than the other measures.

A linear structural equation model relates random variables of interest and corresponding Gaussian noise terms via a linear equation system. Each such model can be represented by a mixed graph in which directed edges encode the linear equations and bidirected edges indicate possible correlations among noise terms. We study parameter identifiability in these models, that is, we ask for conditions that ensure that the edge coefficients and correlations appearing in a linear structural equation model can be uniquely recovered from the covariance matrix of the associated distribution. We treat the case of generic identifiability, where unique recovery is possible for almost every choice of parameters. We give a new graphical condition that is sufficient for generic identifiability and can be verified in time that is polynomial in the size of the graph. It improves criteria from prior work and does not require the directed part of the graph to be acyclic. We also develop a related necessary condition and examine the \"gap\" between sufficient and necessary conditions through simulations on graphs with 25 or 50 nodes, as well as exhaustive algebraic computations for graphs with up to five nodes.

The importance of public participation in the successful implementation of climate change-related policies has been highlighted in previous research. However, existing environmental behavioral studies have not sufficiently addressed the relationship among perceptions of climate change, living conditions, social demographic factors and environmentally friendly behavior. Therefore, this paper investigates whether environmental perception and other social determinants such as living conditions and the subjective evaluation of social inequality affect environmentally friendly behavior. We use survey data (N = 1500) collected in Mumbai, India, and analyze our hypotheses using a structural equation model (SEM). The empirical results confirm the direct and indirect influences of environmentally related perceptions, the subjective evaluation of living environments, social factors and other demographic characteristics on pro-environmental behavior. In particular, we find a robust positive effect of education level on pro-environmental behavior, where we observe both a direct impact and an indirect impact through positive effects on environmental knowledge. Thus, we confirm the importance of living environment, social equality and education in sustainable urban planning and efforts to mitigate climate change.

This article is the second of two parts intended to serve as a primer for structural equations models for the behavioral researcher. The first article introduced the basics: the measurement model, the structural model, and the combined, full structural equations model. In this second article, advanced issues are addressed, including fit indices and sample size, moderators, longitudinal data, mediation, and so forth.

The current decrease in biodiversity affects all ecosystems, and the impacts of diversity on ecosystem functioning need to be resolved. So far, marine studies about diversity–ecosystem productivity-relationships have concentrated on small-scale, controlled experiments, with often limited relevance to natural ecosystems. Here, we provide a real-world study on the effects of microorganismal diversity (measured as the diversity of benthic diatom communities) on ecosystem productivity (using chlorophyll a concentration as a surrogate) in a heterogeneous marine coastal archipelago. We collected 78 sediment cores at 17 sites in the northern Baltic Sea and found exceptionally high diatom diversity (328 observed species). We used structural equation models and quantile regression to explore relationships between diatom diversity and productivity. Previous studies have found contradictory results in the relationship between microorganismal diversity and ecosystem productivity, but we showed a linear and positive basal relationship between diatom diversity and productivity, which indicates that diatom diversity most likely forms the lowest boundary for productivity. Thus, although productivity can be high even when diatom diversity is low, high diatom diversity supports high productivity. The trait composition was more effective than taxonomical composition in showing such a relationship, which could be due to niche complementarity. Our results also indicated that environmental heterogeneity leads to substantial patchiness in the diversity of benthic diatom communities, mainly induced by the variation in sediment organic matter content. Therefore, future changes in precipitation and river runoff and associated changes in the quality and quantity of organic matter in the sea, will also affect diatom communities and, hence, ecosystem productivity. Our study suggests that benthic microorganisms are vital for ecosystem productivity, and together with the substantial heterogeneity of coastal ecosystems, they should be considered when evaluating the potential productivity of coastal areas.