Endogenous Moderator Models: What They are,What They Aren’t,and Why it Matters

Abstract

Models that combine moderation and mediation are increasingly common. One such model is that in which one variable causes another variable that, in turn, moderates the relationship between two other variables. There are many recent examples of these Endogenous Moderator Models (EMMs). They bear little superficial resemblance to second-stage moderation models, and they are almost never conceptualized and tested as such. We use path analytic equations to show that this is precisely what EMMs are. Specifically, we use these path analytic equations and a review of recent EMMs in order to show that these models are seldom conceptualized or tested properly and to understand the best ways to handle such models. We then use Monte Carlo simulation to show the consequences of testing these models as they are typically tested rather than as second-stage moderation models. We end with recommendations and provide example datasets and code for SPSS and R.

As science progresses, the complexity of its models tends to increase. Social and organizational science papers published in the 60s, for example, tended to propose and test additive, direct effect models. Today, such models are quite uncommon. For example, of the first 20 empirical papers published in the Journal of Applied Psychology in 2021, only one, a latent class analysis study, proposed a model that contained neither indirect nor multiplicative effects, and 15 tested one or more models that contained both indirect and multiplicative effects. This phenomenon is certainly not restricted to JAP or even to our field (one sees similar patterns in social psychology, for example).

Clearly, models in which one or more paths of a mediation model are moderated, either by a fourth variable or by the exogenous predictor in the mediation model, are quite common (see Edwards & Lambert, 2007 or Preacher et al., 2007 for detailed treatments of conditional indirect effect, or CIE models, also known as models integrating moderation and mediation (MIMMs: Cortina et al., 2017). Within this class are various types that are distinguished from one another by the location of the moderation. For example, a mediation model in which the first stage of the mediation is moderated is labeled by Edwards and Lambert (2007) as first-stage moderation.

Holland et al. (2017) showed that, in spite of the excellent guidance provided by Edwards and Lambert (2007), Preacher et al. (2007), Hayes (2013) and others, a great deal of confusion exists regarding such models. A related form of model, and the focus on the current paper, is one in which a variable causes a variable that moderates the relationship between two other variables (see Figure 1A for an example). As we show later, confusion exists regarding these as well.

Figure 1.

A, Conceptual diagram for an EMM version of Trougakos et al. (2020). B, The statistical diagram of an Endogenous Moderator Model (EMM). C, Figure 1B reoriented to show the nature of an EMM.

We refer to such models, of which there are many different forms, as Endogenous Moderator Models or EMMs. Although such models are not entirely common at present, they are likely to become so for at least four reasons. First, various authors have pointed out that, as a field develops, it seeks explanatory mechanisms in the form of mediator variables (Baron & Kenny, 1986). Just as a field would want to know the reasons for the effect of X on Y, so would it want to know the reasons for the effect of Z on the X-Y relationship.

Second, there is considerable practical value in identifying the causes of moderators. For example, in a later section, we refer to the model in Trougakos et al. (2020) in which handwashing behavior weakens the relationship between COVID-related health anxiety and emotion suppression. This pattern is useful for organizations if and only if they can cultivate handwashing on the part of their employees. In other words, the practical value of the model is increased if it includes organizational initiatives that increase handwashing, that is, a cause of the moderator.

Third, the increasing emphasis of our field on longitudinal models is causing us to reconsider the stability of factors that used to be considered immutable. For example, personality variables are often hypothesized as moderators of within person relationships (e.g., Ilies et al., 2006). The implication is that stable personality characteristics act as buffers against or accelerators of relationships between within-person variables. But then we must consider a paper such as Wu et al. (2020) in which it was shown that chronic job insecurity increased neuroticism and decreased agreeableness and conscientiousness. If one’s circumstances can change one’s personality, then the nature of models that stipulate personality variables as moderators changes from something like, “This relationship is weaker for agreeable people” to “This relationship is weaker for people whose circumstances haven’t compromised their level of agreeableness.” In other words, what had been simple moderation models must be reconceptualized as EMMs. Finally, the proliferation of multilevel models is bound to lead to more EMMs as scholars seek higher level causes of lower level moderators (e.g., Kim et al., 2021).

A paper that explains the nature of EMMs has value before EMMs become more commonplace because, as we show later, the examples that do exist suggest that EMMs are easily confused with first stage moderation (also known as mediated moderation) models even though, as we show, EMMs are phenomenologically and mathematically distinct from such models. The purposes of the present paper are to identify the nature of EMMs, to allow this to drive recommendations for supporting them, and to demonstrate with Monte Carlo (MC) simulations the consequences of testing them improperly.

Our paper is organized as follows. We begin by offering a general description and illustration of an EMM. Next, we examine several examples of EMMs in the published literature in order to understand how such models are defended and tested. We then use principles of moderated regression analysis and reduced form equations to demonstrate how several of the most common forms of EMM should be justified and tested. We then use MC simulation to demonstrate the consequences of common misspecifications of EMMs. Finally, we offer recommendations for the conceptualization and testing of EMMs.

Conceptualizing the General EMM

The sort of model under consideration is that in which one or more effects are moderated, and at least one cause of the moderator is specified. To illustrate, consider the Trougakos et al. (2020) paper mentioned earlier. These authors hypothesized a chain mediation model in which COVID-related health anxiety compromises outcomes such as goal progress and family engagement through its effects on emotion suppression and psychological need fulfillment. In a nutshell, anxiety triggers the suppression of emotion as a coping mechanism, and this suppression has various unwanted effects. Trougakos et al. (2020) also suggest handwashing frequency as a buffer such that handwashing weakens the relationship between health anxiety and emotion expression because it reduces the need for a counterproductive coping mechanism.

Thus, we have a first-stage moderation model. But what are the organizational implications of the model? In other words, how could an organization use the Trougakos results to improve their organization? As suggested earlier, perhaps the most viable strategy would be to cultivate hand-washing. What an organization would need to know, then, is how to get employees to wash their hands more often than they are used to. More formally, organizations need a model such as that in Figure 1A to be tested. This model still contains a conditional effect in that the anxiety-emotion suppression relationship varies across levels of a behavioral variable, but it also stipulates a cause of that behavior that the organization can manipulate. Although it is true that the anxiety-emotion suppression relationship is also likely to vary across levels of the organizational initiatives variable, the degree to which this is true is likely to be undetectable because the initiatives variable is one step removed from the anxiety-suppression relationship. The interactive effect is really between anxiety and handwashing.

In this example, an EMM was formed by adding to an interaction model a cause of the “moderator” variable. Inclusion of this cause increases the practical value of the model. This is a likely source of EMMs. Alternatively, EMMs can be formed by identifying one or more immediate consequences of a variable, with that consequential variable then moderating a relationship. Consider the model in Wanberg et al. (2020). These authors suggested that SES would moderate the relationship between the pandemic and life satisfaction¹. They then sought the factors that were responsible for this moderating effect and settled on, among others, perceptions of financial resources and COVID-related knowledge. These authors found that the income dimension of SES predicted perceptions of financial resources and the education dimension of SES predictor COVID-related knowledge. Further, their latent change model showed that perceptions of financial resources and COVID-related knowledge were positively related to change in life satisfaction before and after the pandemic. In other words, increases in perceptions and knowledge were associated with increases in the slope of the relationship between COVID and life satisfaction such that the slope was less negative for those higher in perceptions of financial resources and COVID-related knowledge. Thus, the authors identified two variables that are influenced by SES and that, in turn, moderate the relationship between COVID and outcomes.

Here, we have two different geneses of an EMM. In the first, a factor that can be influenced by an organization is identified so that organizations can have an indirect influence on a moderator of relationships that would be of interest to them. In the second, more proximal moderators are sought. Regardless of its genesis, an EMM, unlike the vast majority of interaction models (Cortina et al., 2001), includes a cause of the “moderator.” As we show later, this appears to create various sources of confusion for those who develop such models.

Evidence that Explication Is Needed

The present paper was born of an informal observation by one of the authors that models with endogenous moderators tended to be conceptualized and tested in a particular way, that is, as if they were first-stage moderation models. A first-stage moderation model is a mediation model in which the first stage is moderated (see Figure 1B and Equations 4 and 5 of Edwards & Lambert, 2007 or Figure 2b described below). Predictors X and Z interact in their effect on mediator M, which in turn affects outcome Y. It is not difficult to see why this conceptualization is tempting. For example, consider the model of the effects of abusive supervision on job performance in Ilies et al. (2014). In this model, abusive supervision (X) is negatively related to performance (Y). Further, conscientious people are more likely to engage in active coping strategies (M1) and less likely to engage in avoidance coping strategies (M2). These coping strategies then moderate the X-Y relationship, with active strategies weakening the relationship and avoidance strategies strengthening it (see Figure 2a). The authors explain that, “active and avoidance coping strategies mediate the moderating effects of conscientiousness on the relationship between abusive supervision and job performance” and “this type of mediated moderation model is one in which a moderating variable (conscientiousness) influences the relationship between the independent (abusive supervision) and dependent (job performance) variables” and “the original moderating effect of conscientiousness is mediated through its effects on active and avoidance coping” (p. 141). The term “mediated moderation” is generally used to describe a first-stage moderation model such as that in Figure 2b. As we show later, EMMs such as that depicted in Figure 1A are conceptually distinct from first-stage moderation models, but it may be that the reasoning that leads to EMMs is difficult to distinguish from that needed to defend the more familiar first-stage moderation model.

Figure 2.

A, A reproduction of Figure 1 from *Nandkeolyar et al. (2014). B, The first-stage moderation version.

A similar example can be seen in Priesemuth and Schminke (2019). These authors proposed a model of the effects of observing abusive supervision in which such observations lead to anger which, in turn, lead to coworker protective behavior. Overall fairness of the work environment is then proposed as a moderator such that the anger-protective behavior relationship is stronger in fairer environments. The authors then “propose a mediated moderation model with the intent to delve more deeply into the process by which overall fairness might exert its moderating influence. Specifically, we explore its impact on ethical efficacy and suggestion that it is a person’s confidence in one’s ability to act ethically (Mitchell & Palmer, 2010) that represents the mechanism by which the moderator of overall justice operates” (p. 1226). Priesemuth and Schminke (2019) use the term “mediated moderation” to describe their model. What they have instead is an EMM in which overall fairness causes ethical efficacy, and ethical efficacy moderates the anger-protective behavior relationship. But the first-stage moderation language seems difficult to resist.

As we explain later in our recommendations for conceptualizing EMMs (see Figure 8 and also Appendix B in Supplementary Online Materials), the best way to avoid the temptation to treat an EMM as a first-stage moderation model may be to think of variable Z as a proxy for moderator variable M rather than as a moderator itself. In any case, we wish to make clear that we have absolutely no intention of impugning the work of these authors. But if top-notch scholars and the reviewers of their work struggle to understand the nature of EMMs, surely others struggle as well. These sorts of observations and suppositions led to a search of the literature for papers that included EMMs in order to see whether there was in fact evidence of confusion. This search proved to be quite difficult because labels do not exist for such models at present. A search of journals in our field and in related fields in which mediation models are common using terms mediation and moderation, moderation and indirect, mediation and interaction, transmit and moderation, indirect and interaction, transmit and interaction, moderated mediation, mediated moderation, and conditional indirect effect, as well as a citation search of one of the earliest examples of an EMM that we could find (Grant and Sumanth, 2009) turned up a very large number of papers, which we hand-searched for EMMs.

Figure 8.

A flowchart for identifying the type of Endogenous Moderator Model (EMM).

We found 23 examples of various types of EMM, which three of the authors of the present paper coded with respect to the type of model that was described in the Introduction and the type of model that was tested. Tables 1a and 1b contain a summary of our findings.

Table 1.

Characteristics of 23 Papers That Contain EMMs.

a: Paths defended in Introduction
Was the effect of the moderator fully mediated?	Traditional EMM (Z affecting M affecting the X-Y relationship)?	EMM w/a twist (e.g., X causing Z, M causing Z)?	Was an effect of XZ on Y defended?	Was an effect of XM on Y defended?	Was there an indirect effect of XZ on Y	Was a direct path included in a path diagram but not in the Intro
19/23	14/19	5/19	11/23	17/23	8/23	5/22^*
b: Paths tested in the analyses for the 18 empirical papers that measured M
Z on M	XM on Y	X and M main effects on Y	XZ on Y	2^nd Stage moderation	Causal steps
15/18	15/18	7/18	12/18	1/18	7/18

One paper did not contain a path diagram.

Note. EMM = Endogenous Moderator Model.

Of most importance for our purposes are the following. First, only one (Koopmann et al., 2017) tested an EMM as it should be tested, that is, as a second-stage moderation model (which we explain in detail later). Five of the papers defended the effect of XZ on Y, and eight defended the indirect effect of XZ on Y through M (and sometimes, through XM). Of the 18 empirical papers that included a measure of M, 12 tested for the XZ interaction on Y. Seven used some form of causal steps approach in which the weight for XZ in the prediction of Y was evaluated before and after a mediator was added (either M or strangely XM). As we show later, none of these steps is consistent with an EMM. On the other hand, 3 neglected to test the effect of Z on M, 11 failed to test the main effects of X and M on Y, and 3 omitted the XM interaction effect on Y. All of these are necessary for tests of EMMs. Moreover, inclusion of unnecessary terms and omission of necessary ones would have led to biased estimates of other coefficients.

We do not claim that our search was exhaustive. If nothing else, the lack of established terminology virtually guarantees that we would miss some EMMs. Some papers contained models that were easily identified as EMMs because of the path diagrams included in the papers. Others could only be identified by reading Introduction sections carefully enough to see that the authors were proposing both a moderator and one or more of its causes. In any case, the point of our empirical review was not to estimate parameters but to get a sense of how authors conceptualize and test EMMs. The papers that we did identify are sufficient to make the point that the nature of EMMs is such that they are likely to cultivate confusion both in conceptualization and in testing. Given the precedential nature of citations in our field, this confusion is likely to continue unless some clarity is brought to the topic. The purpose of the next section is to bring such clarity.

Different Forms of EMM

As was mentioned previously, the literature contains a good deal of guidance regarding the justification and testing of CIE models. In spite of this excellent guidance, papers that hypothesize and test such models still reflect a great deal of confusion (Holland et al., 2017). Unlike typical CIE models, almost no guidance exists for the justification and testing of EMMs. The only guidance that we were able to find was a chapter by Liu et al. (2012) in an edited volume in Chinese (of which M. Wang was kind enough to send an English version.) The Liu et al. (2012) chapter focuses primarily on traditional CIE models but does cover a particular type of EMM, namely one in which the predictor X interacts with both exogenous Z and mediator M, and M is endogenous to X. Thus, there is very little in the way of advice for those wishing to justify and test EMMs. Because of the apparent confusion regarding the differences between first-stage moderation models and EMMs, we now break down the EMM and compare it to the findings of Edwards and Lambert (2007).

Figure 1a contains the conceptual version of an EMM. Our examination of published EMMs shows that EMMs are often defended and tested as first-stage moderation models, or what some refer to as mediated moderation. In other words, Z moderates the X-M relationship, and M then causes Y. As we show, an EMM can take several of the forms discussed by Edwards and Lambert (2007), but first-stage moderation is not one of them. In the section that follows, we discuss each of these forms.

Full Mediation EMM

Figure 1b contains the statistical version of a simple EMM in which M is the variable that moderates the X-Y relationship and is endogenous to Z. Path diagrams can be useful, but authors are sometimes misled by the orientation of such diagrams (Holland et al., 2017). If we change the orientation of this diagram, we get Figure 1c. The differences between Figure 1b and Figure 1c are entirely cosmetic, but Figure 1c better demonstrates the nature of an EMM. As can be seen, these are full mediation versions of Hayes (2013) Model 14, which corresponds to Edwards and Lambert (2007) Equations 10-12. In other words, it is a second-stage moderation model with Z as the “predictor” and X as the “moderator” of the relationship between the mediator M and the outcome Y. Figure 1a and 1c (and Figure 1c from Edwards & Lambert, 2007) look quite different, but they are mathematically identical.

This can be seen in the reduced form equation. In the equations that follow, we have omitted error terms because they are not central to the points that we wish to make. The structural equations for this model (using the symbol and subscript system of Edwards & Lambert, 2007) are

M = a_{01} + a_{Z 1} Z

(1)

Y = b_{02} + b_{X 2} X + b_{M 2} M + b_{X M 2} X M

(2)

where the coefficients with 0 subscripts are intercepts, the remaining coefficients are weights, and the other subscripts refer to the equation number or the variable to which a weight is attached. Thus, b_X2 is the weight for variable X in Equation 2, and so on.

At this point, it should be noted that there is debate in the literature about the usefulness of the concept of full mediation. It does seem unlikely that any one set of mediators represents all of the transmitters of a given effect. On the other hand, for reasons of parsimony, it may make sense to allow authors to hypothesize models in which a given set of mediators represent the “primary” transmitters and to ignore other, relatively minor transmitters. In any case, we include full mediation models because they represent a convenient launch point for our coverage of EMMs of increasing complexity and because they remain quite common (Cortina et al. 2017), especially as components of larger causal models.

Returning to Equations 1 and 2, given that M is conditioned by Z, we can substitute for M to create the reduced form equation for Y (i.e., the equation that has been solved for all endogenous variables).

Y = b_{02} + b_{X 2} X + b_{M 2} (a_{01} + a_{Z 1} Z) + b_{X M 2} X (a_{01} + a_{Z 1} Z)

(3)

which can be rewritten in simple path form (i.e., as simple slopes in moderated regression) in the same way that Equation 8 from Edwards and Lambert (2007) could be rewritten into their Equation 9. If we rewrite Equation 3 as

Y = [b_{02} + (a_{01} + a_{Z 1} Z) b_{M 2}] + [b_{X 2} + (a_{01} + a_{Z 1} Z) b_{X M 2}] X

(4)

we see that unlike Equation 9 from Edwards and Lambert (2007), this equation contains no indirect effect of X. It does, however, contain an indirect effect for Z, which can be seen by rewriting Equation 3 in a different simple path form

Y = b_{02} + b_{X 2} X + a_{01} b_{M 2} + a_{01} b_{X M 2} X + a_{Z 1} (b_{M 2} + b_{X M 2} X) Z

(5)

The indirect effect in Equation 5 is that of Z on Y, captured by a_Z1 (b_M2 + b_XM2X).

It is important to note that this equation is inconsistent with the language that many authors use to describe EMMs. As mentioned earlier, authors are tempted to suggest that M transmits the XZ interaction. That language is consistent with a first-stage moderation model. Such models should be represented with figures such as 1b and 1f from Edwards and Lambert (2007) or those associated with Model 7 from Hayes (2013) and should be tested with Equation 8 from Edwards and Lambert (2007).

Thus, it can be seen that a model such as that in Figure 1a is equivalent to a model such as that depicted in Figure 3a, which represents a piece of the second-stage moderation model described by Welsh et al. (2018). In the Welsh et al. (2018) model, sleep deprivation increases unethical behavior via its effect on self-regulatory resources. Further, the relationship between self-regulatory resources and unethical behavior is moderated by contemplation. Using our lettering from Figures 1a-c, sleep deprivation is Z, self-regulatory resources is M, contemplation is X, and unethical behavior is Y.

Figure 3.

A, Welsh et al. (2018) conceptual model. B, Welsh et al. (2018) conceptual model rearranged.

Welsh et al. (2018) clearly identify their model as a second-stage moderation model. If, however, the relationship of primary interest in Welsh et al. (2018) had been that between contemplation and unethical behavior, then the genesis of the paper might have ended in an EMM². The introduction might begin with the notion that contemplation reduces unethical behavior. Next, an argument might be made for self-regulatory resources as a moderator of this relationship. Then sleep deprivation might be stipulated as a cause of self-regulatory resources. The result might be a figure such as 3b.

The danger would be that, instead of the sequence described above (X causes Y, M moderates X-Y, and Z causes M), the sequence would be one that incorrectly suggests first-stage moderation (e.g., X causes Y, Z moderates X-Y, Z causes M, M moderates X-Y, M transmits the moderating effect of Z; see Nandkeolyar et al., 2014). As a result, the model would be tested as a first-stage moderation model instead of as the second-stage moderation model that it actually is. As we show later, this mistake is consequential.

Partial Mediation EMM

Equation 5 reflects full mediation of the effects of Z and can be referred to as a full mediation EMM. Direct effects of Z can also be added, but it isn’t necessarily clear how this should be done. One possibility is to add a direct effect of Z on Y by modifying Equation 2 into

Y = b_{06} + b_{X 6} X + b_{M 6} M + b_{X M 6} X M + b_{Z 6} Z

(6)

with the b_Z6Z term being added to Equations 3 through 5 as well. As before, Equation 1 could be substituted for M to create the reduced form equation³. Here, we would have what might be called a partial mediation EMM. This would result in different values for b_M and b_XM and therefore a different indirect effect of Z on Y. This is a relatively simple addition, however, resulting in nothing more than a second-stage moderation model with partial mediation (of the effect of Z on Y). There are other altogether more complicated possibilities, which we now explore.

Partial Mediation EMM with Direct Moderation

Suppose that one were to add a direct effect of Z onto the X-Y relationship. Grant and Berry (2011) and Figure 1 contains a conceptual version of such a model. In this model, prosocial motivation affects the relationship between intrinsic motivation and creativity both directly and indirectly through perspective-taking. We might refer to this as a partial mediation EMM with direct moderation. The statistical model, however, is not at all clear and has been approached in a variety of ways. The most obvious possibility is perhaps Figure 4.

Figure 4.

Statistical diagram of partial mediation Endogenous Moderator Model (EMM) with direct moderation.

Equation 1 would still apply, but Equation 6 would need to be modified into

Y = b_{07} + b_{X 7} X + b_{M 7} M + b_{X M 7} X M + b_{Z 7} Z + b_{X Z 7} X Z

(7)

If we then substitute Equation 1 for M in Equation 7, we have

Y = b_{07} + b_{X 7} X + b_{M 7} (a_{01} + a_{Z 1} Z) + b_{X M 7} X (a_{01} + a_{Z 1} Z) + b_{Z 7} Z + b_{X Z 7} X Z

(8)

And in simple path form

Y = b_{07} + b_{X 7} X + a_{01} b_{M 7} + a_{01} b_{X M 7} X + a_{Z 1} (b_{M 7} + b_{X M 7} X) Z + (b_{Z 7} + b_{X Z 7} X) Z

(9)

Similar to Equation 5, the indirect effect of Z (which is conditional on X) is captured by

a_{Z 1} (b_{M 7} + b_{X M 7} X)

. In addition, the direct effect of Z on Y is moderated by X, as captured by

b_{Z 7} + b_{X Z 7} X

. Thus, what we end up with is a second-stage moderation model in which the direct effect in the partial mediation (of the Z-Y relationship) is also moderated by X. This equation is equivalent to Equation 22 (and is consistent with their Figure 1G) from Edwards and Lambert (2007). In other words, it reduces to a second-stage and direct effect moderation model.

Double Moderation EMM

Another possibility, a confused version of which we encountered several times in our review, is Figure 5a in which Equation 2 still applies, but Equation 1 must be modified into

M = a_{10} + a_{Z 10} Z + a_{X 10} X + a_{X Z 10} X Z

(10)

If we then substitute for M in Equation 2, the reduced form equation becomes

\begin{aligned} Y = & b_{11} + b_{X 11} X + b_{M 11} (a_{10} + a_{Z 10} Z + a_{X 10} X + a_{X Z 10} X Z) + \\ b_{X M 11} X (a_{10} + a_{Z 10} Z + a_{X 10} X + a_{X Z 10} X Z) \end{aligned}

(11)

Multiplying through, we get

\begin{aligned} Y = & b_{11} + b_{X 11} X + a_{10} b_{M 11} + a_{Z 10} b_{M 11} Z + a_{X 10} X b_{M 11} + a_{X Z 10} b_{M 11} X Z + \\ a_{10} b_{X M 11} X + a_{Z 10} b_{X M 11} X Z + a_{X 10} b_{X M 11} X^{2} + a_{X Z 10} b_{X M 11} X^{2} Z \end{aligned}

(12)

Figure 5.

A, Conceptual diagram of double moderation Endogenous Moderator Model (EMM). B, Statistical diagram of double moderation EMM.

This is, in fact, analogous to Edwards and Lambert’s (2007) Equation 13 (and their Figure 1D) for first- and second-stage moderation, with the indirect effect of Z being captured by (a_Z10+a_XZ10X) (b_M11+b_xm11X). Thus, it can be seen that a model such as that in Figure 5a is equivalent to a model such as that depicted in Figure 5b in which variable X moderates both the Z-M and M-Y paths.

Consider the first- and second-stage moderation model in Avery et al. (2013). In this model, team empowerment affects individual empowerment which in turn affects in-role performance, with both stages being moderated by sex dissimilarity. Using our lettering from Figure 1a, team empowerment is Z, individual empowerment is M, in-role performance is Y, and sex dissimilarity is X. Thus, their Figure 1 could be recast such that sex dissimilarity affects in-role performance and also moderates the relationship between team empowerment and individual empowerment, the latter of which moderates the relationship between sex dissimilarity and in-role performance. Avery et al. (2013) were perfectly clear on the nature of their model, but, as was the case with the Welsh et al. (2018) model, a different emphasis or sequence might have made it difficult to identify the model as the first- and second-stage model that it is.

It should be noted that the double moderation EMM also contains an indirect effect of X, with the first stage moderated by Z and the second stage moderated by X itself. This effect is captured by (a_X10+a_XZ10Z) (b_M11+b_XM11X). Here, we see the curvilinearity inherent in this model.

Moderated EMM

The Z-M path can also be moderated by variables other than X. Thus, we might have a conceptual model like that depicted in Figure 6a. Consider once again the creativity model in Grant and Berry (2011). Suppose that we wished to incorporate a dispositional variable such as agreeableness into the model (see Grant & Berry, 2011, p. 79 for reasons that one might do this). This new variable, W, might moderate the prosocial motivation-perspective taking relationship such that the relationship is weaker when agreeableness is high (perhaps because prosocial motivation is unnecessary for agreeable people). We might label this a moderated EMM. For this model, Equation 1 must be expanded into

M = a_{13} + a_{Z 13} Z + a_{W 13} W + a_{Z W 13} Z W

(13)

Substituting for M in Equation 2, we get

\begin{aligned} Y = & b_{14} + b_{X 14} X + b_{M 14} (a_{13} + a_{Z 13} Z + a_{W 13} W + a_{Z W 13} Z W) \\ + b_{X M 14} X (a_{13} + a_{Z 13} Z + a_{W 13} W + a_{Z W 13} Z W) \end{aligned}

(14)

The simple slope version of which is

\begin{aligned} Y = & b_{14} + b_{X 14} X + a_{13} b_{M 14} + a_{W 13} b_{M 14} W + a_{13} b_{X M 14} X + a_{W 13} b_{X M 14} W \\ + (a_{Z 13} b_{M 14} + a_{Z W 13} b_{M 14} W + a_{Z 13} b_{X M 14} X + a_{Z W 13} b_{X M 14} X W) Z \\ = b_{14} + b_{X 14} X + a_{13} b_{M 14} + a_{W 13} b_{M 14} W + a_{13} b_{X M 14} X + a_{W 13} b_{X M 14} W \\ + [(a_{Z 13} + a_{Z W 13} W) (b_{M 14} + b_{X M 14} X)] Z \end{aligned}

(15)

Figure 6.

A, Conceptual diagram of moderated Endogenous Moderator Model (EMM). B, Statistical diagram of moderated EMM.

Thus, the indirect effect of Z on Y, $(a_{Z 13} + a_{Z W 13} W) (b_{M 14} + b_{X M 14} X)$ , is the product of two conditional effects, one upon the value of W and the other upon the value of X. Although it may be tempting to view this model as equivalent to Hayes’ model 11 or perhaps Hayes’ model 18, this is in fact equivalent to Hayes’ model 21, with different variables moderating the first and second stages of the indirect effect of Z on Y.

It should also be noted that this model contains a second indirect effect, that of W on Y. Like the indirect effect of Z, the indirect effect of W would be conditional upon the values of two variables, in this case, Z and X. Specifically, the indirect effect of W would be

[(a_{W 13} + a_{Z W 13} Z) (b_{M 14} + b_{X M 14} X)]

(16)

Thus, it can be seen that an EMM such as this is equivalent to the model in Figure 6b in that the first stage of the Z-M-Y mediation is moderated by W, and the second stage is moderated by X.

EMM and Three-Way Interactions

The interactions in an EMM can of course be moderated by other variables. We might call this a higher order EMM. Consider the conceptual diagram in Figure 7a. Here, we have a partial mediation EMM with direct moderation, but we also have a new variable, W, that moderates the XM interaction on Y.

Figure 7.

A, *Grant and Berry (2011) conceptual model. B, Figure 7a reoriented to show the nature of the Endogenous Moderator Model (EMM).

For example, suppose that the Grant and Berry (2011) model were expanded so that the moderating effect of perspective-taking was itself moderated by norms for risk avoidance. If there were strong norms for risk avoidance, then creativity might be low regardless of motivation, perspective-taking, or anything else. As a result, the motivation by perspective-taking interaction would exist only when norms for risk avoidance were low. Equation 1 would still apply, but Equation 7 would need to be expanded into

Y = b_{017} + b_{X 17} X + b_{M 17} M + b_{X M 17} X M + b_{Z 17} Z + b_{X Z 17} X Z + b_{W 17} W + b_{M W 17} M W + b_{X W 17} X W + b_{X M W 17} X W M

(17)

If we then substitute Equation 1 for M in Equation 17, we have

Y = b_{017} + b_{X 17} X + b_{M 17} (a_{01} + a_{Z 1} Z) + b_{X M 17} X (a_{01} + a_{Z 1} Z) + b_{Z 17} Z + b_{X Z 17} X Z + b_{W 17} W + b_{M W 17} (a_{01} + a_{Z 1} Z) W + b_{X W 17} X W + b_{X M W 17} X W (a_{01} + a_{Z 1} Z)

(18)

And in simple path form

\begin{aligned} Y = & b_{017} + b_{X 17} X + b_{M 17} a_{01} + b_{X M 17} X a_{01} + b_{W 17} W \\ + b_{M W 17} a_{01} W + b_{X W 17} X W + b_{X M W 17} X W a_{01} \\ + {b_{Z 17} + b_{X Z 17} X + [a_{Z 1} (b_{M 17} + b_{X M 17} X + b_{M W 17} W + b_{X M W 17} X W)]} Z \end{aligned}

(19)

The indirect effect of Z on Y is captured by

a_{Z 1} (b_{M 17} + b_{X M 17} X + b_{M W 17} W + b_{X M W 17} X W)

. As was the case with Equation 15, the indirect effect is conditional upon both X and W. Unlike Equation 15, it is only the M-Y path that is conditional. Equations 17–19 are in fact the equations that would be associated with Hayes’s model 18 in which there is a three-way interaction in the second stage of the mediation model. Returning to the Grant and Berry (2011) example, we could say that prosocial motivation influences perspective-taking which in turn influences creativity, that intrinsic motivation moderates the effect of perspective-taking on creativity, and that norms for risk avoidance moderate this interaction effect (Figure 7b).

Double Partial Mediation EMMs with Direct Moderation

A more complex sort of EMM can be found in Nandkeolyar et al. (2014). In this model, which we mentioned at the outset, abusive supervision (X) affects job performance (Y), and conscientiousness (Z) affects two different forms of coping (M₁ and M₂), each of which moderates the X-Y relationship. Conscientiousness also has a direct link to the X-Y path. This model is similar to the Grant and Berry (2011) model except that it contains two transmitters of the effect of Z (see Figure 2a). This double partial mediation EMM requires a version of Equation 1 for each of the two mediators.

M_{1} = a_{020} + a_{Z 20} Z

(20)

M_{2} = a_{021} + a_{Z 21} Z

(21)

This model also requires an expanded version of Equation 7

Y = b_{022} + b_{X 22} X + b_{M 1_{22}} M_{1} + b_{X M 1_{22}} X M_{1} + b_{M 2_{22}} M_{2} + b_{X M 2_{22}} X M_{2} + b_{Z 22} Z + b_{X Z 22} X Z

(22)

Equations 20 and 21 could then be substituted into Equation 22 to yield the reduced form for an expanded direct and second-stage moderation model. This equation would, among other things, show both of the indirect effects of Z onto Y: a_Z20 (

b_{M 1_{22}} + b_{X M 1_{22}} X

) and a_Z21 (

b_{M 2_{22}} + b_{X M 2_{22}} X

). Such a model is an expanded version of the partial mediation or the partial mediation with direct moderation EMMs described earlier.

Wang et al. (2015) contain a double partial mediation EMM that is more complicated still. In addition to a Z variable and two transmitters, it has three different X variables. Thus, the indirect effect of Z on Y would be conditional upon X₁, X₂, and X₃. Although it isn’t entirely clear whether a direct effect of Z is also intended, their Figure 1 does not contain such an effect. With such a direct effect, this model would require Equations 20 and 21 as well as an expanded version of Equation 22 that includes all six (i.e., 3*2) XM products, resulting in a very complicated direct and second-stage moderation model. Without direct effects, the model would be a second-stage moderation model with three moderators of each of the two second stages and would require Equations 20 and 21 as well as an expanded version of Equation 2.

Other Approaches to EMMs

Some authors have taken the pattern in Figure 4 to mean that the XZ interaction is transmitted by XM. In other words, the XM product is the mediator. This is reflected in some of the language that is used to describe EMMs. It is also implied by the way that such models are sometimes tested. For example, in a few of the papers that we reviewed, the authors used a form of causal steps analysis in which they entered the XZ product and then entered the XM product to see what happened to the XZ weight. Put another way, they conducted a Baron and Kenny (1986) test of their EMM with XM as the mediator.

If we consider the equations for this type of model, we run into difficulties. Such a model might take different forms, but the most obvious would be to use Equation 10 for M, Equation 7 for Y, and add a third Equation for XM, which is now conditional upon XZ.

X M = a_{023} + a_{M 23} M + a_{Z 23} Z + a_{X 23} X + a_{X Z 23} X Z

(23)

The use of this model creates a variety of problems. First, it makes a product endogenous. Given that the product has no conceptual value and is only a vehicle for testing interactions (Cortina et al., 2001; see also Saunders, 1956), there is no conceptual value in explaining its variance. This also requires that X be made an endogenous variable, which results in a geometric increase in the complexity of the model. Third, it is odd to substitute for a variable (e.g., M) on one side of the model but not the other. If we substitute on both sides, however, we are left with no endogenous variables at all. Finally, the weight that is presumably of most interest (that for XM to Y) represents the rate of change in Y per unit increase in XM holding other predictors in the equation constant. The other predictors in Equation 2 are X and M. In Equation 6, Z is added to the mix. Once XM becomes endogenous, many other things would need to be held constant as well, and the meaning of the weight for XM becomes very difficult to understand or to link back to theory. For these reasons, (not to mention the fact that the Baron & Kenny, 1986 approach has fallen out of favor), we would not recommend treating XM as endogenous, either conceptually or empirically.

Implications of Linkages Between EMM and CIE

Thus, we see that EMMs take different forms, which we label full mediation EMM, partial mediation EMM, partial mediation EMM with direct moderation, double moderation EMM, moderated EMM, higher order EMM, and double partial mediation EMM. These forms reduce to one of the various types of CIE model: (1) Second-stage moderation with full mediation, (2) Second-stage moderation with partial mediation, (3) Direct effect and second-stage moderation, and (4) First and second-stage moderation, (5) Hayes’ model 21, and (6) Hayes’ model 18, which is one of the various types of model that would fall under the heading of higher order EMM. We would, however, differentiate EMMs from CIE models in the same way that “moderator” variables are distinguished from “predictor” variables.

It is well known that there is no mathematical reason to label one variable the moderator and another the predictor in an interaction model. Suppose that we have predictors X and Z and outcome variable Y. The weight for the XZ product gives the rate of change in the slope representing the relationship between Y and one of the variables in the product per single point increase in the other. The way that these letters are used conventionally, we might say that the weight for XZ is the rate of change in the X-Y slope per single point increase in Z. But we can just as easily (and accurately) say that it is the rate of change in the Z-Y slope per single point increase in X.

Nevertheless, there may be good phenomenological reasons to label one variable the moderator. Consider Mischel’s situation strength arguments (e.g., Mischel, 1973). In strong situations, that is, those with clear and powerful norms for behavior, everyone behaves in the same manner regardless of their personalities. In weak situations, behavior isn’t constrained by norms and is then determined by personality. Mathematically, it makes no difference if we draw conclusions about the rate of change in the personality-behavior slope or the situation strength-behavior slope. Given the phenomenon of interest, however, it makes a great deal of difference. The genesis of Mischel’s work lay in the inability of personality researchers to find empirical linkages between personality and behavior. Mischel identified a class of situational characteristics that might explain these previous findings. He hypothesized that personality drives behavior if and only if the situation allows it to do so. If we then add a cause of situation strength, that is, a reason for the development of norms in a given situation, we have an EMM. As we have demonstrated, this should be tested in the same way that the appropriate second-stage model is tested, but the conceptual model remains one in which M is the moderator of the X-Y relationship, and the conceptual diagram that is most likely to be useful to the reader is probably one similar to Figure 1a.

Having established the nature of different types of EMM as well as some of the ways that such models are likely to be misspecified, we now use MC simulation to examine the consequences of such misspecification.

Consequences of Testing an EMM as a First-Stage Moderation Model

In our MC simulations, we generated a series of data sets reflecting different parameter values from different sorts of EMM. We then analyzed them in a series of “correct models” and “misspecified” models (i.e., first-stage moderation). Correct models were those described by Equation 1 as well as Equation 2 (EMM), Equation 6 (Partial Mediation EMM), Equation 7 (Partial Mediation EMM with direct moderation), and Equation 8 (Double moderation EMM).

Although there are, potentially, many ways to misspecify an EMM, we chose variants on first-stage moderation with XZ predicting M because we had found this to be very common misspecification. It appears, in short, to be an easy mistake to make because of the typical flow of arguments made in support of EMMs (i.e., Z moderates X-Y followed by M transmits the moderating effect of Z on X-Y). In total, we had 4 types of models (EMM, Partial Mediation EMM, Partial Mediation EMM with direct moderation, and Double moderation EMM) each with 2 equations (one for predicting the mediator M, one for predicting the dependent variable Y), and 4 types of misspecified models (i.e., the FSM versions of Equations 2, 6, 7, and 8), for a total of 2*8 = 16 equations used in our MC simulations. Table 2 contains simplified versions of the equations (i.e., equations without the precise subscripts used earlier in the paper).

Table 2.

Equations Used for Correct and Misspecified Models.

EMM correct 1	M = a₀ + a_Z × Z
EMM correct 2	Y = b₀ + b_M × M + b_X × X + b_XM × XM
EMM misspecified 1	M = a₀ + a_Z × Z + a_X × X + a_XZ × XZ
EMM misspecified 2	Y = b₀ + b_M × M
Partial Mediation EMM correct 1	M = a₀ + a_Z × Z
Partial Mediation EMM correct 2	Y = b₀ + b_M × M + b_X × X + b_XM × XM + b_Z × Z
Partial Mediation EMM misspecified 1	M = a₀ + a_Z × Z + aX × X + a_XZ × XZ
Partial Mediation EMM misspecified 2	Y = b₀ + b_M × M + b_X × X
Partial Mediation EMM with direct moderation correct 1	M = a₀ + a_Z × Z
Partial Mediation EMM with direct moderation correct 2	Y = b₀ + b_M × M + b_X × X + b_XM × XM + b_Z × Z + b_XZ × XZ
Partial Mediation EMM with direct moderation misspecified 1	M = a₀ + a_Z × Z + a_X × X + a_XZ × XZ
Partial Mediation EMM with direct moderation misspecified 2	Y = b₀ + b_M × M + b_X × X + b_XZ × XZ + b_Z × Z
Double moderation EMM correct 1	M = a₀ + a_Z × Z + a_X × X + a_XZ × XZ
Double moderation EMM correct 2	Y = b₀ + b_M × M + b_X × X + b_XM × XM
Double moderation EMM misspecified 1	M = a₀ + a_Z × Z + a_X × X + a_XZ × XZ
Double moderation EMM misspecified 2	Y = b₀ + b_M × M + b_X × X + b_Z × Z + b_MZ × MZ

Note. EMM = Endogenous Moderator Model.

The details of our MC processes, analyses, and results can be found in Appendix A of Supplementary Online Materials. Our MC findings can be summarized as follows. For EMM and Partial Mediation EMM, two general observations are most important. First, the misspecified model estimates, among other things, an effect for X on M and an effect for the XZ product on M. Because neither effect is appropriate for an EMM, the estimation of these paths, at best, results in lack of parsimony, and at worst, results in a Type I error.

More importantly, the misspecified model replaces the correct interaction (i.e., the XM interaction on Y), with an interaction that has little basis (i.e., the XZ interaction on M). Thus, it is likely that one would conclude from the misspecified model that there is no interaction and that the indirect effect is constant even when both conclusions are incorrect. The omissions in the misspecified model also result in overestimation of residual variance for Y. Formally, the misspecified model leads to incorrect conclusions regarding b_X and b_XM, because they are omitted, and also the indirect effects, and their difference, as a result of these omissions. Because the misspecified model estimates not only the wrong interaction but also one that is unlikely to be present (based on the arguments needed to support an EMM), it is likely to result in Type II errors with regard to interaction and with regard to differences in indirect effects. It will also sometimes produce Type I errors with regard to the effects of X and XZ on M, in which case it will also produce biased estimates of a_Z.

In addition, for Partial Mediation EMM, the misspecified model omits the direct effect of Z on Y (b_Z), and it produces biased estimates of the coefficients for X on Y (b_X) and M on Y (b_M). The larger the direct effect of Z on Y (b_Z), the worse the performance of the misspecified model. And once again, larger true values of a_Z, b_M, and b_XZ result in larger discrepancies in the misspecified model. Combinations such as high b_Z, a_Z, b_M, and b_XM are particularly problematic.

For Partial Mediation EMM with direct moderation, the first general observation holds, but now the correct interaction is included in the misspecified model. The estimate of its weight (b_XZ), however, is biased because the list of variables that are partialled in the equation for Y is incomplete (see Appendix Table 3). This is particularly true for larger values of b_XM, a_Z, and b_M.

Table 3.

Results of EMM Analyses Using Example Data Sets.

Full Mediation EMM Weights and Standard Errors
SPSS (using PROCESS)
a_Z	b_X	b_M		b_XM		IE_low-X	IE_high-X
0.3 (0.065)	0.3 (0.077)	0.3 (0.149)		0.3 (0.140)		0.015 (0.061)	0.160 (0.059)
R-Scripts (using lavaan)
a_Z	b_X	b_M		b_XM		IE_low-X	IE_high-X
0.3 (0.054)	0.3 (0.064)	0.3 (0.123)		0.3 (0.154)		0.001 (0.057)	0.179 (0.066)
Partial Mediation EMM Weights and Standard Errors
SPSS (using PROCESS)
a_Z	b_X	b_M	b_Z	b_XM		IE_low-X	IE_high-X
0.3 (0.065)	0.3 (0.077)	0.3 (0.149)	0.3 (0.083)	0.3 (0.140)		0.015 (0.061)	0.160 (0.059)
R (using lavaan)
a_Z	b_X	b_M	b_Z	b_XM		IE_low-X	IE_high-X
0.3 (0.054)	0.3 (0.073)	0.3 (0.146)	0.3 (0.082)	0.3 (0.158)		0.001 (0.061)	0.179 (0.073)
Partial Mediation EMM with Direct Moderation Weights and Standard Errors
SPSS (using PROCESS)
a_Z	b_X	b_M	b_Z	b_XM	b_XZ	IE_low-X	IE_high-X
0.3 (0.065)	0.3 (0.078)	0.3 (0.151)	0.3 (0.084)	0.3 (0.150)	0.3 (0.077)	0.015 (0.068)	0.160 (0.061)
R (using lavaan)
a_Z	b_X	b_M	b_Z	b_XM	b_XZ	IE_low-X	IE_high-X
0.3 (0.054)	0.3 (0.077)	0.3 (0.149)	0.3 (0.085)	0.3 (0.165)	0.3 (0.070)	0.001 (0.064)	0.179 (0.074)
Double Moderation EMM Weights and Standard Errors
SPSS (using PROCESS)
a_Z	a_X	a_XZ	b_X	b_M	b_XM	IE_low-X	IE_high-X
0.3 (0.069)	0.3 (0.070)	0.3 (0.071)	0.3 (0.077)	0.3 (0.127)	0.3 (0.112)	0.003 (0.022)	0.284 (0.095)
R (using lavaan)
a_Z	a_X	a_XZ	b_X	b_M	b_XM	IE_low-X	IE_high-X
0.3 (0.062)	0.3 (0.082)	0.3 (0.079)	0.3 (0.068)	0.3 (0.110)	0.3 (0.129)	0.000 (0.019)	0.356 (0.126)

Note. EMM = Endogenous Moderator Model.

For Double Moderation EMM, the first general observation no longer holds, but the second does. Because the misspecified model correctly includes a_XZ, there will be indirect effect moderation. As before, however, the indirect effect of Z on Y through M is missed. In addition, the replacement of the correct second-stage interaction effect (b_XM) with one that has no basis (b_XZ) results in inaccurate estimates of the overall indirect effect and in estimates of indirect effects at different levels of moderators. The misspecified model is especially problematic when values such as a_Z, a_X, and b_XM are high (see Appendix Table 4).

In short, although the inclusion of the wrong interaction term does not bias parameter estimates to any great extent, omitting the correct terms is consequential. For example, in the full mediation EMM model, omission of b_X coupled with replacement of b_XM with a_XZ leads one not only to miss the interaction altogether but also to miss the fact that there is an indirect effect (of Z on Y) when X is high, particularly at large values of a_Z and b_XM. But such is obvious from Appendix Table 1. Perhaps less obvious are the consequences of misspecification for the more complicated models. In the case of the double moderation EMM, which contains both a_XZ and b_XM, when b_XM was replaced by b_ZM in the misspecified analysis, estimates of the truly zero b_ZM were generally in the range of 0.004 to 0.084. Not only is this small compared to the true effects for XM, which ranged from 0.0 to 0.30, it also led to considerable over- or underestimation of the indirect effect. In some cases, the misspecified model would lead one to conclude that there is a consistent lack of indirect effect when, in fact, there is a rather strong indirect effect at one level of the moderator and a null effect at another. In short, common overparameterizations of EMMs do not compromise conclusions to any great degree, but common underparameterizations do.

Recommendations for Conceptualizing and Testing EMMs

Our mathematical analysis of EMMs, combined with our review of papers that have hypothesized such models and our simulations of the consequences of common misapprehensions, suggests that the field might benefit from guidance regarding the conceptualization and testing of such models.

There is a series of questions that one should ask oneself when crafting an Introduction section, the answers to which drive reporting and analysis for an EMM. Figure 8 contains a flowchart of these questions and the actions that one should take depending on the answers. The “action” rectangles describe the appropriate EMM and the equations that are relevant for them. Appendix B contains a more detailed treatment of these questions and answers. If one follows these steps, one maximizes the probability of conceptualizing and analyzing an EMM properly.

In order to demonstrate analysis and interpretation of EMMs, we created four N=50 datasets—one for full mediation EMM, one for partial mediation EMM, one for partial mediation EMM with direct moderation, and one for double moderation EMM. We then created SPSS and R code to run them. The SPSS code relies on PROCESS (Hayes, 2013) for indirect effects while the R code uses lavaan⁴. Readers unfamiliar with both PROCESS and lavaan will need to rectify the situation. The datasets and code can be found at josecortina.com. Note that all of the sets of code include a “seed” for the bootstrapping portion, which should allow the reader to reproduce our results exactly.

Table 3 contains the results from PROCESS and lavaan for all four models. Although there are slight differences between the PROCESS and lavaan results, they are very small, so the following applies to either analytic approach. For full mediation EMM, the table shows the weights for Equations 1 and 2 and the indirect effect as per Equation 5. The effect of Z on M is significant, even with modest sample size. The same is true of the main effects of X and M on Y. The direct effect of Z on Y is zero, thus confirming the full mediation portion. Finally, the indirect effect of Z on Y varies as a function of X—when X is low (i.e., −1SD), the indirect effect is essentially zero whereas when X is high (i.e., +1SD), the indirect effect is positive and significant. All components of the full mediation EMM were confirmed.

Results were similar for the partial mediation EMM (Equations 1 and 6 along with indirect effects as per footnote 4). The only important difference is that there is now a direct effect of Z on Y. All components of the partial mediation EMM were confirmed.

The partial mediation EMM with direct moderation involves the addition of an X*Z interaction (Equations 1 and 7 along with the indirect effect components of Equation 9). Here, the X*Z interaction is significant (i.e., direct effect moderation), but the X*M interaction is not, although once again, the sample size is quite modest. The indirect effect of Z on Y is essentially zero when X is low but positive and significant when X is high. Thus, the partial mediation EMM with direct moderation is largely confirmed.

Finally, we have the results for double moderation EMM (Equations 10 and 2 along with the indirect effect component of Equation 12). Here, in addition to the effect of Z on M, we now have a significant main effect of X on M along with an X*Z interaction on M. The indirect effect is also very different at low versus high X. This is not surprising given that X moderates both the first and second stages as per the double moderation EMM.

Given that the data were generated from each respective EMM, it is not surprising that each type of EMM was confirmed. Real data are seldom so cooperative. If a_Z doesn’t pan out in any of the first three models, then the EMM fails. The main effect of Z on M is less important in the double moderation EMM because the effect of Z is moderated by X. If there were a disordinal X*Z interaction, then the main effect of Z could be quite small while the indirect effect of Z on Y would differ substantially between low and high X. For the same reason, b_M is less important in all of the models.

The indirect effects are perhaps of most importance. These should differ between high and low values of X with one exception. In the double moderation case where X moderates the Z-M relationship in one direction and the M-Y relationship in the other direction, the changes in the two stages cancel one another out. The only time that such a pattern is likely is when the reason for the first- and second-stage interactions is compression of M by X (i.e., a restricted variance interaction). In that case, as X goes up, the Z-M relationship is weakened, and the M-Y relationship is strengthened by the same amount (Cortina et al., 2019). The indirect effect, therefore, does not vary as a function of X.

Discussion

Moderation has become quite common in our field. Traditionally, moderators have been exogenous variables. Like other variables, moderators are caused by something, but our field has tended to leave those causes outside of models. This is beginning to change. Just as we wish to explain variance in relationships with moderator variables, so might we wish to explain variance in the moderators themselves. Thus, we are seeing more examples of endogenous moderator models, or EMMs.

EMMs are far more complicated than they might seem. Although they are easily confused with first-stage moderation models, each different type of EMM is in fact a form of second-stage moderation model. In the present paper, we used reduced form equations to understand the true nature of a variety of EMMs. We then used MC simulation to demonstrate the consequences of common misspecifications of such models.

Although we describe what we consider to be the most likely forms of EMM, our list is not exhaustive. For example, endogenous moderators in chain mediation would create new complexities, with the specifics depending on where in the chain the moderation occurred. Different complexities would be formed if the cause of the moderator were itself made endogenous (e.g., W causes Z which causes M). Among other things, this would create some complex indirect effects of W, with the specifics depending on the presence of other moderators, the location of direct effects, etc. As another example, multilevel EMMs would create their own complexities. Future research would do well to use a similar approach to that used in the present paper to breakdown various forms of multilevel EMM. Finally, all of our models are recursive. But what if Z causes M and then M has the audacity to cause Z right back? Feedback loops always create unique challenges, especially when they are tested with cross-sectional data. In an EMM, there would be additional conceptual, analytic, and reporting challenges. In any case, the present paper is only a beginning. As EMMs become more common, our hope is that this paper will help researchers to think through the nature and requirements of their models so that those models can be adequately justified and tested.

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Supplemental material, sj-R-14-orm-10.1177_10944281211065111 for Endogenous Moderator Models: What They are, What They Aren’t, and Why it Matters by Jose M. Cortina, Christian Dormann, Hannah M. Markell and Sheila K. Keener in Organizational Research Methods

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Jose M. Cortina

Christian Dormann

Sheila K. Keener

Supplemental Material

Supplemental material for this article is available online.

Notes

Author Biographies

Jose M. Cortina is a Professor of Management and Entrepreneurship in the School of Business at Virginia Commonwealth University. He is a past Editor of Organizational Research Methods and Associate Editor of the Journal of Applied Psychology. Dr. Cortina was honored by the Society for Industrial and Organizational Psychology with the 2001 Distinguished Early Career Contributions Award, by the Research Methods Division of AOM with the 2004 Best Paper Award and by the ORM Editorial Board with the 2012 and 2017 Best Paper Awards. He was honored by GMU with a 2010 Teaching Excellence Award and by SIOP with the 2011 Distinguished Teaching Award.

Dr. Cortina recently served as President of SIOP. In 2020, he was honored by the Research Methods Division of AOM with the Distinguished Career Award. Among his current research interests are the improvement of research methods in the organizational sciences and the use of restricted variance to hypothesize and test interactions.

Christian Dormann is Professor of Business Education & Management at the Faculty of Economics & Law of the Johannes Gutenberg-University Mainz, Germany, and adjunct research professor at the School of Psychology at the University of South Australia. He served as editor-in-chief of the European Journal of Work and Organizational Psychology and as associate and consulting editor of several other journals, including the Journal of Occupational and Organizational Psychology, Journal of Organizational Behavior, and Journal of Occupational Health Psychology. His research focus is on stress in organizations with special focus on methodological considerations including moderating effects, continuous time modeling, and meta-analysis.

Hannah Markell-Goldstein earned her PhD in Industrial-Organizational Psychology at George Mason University, where she focused on advanced methodological techniques and diversity/discrimination among women and people of color in the workplace. She worked as a Business Manager on the People Strategy & Analytics team at Capital One, where she focused on Enterprise Diversity, Inclusion, & Belonging Analytics. Some of her projects included understanding new-hire inclusion over time, developing additional measures of inclusion indicators, such as belonging, and presenting representation data to inform executive-level goals for representation. She recently accepted a new role as a Functional Business Analytics Partner in Diversity Recruiting Analytics at Meta (FKA Facebook).

Sheila K. Keener is an Assistant Professor of Management in the Strome College of Business at Old Dominion University. Her main research interests include the improvement of research quality (e.g., research methods, research integrity), employee selection, and employee compensation. Her work has been published in outlets such as the Journal of Applied Psychology, Industrial and Organizational Psychology: Perspectives on Science and Practice, and Intelligence.

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