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2 Identification Estimation in Path Analysis with Unmeasured Variables' Charles E. Werts, Karl G. Joreskog, Robert L. Linn Educational Testing Service The application of Joreskog's (1970a) general model for the analysis of covariance structures to several path models involving unmeasured variables is discussed. When these path models have a factor-analytic structure, a simple heuristic rule derived from factor analysis may be helpful in determining the identification status of parameters. Joreskog (1970a) developed a method for estimating parameters of models involving structures of a very general form on means, variances, covariances of multivariate observations. A great deal of generality flexibility is achieved in that the method is capable of hling most stard statistical models as well as many nonstard complicated ones. For example, the program may be used to hle such stard problems as multivariate regression, analysis of congeneric tests, factor analysis, analysis of multitrait-multimethod data, analysis of simplexes circumplexes, analysis of multitest-multioccasion data growth data in general, analysis of mixed rom effects ANOVA MANOVA, path analysis, linear structural equations, various other models involving correlated errors of measurement (Joreskog 1969, 1970a, 1970b, 1971).2 The main purpose of this paper is to demonstrate the application of Joreskog's (1970a) general model to path analysis problems involving multiple indicators of underlying constructs. Hauser Goldberger (1970) have provided an introduction to this subject especially in comparing the ad hoc estimating procedures previously used by path analysts to J6reskog's more efficient maximum likelihood approach. We propose to extend the Hauser-Goldberger discussion by considering the formulation of a wider range of models from the path analysis literature. JORESKOG'S GENERAL MODEL FOR THE ANALYSIS OF COVARIANCE STRUCTURES The general model considers a data matrix X(Nxp) of N observations on p variates assumes that the rows of X are independently distributed, 1 The research reported herein was performed pursuant to grant no. OEG (509) with the U.S. Department of Health, Education, Welfare the Office of Education. 2 The relevant computer programs (J5reskog, Gruvaeus, van Thillo 1970 J6reskog, van Thillo, Gruvaeus 1971) may be obtained from Marielle van Thillo at Educational Testing Service, Princeton, New Jersey AJS Volume 78 Number

3 American Journal of Sociology each having a multivariate normal distribution with the same variancecovariance matrix X. It is assumed that E(X)=A*P, (1) where A(NXg) - (a,) P(hXp) = (pti) are known matrices of ranks g h, respectively, g - N, h p t(g X h) = ({st) is a matrix of parameters; that X has the form X, = B(A(DIV? + *2) B` + 02 (2) where the matrices B(p X q) = (10k), A(q X r) = (Xkm), the symmetric matrix ID(r X r) = ((mn) the diagonal matrices 4(q X q)- (8klPk) 0(p X p) = (8ij01) are parameter matrices. Thus the general model is one where means, variances covariances are structured in terms of other sets of parameters that are to be estimated. In any application of this model, p, N X will be given by the data, g, h, q, r, A P will be given by the particular application. In any such application we shall allow for any one of the parameters in Z, B, A, 0, qp E to be known a priori for one or more subsets of the remaining parameters to have identical but unknown values. Thus parameters are of three kinds: (i) fixed parameters that have been assigned given values, (ii) constrained parameters that are unknown but equal to one or more other parameters, (iii) free parameters that are unknown not constrained to be equal to any other parameter. The computer program estimates the free constrained parameters of any such model by the maximum likelihood method provides a test of goodness of fit of the whole model. [J6RESKOG, VAN THILLO, AND GRUVAEUS 1971, pp. 2-3] In large samples, assuming that the observed distributions are normally distributed, the maximum likelihood estimation procedure yields a X2 statistic which measures the fit of the model to the data. This X2 has degrees of freedom equal to the degrees of overidentification. The superiority of Joreskog's approach (aside from its generality) lies in the fact that: (a) the estimates are statistically efficient, (b) an overidentification test statistic which can be used in conjunction with estimates from alternative models to test a wide variety of hypotheses is routinely generated, for example, whether two parameters are equal, (c) stard errors for the estimated parameters are available. The estimated value of X generated by the parameter estimates is routinely computed may be compared with the actually observed variance-covariance matrix to help judge the fit of the model to the data. THE FACTOR-ANALYTIC MODEL In order to simplify the presentation, only models which fit a factoranalytic model will be discussed. The factor model was chosen because so 1470

4 Path Analysis with Unmeasured Variables many of the models in the path-analysis literature on unmeasured variables (Blalock 1963, 1969; Costner 1969; Wiley Wiley 1970) fit this formulation. The discussion in this section is drawn primarily from Jdreskog's (1969) paper on confirmatory factor analysis. To simplify the presentation we shall assume that all variables (except errors or residuals), measured unmeasured, are stardized with mean of zero stard deviation of 1. This assumption is not required by Joreskog's model but is made so that parameters will be stardized path coefficients. In matrix terminology the basic factor model is X - AF + e, (3) where X is a column vector of observed variables F is a column vector of factors, e is a column vector of residuals, A is a matrix of factor loadings. If E is the population matrix of correlations among the observed variables X, then X=A(PA?T2 (4) where A' is the transpose of A, (D is the correlation matrix of the factors F, It2 is a diagonal matrix whose elements are the error variances (Ve,). Because the matrix terminology may be unfamiliar to many users of path analysis, let us consider an example (fig. 1) in which there are three ob- x b* b* FIG. 1 served variables X1, X2, X3 which have a single underlying factor F1. The structural equations are 1 X= bj*fj + el, X2 - b2*fj + e2, (5) X3 5b3*FL + e3, 1471

5 American Journal of Sociology where bi* are path coefficients ei are errors or residuals assumed uncorrelated with each other F1. Defining p as the population correlation, then from path-analysis procedures p(xlx2) = b1*b2*, (6a) p(xlx3) = b1*b3*, (6b) p(x2x3) b2*b3*, (6c) 1 (b1*)2 +Vel, (6d) 1 (b2*)2 + Ve2, (6e) 1 (b3*)2 + Ve3. (6f) In the matrix terminology of equations (3) (4) [ 1 p(x1x2) p(x1x3) X= j p(xix2) 1 p(x2x3) J p(xix3) p(x2x3) A' _ - bl*" A= b2* L_ b3* _j [bl*, b2*, b3*], ID= [1], Ve Y2 = O Ve2 0.? O Ve3 By matrix multiplication it can be shown that (bl*)2 b *b2* bl*b3* ADA' j bl*b2* (b2*)2 b2*b3*. bl*b3* b2*b3* (b3*)2 Thus we see that equation (3) is merely a concise way of writing the structural equations given by (5), equation (4) corresponds to the pathanalysis equations (6a)-(6f) relating expected model correlations to model 1472

6 Path Analysis with Unmeasured Variables parameters. The diagonal unities in X (D correspond to the assumption that the X F are stardized. The factor-analytic model may be obtained from equation (2) by specifying that the matrix 02 be a null matrix (i.e., all elements zero) that the matrix B be the identity matrix (i.e., unities in the diagonal, zero elsewhere). In the three-measure single-factor example above I 0 0 B= l- 0 {O O The full model given by equation (2) is necessary to do second-order factor analyses to deal with other path-analysis problems (Joreskog 1970a, sec ). If in fact the data for figure 1 were given, the parameter values could be computed directly from the observed correlations, since the model is just identified. If, however, we desire to test the tenability of additional restrictions, J6reskog's technique becomes quite useful. If, for example, the restriction is made that bl* - b2*- b3*, then the resulting X2 (with 2 df corresponding to the two restraints) would be a test of this hypothesis. If the observed measure variances were not stardized, we could also test equality of error variances /or unstardized factor loadings. In general the increase in x2 resulting from imposition of additional restrictions is a measure of the tenability of those restrictions (with degrees of freedom corresponding to the number of additional restrictions being made). IDENTIFICATION As with any structural model, it is important in Joreskog's technique to know the identification status of each parameter. This does not mean that it is necessary to solve the path equations (e.g., eqq. [ 6a]-[ 6f] ) for each parameter as a function of observed correlations but only that it be known whether or not a parameter is identified. If the path equations can be solved for a parameter in at least one way, then that parameter is identified. Another approach to the identification question is to examine the matrices in equation (2) to see if they satisfy one of several known sufficient condi- 1473

7 American Journal of Socioloyv tions for identifiability (Joreskog 1969, p. 186). The degrees of freedom for the x2 statistic correspond to the number of distinct elements in E less the number of independent parameters to be estimated, that is, the number of path equations less the number of unknown path coefficients residual variances. For example, in the single-factor case with four observed measures there are (4 X 5) distinct elements in I as compared with four, zero, four "free" parameters in A, F, T2, respectively, which results in df. Even if the degrees of freedom are positive, it is quite possible for one or more of the parameters to be unidentified, for example, the product of two parameters may be identified. In such cases it is necessary to add restrictions until each element in the parameter matrices is identified. If, for example, only the product of two weights is identified, then for estimation purposes one of them may be given an arbitrary fixed value which would identify the other weight (in interpreting the results only the estimate of the product is meaningful). It is always necessary to know what the fundamental (as opposed to arbitrary) restrictions on a model are in order to interpret exactly what the x2 statistic is testing. OTHER SINGLE-FACTOR MODELS In order to deal with the identification problem it is helpful to recognize variations of the single-factor model. The most important of these is illustrated in figure 2, in which F1 - bj*xj + 01, where 01 is independent of xl * bl e2 X2 + _!F e3 FIG. 2 e2 e3. A path analysis shows that equations (6a)-(6c) hold for this model. Furthermore, given the same set of data, it can be seen that the same correlations of F1 with Xi are obtained in both figures 1 2, that is, for estimation purposes we could use a single-factor procedure to estimate the parameters in figure 2. Notice that if two of the observed variables 1474

8 Path Analysis with Unmeasured Variables influenced F1, for example, X1 -- F1 *- X2, then equation (6a) would not hold, that is, the bi* p(xifl) would not be identified. In figure 3 is shown a simple variation of figure 2 discussed by Blalock (1963, fig. 1). The common factor in figure 3 is F2, the only difference from the bl * p(x2f2), p(x3f2) are just identified, that is, b2i the products (bi* b4*) (b3* b5*) are just identified. The estimating equations (R denotes observed correlations) are: b134* -\/R(XlX2)R(XlX3) R(X2x3), g2*-\/r(xlx2)r(x2x3).r(xlx3), g345*= -\R (X1X3)R (X2X3) R (X1X2). In addition to the relationships detailed by Blalock (1963), our analysis: (a) indicates that this model should be rejected if any of the estimated correlations p(xif2) exceed unity, (b) sets limits on the underidentified parameters, that is, in general both bl* b4* will be less than unity but greater than R(X1X2) R(X1X3) - R(X2X3), b3* b5* will be less than unity but greater than R(X,X3) R(X2X3) - R(XlX2). 1475

9 American Journal of Sociology Another example of a single-factor model, discussed by Wiley Wiley (1970, fig. 1) is shown in figure 4. The common factor is F2, the correlations of the observed variables with this factor are identified, that is, b2*, [bl* b4*], [b3* b5*] are identified. Joreskog (1970b) shows that this model may be estimated by a single-factor model with F2 as the common factor. el e2 e3 b1 b2 b3 b4 ~b FIG. 4 The distinguishing feature of the models in figures 1, 2, 3, 4 is that there are three observed variables Xi, Xj, Xk whose correlations with the common factor F are identified. For purposes of generalization it may be noted that if there are three observed variables, Xi, Xj, Xk, an unmeasured variable F, such that each observable correlation p(xx1j) is equal to the product of the corresponding correlations p(xjf) p(xjf) with the factor, that is, p(xxj) -_ p(xjf) p(xjf), then the correlations with the factor are all identified. That is, p (XJF) /p (XiXj) p (XiXk) /p (XjXk), (7) similarly for p(xjf) p(xkf). An important qualification to this rule is that the correlation between observed variables cannot be zero since equation (7) would not be defined. Note that the rule intentionally states that the factor correlations, not path coefficients, are identified. The general characteristic of these models is that each factor correlation p(xjf) arises from a causal linkage, the parameters of which do not enter into the factor correlations with the other two or more observed variables. To illustrate the usefulness of the above rule in determining the identification status of more complicated models, consider figure 5 which is taken from Costner (1969, fig. 4) which has been further analyzed by Hauser 1476

10 Path Analysis with Unmeasured Variables el 13 rt t *, * b5 b2 b4 e2 e FIG. 5 Goldberger (1970, fig. 4.1). From path-analysis principles it can be immediately seen that p(xifl) = b1*, p(x2f1) = b2*, p(x3f1) - b3* b5*, which means that these factor correlations, therefore b1* W, are identified. Analogously, using X1, X3, X4 in relation to F2, p(xlf2)= bl* b5*, p(x3f2) b3*, p(x4f2) - b4*, which means that b3* b4* are identified. Path analysis also gives p(x1x3) bl* b5* b3*, since bi* b3* are identified, it follows that b5* is identified. If the figure 5 model were complicated by e2 being correlated with e4 (corresponding to fig. 5a in Costner 1969), the analysis above would still hold true p(x2x4) - b2*b5*b4* + A/J1- (b2*)2 p(e2e4) V1 (b4*)2, implying that p(e2e4) is identified since b2*, b4*, b5* are identified. Because the number of parameters equals the number of observed correlations, this correlated error model is just identified. EXAMPLES For the purpose of illustrating in detail identification procedures formulation in terms of Jdreskog's general model, we have chosen three examples from the path-analysis literature which are more complicated than the models previously discussed which involve special features. Our first example, shown in figure 6, corresponds to figure 2 in Blalock (1963). Path analysis of this model indicates that p(xifl) b1*, p(x2f,) = b2*, p(x3f,) b5* b3*, that is, bl*, b2*, the three 1477

11 Path Analysis with Unmeasured Variables bdl b4u b2 Ib3 e2 e3 FIG is the residual of F2 on X4 F1 correlations with F1 are identified. According to the model p(xlx4) p(x2x4) - 0, p(x3x4) = b3* 54*. This means that only the products (b3* b4*) (b3* b5*) are identified. For estimation purposes we can assign b3* an arbitrary value (e.g., 0.5) which would in turn fix b4* b5*. It would appear that all the parameters would be just identified then, we may well ask why J6reskog's program should be used. The answer is that the model may fail to fit the data because of the two restrictions that the population values p(x1x4) p(x2x4) be zero. The formulation problem is that we wish to impose the restriction that X4 be independent of X1 X2 even though the observed sample correlation may differ from zero, presumably because of sampling error. J6reskog's model, however, only allows us to specify correlations between factors as zero. This difficulty can be hled by creating a new factor, X4, which is identical with the observed X4, that is, X4 - X4. The structural equations are F1 - bi*x1 + 01, X2 = b2*fi + e2, F2 b4*x4 + b5*fl + 02, X3 - b3*f2 + e3. Recalling from our analysis of figure 2 that F1 b1*xi + 0 is from an estimation viewpoint indistinguishable from X1 = b1* F1 + Oi, it follows that X' F' - (X12 X2, X3, X4), (Fi, F2, x4), e'= (81, e2, e3, 0), 1478

12 Path Analysis with Unmeasured Variables A- bl* 0 0 b2* 0 0 O b3* i where b3* is an arbitrary fixed value with an absolute value less than or equal to one, I b5* 0- (D - b5* I b4*, _ b4* VX4 Ve, O O O IP2r 0 Ve2 0 0 O O Ve3 0 The 1 in the fourth row of A the 0 in the fourth row diagonal of fl2 indicate the identity X4 - X4 without residuals. If the matrix f, is computed, that is, X AFDA' + TJ2, we find 1 (1(bl*b2*) (bl*b3*b5*) 0 X (bj*b2*) 1 (b2*b3*b5*) 0 (bl*b3*b5*) (02*b3*b5*) 1 (b3*b4*) 0 0 (b3*b4*) 1 This shows that the expected correlations of X1 X2 with X4 are zero. This follows from the specification in 1D that X4 is uncorrelated with F1. This model has eight elements in A, 1D, P2 to be estimated, as opposed to 10 distinct elements in l. The 2 df correspond to the two restrictions that X1 X2 be independent of X4 (i.e., the X2 is a test of this assumption). This analysis clearly shows that identification status of parameters must be known if x2 results are to be correctly interpreted. The second example, shown in figure 7, has the special feature of two observed correlated variables influencing an unobserved variable. It corresponds to figure 4 in Blalock (1969) to figure 5.1 in Hauser Goldberger (1970) who approached the analysis from an econometric per- 1479

13 American Journal of Sociology F _ = _e4 FIG. 7.-O is the residual of F1 on X1 X2 regression spective. In contrast to the latter authors' approach we will not employ reduced-form equations. Using the shortcut identification procedure, the fact that p(x1fi) bi* + p(xlx2)b*, p(x3fi) - b3*, p(x4f,) b4* implies that these correlations, b3* b4*, are identified. Using X2, X3, X4, it follows similarly that p(x2f,) is also identified. Given the correlations among X1, X2, F1, the path coefficients bi* b2* may be identified, since p(x1f1) - p(xlx2)p(x2f,) 1 - p2(x1x2) p (X2F,) - p(xlx2)p(x1fl) 1 - p2(x1x2) The simplest procedure is to estimate the correlations among X1, X2, F1 then compute b1* b,j* from the estimated correlations. This problem can be hled by defining two factors x1 X1 x2 X2. The structural equations are: X1 X1, X2 X2, X3 b3*fi + e3, X4 - b4*f1 + e4. The factors are F' - (x1, X2, F1), 1480

14 Path Analysis with Unmeasured Variables A- I o 0 b3* o o b4* [ Vx1 C(xlx2) C(x,Fi) (D C(xlx2) Vx2 C(x2F1) C(xiFi) C(x2F,) 1 where C denotes covariances rather than correlations, IP2 [I O O Ve3 0 L O O Ve4 There are 10 distinct elements in X nine parameters to be estimated (b3*, b4*, VX1, VX2, C(X1X2), C(xiFi), C(x2F,), Ve3, Ve4), so that the model has one overidentifying restriction. Note that the estimated elements of D can be used to compute p(xlx2), p(xifi), p(x1f1) then to estimate bl* b2*. It is possible to estimate bi* b2* directly using equation (2); however, this will not be explained here because it is the subject of a forthcoming paper by Goldberger Joreskog (1972). In addition to the rom case considered here, Goldberger Joreskog consider the fixed case, that is, when X1 X2 are fixed predetermined variables. Our last example, shown in figure 8, corresponds to figure 9.b. in Costner (1969): X1, X2, X3 correspond to the single-factor model of figure 1, that is, bl*, b2*, b3* are identified. Since p(x6f2) = b6*, p(x5f2) - b5*, p(x4f2) - b4* + b7* b8*, it follows from equation (7) that these correlations, b5* b6*, are identified. Because p(x1x6) - bi* b7* b6*, b7* is identified, since bi* b6* are. Since X1, X2, X4 identify p(x4f,) (i.e., p(x4f,) - p(xlx4)? bl*), it follows that b4* b8* are identified since these parameters are functions of p(x4f,), p(x4f2), b7*. Notice how important it is to the model to have both X5 X6, for example, without X6 the p(xifl) are identified, but the p(xif2) p(fif2) are not. In formulating figure 8 in terms of equation (2) we shall treat the residuals (ei) as factors in order to facilitate comparisons with Costner's (1969) analysis. In this case 1481

15 American Journal of Sociology X' (X1, X2, X3, X4, X5, X6), F' (F1, F2, e1, e22 e32 e4, e5, e6)2 bl* b2* b3* b8* b4* O b5* O b6* L 1 p(flf2) p(flf2) O 0 Ve O 0 0 Ve O Ve O Ve4 0 0 O O Ve5 0 0 O Ve6 - A1'A' (i.e., T2 = 0). In this model p(flf2) - b7*, both F1 F2 are stardized [V(F1) - V(F2) = 1]. There are 6 X distinct elements in X, less 14 parameters to be estimated, resulting in 7 df. Costner raised the question as to whether b8* - 0. To test this hypothesis we would rerun the computer program, changing b8* to a fixed value of zero. This revised model is more restrictive will therefore typically have a poorer fit with the data, that is, the X2 will increase. In large samples, the difference in x2 between these two models, with degrees of freedom equal to the difference in number of restrictions, can be used to test the hypothesis that b8* - 0. Costner was also concerned with the model when b8* = 0 but e3 e4 were correlated. To allow for this possibility the corresponding element in 1D (i.e., fifth col., sixth row), should be left free instead of fixed - 0 as above. The increase in X2 from the b8* = 0 model with 1 df is a test of the hypothesis that p(e3e4) = 0. Because the error variances are not stardized, the program estimates the covariance of el e2 [C(ele2)] from which the correlation may be estimated 1482 P(ele2)- lnele).. Ve3Ve4.

16 Path Analysis with Unmeasured Variables el Xe XI * X6 * FIG. 8 A comparison of the x2 for b8* 0 with b6 that for p(e3e4) # 0 gives an indication of which is the better fitting model. Costner (1969, fig. 10) also raises the question of whether e1 e2 are correlated. This hypothesis is tested by allowing the covariance between e2 e2 in I5 to be "free." The change in x2 with 1 df provides the appropriate statistical test. Hypotheses involving "constrained" parameters may be tested similarly, for example, bi* - b6* (Heise 1969) or Vel - Ve6 (Wiley Wiley 1970). REFERENCES Blalock, H. M "Making Causal Inferences for Unmeasured Variables from Correlations among Indicators." American Journal of Sociology 69 (July): "Multiple Indicators the Causal Approach to Measurement Error." American Journal of Sociology 75 (September): Costner, H. L "Theory, Deduction, Rules of Correspondence." American Journal of Sociology 75 (September): Goldberger, A. S., K. G. Jioreskog. In preparation. "Estimation of a Model with Multiple Causes Multiple Indicators at a Single Unobservable Variable." Hauser, R. M., A. S. Goldberger "The Treatment of Unobservable Variables in Path Analysis." In Sociological Methodology, edited by E. F. Borgatta G. W. Bohrnstedt. San Francisco: Jossey-Bass. Heise, D. R "Separating Reliability Stability in Test-Retest Correlations." American Sociological Review 34 (February): J6reskog, K. G "A General Approach to Confirmatory Maximum Likelihood Factor Analysis." Psychometrica 34 (June): a. "A General Method for Analysis of Covariance Structures." Biometrika 57(2) :

17 American Journal of Sociology. 1970b. "Estimation Testing of Simplex Models." British Journal of Mathematical Statistical Psychology 23, pt. 2 (November): "Statistical Analysis of Sets of Congeneric Tests." Psychometrica 36 (June): J6reskog, K. G., G. T. Gruvaeus, M. van Thillo ACOVS. A General Computer Program for Analysis of Covariance Structures. Research Bulletin Princeton, N.J.: Educational Testing Service. Jbreskog, K. G., M. van Thillo, G. T. Gruvaeus ACOVSM, A General Program for Analysis of Covariance Structures Including Generalized MANOVA. Research Bulletin Princeton, N.J.: Educational Testing Service. Wiley, D. E., J. A. Wiley "The Estimation of Measurement Error in Panel Data." American Sociological Review 35 (February):