# Using Moderator Variables in Structural Equation Models

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Paul L. Sauer and Alan Dick (1993) ,"Using Moderator Variables in Structural Equation Models", in NA - Advances in Consumer Research Volume 20, eds. Leigh McAlister and Michael L. Rothschild, Provo, UT : Association for Consumer Research, Pages: 636-640.

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http://acrwebsite.org/volumes/7532/volumes/v20/NA-20

This article provides a brief tutorial on extending the concept of moderator variables to structural equation model systems. Procedures and theoretical rationale for application of the moderator concept to structural equation systems are described. Appropriate tests of discrete and continuous moderator variables are discussed. Tests of discrete (categorical) moderator variable effects can be performed by utilizing the moderator to divide the sample into groups and performing a Chi-square test of the significance of the difference between designated structural parameters across groups. Tests of continuous moderator variable effects can be performed by specifying interaction effects within the structural equation model context.

INTRODUCTION

There is increasing concern among researchers for distinguishing between moderator and mediator effects of variables in tests of alternative theories in marketing (e.g., Batra and Stayman 1990; Chattopadhyay and Basu 1990; Olney, Holbrook and Batra 1991). Furthermore, the use of structural equation models to statistically specify and test causal processes in marketing contexts has strong theoretical appeal from a comprehensive theory testing perspective (Anderson and Gerbing 1988; Bagozzi 1984). The application of structural equation methodologies is becoming more widespread and prevalent (e.g., Bagozzi 1990; Bagozzi and Yi 1991; Mittal 1990; Wind, Rao, and Green 1991). This is in part because application of structural equation techniques is made simpler with the incorporation of structural equation algorithms in standard PC and mainframe based statistical packages (e.g., LISREL (J÷reskog and S÷rbom 1989) in SPSS and EQS (Bentler 1985) in BMDP). Structural equation techniques are also becoming applicable to a wider variety of theory and measurement configurations through such methodological improvements as generalizations and adaptations in statistical procedures (Bentler 1986; Bentler and Dijkstra 1985; Browne 1984), incorporation of discrete rather than continuous measures (Mnthen 1984) and accommodation of higher-order factor structures (J÷reskog 1971b; Weeks 1980). The purpose of this work is to describe a methodological approach for incorporating tests of moderator variable effects in the context of structural equation models.

Though Baron and Kenny (1986) suggest that multi-group analysis be used in LISREL to model moderator variable effects in a structural model context, the procedure is not described. This article builds on this suggestion by describing a procedure for incorporating moderator variable effects on relationships between constructs into a structural equation model and statistically testing these effects using multi-group analysis (J÷reskog 1971a; J÷reskog and S÷rbom 1989).

Moderator and Mediator Variables

A literature base has developed regarding differentiation between mediator and moderator variables, primarily in the psychology literature (e.g., Abrahams and Alf 1972; Baron and Kenny 1986; James and Brett 1984; Judd and Kenny 1981). A brief review of terms and concepts is provided here to set the stage for the description of application to structural equations systems which follows. The interested reader is referred to Baron and Kenny (1986) and James and Brett (1984) for a more thorough development of the distinction between moderator and mediator variables.

Baron and Kenny define a mediator as any variable which "accounts for the relation between the predictor and the criterion" (1986 p. 1176). That is, if Y=f(X) and Z=f(Y), but Z_f(X), then variable Y is a mediator of the effect of variable X on variable Z (James and Brett 1984). Baron and Kenny define a moderator as "a qualitative or quantitative variable that affects the direction and/or strength of the relation between an independent and dependent or criterion variable" (1986, p. 1174). If Z=f(X) and W is a moderator variable, then for different values of W, the form and/or strength and/or sign of the Z=f(X) relationship may vary depending upon the value of W (James and Brett 1984; Sharma, Durand and Gur-Arie 1981). It is also possible that a variable may be a hybrid in which a it acts as both a mediator and a moderator. If the moderator effect of W on the effect of X on Z is mediated by the variable Y, this effect is called "Mediated Moderation" (Baron and Kenny 1986). If the moderator variable W interacts with the mediator variable Y to effect Z and the variable Y has a mediation effect on the relationship between X and Z, the interaction effect is termed "Moderated Mediation" (James and Brett 1984).

In addition to these categories of mediator/moderator variables, Sharma et al. (1981) define a typology of moderator variables which distinguishes between moderator variables which affect the strength of the relationship between predictor and criterion, and moderator variables which affect the form of the relationship. Moderator variables which affect the form of the relationship can be either quasi moderator variables or pure moderator variables. If a moderator variable both interacts with the predictor variable and is directly related to the criterion variable it is a quasi moderator. If a moderator variable only interacts with the predictor variable it is a pure moderator (Sharma et al. 1981). Sharma et al. (1981) contended that quasi moderators were not considered moderator variables in the psychometric literature. That restriction appears to have been relaxed, as Baron and Kenny note that "there may also be significant main effects for the predictor and the moderator, but these are not directly relevant conceptually to testing the moderator hypothesis" (1986, p. 1174). While it is possible to statistically evaluate whether a variable is functioning as a mediator, a moderator, or a hybrid, the overriding concern should be whether the theory being tested supports a moderator or mediator role for the variable in question (James and Brett 1984; Sharma et al. 1981). That is, theory should be used to define the functional form of the model (James and Brett 1984).

Theoretical and Methodological Considerations

The ability of structural equation models to address the theoretical testing needs of marketing science will depend upon the robustness and flexibility of structural equation methodologies in addressing both measurement and structural form varieties encountered in theory development. The realms of structural equation model applications are expanded by the ability to include alternative functional forms containing both moderator and mediator variables within a global model (cf., Baron and Kenny 1986). This expansion addresses both the metatheoretical issues relevant to evaluative tests of competing theories and the metamethodological issues relevant to competing statistical approaches to modeling alternative functional forms (cf., Leong 1985). The ability of structural equation systems to model a variety of functional forms, ranging from simple regression forms to complex latent variable regression forms (J÷reskog and S÷rbom 1989) is the primary reason that such systems appear to offer metamethodological promise for metatheoretical investigations into the superiority of competing theories of marketing phenomena. The greater the variety of functional forms which can be included in a single system, the greater the variety of theories which can be statistically tested. The inclusion of variables as moderators rather than antecedents or mediators when such form would be more theoretically appealing affords an expansion of the metamethodological as well as metatheoretical role which structural equations can have in tests of competing theories.

A methodology which has the ability to embrace within itself a set of other methodologies offers the metamethodological framework which enables the metatheoretical evaluation of competing theories, specifying perhaps competing measurement and/or structural functional forms, within the umbrella of a single, global methodology. This is perhaps why Bagozzi (1984) argued for structural equation modeling as the dominant form of methodological testing in marketing. To this end, structural equation techniques are increasingly being generalized to offer the metamethodological advantages over other statistical methodologies. Flexibilities such as non-continuous variable indicators (Mnthen 1984), reflective and formative indicators (Fornell and Bookstein 1982; J÷reskog and Wold 1982), two-step model testing (Anderson and Gerbing 1988); higher-order factor structures (J÷reskog 1971b; Weeks 1980) and the moderator variable approaches discussed in this article are examples of improvements in methodological flexibility afforded by structural equation techniques such as LISREL which facilitate more global testing of competing theories.

LISREL provides the ability to model latent variable constructs and to estimate the parameters for both the observed variable - latent variable relationships and the structural relationships simultaneously using the full information contained in the observed variance-covariance matrix.

When using a more restrictive methodology such as regression, the tendency has been to add or average multiple items in a formative manner to yield a single composite value. This may not always be appropriate as latent constructs such as attitude, for example, theoretically lend themselves to a representation in which the latent variable is reflected by observed indicator variables. The distinction is exemplified by comparing the reflective approach to modeling of LISREL and EQS to the formative approach used in multiple regression. Theoretical considerations should prevail over measurement issues in determining the functional form of the model.

Though such a decision should be made a priori using a well-developed theoretical rationale, a problem that may arise is that theoretical rationales may differ depending upon the philosophical approach to theory development taken by the researcher. Philosophical approaches to theory development and testing range from the logical positivist / empiricist approach on one extreme to the relativist / constructionist approach on the other extreme (See, for example, Leong 1985 and the special issue on Marketing Theory in The Journal of Marketing, Fall 1983). While it is beyond the scope of this article to deal with all the issues surrounding these and other philosophical approaches, it is important to note that the role of theory and methodology in determining the status of a variable as either a mediator or moderator is dependent upon the adopted philosophy.

If one were to prefer to base the model specification on methodology, techniques capable of capturing both roles and providing a test of the competing merits of each would be required (cf., Baron and Kenny 1986). Techniques should be employed which provide capabilities for capturing both the moderator and mediator roles of a variable. To this end statistical methodologies which provide the most global approach to embracing alternative functional forms would be more appealing (Leong 1985). Because linear models such as ANOVA, MANOVA and regression are special cases of the more general structural equation models, structural equation models offer a metamethodologically superior tool for testing marketing theory.

Moderator Variables in Structural Equation Models

In this paper structural equation models refer to causal models containing reflective measured variables as indicators of constructs which are structurally linked to one another. These models are frequently tested using full information likelihood techniques such as LISREL (J÷reskog and S÷rbom 1989) or EQS (Bentler 1985).

Assuming the moderator variable is defined based on a priori theory, moderator variables may affect relationships between observed and/or latent variables in a variety of ways as previously discussed. We do not consider quasi moderators which are equivalent to hybrid types of moderated-mediator or mediated-moderator variable effects, but rather focus on the evaluation of what Sharma et al. (1981) label pure moderator effects.

Discrete Moderator Variables

Many variables which have potential for use as moderators in marketing are by their very nature discrete (e.g., nominally scaled demographic variables such as gender or occupation). Moderator variables which are at least ordinally scaledcould be made discrete by using theoretically appealing cutpoints (Baron and Kenny 1986, James and Brett 1984). In such cases multi-group analysis is the recommended approach (Baron and Kenny 1986).

Statistical programs such as LISREL estimate the parameters of a structural equation model by minimizing a fitting function which compares the observed sample covariance matrix of the measured indicator variables with the covariance matrix from the parameters estimated for the entire model. In the application of LISREL to discrete moderator variable effects, the covariance matrices or any set or subset of parameters may be constrained to be equal in a multi-group analysis (J÷reskog 1971a; J÷reskog and S÷rbom 1989). The test for a pure moderator effect can be performed using a multi-group specification of the structural equation model in which the structural and/or measurement parameters (linking the observed indicators to the latent constructs) are constrained equal across groups. In this presentation of the pure moderator effect we limit constraints to the structural parameters, although it is simple to constrain any or all parameters, whether measurement or structural, to be equal. The discrete moderator variable is used to form theoretically homogeneous groups of observations from the overall sample.

Two runs of the data are required. In the first run the appropriate structural parameters (linking the latent constructs) are constrained to be equal across groups (see J÷reskog and S÷rbom 1989 for a description of how to set up the LISREL commands). This generates an estimated covariance matrix for each group and an overall Chi-Square value (c^{2}H) for the sets of submodels as part of a single structural system.

In the second LISREL run the structural parameter constraints are removed, resulting in a Chi-Square (c^{2}N) with additional degrees of freedom. It is important to note that this is not a LISREL run using aggregate data with no groups. Rather, groups (e.g. male, female) are specified with all parameters free to be unique to each group.The moderator effect is then statistically tested by taking the difference in the two Chi-Square values (i.e., c^{2}N - c^{2}H) which is itself a Chi-Square value with degrees of freedom equal to the difference in degrees of freedom of the (c^{2}H) and (c^{2}N) values. This difference is distributed as c^{2}, therefore, the test of the null hypothesis proceeds according to classical test theory.

Though all structural parameters may be constrained to be equal simultaneously, it may be more desirable to constrain only one of these parameter to be equal during each LISREL run. Though this will depend upon the theory being tested, the primary reason is the interpretability of the Chi-square test results with respect to the parameters which were constrained equal.

In conventional tests of structural equations models one typically tries to accept rather than reject the null hypothesis (i.e., the model matches the data). This approach creates some statistical problems with the use of c^{2}, therefore, alternative measures of fit such as the goodness of fit index (Bentler and Bonett 1980) and the root mean square of the residual are recommended. The c^{2} difference test, however, eliminates the need to compute alternative measures of fit since it relies on classical test theory. The comparison of the effect size of the constrained versus unconstrained structural models can then be made across groups using the estimated standardized coefficients values. The significance of these coefficients can in turn be tested using the standard error of each coefficient which is provided in the LISREL output.

Continuous Moderator Variables

Though a moderator variable may be either discrete or continuous, James and Brett (1984) suggest that if a moderator variable is continuous it be rescaled as discrete. Bagozzi, Baumgartner, and Yi (1992), however, argue that when variables are measured as continuous it is preferable to model moderated variable effects as multiplicative interactions to retain the full information contained in continuous variables. Specifically they contend that modeled interactions are favored over multi-group analysis because: 1. multi-group analysis may have lower statistical power and may confound group variance differences with true moderator effects while moderated regression maintains original scores on a moderator variable and avoids loss of information resulting from transformation of a continuous variable to a qualitative (discrete) one; 2. a median split into groups may create groups which do not exist at least for the present sample; and, 3. observed relationships can sometimes be very sensitive to cutoff points used to form groups, especially when there is no natural cutoff point.

If moderator variables are continuous, it is possible to model their effect using interaction terms (Bollen 1989; Hayduk 1987) in a manner similar to that used in multiple regression (Baron and Kenny 1986). If the moderator variable is simply an observed variable, one approach to modeling involves forming a new variable which is the product of two variables, one of which is the moderator variable. The reader is referred to texts such as Hayduk (1987) and Bollen (1989) for a detailed discussion of how interaction effects in the structural component may be modeled using structural equation techniques such as LISREL. The primary caution with modeling interaction effects is that the observed variables which interact must themselves be multinormally distributed (Bollen 1989).

Assuming one of the models is nested in the other, two LISREL runs would be necessary, one in which the formed interaction variable is present as an indicator, the other in which it is absent. The difference in Chi-square values may then be computed and the test using the Chi-square difference would proceed as with the case for discrete moderator variables previously described. Structural parameters may similarly be compared across models.

If moderation is a function of a latent construct rather than a measured variable, the structural positioning of the latent construct determines its function as either a mediator or moderator variable. Essentially the alternative forms presented by James and Brett (1984) for linear regression models can be extended to the forms for the relationships between structural parameters. One approach would involve adding a link in which the moderating latent variable serves as a mediator while the direct link between the exogenous and endogenous latent constructs is retained. If the mediational role of the latent construct is non-significant, but the value of the direct link is significantly altered, the latent variable is functioning as a pure moderator variable. On the other hand, if the direct link is rendered insignificant, the latent construct is actually not moderating the relationship, but rather is functioning as a mediator variable. If both the direct link and the mediated link are significant and the parameter value of the direct link is significantly different than when the moderating latent construct is present, a type of hybrid moderator variable effect is occurring (Baron and Kenny 1986; James and Brett 1984). Assuming one of the models is nested in the other it is possible to again perform a Chi-square difference test by making two LISREL runs, one of which constrains the direct link between exogenous and endogenous variable to be zero.

Conclusion

This paper has presented a brief tutorial overview of potential procedures for treating moderator variable effects in structural equation models. While in this brief tutorial it was not possible to provide the detail required for a complete understanding of these techniques, it is hoped that the reader will pursue references noted here in constructing tests and will also include hybrid modeling efforts such as moderated-mediation and mediated-moderation in designing future structural equation analyses in empirical research.

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