Stanley A. Mulaik

A Tale of Two Methods

The structural equation modeling approach to testing for mediation is compared to the Baron and Kenny approach. The approaches are essentially the same when the hypothesis being tested predicts partial mediation. The approaches differ, however, in how each tests for complete mediation. Disparities in both theory and statistical estimators are identified and discussed. A strategy for future tests of mediation is recommended.

Doing the Four-Step Right

Our response to Hayduk and Glaser will principally focus on their critique of the four-step procedure. Hayduk and Glaser project things into the four-step procedure that are not part of its conception. They fail to see the implicit context in which those who use this particular four-step procedure operate, which qualifies its application. They also have misunderstandings about the rationale for the procedure and read too much of exploratory factor analysis into its use. Hayduk (1996) proposed a method for doing structural equation modeling with as few as one or two indicators per latent variable, which he feels is incompatible with the factor-analytic underpinnings of the four-step procedure and which motivates him further to seek its overthrow. He did not clarify this sufficiently on SEMNET, so we have been compelled to read Hayduk (1996) to better understand his position, and we will point out some limitations of it from our own point of view. We further argue that Hayduk’s (1996) advocacy of the use of fixed parameters in sparse measurement models is not a viable general alternative to the use of multiple indicators. Hayduk also believed the usual .05 level of significance in testing the exact fit of models favors the null hypothesis. He recommended that the significance level for a chi-square test be set at .75. We show this recommendation to be incoherent with the idea of a significance test and further show it to be unnecessary because, on the contrary, in most studies the null hypothesis is likely to be rejected. STRUCTURAL EQUATION MODELING,7(1), 36–73 Copyright

Causation issues in structural equation modeling research

As the use of structural equation modeling (SEM) has increased, confusion has grown concerning the correct use of and the conclusions that can be legitimately drawn from these methodologies. It appears that much of the controversy surrounding SEM is related to the degree of certainty with which causal statements can be drawn from these procedures. SEM is discussed in relation to the conditions necessary for providing causal evidence. Both the weaknesses and the strengths of SEM are examined. Although structural modeling cannot ensure that necessary causal conditions have been met, it is argued that SEM methods may offer the potential for tentative causal inferences to be drawn when used with carefully specified and controlled designs. Keeping in mind that no statistical methodology can in and of itself determine causality, specific guidelines are suggested to help researchers approach a potential for providing causal evidence with SEM procedures.

A Brief History of the Philosophical Foundations of Exploratory Factor Analysis

Exploratory factor analysis derives its key ideas from many sources. From the Greek rationalists and atomists comes the idea that appearance is to be explained by something not observed. From Aristotle comes the idea of induction and seeking common features of things as explanations of them. From Francis Bacon comes the idea of an automatic algorithm for inductively discovering common causes. From Descartes come the ideas of analysis and synthesis that underlie the emphasis on analysis of variables into orthogonal or linearly independent factors and focus on reproducing (synthesizing) the correlation matrix from the factors. From empiricist statisticians like Pearson and Yule comes the idea of exploratory, descriptive statistics. Also from the empiricist heritage comes the false expectation some have that factor analysis yields unique and unambiguous knowledge without prior assumptions -- the inductivist fallacy. This expectation founders on the indeterminacy of factors, even after their loadings are defined by rotation. Indeterminacy is unavoidable in the interpretation of common factors because the process of interpretation is inductive and inductive inferences are not uniquely determined by the data on which they are based. But from Kant we learn not to discard inductive inferences but to treat them as hypotheses that must be tested against additional data to establish their objectivity. And so the conclusions of exploratory factor analyses are never complete without a subsequent confirmatory analysis with additional variables and new data.

First order or higher order general factor

The Schmid‐Leiman decomposition of a hierarchical factor model converts the model to a constrained case of a bifactor model with orthogonal common factors that is equivalent to the hierarchical model. This article discusses the equivalence and near‐equivalence of the hierarchical and bifactor models and the implications of the difficulty of distinguishing between these models because of low power in samples commonly found in academic research.

The weighted varimax rotation and the promax rotation

Kaisers iterative algorithm for the varimax rotation fails when (a) there is a substantial cluster of test vectors near the middle of each bounding hyperplane, leading to non-bounding hyperplanes more heavily overdetermined than those at the boundaries of the configuration of test vectors, and/or (b) there are appreciably more thanm (m factors) tests whose loadings on one of the factors of the initialF-matrix, usually the first, are near-zero, leading to overdetermination of the hyperplane orthogonal to this initialF-axis before rotation. These difficulties are overcome by weighting the test vectors, giving maximum weights to those likely to be near the primary axes, intermediate weights to those likely to be near hyperplanes but not near primary axes, and near-zero weights to those almost collinear with or almost orthogonal to the first initialF-axis. Applications to the Promax rotation are discussed, and it is shown that these procedures solve Thurstones hitherto intractable “invariant” box problem as well as other more common problems based on real data.

The effect of additional variables on factor indeterminacy in models with a single common factor

Abstract“Determinate” solutions for the indeterminate common factor ofp variables satisfying the single common factor model are not unique. Therefore an infinite sequence of additional variables that conform jointly with the originalp variables to the original single common factor model does not determine a unique solution for the indeterminate factor of thep variables (although the solution is unique for the factor of the infinite sequence). Other infinite sequences may be found to determine different solutions for the factor of the originalp variables. The paper discusses a number of theorems about the effects of additional variables on factor indeterminacy in a model with a single common factor and draws conclusions from them for factor theory in general.

Trait Ratings from Descriptions of Behavior As Mediated by Components of Meaning

This study examined the role language plays in mediating the influence of verbal descriptions of persons on trait ratings of those persons. Subjects were given written descriptions of the behavior of fictitious persons in a work situation and were asked to rate them on fifteen trait- adjective scales. In one condition of the experiment, specific information about certain traits was withheld, forcing subjects to rate persons on traits for which they had no direct behavioral clues. In the other two conditions, the specific information was provided. Providing specific information about a trait directly influenced ratings on that trait even when sufficient general information on that trait was given. In one condition, the influence on the ratings of the additional behavioral clues was such that a new latent variable representing an additional component of meaning was called for in the structural equation model.

Confirmatory Factor Analysis

Exploratory and confirmatory factor analysis reflect respectively two different approaches to the philosophy of science. To understand the differences between these two approaches, we must first consider their philosophical background.

Limited Information Parameter Estimates for Latent or Mixed Manifest and Latent Variable Models

We argue for separate analyses of the measurement and structural portions of latent or mixed manifest and latent variable models. We present limited information (single equation) procedures for estimating parameters in the structural portion of these models. These include parameter estimation procedures for recursive or nonrecursive relations, and procedures for testing zero-effect hypotheses. We then compare full and limited information estimates in a Monte Carlo analysis of sample correlation matrices that contained structural model misspecifications. Both full and limited information estimates identified misspecified nonzero effects reasonably well. However, limited information estimates were far superior in detecting misspecified zero-effect hypotheses. We recommend limited information parameter estimation procedures over full information techniques for (a) testing specific causal hypotheses and (b) locating specific structural model misspecifications.