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plausible solutions, and no one of them can be taken as optimal over the others. Thus, we need to have some criteria for deciding what the optimal solution is likely to be. Making this determination is the focus of Chapter 5. First and foremost, we must be sure that the solution we ultimately decide upon is conceptually meaningful. In other words, the factor model must make sense and have a basis in theory in order for us to accept it. Practically speaking, this means that the way in which the variables group together in the factors is reasonable. In addition to this theoretically based determination, there are also a number of statistical tools available to us when deciding on the number of factors to retain. Several of these are ad hoc in nature and may not provide terribly useful information. Others, however, are based in statistical theory and can provide useful inference regarding the nature of the final factor analysis model. We will devote time to a wide array of approaches, some more proven than others, but all useful to a degree. We close the chapter with a full example and some discussion regarding how the researcher should employ these various methods together in order to make the most informed decision possible regarding the number of factors to retain.

      We conclude the book with a chapter designed to deal with a variety of ancillary issues associated with factor analysis. These include the calculation and use of factor scores, which is somewhat controversial. Factor scores are simply individual estimates of the latent trait being measured by the observed indicator variables. They can be calculated for each member of the sample and then used in subsequent analyses, such as linear regression. Given the indeterminacy of the exploratory factor model, however, there is disagreement regarding the utility of factor scores. We will examine different methods for calculating them and delve a bit into the issue of whether or not they are useful in practice. We will then consider important issues such as a priori power analysis and sample size determination, as well as the problem of missing data. These are both common issues throughout statistics and are important in exploratory factor analysis as well. We will then focus our attention on two extensions of EFA, one for cases in which we would like to investigate relationships among latent variables, but where we do not have a clear sense for what the factors should be. This exploratory structural equation modeling merges the flexibility of EFA with the ability to estimate relationships among latent variables. We will then turn our attention to the case when we have multilevel data, such that individuals are nested within some collective, such as schools or nations. We will see how ignoring this structure can result in estimation problems for the factor model parameters, but that there is a multilevel factor model available to deal with such situations. We will conclude the chapter and the book with discussions on best practices for reporting factor analysis results and where exploratory factor analysis sits within the broader framework of statistical data reduction. This discussion will include tools such as discriminant analysis, canonical correlation, and partial least squares regression.

      Upon completing this book, I hope that you are comfortable with the basics of exploratory factor analysis, and that you are aware of some of the exciting extensions available for use with it. Factor analysis is a powerful tool that can help us understand the latent structure underlying a set of observed data. It includes a set of statistical procedures that can be quite subtle to use and interpret. Indeed, it is not hyperbole to say that successfully using factor analysis involves as much art as it does science. Thus, it is important that when we do make use of this tool, we do so with a good sense for what it can and cannot do, and with one eye fixed firmly on the theoretical underpinnings that should serve as our foundation. With these caveats in mind, let’s dive in.

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