= Value of canonical variate v for individual i
wcp = Canonical weight relating variate v and variable p
xpi = Value of variable p for individual i.
Note how similar Equation 1.1 is to Equation 1.2. In both cases, the observed variables are combined to create one or more linear combination scores. The difference in the two approaches is in the criteria used to obtain the weights. As noted above, for DA the criteria involve maximizing group separation on the means of Df, whereas for CC the criteria is the maximization of correlation between Cv for the two sets of variables.
The final statistical model that we will contrast with EFA is partial least squares (PLS), which is similar to CC in that it seeks to find linear combinations of two sets of variables such that the relationship between the sets will be maximized. This goal stands in contrast to EFA, in which the criterion for determining factor loadings is the optimization of accuracy in reproducing the observed variable covariance/correlation matrix. PLS differs from CC in that the criterion it uses to obtain weights involves both the maximization of the relationship between the two sets of variables as well as maximizing the explanation of variance for the variables within each set. CC does not involve this latter goal. Note that PCA, which we discuss in Chapter 3, also involved the maximization of variance explained within a set of observed variables. Thus, PLS combines, in a sense, the criteria of both CC and PCA (maximizing relationships among variable sets and maximizing explained variance within variable sets) in order to obtain linear combinations of each set of variables.
A Brief Word About Software
There are a large number of computer software packages that can be used to conduct exploratory factor analysis. Many of these are general statistical software packages, such as SPSS, SAS, and R. Others are specifically designed for latent variable modeling, including Mplus and EQS. For many exploratory factor analysis problems, these various software packages are all equally useful. Therefore, you should select the one with which you are most comfortable, and to which you have access. On the other hand, when faced with a nonstandard factor analysis problem, such as having multilevel data, the use of specialized software designed for these cases might be necessary. In order to make this text as useful as possible, on the book website at study.sagepub.com/researchmethods/qass/finch-exploratory-factor-analysis, I have included example computer code and the annotated output for all of the examples included in the text, as well as additional examples designed to demonstrate the various analyses described here. I have attempted to avoid including computer code and output in the book itself so that we can keep our focus on the theoretical and applied aspects of exploratory factor analysis, without getting too bogged down in computer programming. However, this computer-related information does appear on the book website, and I hope that it will prove helpful to you.
Outline of the Book
The focus of this book is on the various aspects of conducting and interpreting exploratory factor analysis. It is designed to serve as an accessible introduction to this topic for readers who are wholly unfamiliar with factor analysis and as a reference to those who are familiar with it and who need a primer on some aspect of the method. In Chapter 2, we will lay out the mathematical foundations of factor analysis. This discussion will start with the correlation and covariance matrices for the observed variables, which serves as the basis upon which the parameters associated with the factor analysis model are estimated. We will then turn our attention to the common factor model, which expresses mathematically what we see in Figure 1.1. We will conclude Chapter 2 with a discussion of some important statistics that will be used throughout the book to characterize the quality of a particular factor solution, including eigenvalues, communalities, and error variances.
Chapter 3 presents the first major step in conducting a factor analysis, extraction of the factors themselves. Factor extraction involves the initial estimation of the latent variables that underlie a set of observed indicators. We will see that there are a wide range of methods for extracting the initial factor structure, all with the goal of characterizing the latent variables in terms of the observed ones. The relationships between the observed and latent variables are expressed in the form of factor loadings, which can be interpreted as correlations between the observed and latent variables. The chapter describes various approaches for estimating these loadings, with a focus on how they differ from one another. Finally, we conclude Chapter 3 with an example. Chapter 4 picks up with the initially extracted factor loadings, with a discussion of the fact that the initially extracted loadings are rarely interpretable. In order to render them more useful in practice, we must transform them using a process known as rotation. We will see that there are two general types of rotation: one allowing factors to be correlated (oblique) and the other which restricts the correlations among the factors to be 0 (orthogonal). We will then describe how several of the more popular of these rotations work, after which we present a full example, and then conclude the chapter with a discussion of how to decide which rotation we should use.
One of the truths about exploratory factor analysis is that the model is indeterminate in nature. This means that there are an infinite number of mathematically 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
