Some Methods for Respecifying Measurement Models to Obtain Unidimensional Construct Measurement

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1 JAMES C. ANDERSON and DAVID W. GERBING* Lack of unidimensionality in structural equation models most often represents misspecification. The authors review the necessary conditions for unidimensional measurement of constructs. Two methods which assist the researcher in measurement model respecification, multiple-groups analysis and similarity coefficients, are presented and an example illustrating the use of these methods is given. Some Methods for Respecifying Measurement Models to Obtain Unidimensional Construct Measurement The development of methods of analysis for linear structural equation models with latent variables has provided researchers considerable means to construct, test, and modify theories. However, the initial analysis almost invariably indicates the need for a revised model, either in the measurement of the latent variables or in the causal relations among the latent variables. The purpose of this article is to present some methods for respecifying measurement models. By far the most widely used approach is the full information maximum likelihood (MLE) method developed by Joreskog (0, ) and implemented in the program LISREL (Joreskog and Sorbom ), This approach was introduced to the marketing discipline by Bagozzi (), who gave it a thorough presentation for marketers in a subsequent book (Bagozzi 0), With a full information method, one obtains estimates of all the parameters in a model simultaneously from the observed correlation (or covariance) matrix. In contrast, with a limited information approach one obtains the parameter estimates for one structural equation at a time using only the corresponding elements of the matrix (Heise 5). A full information maximum likelihood estimation method has several desirable statistical properties (Lawley and Maxwell ; see Browne for * James C, Anderson is Assistant Professor of Marketing, The University of Texas at Austin, David W, Gerbing is Assistant Professor of Psychology, Baylor University, As the authors contributed equally to the article, their names are in alphabetical order. The authors gratefully acknowledge the influence of John E, Hunter of Michigan State University upon their work. discussion of a generalized least squares full information altemative). Though simultaneous estimation of all parameters in a model with LISREL is possible, the modeling process can be thought of as the analysis of two conceptually distinct models: measurement and structural (Gerbing ; Joreskog and Sorbon ), The measurement model specifies the causal relations between the observed variables or indicators and the underlying latent variables or theoretical constructs, which are presumed to determine responses to the observed measures. The structural model specifies the causal relations among the theoretical constructs. To estimate the parameters and assess the fit of a hypothesized measurement model to the observed correlations, one can use a confirmatory factor analysis (Holzinger ; Joreskog, ). The reason for drawing a distinction between the measurement model and the structural model is that proper specification of the measurement model is necessary before meaning can be assigned to the analysis of the structural model. That is, good measurement of the latent variables is prerequisite to the analysis of the causal relations among the latent variables. Our article concentrates on the measurement of latent variables or constructs embedded in structural equation models. Each construct is measured by multiple indicators and each indicator measures only a single construct. That is, the set of indicators defining each construct are unidimensional or congeneric (Aaker and Bagozzi, p, ; Bagozzi 0, p. 5-; Joreskog 0). We refer to these models as multiple-indicator measurement models throughout this article. Lack of unidimensionality most often represents a 5 Journal of Marketing Research Vol, XIX (November ), 5-0

2 5 JOURNAL OF AAARKETING RESEARCH, NOVEMBER measurement model misspecification and, unfortunately, a number of misspecifications of this kind typically occur with initial models. Drawing on early psychometric literature, we present two methods which will help the researcher in respecifying a measurement model to obtain unidimensional measurement of constructs embedded in structural equation models. These methods are used to build measurement models which are good approximations and, when properly specified, are then submitted to full information estimation. We first review the necessary conditions for unidimensional measurement and usefulness of a construct, then briefly review the kinds of information that have been used for respecifying misspecified models. NECESSARY CONDITIONS FOR UNIDIMENSIONAL MEASUREMENT AND USEFULNESS Internal and External Consistency, and Reliability The first condition to be met if a covariance structure of measures is to be considered unidimensional can be called "intemal consistency," Though intemal consistency has been presented in several equivalent forms, the "fundamental equation" (Hart and Spearman, p. 5; Spearman, p. ) for intemal consistency is () where a, b, p, and q are observed measures of the same theoretical construct, A related equation is the product rule for intemal consistency: () r^ = r^jr^ where a and b are observed measures of construct T. Depending on the nature of the observed measure-construct correlations (r,j,), two basic pattems can be observed for unidimensional matrices. First, if all r^j, are equal, the correlations among the measures will be uniform (to within sampling error). Altemately, if the r,,- differ from one another, the correlation matrix can be reordered (based on size of r^) so as to show a "hierarchical ordering" (Spearman 0) of observed correlations (again, to within sampling error). Thus, convergence in measurement (Bagozzi, p. 5) represents a sufficient though not necessary condition for intemal consistency. The second condition for a covariance structure of altemate indicators can be called "extemal consistency" or "parallelism" (Hunter and Gerbing ; Tryon ). The equation for extemal consistency can be given as () where x, and X are any two measures of a given construct and yj and y^ are measures of another construct. A related equation is the product rule for extemal consistency (Tucker 0) () 'xt'tv'uy where x is any measure of construct T and y is any measure of construct U. This product rule can also be used to reproduce the correlations of an observed measure for a given construct with other constructs in the model. Extemal consistency is similar to the concept of a point variable discussed by Wright (0). It should be noted from equation that the correlations of measures for a given construct with measures of other constructs need not be uniform, only proportional. Thus, differentiation in constructs (Bagozzi, p. ) represents a sufficient though not necessary condition for extemal consistency (Fomell and Larcker ). It should also be noted that when there are less than four measures of a construct, it is not possible to test for intemal consistency (Spearman and Holzinger ). In this case, extemal consistency becomes the sole condition for assessing unidimensionality. Though it is neither necessary nor sufficient that a covariance structure of measures be reliable to be unidimensional, high composite reliability is nonetheless important for a measured construct to be useful. Of the several coefficients of reliability (Cronbach ), the coefficient of equivalence seems most appropriate for assessing construct reliability. General factor variance (if any) and group factor variances (there should be just one group factor for unidimensional measures) contribute to the size of the coefficient whereas specific variances (the part of each measure's reliable variance not due to common factors) and residual variance are treated as "error variance." The stability over time of scores provided by a measure is not considered. Estimates of the coefficient of equivalence are provided by coefficient alpha (Cronbach 5) or, as recommended by Smith (), coefficient omega (Heise and Bohmstedt 0). Examples of Lack of Unidimensionality An example of problems that occur when the conditions for unidimensionality and usefulness are not met is interpretational confounding (Bagozzi, 0, p. -; Burt, ), Interpretational confounding "occurs as the assignment of empirical meaning to an unobserved variable which is other than the meaning assigned to it by an individual a priori to estimating unknown parameters" (Burt, p. ). Interpretational confounding occurs "when two conditions arise: () the indicants of the unobserved variable have low covariance among themselves, and () the covariances of the indicants of the unobserved variable with the indicants of other unobserved variables in the model are widely different" (Burt, p. ). It follows from the preceding discussion that the presence of the first condition means the intemal consistency condition has not been met or the composite reliability is low, and that the presence of the second

3 UNIDIMENSIONAL CONSTRUCT MEASUREMENT 55 condition means the extemal consistency condition has not been met. A second example is provided in Table of Fomell and Larcker (, p. ). These researchers present an example of "a model based on a correlation matrix where correlation coefficients are significant, of the same sign, and fairly uniform in magnitude" (Table title) to demonstrate that a poor fit can result when Bagozzi's () criteria of convergence in measurement and differentiation in constructs are met. As discussed before, intemal consistency cannot be assessed properly if there are only two indicators for each of the two constructs (r^i^j =,0, r^i^ =,5), However, the external consistency of the model can be assessed as follows. (5) If this equality holds (to within sampling error), extemal consistency is met. Substituting from Table into equation 5, we have ().,, '^. The value obtained from a vanishing tetrad (Spearman and Holzinger ), which can be generalized to the case of extemal consistency, is found to be greater than that expected from probable errors of sampling (with A^ = ). This is as expected, because the significant chi square statistic indicates a lack of fit which must be due to extemal consistency not being met. Thus the cross-construct correlations are found to be neither uniform nor proportional. When Bagozzi's criteria are properly met, unidimensional measurement is assured and the resulting measurement model will fit perfectly. SOME PROCEDURES FOR RESPECIFICATION OF MULTIPLE-INDICATOR MEASUREMENT MODELS Misspecification Information Though the desirable statistical properties of FIML have been well established, these properties are based on the assumption that the model is correctly specified. An important task for the researcher, however, is to respecify a model given the almost inevitable global or local failure of the initial model. Little advice on model respecification is given in the literature. One obvious set of information is provided by the residuals, the difference between the predicted and observed variable covariances. However, Costner and Schoenberg () have argued against the use of residuals in an MLE analysis, Sorbom (5) and Saris, de Pijper, and Zegwaart () suggested the use of the fu-st-order partial derivatives of the likelihood function with respect to each of the parameter values as an aid to the respecification of the model. Free parameters have derivatives equal to zero in a solution which has converged, but large derivatives for fixed parameters indicate "stress" or lack of fit. Therefore, freeing a parameter most often leads to a better fit because the MLE method is then able to adjust the value of the parameter as it maximizes the likelihood function. Joreskog () and Bentler and Bonett (0) describe procedures for evaluating the fit of an initial model embedded or nested within a more general respecified model. However, they give no indication of how to locate the parameters which are to be freed. Furthermore, nesting is not applicable in the analysis of multiple-indicator measurement models. The lack of guidelines in the literature on model respecification indicates more work in this area is needed. We therefore present two techniques for developing respecifications of multiple-indicator measurement models. One technique is based on an altemate method for confirmatory factor analysis and the other technique is an exploratory guide for respecification. Confirmatory Analysis The phrase "confirmatory factor analysis" has often been applied exclusively to Joreskog's MLE method. Building on the work of Lawley () and Bock and Bargmann (), Joreskog (, ) solved some of the practical problems of obtaining MLE solutions, implemented MLE on the computer, and introduced the phrase "confirmatory factor analysis." However, though Joreskog's groundbreaking work has stimulated the current interest in latent variables embedded in structural equation models, Gerbing and Hunter (0) have noted that the concept of confirmatory factor analysis was introduced in the early part of this century. The beginnings of factor analysis as a technique for data analysis are found in the research of Spearman (0, 0) and of Burt (0). Harman () reports that the first computation of factor loading was by Burt in with the centroid method. All of these early factor analyses were confirmatory analyses of Spearman's hypothesis that intelligence could be largely accounted for with a single general factor. After the introduction of the multiple-factor approach (Thurstone ), an altemate conceptualization of multidimensionality was advocated (e.g., Burt ). Rather than factoring the matrix as a whole, one could partition the matrix on the basis of content into submatrices and analyze each submatrix separately. If the model failed, one could respecify it by defining a new grouping of the variables. Tryon () advocated this partitioning or grouping of observed variables into unidimensional clusters and applied this approach to data analysis. Tucker (0) formally introduced the correlated factors model. Holzinger () tied these developments together with oblique centroid multiple groups analysis (MGRP). Guttman (5) extended the statistical development of the analysis technique and Harman (5) reviewed these developments. MGRP is a confirmatory factor analytic technique which requires that the variables be partitioned a priori (for instance, on the basis of theory or

4 5 JOURNAL OF AAARKETING RESEARCH, NOVEMBER content) into hypothesized unidimensional groupings with uncorrelated measurement errors. In contrast to other extraction techniques such as principal components analysis, each factor is defined as simply the unitweighted sum of the observed variables which define the factor. The factor loadings are simply the correlations of each item with the composite (factor). The factor intercorrelations are obtained by correlating the composites. Communalities are computed within each group by iteration. With communalities in the diagonal, the resultant item-factor and factor-factor correlations are corrected for attenuation due to measurement error by means of the attenuation formula from classical reliability theory. Lawley and Maxwell (, p. ) state that the centroid method "provides estimates of the factor loadings which generally correspond fairly closely with maximum likelihood estimates and are adequate for most practical purposes." Thus, MGRP is a confirmatory technique complimentary to MLE for analyzing multiple-indicator measurement models. The value of MGRP for detecting misspecification is that it is a limited-information technique. Unlike MLE, which uses all the covariances in the estimation of each parameter, MGRP uses only those covariances of the variables in any one equation of the model to estimate the parameters in that equation. This theoretical limitation of MGRP (less efficient estimators in large samples) is its strength in the presence of misspecification. Separate estimation of the parameters, the factor loadings and factor intercorrelations, for each factor of the model means that misspecification in one part of the model does not affect the estimation of parameters in any other part of the model. The covariance structure of unidimensional models, intemal and extemal consistency, is not forced in the computation of the parameter estimates. MLE analysis computes parameter estimates to minimize a function of the residuals. As a result, for multiple-indicator measurement models MLE produces parameter estimates that best conform to the constraints of intemal and extemal consistency. As this is not the case for limited-information techniques such as MGRP, the parameter estimates themselves can be used to detect misspecification. Moreover, in the LISREL model the correlations of indicators with factors other than their defined factor are not considered to be parameters of the model; that is, they are not computed directly by the MLE method. Rather, they are derived from the factor pattem coefficients and factor correlations using the product rule for extemal consistency applied to an indicator and a factor. MGRP, in contrast, treats all factor loadings as parameters, providing a separately computed estimate for each loading. As an example of an MGRP analysis, consider a recent application of structural equation modeling to the measurement of latent variables. Hunter, Gerbing, and Boster (in press) investigated the constmct validity of Machiavellianism as measured by the Mach IV Scale (Christie and Geis 0). The large residuals in both the MGRP and MLE analyses testing the unidimensionality of the Mach IV scale, and the MLE likelihood ratio test of fit, clearly indicated a lack of unidimensionality. A subsequent four-factor confirmatory analysis of Mach IV supported the multidimensional hypothesis. Machiavellianism was found to be represented by the four related but distinct factors: the advocation of deceit and flattery and the belief in the cynical and immoral nature of others (for further discussion of this model, see Hunter, Gerbing, and Boster, in press). Of interest here is the method used to respecify the model. MGRP was employed throughout except for the final model which was estimated by MLE and found to fit successfully according to the likelihood ratio chi square test of fit. The MGRP computations were accomplished with the computer program PACKAGE (Hunter and Cohen ; Hunter et al. 0). The MLE computations were accomplished with the program LISREL IV. Although Christie and Geis (0) defined Machiavellianism as a unidimensional trait, they did classify the Mach IV items into three groups on the basis of content: tactics, human nature, and morality. This clustering served as the initial multifactor multiple-indicator measurement model. The three hypothesized unidimensional scales were analyzed with confirmatory analysis using both MGRP and MLE methods. The number of large residuals in both MGRP and MLE analyses and the significant likelihood ratio chi square indicated that the three-factor model did not fit the data. Indices of misspecification therefore were used to respecify the model until the successful four-factor model was obtained. The factor loadings computed by MGRP and MLE for six of the Mach IV items are reported in Table along with the MLE first-order derivatives of the factor loadings. These items were selected because they satisfied one of the MLE misspecification criteria: they had the largest derivatives. In this case, all derivatives were larger than.05. Table also shows some of the relationships between the initial grouping of these six items and the grouping in the final model. Because the structures of the initial and final models are so disparate, the indices of misspecification for the initial model do not immediately lead to the final specification. However, the indices of misspecification should begin to lead to the correctly specified model. In the final solution, only the first two factors in Table were retained with the modification that they were partitioned further into a total of four clusters. Thus the first two factors in Table were intermediate factors which eventually led to the correct solution. Each item can be respecified to be placed in the correct intermediate factor or placed incorrectly on the wrong intermediate factor (item 0 did not appear on any of the four final factors). In this analysis, both MGRP and MLE correctly indicated that items,, and should be included in

5 UNIDIMENSIONAL CONSTRUCT MEASUREMENT 5 Table A COMPARISON OF THREE RESPECIFICATION INDICES MGRP' factor loadings FIML derivatives FIML factor loadings Item 0 i ic 5c 5i i c 5c c i Fs i c 00c 05 -c c -05c Fs i ic 5c i i 5 c c c i Fs 5i 0 "Computed with four iterations, i: initially specified factor, c: correct factor for the intermediate solution, the second factor. In the MGRP solution, all three items correlate more highly with the second factor (Fj) than the first factor (F^). The items also have comparatively large derivatives in the MLE solution though no comparison is possible with the derivatives of the first factor (which by definition should be zero). A shortcoming of MLE is illustrated by items and. Item is already specified correctly on the first intermediate factor, but there is a relatively large derivative on the second factor. Item belongs in the first factor, but the derivatives are all zero, providing no information for respecification. In the MGRP analysis, however, the largest loadings of both of these items are correctly located on the first factor. Note that the MLE factor loadings of items on factors other than their own provides no information for respecification. By definition, the largest loadings are on the item's own specified factor. In addition, the factor loadings of items on their defined factors given by MGRP and MLE are found to be approximately the same, the largest discrepancy being only.0. Respecification does not come directly from any computer output; rather the respecification is suggested by information contained in the output. In this respect, for a given analysis the pattem of LISREL derivatives may contain the information which would suggest the proper respecification. The point, however, is that the respecification information provided by MGRP is complementary to the information provided by LISREL and, under some circumstances, provides information that more readily suggests model respecification. Further, the MGRP program in PACKAGE is easier to use than LISREL (or COFAMM, Sorbom and Joreskog ) for confirmatory factor analysis. For example, the analysis of the three-factor model suggested by Christie and Geis (0) was accomplished with only four PACKAGE program control statements. The MGRP program additionally provides standard score coefficient alphas for each group of multiple indicators. Similarity Coefficients Respecification would be aided by an index of the "similarity" of two indicators in terms of the extent to which they serve as altemate indicators of the same factor. Such an index, which Hunter () calls a "similarity coefficient," already exists. The basis of the index is the extemal consistency property of the covariance structure of multiple-indicator measurement models. Pairs of items with extemally consistent correlations across other altemate indicators of the same factor and indicators of other factors will conform to the criteria of fit in a multiple-indicator model. A measurement model can then be specified on the basis of grouping indicators together which share high similarity coefficients. The historical basis of this index can be traced to Hart and Spearman () and Spearman (), Spearman noted that if two columns of correlations are proportional they should correlate perfectly, although the application was limited to correlations of variables hypothesized as altemate indicators of the same factor. The problem with correlation coefficients is that they are too general. Any two variables related by a linear transformation correlate perfectly; but proportionality is described only by those iinear transformations with an intercept of zero, T'hus an index is needed "that does not reduce the data to deviation scores" (Hunter ). The result () / J ^xixl, ^x} yk is the similarity coefficient. The square of this coefficient is called the "index of proportionality" by Tryon and Bailey (0), The value of this index ranges from - to with these extreme values representing perfect intemal and extemal consistency. The usefulness of this coefficient for exploratory analysis of multiple-indicator measurement models is outlined by Hunter (), A matrix of indicator correlation coefficients can be transformed to similarity coefficients and then ordered according to the following criterion. The first variable has the highest sum of squared coefficients with the remaining variables. The second variable has the highest coefficient with the first, the third

6 5 JOURNAL OF MARKETING RESEARCH, NOVEMBER Table ORDERED SIMILARITY AND CORRELATION COEFFICIENTS OF AAACH IV" Item 0" " " " 5 " " " " " " 5 5 " " " " 'Similarity coefficients appear below the diagnonal "Not included in the final measurement model. correlation coefficients are above the diagonal. has the highest coefficient with the second, etc. The result is an ordering of the variables with relatively large drops in adjacent similarity coefficients indicating the cluster boundaries. Continuing with the same example, we show the ordering of the similarity coefficients of the 0 Mach IV items in Table. The computations were provided by the PACKAGE programs SMLR and ORDER. An interesting feature of this table is that although this information was not used to respecify the model, the ordering of the variables is perfectly consistent with the final measurement model proposed by Hunter, Gerbing, and Boster (in press). The partitioning of the ordered variables in Table delineates the final model defined by the use of MGRP, along with some interspersed variables that were not retained in the model. The similarity coefficients are summary descriptions of the extemal consistency criterion condition which multiple-indicator measurement models must fulfill to fit the data successfully. Violation of extemal consistency is what leads to nonzero residuals, whether computed by MGRP or MLE. For example, items 0,,, and in Table were deleted from the model for lack of fit in the MGRP analysis. For the model as a whole, all of these items also have similarity coefficients less than.0. All items retained in the model have similarity coefficients no lower than.0 with altemate indicators of their respective factor as specified by the measurement model. This particular cutoff is not meant to be interpreted as a rigid value, but may serve as a useful guideline for model construction. CONCLUSIONS The measurement of latent variables with multiple indicators began with the work of Spearman early in this century. More recently, the FIML approach developed by Joreskog has been used to analyze latent variable structural equation models. However, some altemative methods and concepts from the early psychometric literature are useful to present-day researchers in building and modifying models. A prerequisite to the causal analysis of constructs is satisfactory measurement of the constructs themselves. The dual constraints of unidimensionality and reliability must be satisfied. Unidimensionality is defined by both intemal consistency and extemal consistency. One of the foremost problems encountered by the researcher is the respecification of a misspecified measurement model. We present two methods, one confirmatory and one exploratory, to guide the researcher in this respecification task. Multiple-groups analysis provides a confirmatory factor analysis complementary to the analysis provided by LISREL. In particular, the parameter estimates given by multiple-groups analysis can be used directly to respecify a measurement model. An exploratory analysis is provided by similarity coefficients. The size and pattems of these coefficients can guide a researcher in specifying a measurement model. A recommended strategy for the construction of multiple-indicator measurement models is to begin with confirmatory multiple-groups analyses. The initial models can be constructed on the basis of theory and the order-

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8 0 JOURNAL OF AAARKETING RESEARCH, NOVEMBER Schuesster, ed. San Francisco: Jossey-Bass. Smith, K. W. (), "On Estimating the Reliability of Composite Indices Through Factor Analysis," Sociological Methods & Research, (May), 5-5. Sorbom, D. (5), "Detection of Correlated Errors in Longitudinal Data," British Journal of Mathematical and Statistical Psychology,, -5. and K. G. Joreskog (), COFAMM: Confirmatory Factor Analysis with Model Modification. User's Guide. Chicago: Intemational Educational Services. Spearman, C. (0), "'General Intelligence,' Objectively Determined and Measured," American Journal of Psychology,, 5-. (0), "Demonstration of Formulae for True Measurement of Correlation,'' American Journal of Psychology,, -. (), "Theory of Two Factors," Psychological Review,, and K. Holzinger (), "The Sampling Error in the Theory of Two Factors,'' British Journal of Psychology,, -. Thurstone, L. L. (), "Multiple Factor Analysis," Psychological Review,, 0-. Tryon, R. C. (), Cluster Analysis. Ann Arbor, MI: Edwards Brothers, Inc. and D. E. Bailey (0) Cluster Analysis. New York: McGraw-Hill Book Company. Tucker, L. R. (0), "The Role of Correlated Factors in Factor Analysis," Psychometrika, 5 (June), -5. Wright, S. (0), "Path Coefficients and Path Regressions: Altemative or Complementary Concepts," Biometrics, (June), ^0. Our B I G G E S T Books of Any Year! The Changing Marketing Environment: New Theories And Applications Educators' Conference Proceedings, Series * Kenneth Bernhardt, Ira Dolich, Michael Etzel, William Kehoe, Thomas Kinnear, William Perreault, Jr., and Kenneth Roering, editors pp. $5.00/Members $0.00/Nonmembers Proceedings of the Conference held in Washington, D.C. August -,, this book contains papers reflecting changes in the discipline as we move toward the st century. Explored are the primary subject areas of Management, Buyer Behavior, Public Policy, Research Methodology, and Education, with author and subject indices, and a list of reviewers included. As Assessment Of Marketing Thought L Practice Educators' Conference Proceedings, Series * Bruce J. Walker, William O. Bearden, William R. Darden, Patrick E. Murphy, John R. Nevin, Jerry C. Olson, and Barton A. Weitz, editors 5 pp. $5.00/Members $0.00/Nonmembers The edition of this annual event, this Proceedings contains reviewed papers presented in Chicago in August, with a listing of the Special Sessions held at the Conference, and both Author and Subject indices. Published papers present the current state of the art for student, practitioner and professor alike! A reviewers' list is also included. Order Your Copies Today! Contact: Order Department, American Marketing Association, 50 S. Wacker Drive, Chicago, il 00, () -05.

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