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

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.

This book describes a family of statistical methods known as structural equation modeling (SEM). SEM is used in a variety of techniques known as “covariance structure analysis”, “latent variable modeling”, “path modeling”, “path modeling with LISREL”, and sometimes it is mistaken for path analysis. This book will help biologists to understand the distinction between SEM and path analysis. The book consists of contributed chapters from biologists as well as leading methodologists in other research fields. We have organized the chapters and their content with the intent of providing a volume that readers may use to learn the methodology and apply it themselves to their research problems. We give the basic formulation of the method as well as technical details on data analysis, interpretation, and reporting. In addition, we provide numerous examples of research designs and applications that are germane to the research needs and interests of organismal biologists. We also provide, as a learning aide, the simulation programs, analysis programs, and data matrices, presented in the book at a website (http://www.usgs.gov/) so that readers may download and run them.

The book is divided into three sections. The first section, “Theory”, describes the SEM model and practical matters of its application. Chapter 1 lays out the mathematics of SEM in a comprehensible fashion. Using an example from behavioral genetics, the authors express their model in what is called LISREL notation, a symbolic language that is commonly used to express SEM models.

Abstract

Although distinctions are commonly made between exploratory and confirmatory models, structural equation modeling (SEM) is not an exploratory statistical method per se. The successful implementation of a structural equation model requires considerable a priori knowledge of the subject matter under investigation. The researcher will usually have a theoretical model in mind to test, and measurement instruments, including latent variables, devised to measure and relate constructs within the model. Research with SEM is a process in which theory is devised, data are collected and analyzed, and models are tested, modified, and confirmed with new data in an iterative fashion. In this context, SEM is rightly viewed as a confirmatory method. As a consequence, a number of epistemological and technical issues require consideration over and above the pure mathematics of the SEM model. In this chapter, we provide background on the philosophical aspects of the study of dependence relationships with SEM, the formulation of latent constructs, model justification, model identification, sample size and power, estimation methods, evaluation of model fit, model modification, interpretation of results, and publication of results. Our objective is to provide a guide that researchers can use to successfully devise and report the results of an SEM study.