Encyclopedia of Sociology Vol

scaling is not a requirement of the model. In fact, in confirmatory factor models, it is often desirable to leave the variables in their original metric, especially when comparing the properties of these models across populations or subpopulations (see Alwin 1988b).

According to the common factor model, each observed z variable is, then, a linear function of a set of k latent or unobserved variables and a residual variable, ui, (also unobserved), which contains variation specific to that particular variable and random measurement error. The a coefficients in equation 1 are ‘‘factor loadings.’’ They reflect the linkage between the ‘‘observed’’ variables and the ‘‘unobserved’’ factors. In the case of uncorrelated factor these loadings equal the correlations of the variables with the factors. The loadings thus provide a basis for interpreting the factors in the model; factors obtain their meaning from the variables to which they are linked and vice versa. Thus, in many investigations the primary objective is to estimate the magnitudes of these factor loadings in order to obtain a meaningful understanding of the nature of the data.

The k latent variables are called common factors because they represent common sources of variation in the observed variables. As such, these common factors are thought to be responsible for covariation among the variables. The unique parts of the variables, by contrast, contribute to lack of covariation among the variables. Covariation among the variables is greater when they measure the same factors, whereas covariation is less when the unique parts of the variables dominate. Indeed, this is the essence of the model—variables correlate because they measure the same things. This was the basis of Spearman’s original reasoning about the correlations among tests of mental ability. Those tests correlated because they measured the same general factor. In the general case those variables that correlate do so because of their multiple sources of common variation.

Common variation in the aggregate is referred to as communality. More precisely, a variable’s communality is the proportion of its total variation that is due to its common sources of variation. The communality of variable i is denoted h2i. A variable’s uniqueness, denoted u2i, is the complement of the communality; that is, u2i = 1.0 - h2i. The

uniqueness is thought of as being composed of two independent parts, one representing specific variation and one representing random measurement error variation; that is, u2i = s2i + e2i. (This notation follows traditional psychometric factor analysis notation. Each of these quantities is a variance, and, thus, in the covariance modeling or structural equation modeling tradition these quantities would be represented as variances, such that σ2ui = σ21i + σ2ei [see below].) Specific variance is reliable variance, and thus the reliability of variable i can be expressed as r2i = h2i + s2i. Unfortunately, because of the presence of specific variance in most variables, it is virtually impossible to use the traditional form of the common factor model as a basis for reliability estimation (see Alwin 1989; Alwin and Jackson 1979). The problem is that the common factor model typically does not permit the partitioning of u2i into its components, s2i and e2i. In the absence of specific variance, classicial reliability models may be viewed as a special case of the common factor model, but in general it is risky to assume that e2i = u2i. Alwin and Jackson (1979) discuss this issue in detail. Some attempts have been made to augment the traditional latent ‘‘trait’’ model inherent in the common factor model by adding ‘‘method’’ factors based on the multitrait–multimethod design of measurement within the framework of confirmatory factor models (see Alwin 1974; Alwin and Jackson 1979; Werts and Linn 1970). This provides a partitioning of the specific variance due to method, but it does not provide a general solution to the problem of handling specific variance.

Returning to the above example (in equation 1), in Spearman’s case (the one-factor case) each variable contains a common factor and a specific factor, as shown in equation 2.

zp = ap1ƒ1 + sp

In this case h2i = a2i and u2i = s2i. Spearman’s (1927) theory in essence assumes perfect measurement, not unlike most common factor models. However, unlike researchers of today, Spearman was very

concerned about measurement errors, and he went to great lengths to correct his observed correlations for imperfections due to random errors of measurement (Spearman 1904b). Thus, when applied to such corrected correlational data, these assumptions may be appropriate.

As can be seen from the equations for Spearman’s model (equation 2), the correlations among variables zi, and zj ry = E[zi zj] (the expected value of the cross-products of the z scores for the two variables) may be written as ry = aiaj . For example, if p = 3, the correlations among the variables can be written as

r12 = aia2, r13 = aia3, and r23 = a2a3. In vector notation (introduced in greater detail below), the common

parts of the correlations among the variables of the model are composed of the matrix product AA′. In the case where p = 3, the matrix A is written as in equation 3,

a1

a3

and the product AA′ is written as in equation 4.

The variances of the variables are also affected by the common factors, but, as indicated in the foregoing, there is a residual portion of variance containing specific and unreliable variance. In Spearman’s model the variance of variables i is as shown in equation 5.

Then it can be seen that the correlation matrix is equal to R = AA′ + U2, where the matrix U2 for the p = 3 case is written in matrix form as in equation 6.

These results have general applicability, as will be seen below.

ESTIMATION AND TESTING OF THE

FACTOR MODEL

Before proceeding to the more general uses of the model, it is important to review the logic behind Spearman’s approach. In the general Spearman case, the correlation of two variables is equal to the product of their loadings on the general factor; that is, rij = aiaj . Recall that under this model the a coefficients represent the correlations of the variables with the factor. Spearman reasoned therefore that if the model were true (that is, if a single unobserved common factor could account for the correlations among the observed variables), then certain things had to hold in the empirical data.

Spearman reasoned that, if the single factor model holds, the partial correlation between any two variables, holding constant the underlying common factor, rijf should be zero. This stems from the fact that the numerator of this partial correlation, rij - rifrjf , is zero, because under the model

rij - aiaj = aiaj - aiaj = 0. Of course, it is not possible to calculate such a partial correlation from the data

because the factor score, f, does not exist except in the theory. Spearman, however, recognized a specific pattern to the components of the correlations under the model. He noted that, if the single factor model held for p = 4, the intercorrelations of the variables had to satisfy two independent conditions, referred to by Spearman (1927) as vanishing tetrads, shown in equation 7.

r12 r34 – r14 r23 = 0

(7)

r13 r24 – r14 r23 = 0

Note that the case of p = 3 is a trivial case, since a one-factor model can always be used to describe the intercorrelations among three variables. For p = 5 there are [p(p - 1) (p - 2) (p - 3)]/8 different tetrads (see Harman 1976), which equals fifteen. Not all of the possible tetrad differences formed from these fifteen are independent, and for one factor to explain the correlations, there are p (p - 3)/2 independent tetrad differences. Thus, in the case of five variables there are five tetrad differences that must vanish, and for six there are nine, and so forth.

Although in recent years there has been a revival of interest in Spearman’s vanishing tetrads for sets of four variables (Glymour et al. 1987), at

the time he developed this logic there was little that could be done computationally with very large problems. Thurstone (1947) developed the centroid method as an approximation to the principal axes approach involved in Spearman’s early work, which was in common use during the 1940s and 1950s, but with the development of the high-speed computer, principal axes methods became (and remain) quite common in many applications of the model.

In exploratory factor analysis, where the number of factors of the model is not known beforehand, estimation is carried out by way of an eigen- value/eigen-vector decomposition of some matrix, either R or some estimate of R - U2. There is a wide variety of types of factor analyses that can be done—principal component factor analysis (which analyzes the p first nonzero components of R), communality-based factor analysis (which analyzes R with a communality estimate in the diagonal), alpha factor analysis, canonical factor analysis, or image analysis (see Harris 1963, 1964). Few developments have been made in these approaches since the 1960s, although there continues to be considerable debate about the desireable properties of these various approaches (e.g., see Widaman 1991).

Perhaps the most important development affecting exploratory factor analysis since the 1960s has been the development of maximum-likelihood factor analysis. Maximum-likelihood estimation, however, requires the prior estimate of the number of factors. These methods are most often discussed in connection with confirmatory factor analysis, although the approach to exploratory factor analysis discussed by Lawley and Maxwell (1971) illustrates how a form of traditional exploratory factor analysis can be done by setting minimal constraints on the model and testing successive hypotheses about the number of factors. A discussion of these models occurs in a subsequent section on confirmatory factor analysis. Before this, a more formal presentation of the factor model in matrix form is given, along with a discussion of several of the longstanding problems that dominate the literature on factor analysis, specifically the problem of estimating communality, the problem of estimating factor scores, and the problem of determining factorial invariance.

THE FACTOR MODEL IN MATRIX

NOTATION

We can generalize the model given above for the case of multiple factors, k, in matrix notation. And again, the factor model can be easily represented in terms of the data matrix at hand. (The model can also be written compactly in vector notation for populations of interest. This is the approach taken in the subsequent discussion of confirmatory factor analysis.) The data matrix in this case can be represented as a p by n array of variable scores. Let Z′ symbolize this p by n data matrix. Using this notation, write the common factor model for a set of p variables as Z′ = AF′ + UW′ where Z′ is as defined above, A is a p by k factor pattern matrix (in the case of uncorrelated factors A is called the factor structure matrix), F′ is a k by n matrix of hypothetical factor scores, U is a p by p diagonal matrix of unique-score standard deviations (defined such that the element ui is the square root of the unique variances, σ2ui), and W′ is a p by n matrix of hypothetical unique scores. Note that the factors (both common and unique) are never observed—they exist purely at the hypothetical level. Note also that because we have standardized the variables (the z’s) to be centered about the mean and to have standard deviations of unity, the factor scores in this model are theoretically standardized in the same fashion. In other words, E(F′F) = Ik , and the variances of the unique scores are equal to E(W′W) = U2, assumed to be a diagonal matrix.

Traditionally, the factor model assumed that the factors of this model were uncorrelated, that the unique parts of the data (the W) are uncorrelated with the common parts (the F), and that the unique variation in variable i is uncorrelated with the unique variation in variable j, for all i and j. In matrix notation, the factor model assumes that

E(F′F) = Ik , E(W′W) = U2, and E(F′W) = E(W′F) = 0. In other words, the factors of the model are

uncorrelated with one another and have variances of unity. Also, the common factors and unique factors are uncorrelated, and the unique factors are uncorrelated among themselves.

This type of notation helps clarify the fact that factor analysis is in effect interested in the ‘‘reduced’’ data matrix, Z′ - UW′, rather than Z′. Consequently, the factor model is concerned with

the decomposition of the matrix R - U2 (the correlation matrix with communalities in the diagonal) rather than R (the correlation matrix with unities in the diagonal), since in equation 8 we demonstrate the following:

This demonstrates an often misunderstood fact —namely, that factor analysis focuses on the reduced-correlation matrix, R - U2, rather than on the correlation matrix (with 1s in the diagonal). As will be clarified below, this is the fact that differentiates factor analysis from principal components analysis—the latter operates on the correlation matrix. In factor analysis, then, one must begin with some estimate of I - U2 or H2, the matrix of communalities, and then work on the decomposition of R - U2. This poses a dilemma, since neither the common nor unique factors are observed, and it is therefore not possible to know U2 and H2 beforehand. The objective is to come up with an estimate of H2 that retains a positive semidefinite property to R - U2. At the same time, H2 is one aspect of what one wants to discover from the analysis, and yet in order to estimate the model one must know this matrix beforehand. The solution to this problem is to begin with an ‘‘estimate’’ of the communalities of the variables, and then through an iterative procedure new estimates are obtained and the solution is reached through convergence to some criterion of fit.

COMMUNALITY ESTIMATION AND THE

NUMBER OF FACTORS

Determining the number of factors in exploratory factor analysis is one of the fundamental problems involved in arriving at a solution to the parameters of the factor model. The problem essentially involves determining the rank, k, of the matrix R - U2, where these matrices are as defined above. Technically, we want to find a matrix U2 that will retain the property of positive semidefiniteness in R - U2 with the smallest possible rank (Guttman 1954). The rank of this matrix in this case is the minimum number of factors necessary to reproduce the offdiagonal elements of R. Thus, the problem of

determining k is closely related to the communality estimation problem, that is, determining an estimate for the diagonal of R - U2, that is, H2.

Guttman (1954) outlined the problem of deciding on the number of factors and compared three principles for estimating k via solutions to the communality estimation problem. He described a ‘‘weak lower bound’’ on the number of factors, k1, as the nonnegative roots (eigen values) of the matrix R - I. This is equivalent to the number of roots of R greater or equal to unity, since R and R - I differ only by I, and their roots differ therefore only by unity. Guttman shows that k1 is a lower bound to k, that is, k ≥ k1. A second principle, one that also implies another approach to estimating communality, is based on the matrix R - D, where D is a diagonal matrix whose elements, l - r2j (j = 1, p), are the result of unity minus the square of the largest correlation of variable j with any of the p - 1 other variables. Guttman shows that k2 is also a lower bound to k, such that k ≥ k2. A third and perhaps the most common approach to estimating communalities is based on the idea that the squared multiple correlation for each variable predicted on the basis of all the other variables in the model is the upper limit on what the factors of the model might reasonably explain. If we define the matrix R - C2, where C2 is a diagonal matrix whose elements C2j (1, p) are equal to 1 - r2j, where r2j is the squared multiple correlation of variable j with the remaining p - 1 variables. Guttman shows that k3 is also a lower bound to k. This third lower bound is often referred to as Guttman’s strong lower bound since he showed the following relationships among the lower bounds: k ≥ k3 ≥ k2 ≥ k1. In practice, k1 may be adequate but it could be wrong, and, if wrong, it is likely to be too small. The use of k1 is probably questionable in the general case. The use of k2 is obsolete and not practicable. It estimates communality in the way of the Thurstone centroid method, which is only a rough approximation to a leastsquares approach. Perhaps the best solution is the choice of k3. It is less likely to overlook common factors, as k1 might, since 1 - s2j is a lower bound to h2j. It should be pointed out that the lower bounds k1 and k3 are what distinguish the two main approaches to factor analysis, namely an incomplete principal components decomposition (referred to as principal components factor analysis) and the principal factor method of analysis.

FACTOR ANALYSIS VERSUS PRINCIPAL

COMPONENTS ANALYSIS

It was mentioned above that it is not understood well enough that factor analysis is concerned mainly with the matrix R - U2 rather than with R. This is in fact one of the things that distinguishes factor analysis from principal components analysis. However, the differences between the two are more fundamental. Factor analysis is based on a model, a particular theoretical view (hypothesis, if you like) about the covariance (or correlational) structure of the variables. This model states (as given above) that the correlation matrix for a set of variables can be partitioned into two parts—one representing the common parts of the data and one representing uniqueness—that is, R = A′A + U2. The factor model states, first, that the off-diagonal elements of A′A equal the off-diagonal elements of R and, second, that the elements of U2 (a diagonal matrix) when added to the diagonal elements of A′A give the diagonal elements of R. Thus, the factor model posits a set of k hypothetical variables (k < < p) that can account for the interrelationships (or correlations) of the variables but not for their total variances.

In contrast to this, principal components is not a model in the same sense—it is best viewed as a method. It is one method for obtaining an initial approximation to the common factor model (see Guttman’s weak lower bound, discussed above), but it is extremely important to distinguish such an ‘‘incomplete’’ principal components solution (one associated with the roots of R that are equal to or greater than unity) from the full-rank principal components decomposition of R (see Maxwell 1977).

Any square symmetric nonsingular matrix, for example, R = R′, can be written in the form R = QD2Q′, where D2 is a diagonal matrix of order p containing eigen values ordered according to decreasing magnitude, and Q is a matrix of unitlength eigen vectors (as columns) associated with the eigen values. Q is an orthonormal matrix, Q′Q = I = QQ′. This model is referred to as the principal components decomposition of R. Typically, one either analyzes a correlation matrix with 1s in the diagonal, or a covariance matrix with variances in the diagonal, in the application of this decomposition. In this model the correlation matrix, R, is formed by a centered or deviation-score data matrix scaled so that the variables have variances of

unity. Let Z′ be the p × n data matrix, as above. Note that the expected value of Z′Z = QD2Q′, since

Z′ = QDY′.

If the correlation matrix is of full rank, then there will be p columns in Q. This means that in this case the principal components model involves a transformation of p variables into a set of p orthogonal components. When the correlation matrix is singular, meaning that the rank of the matrix is less than p, the principal components decomposition is said to be incomplete, but from the point of view of factor analysis this is often irrelevant since it is the matrix R - U2 that is of interest to the factor analyst.

If P is an r × p components matrix (P = QD), and r = p, then it is well known that Y′ = (P′P)-1;P′Z′ = P-1 Z′, where Y′ is a set of r component scores, P is as defined above, and Z′ is a p × n data matrix involving p variables and n units of observation (e.g., persons, cities, social organizations). In other words, component scores (in contrast to factor scores) are directly calculable.

THE ROTATION PROBLEM–CORRELATED VERSUS UNCORRELATED FACTORS

Principal components are by definition uncorrelated with one another. The basic objective of the method is to obtain a set of p orthogonal (uncorrelated) new variables via a linear transformation of the p original variables. Factors are different. Factors may be uncorrelated, and in classical exploratory factor analysis one always begins with a set of uncorrelated factors, but in general this is not a requirement. Indeed, in exploratory factor analysis the factors one obtains are uncorrelated because of the nature of the methods used, but normally one performs a transformation or rotation of these factors to achieve a more pleasing representation for interpretation purposes.

Two types of rotations are available—those that preserve the uncorrelated nature of the factors, such as the varimax and quartimax rotations (see Kaiser 1958; Neuhaus and Wrigley 1954), and those that allow the factors to be correlated. The latter are called ‘‘oblique’’ rotations because they move the factors out of the orthogonal reference into a vector space that reduces the geometric

angles between them. Using either of these approaches, the basic goal of rotation is to achieve what Thurstone called simple structure, the principle that variables should simultaneously load highly on one factor and low on all other factors. These rotational approaches are relatively straightforward and discussed in all of the textbook descriptions of factor analysis.

FACTORIAL INVARIANCE

Following Thurstone’s (1938, 1947) discussions of factor analysis, students of the method have frequently been concerned with the problem of the correspondence between factors identified in separate studies or in subgroups of the same study. Using Thurstone’s terminology, a concern with the correspondence of factors refers to the invariance of factors. The concern with factorial invariance has generated an array of methods for comparing factors (see Mulaik 1972). The most common approach to the problem involves the computation of an index of factor similarity for corresponding factors given estimates of a factor model using the same variables in two or more samples. The details of various strategies for estimating factor similarity will not be covered here, as descriptions can be found in a variety of factor analysis textbooks.

These approaches were developed primarily for results obtained from exploratory factor analysis, and it can be argued that the issues of factorial invariance can be more fruitfully addressed using the methods of confirmatory factor analysis (see Jöreskog 1971b; Lawley and Maxwell 1971). The technical aspects of these methods will not be reviewed here, as they have been exposited elsewhere (see Alwin and Jackson 1979, 1981). Suffice it to say that issues of factorial invariance can be phrased, not only with respect to the correspondence of the factor pattern coefficients (the A matrix) across populations, but also with respect to other parameters of the model as well, particularly the matrix of factor interrelationships (correlations and covariances) and the matrix of disturbance covariances.

It is perhaps useful in this context to raise a more general question regarding the nature of factorial invariance that is sought in the analysis of the factorial content of measures. In general there

is no consensus regarding whether stronger or weaker forms of invariance are necessary for comparisons across populations or subpopulations. Horn and associates (1983), for example, suggest that rather than the ‘‘equivalence’’ of factor structures across populations, weaker ‘‘configurational’’ forms of invariance are ‘‘more interesting’’ and ‘‘more accurate representations of the true complexity of nature.’’ By contrast, Schaie and Hertzog (1985, pp. 83–85) argue that the condition of factorial invariance, that is, ‘‘the equivalence of unstandardized factor loadings across multiple groups,’’ is critical to the analysis of differences among groups and developmental changes within groups.

Clearly, these represent extremes along a continuum of what is meant by the question of factorial invariance. On the one hand, for strict comparison of content across groups, it is necessary to have the same units of measurement, that is, invariance of metric. This requires the same variables measured across populations, and some would also argue that such invariance of metric requires that the relationships of the variables and the factors be equivalent across populations (see Jöreskog 1971a). On the other hand, if the same pattern of loadings seems to exist, it may be an example of misplaced precision to require equivalence in the strictest sense. Of course, the resolution of these issues has implications for other uses to which factor analysis is typically put, especially the construction of factor scores and the use of causal modeling strategies to compare substantive processes across groups.

THE PROBLEM OF FACTOR SCORE

ESTIMATION

Researchers using the common factor model as a data reduction tool typically engage in the estimation of such models in order to obtain scores based on the factors of the model, which can then be used to represent those factors in further research. As will be shown here, factor scores can never be computed directly (as in the case of component scores). Factor scores are always estimated, and, due to the nature of the factor model, ‘‘estimated factor scores’’ never correlate perfectly with the underlying factors of the model. An important

alternative to factor scores is what have come to be called ‘‘factor-based’’ scores, which are scores derived from the results of the factor analysis, using unit versus zero weightings for the variables instead of the factor score weights derived from one or another method of estimating factor scores. Factor-based scores, which are frequently more easy to justify and much more practical, typically correlate so highly with factor score estimates as to make one skeptical of the need for factor score estimates at all (see, e.g., Alwin 1973b).

However, it is important that factor analysts understand the nature of the factor score estimation problem, regardless of whether factor score estimates become any more appealing than the simpler and more stable factor-based scores. The factor score estimation problem can best be seen in terms of an interest in solving for the matrix F′ in the above matrix representation of the common factor model, Z′ = AF′ + UW′. This can be done analytically, but, as will be seen, it is not possible to do so empirically because of the nature of the model. To solve for F′ in this model we arrive at the following representation (without going through all of the necessary steps): F′ = (A′A)-1 A′ [Z′ - UW′]. The calculations implied by this expression cannot actually be carried out because one never knows W′. This is known as the ‘‘factor measurement problem,’’ which results in the fact that factor scores cannot be computed directly and must therefore be estimated.

The question, then, becomes whether it is possible to estimate factor scores in a manner that is useful, given what is known—Z, A, and U2. Several approaches have been set forth for estimating the factors, all of which involve some transformation of the data matrix Z′ into a set of k scores that vary in their properties (see Harris 1967; McDonald and Burr 1967). Most of these methods bear some resemblance to the analytic solution for F′ above, but there are some technical differences. One of the most commonly misunderstood facts involved in the estimation of factor scores is that the factor pattern coefficient matrix, A, cannot be applied directly to the estimation of factors; that is, F′ cannot be estimated by A′Z′. This, of course, should be clear from the above representation, but it is often used, probably due to ignorance of the more ‘‘correct’’ factor score estimation strategies.

There are four recognized strategies for estimating scores representing the common factors of the model, given Z, A, and U2 (Alwin 1973b). All of these approaches are typically discussed for a model such as that discussed above, namely a set of uncorrelated factors scaled to have 0 means and standard deviations of 1. It is not the purpose of the present discussion to evaluate the properties of these various approaches to factor score estimation, but a brief summary can perhaps provide a guide to the technical literature on this topic. It is important to emphasize that none of these approaches produces factor score ‘‘estimates’’ that are perfectly correlated with the underlying factors of the model. Some of these approaches produce univocal score estimates, meaning that each factor score correlates only with the factors they are intended to measure and not with factors they are not intended to measure. Only one of the approaches produces a set of factor score estimates that reflect the property of uncorrelated factors with unit standard deviations. But it is difficult in practice to evaluate the desirability of any of the properties of factor score estimates.

METHODS OF CONFIRMATORY FACTOR

ANALYSIS

Confirmatory factor analysis, unlike the methods of exploratory factor analysis, begins with prior knowledge regarding the number of factors and something about the nature of their relationships to the observed variables. In the absence of such knowledge, confirmatory factor analysis is not appropriate. In the typical situation of confirmatory factor analysis, then, the investigator begins with some specific theoretical hypotheses involving a model that can be tested against the data. Naturally, there is an extremely large number of such possible models, so it should be obvious that the techniques cannot easily be used to ‘‘search’’ for the best possible set of restrictions involving k factors (but see Glymour et al. 1987).

The bulk of this review has been devoted to the use of exploratory techniques of factor analysis. This imbalance is perhaps justified, given what is known within sociology about the common factors in our data. Exploratory factor analysis techniques are likely to be much more useful,

especially at a stage where less knowledge has been developed. And within a field like sociology, where there is a broad variety of competing concepts and paradigms, exploration of data may often be the most salutary strategy. There are, however, clearcut instances where the use of confirmatory factor analysis techniques is in order, and the remainder of this discussion focuses on these situations.

Consider the following model for a set of p variables: y = ν + λ η + ε, where ν is a vector of location parameters or constants representing the origins of measurement of the p observed variables, η is a vector of k latent variables or factors, and ε is a vector of random disturbances for the p observed variables. The covariance properties associated with η and ε are basically the same as those discussed in the section on exploratory factor analysis for F and W, except that in general these are not required to be uncorrelated within the common factor set. And, of course, there is no restriction on the metric of the variables; that is, the p variables and k factors are not necessarily standardized to have 0 means and standard deviations of unity. The coefficient matrix, λ, is a matrix of regression coefficients relating the p observed variables to the k latent factors. In the case of a single population, one can easily consider the p variables centered (which would remove the vector of location constants) and scaled to have unit variance, but in the situation where one wants to compare populations neither of these constraints is probably desirable.

The use of these models requires the distinction between constrained and unconstrained parameters. Typically, one refers to parameters of the model as fixed if they are constrained to have a particular value, such as a factor loading of 0 or, in the case of the variance of a latent factor, a variance of unity. By contrast, the unknown parameters of the model, for example the λs, are referred to as free parameters, which means that they are estimated in the model under the constraints specified for fixed parameters. Thus, one speaks of fixed or constrained parameters on the one hand and free or estimable parameters on the other. The major breakthrough in the use of this type of model was the development of computer programs that allow one to fix certain parameters of the model to known quantities while estimating

the free parameters under these constraints. The general approach also allows one to specify causal connections among the latent factors of the model, and it allows one to specify correlations among the errors in the variables and the errors in the equations connecting the latent factors.

Consider the factor model for the situation where there are p = 4 variables and k = 2 factors, with the pattern of factor pattern coefficients shown in equation 9, where the first two variables are believed to measure η1 and the third and fourth variables are said to measure η2. This is the kind of situation described by Costner (1969), who developed an approach to confirmatory factor analysis using Spearman’s tetrad differences. Of course, there are more efficient estimation strategies than those proposed by Costner. In any event, the point of this example is that the investigator begins not only with a specific number of factors in mind but also with a specific set of assumptions about the pattern of loadings.

In the general case the covariances and correlations of the common factors of the model, E(η′η), can be symbolized by ψ (sometimes this matrix is denoted as Φ, but there are many ways in which to symbolize these quantities), and the covariances of the disturbances (or errors) on the variables can be symbolized by θe. Neither of these two matrices, ψ and θe, is required by the model to be diagonal, although in the simplest form of the confirmatory model, θe is often assumed to represent a set of uncorrelated disturbances. In more complicated forms of the model, within constraints placed by the identification problem, both of these matrices can be nondiagonal. In either case, the model here is written with the assumption that the investigator has prior theoretical knowledge regarding the number of sources of common variation and that the η vector exhausts those sources.

Any application of this model requires that it be identified, which essentially means that there must be enough independent information within the covariance and correlation structure being

analyzed sufficient to solve for the unknown parameters of the model. In general, there need to be k2 constraints on a particular common factor model, that is, in the parameters in λ and ψ. Other constraints, of course, are possible. Space does not permit the discussion of these matters, but a detailed treatment of these issues can be found elsewhere (see Alwin 1988a; Alwin and Jackson 1979).

The True-Score Models. It can be shown that a well-known class of measurement models that form the basis for classical test theory (Lord and Novick 1968) can be specified as a special case of confirmatory factor analysis (see Alwin and Jackson 1979; Jöreskog 1971a). In brief, by placing particular constraints on the λ and θ, matrices of the model, one can estimate the parameters of models that assume the measures are parallel, tauequivalent, or congeneric. Of course, as indicated earlier, in order to use the common factor model in such a fashion, one must be reasonably sure that there is little or no specific variance in the measures. Otherwise, one runs the risk of confusing reliable specific variance with measurement error variance.

Multitrait–Multimethod Models. In addition to the application of confirmatory factor analysis to the estimation of classical true-score models, several attempts have been made to augment the traditional latent ‘‘trait’’ model, inherent in the classical model, by the addition of ‘‘method’’ factors based on the multitrait–multimethod design of measurement within the framework of confirmatory factor models (see Alwin 1974; Alwin and Jackson 1979; Werts and Linn 1970). This provides a partitioning of the specific variance due to method, but it does not provide a general solution to the problem of handling specific variance. While these models can be very useful for partitioning item-level variance into components due to trait, method, and error, they place relatively high demands on the measurement design. And while the required designs are relatively rare in practice, these models help sensitize the researcher to problems of correlated method error (see e.g., Costner 1969). Recent work in this area has shown that the multitrait-multimethod model can be fruitfully applied to questions of survey measurement quality, assessing the extent to which correlated measurement errors account for covariation among survey measures (see Alwin, 1997; Scherpenzeel, 1995).

Multiple-Indicator, Multiple-Cause Models.

One of the simplest forms of causal models involving latent common factor models is one in which a single latent endogenous variable having several indicators is determined by several perfectly having measured exogenous variables. Jöreskog (1974) and Jöreskog and Goldberger (1975) refer to this as a multiple-indicator, multiple-cause (MIMC) model. This kind of model has certain similarities to the canonical correlation problem (see Hauser and Goldberger 1971).

Analysis of Change—Simplex Models. One type of model that can be viewed as a confirmatory factor model, and is useful in analyzing change with respect to the latent common factors over time, falls under the rubric of simplex models (Jöreskog 1974). Such models are characterized by a series of measures of the same variables separated in time, positing a Markovian (lag-1) process to describe change and stability in the underlying latent variable. This model can be used in situations where there is a single variable measured over time (see Heise 1969) or in situations where there are multiple measures of the latent variable at each time (see Wheaton et al. 1977).

These models have proven to be valuable in the study of human development and change, especially applied to panel studies of individual lives over time (Alwin 1994, 1995a; Alwin and Krosnick 1991; Alwin et al. 1991; Asendorf 1992.); A related set of models in this area are latent growth curves, in which levels and trajectories of the growth in latent factors can be conceptualized and estimated (e.g., Karney and Bradbury 1995; McArdle 1991; McArdle and Anderson 1990; Willett and Sayer 1994). In such applications the focus is on the nature of growth processes and the correlates/predictors of individual change in latent factors over time. This is a natural extension of traditional methods of confirmatory factor analysis and causal modeling strategies within the context of longitudinal data.

Causal Modeling of Factors. If one obtains multiple measures of factors, for which common factor models are believed to hold, and the factors can be specified to be causally related, then it is possible to use confirmatory techniques to estimate the causal influences of factors on one another. Of course, one must be able to justify these

models strongly in terms of theoretical considerations, and there must be considerable prior knowledge (as in the use of confirmatory factor analysis) that justifies the specification of such measurement models. The logic involved in dealing with the linkage between observed and unobserved variables is essentially that involved in confirmatory factor analysis, while the logic applied in dealing with the causal linkages among factors is that involved in path analysis and structural equation modeling. The main point is that the parameters of models that essentially contain two parts—a measurement part specifying a model linking observed and latent variables and a structural model linking the latent variables—can be estimated within the framework of LISREL-type models. The measurement part can typically be viewed within the framework of confirmatory factor analysis, although in some cases an ‘‘induced variable’’ model is more appropriate (Alwin, 1988b).

CONCLUSION

This review is designed to provide an overview of the major issues involved in the use of factor analysis as a research tool, including both exploratory and confirmatory techniques. There are several useful textbook discussions of factor analysis that will aid those who desire further study. Among these, the texts by Gorsuch (1984), Harman (1976), Mulaik (1972), McDonald (1985), and Lawley and Maxwell (1971) are some of the best sources on exploratory factor analysis. There are some recent texts offering instruction on the conceptual understanding and practical guidance in the use of these confirmatory factor analysis and causal modeling strategies (e.g., Bollen 1989; Loehlin 1992). The newest developments in the area involve advances in the analysis of categorical and ordinal data, statistical estimation in the presence of incomplete data, the provision of graphic interfaces for ease of specification of causal models. There are now several major competitors in the area of software packages that can be used to estimate many of the confirmatory factor models discussed here. Although these several packages offer largely comparable material, each offers a somewhat special approach. The LISREL approach to the analysis of covariance structures was first made available in the early 1970s. It is now in its

eighth version and offers many improvements to its use (Jöreskog and Sörbom 1996a). The LISREL8 program is now supplemented by PRELIS (Jöreskog and Sörbom 1996b), which provides data transformaton capabilities, and by SIMPLIS (Jöreskog and Sörbom 1993), which provides a simple command language for formulating LISRELtype models. There are several alternatives to the LISREL approach, including EQS (Bentler 1992), AMOS (Arbuckle 1997), and Mplus (Muthén and Muthén 1998), among others, which provide many attractive features (see West, 1997).

(SEE ALSO: Causal Inference Models; Measurement; Multiple

Indicator Models; Validity).

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———1988a ‘‘Structural Equation Models in Research on Human Development and Aging.’’ In K. Warner Schaie, Richard T. Campbell, William Meredith, and Samuel C. Rawlings, eds., Methodological Issues in Aging Research. New York: Springer.

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