Abstract

Examinations of the relationships between environmental variables and ordination results often give little consideration to the complex relationships among environmental factors. In this chapter I consider the utility of structural equation modeling (SEM) with latent variables for evaluating the relationships among environmental variables and ordination axis scores. Using an example data set, I compare the efficiency of three approaches – (1) multiple regression, (2) principle component analysis, and (3) SEM – as methods for extracting information from multivariate data. All three approaches were found to be equivalent in their ability to explain variance in response variables but differ in their ability to explain the covariation among predictor variables. In general, when sufficient theoretical knowledge exists to permit the formulation of hypotheses about the relationships among variables, structural equation modeling can provide for a more comprehensive analysis. It is suggested that the analysis of latent variables using SEM may advance our understanding of environmental effects on vegetation data in many cases.

Abstract

This chapter presents an example of how structural equation modeling (SEM) could be used to test a classic theoretical model of population dynamics of the Shiras moose (Alces alces). A longitudinal model is developed in which population density is measured in two waves. The change in population density between the two periods of measure is modeled in relation to a complex set of interrelationships among environmental and population level variables. Included in the model are examples of composite variables and nonzero fixed parameters. Analysis of a simulated data set demonstrates the procedures of a typical SEM study that begins first with a measurement model and proceeds with a series of exploratory and confirmatory analyses. The use and pitfalls of fit statistics, t-values, modification indices, and Q–Q plots as diagnostic tools are demonstrated. Two types of estimate, covariance estimates and standardized estimates, are contrasted. Examples of the calculation of total effects from direct and indirect effects are presented. Results demonstrate a significant potential for using SEM to develop expert systems and ecological models.

Abstract

This chapter reviews, in a nonmathematical way, some of the key concepts that set structural equation modeling (SEM) apart from conventional parametric multivariate methods. The chapter is sectioned by the somewhat overlapping topics of measurement error, latent variables, hypothesis testing, and complex models. In the section dealing with measurement error, I describe some of the sources of measurement error found in non-experimental field data and how measurement error in concert with collinearity among independent variables can bias and distort regression path coefficients, factor loadings, and other outputs of conventional statistical methods. In the next section, latent variables, I introduce the latent variable with multiple indicators and demonstrate their efficacy for dealing with collinearity and reducing measurement error. As a result, higher precision of estimates of path coefficients may be obtained with SEM analysis over conventional methods. In the following section, hypothesis testing, I demonstrate how the use of latent variables introduces a new dimension to the analysis in the form of model fitting and hypothesis testing. Finally, I provide examples of how conventional methodologies can be extended with SEM for the analysis of complex systems.

Abstract

Increasingly, the analysis of biological data involves collection of data over time, especially when the phenomena of interest concern identification of patterns of differential growth or response to long-term environmental stressors. When different patterns over time can be identified, they may indicate the presence of different survival strategies in a population of interest, species-specific responses to stressors, or may be markers for as yet unidentified subspecies in the population. In this chapter, two techniques for the analysis of biological data over time are described. The first technique is a two-stage analysis of data over time in which polynomial regressions are conducted to identify distinct growth patterns in the data. These growth curve patterns are then used in subsequent structural models to identify the correlates of growth curves. The second technique, the method of “Tuckerized” growth curves, proposes identification of growth curves by an eigenvalue decomposition. This solution is then rotated to make the solution conceptually interpretable within the context of overall growth over time. Factor scores from this rotated solution are then used to investigate whether such growth patterns differ as a function of environmental or species classification. Both techniques are compared and contrasted using longitudinal data on tree circumference taken from three species of trees located in the wetlands of two North American states, South Carolina and Louisiana. Rather than being competing models of growth over time, these two techniques represent complementary analyses, which differ in terms of their assumptions of categorization and growth curve processes over time.

Abstract

Structural equation modeling (SEM) is a method of testing systems of equations against observed covariance matrices while system dynamic modeling (SDM) simulates systems of differential equations representing nonlinear feedback loops. SEM has typically been constrained to linear systems of equations, but more recent developments have allowed SEM to represent nonlinear as well as nonrecursive (feedback) relationships. In this sense, SEM path diagrams with nonrecursive relationships and SDM causal loop diagrams look similar. This chapter analyzes the relationship between system dynamics modeling and structural equation modeling. A formal notion of equivalence between system dynamic models and structural equation models is developed. This is then analyzed through an illustration comparing the implied covariance matrices from one system dynamics model against the implied covariance matrices of two “equivalent” LISREL models. The first LISREL model tests whether an “equivalent” model can explain the entire family of covariance matrices generated by a system dynamics model, while the second LISREL model is developed to fit the data. The results suggest that structural equation models are able to explain the covariance matrices of a system dynamics model with changes in feedback loop dominance, but only by sacrificing a substantial portion of the latent structure representing feedback loops.

Abstract

It is possible to test a multivariate biological hypothesis concerning cause–effect relationships using structural equation modeling (SEM) applied to observational data. However, to do this we must translate from the language of causality to the language of probability distributions and this process of translation is almost always imperfect. One consequence of this imperfection of causal translation is the existence of equivalent SEM models; that is, different causal models that produce exactly equivalent statistical structural equation models. In this chapter I describe how such equivalent models arise, how to find equivalent models based on path diagrams, and why their existence complicates our interpretation of standard statistical tests in SEM. I illustrate these concepts using two actual path models taken from plant ecology.

Abstract

This chapter describes theory and methods for measuring mean changes in phenotypes when the phenotypic variable is considered as a latent construct with multiple indicators. Data for exposition of the methods are generated using EQS simulation techniques. Two datasets are used, one simulating the population prior to a selection event, the pre-selection group, and another simulating the population after the selection event, the post-selection group. The example includes an environmental construct that may also influence phenotypic variable means. The example demonstrates how structural equation modeling (SEM) may be used to statistically control for such environmental influences to arrive at a more accurate estimate of the effect of the selection event. Data are analyzed with LISREL version 8.30. The analysis includes a measurement model and stacked model of covariances to test model assumptions, and a final means model to estimate the mean phenotypic response to selection.

Abstract

This chapter provides an introduction to structural equation modeling (SEM), a statistical technique that allows scientists and researchers to quantify and test scientific theories. As an example, a model from behavioral genetics is examined, in which genetic and environmental influences on a trait are determined. The many procedures and considerations involved in SEM are outlined and described, including defining and specifying a model diagrammatically and algebraically, determining the identification status of the model, estimating the model parameters, assessing the fit of the model to the data, and respecifying the model to achieve a better fit to the data. Since behavioral genetic models typically require family members of differing genetic relatedness, multisample SEM is introduced. All of the steps involved in evaluating the behavioral genetic model are accomplished with the assistance of LISREL, a popular software program used in SEM.

Abstract

The goal of this chapter is to provide a historical account of structural equation modeling (SEM). To do so I followed the development throughout the twentieth century of the main ideas in the four disciplines that contributed to the development of SEM: biometrics, econometrics, psychometrics, and sociometrics. Special attention is paid to the development of path analysis by Wright and its sociological applications. Also presented are the development of estimation procedures and the formulation of identification issues in econometrics and the beginning of exploratory and confirmatory factor analysis in psychology. The last part of this account presents the synthesis of these ideas in the LISREL model as well as extensions of the original model.