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piecewiseSEM: Piecewise structural equation modeling in R for ecology, evolution, and systematics
Jonathan S. Lefcheck
TL;DR
Complex ecological datasets require flexible structural equation modeling, so this paper introduces the open-source piecewiseSEM R package and worked examples. The approach produced substantially different inferences from multiple regression and traditional variance-covariance SEM.
Problem
Increasingly multifaceted ecological datasets motivate statistical tools for describing complex natural systems.
Method
The paper presents the open-source piecewiseSEM R package and demonstrates it through two worked examples using complex ecological data.
Results
The piecewise approach yielded substantially different inferences than multiple regression and traditional variance-covariance SEM.
Takeaways & Limitations
piecewiseSEM provides a quick and easy way to implement structural equation models for ecological data.
Takeaways & Limitations
The package lacks implementations for correlated errors and bidirectional relationships.
Abstract
from arXiv · showhide
Ecologists and evolutionary biologists are relying on an increasingly sophisticated set of statistical tools to describe complex natural systems. One such tool that has gained increasing traction in the life sciences is structural equation modeling (SEM), a variant of path analysis that resolves complex multivariate relationships among a suite of interrelated variables. SEM has historically relied on covariances among variables, rather than the values of the data points themselves. While this approach permits a wide variety of model forms, it limits the incorporation of detailed specifications. Here, I present a fully-documented, open-source R package piecewiseSEM that builds on the base R syntax for all current generalized linear, least-square, and mixed effects models. I also provide two worked examples: one involving a hierarchical dataset with non-normally distributed variables, and a second involving phylogenetically-independent contrasts. My goal is to provide a user-friendly and tractable implementation of SEM that also reflects the ecological and methodological processes generating data.
1. Ecologists and evolutionary biologists are relying on an increasingly sophisticated set of statistical
The paper presents piecewiseSEM, a fully documented, open-source R package for structural equation modeling. It aims to provide a user-friendly, tractable implementation that reflects ecological and methodological data-generating processes.
- piecewiseSEM is a fully documented, open-source R package for structural equation modeling.
- The package builds on base R syntax for generalized linear, least-square, and mixed-effects models.
- Two worked examples cover hierarchical data with non-normally distributed variables and phylogenetically independent contrasts.
- The implementation is intended to be user-friendly and tractable while reflecting ecological and methodological processes generating data.
Introduction
The paper motivates piecewise structural equation modeling as a flexible framework for analyzing complex ecological datasets while accommodating distributions, hierarchical structures, and estimation procedures that constrain traditional SEM. Worked examples show that modeling data structure can alter ecological interpretations and reveal indirect pathways.
- Motivation: Increasingly multifaceted ecological datasets require sophisticated statistical tools to uncover relationships in complex natural systems.
- Structural equation modeling: Structural equation models unite multiple predictors and responses in causal networks represented by directional paths, allowing variables to serve as both predictors and responses.This structure supports testing and quantifying indirect relationships.
- Limitations of traditional SEM: Traditional SEM assumes independent, multivariate-normal variables and requires enough observations to estimate the complete variance-covariance matrix.These assumptions restrict distributions, sampling designs, and dataset size.
- Piecewise SEM: Piecewise SEM permits varied distributions and sampling designs, can fit smaller datasets, and accommodates non-normal distributions, hierarchical structures, and different estimation procedures.The approach extends directed-separation tests and is implemented in a fully documented, open-source R package.
- Worked examples: In the kelp-forest example, hierarchical piecewise SEM changed the interpretation by indicating that wave disturbance decreases food-web complexity principally by removing habitat.The analysis found that temporal autocorrelation did not improve understanding of the interaction system.
- Worked examples: In Synalpheus shrimp, eusociality positively affected host range after accounting for body mass (standardized regression coefficient = 0.58), while host range positively affected relative abundance (0.47).Eusociality had no significant direct relationship with abundance.
Discussion
The discussion emphasizes that piecewise SEM can yield substantially different inferences when analyses account for nonindependence through random variation or phylogenetic contrasts. It also presents piecewiseSEM as a quick, easy way to implement complex local estimation, with applications in several ecological studies.
- Discussion: Accounting for data-point nonindependence with random variation or phylogenetic contrasts yielded substantially different inferences than multiple regression or traditional variance-covariance SEM.The paper applies piecewise SEM to two existing analyses.
- Discussion: The piecewiseSEM R package enables quick and easy implementation of complex local estimation.The package builds on the paper’s introduction and application of piecewise structural equation modeling.
- Discussion: The package has been used to explore eelgrass ecosystem-functioning drivers, trophic-level functional diversity, and grassland multifunctionality drivers.These applications span eelgrass beds, experimental estuarine mesocosms, and grasslands.
Broader Applications
The piecewiseSEM package includes additional functions for evaluating component-model fit, generating predictions, and visualizing partial relationships. These tools also support prediction uncertainty and can reveal nonlinear relationships for incorporation into model structure.
- Model evaluation: Additional piecewiseSEM functions provide model-fit statistics, including R2, pseudo-R2, and AIC values for component models.sem.model.fits uses methods from Nakagawa & Schielzeth (2012) and Johnson (2014).
- Prediction: predict.sem wraps R’s generic predict function and adds standard errors for predictions from lmer-based models.The standard errors are based on the variance of fixed effects only.
- Partial relationships: partial.resid produces partial-correlation plots between two variables after accounting for covariate effects.The plots provide an intuitive graphical way to emphasize partial regression coefficients returned by sem.coefs or summary.
- Partial relationships: Exploring partial correlations can identify previously unrecognized nonlinear relationships that can then be incorporated into the model structure.This extends partial-residual visualization from coefficient emphasis to model-structure exploration.
Limitations
piecewiseSEM remains vulnerable to inadequate or excessive sample sizes, unstable model-fitting inputs, and weak component-model performance. Its developing framework also cannot represent several causal structures, including correlated errors, feedbacks, reciprocal relationships, and latent variables.
- Sample size and model fit: Adequate overall SEM fit is insufficient when component models have low explanatory power, and large samples may make everything significant at α = 0.05.In the latter case, a more stringent significance cutoff may be needed.
- Implementation: P-values from lmerTest were unstable and could cause errors in sem.fit, whereas nlme estimates appeared more reliable.Users were recommended to construct models with nlme when lmerTest produced an error.
- Structural limitations: piecewiseSEM lacks a real implementation of correlated errors and cannot represent relationships that are bidirectional or caused by a shared underlying driver.It approximates correlated errors by excluding them from the basis set and testing the bivariate correlation.
- Structural limitations: The method cannot disentangle cyclic relationships, evaluate feedbacks, or assess reciprocal relationships within the same model.Examples include A -> B -> C -> A and simultaneous A -> B and B -> A relationships.
- Structural limitations: piecewiseSEM has no formal integration of latent variables, and factor-analysis approaches have not yet been thoroughly investigated or applied.Future developments were expected to relax some of these limitations.
Figure Captions:
The captions explain how relationships, significance, path strength, and model fit are represented in the SEM diagrams. They compare variance-covariance SEM with piecewise SEM extensions for nested, temporal, and phylogenetic structure.
- Figure Captions:: Boxes represent measured variables, while arrows represent unidirectional relationships among variables.
- Figure Captions:: Black arrows indicate positive relationships, red arrows indicate negative relationships, and semi-transparent arrows denote non-significant paths (P ≥ 0.05).
- Figure Captions:: Significant path thickness reflects the magnitude of the standardized regression coefficient, which is reported in the associated box.
- Figure Captions:: Component-model R2 values are shown in response-variable boxes, with conditional R2c used for panels b and c and R2 omitted for panel 2b gls models.
- Figure Captions:: Panel 1 compares variance-covariance SEM with piecewise SEM incorporating Transect nested within Site and, additionally, an autocorrelation term for Year.
- Figure Captions:: Panel 2 compares variance-covariance SEM with piecewise SEM incorporating a fixed correlation structure based on phylogenetic distances from a molecular phylogeny.