Problematic and How This Might Be Solved. Johann Jacoby & Kai Sassenberg. Knowledge Media Research Center, Tübingen

Transcription

1 Conditional Indirect Effects Among X, M, and Y 1 Why an Interaction Term in a Three Variable Mediation Model Suggests That The Model is Problematic and How This Might Be Solved Johann Jacoby & Kai Sassenberg Knowledge Media Research Center, Tübingen words (including Abstract, References, Figure captions, Appendices) Corresponding author: Johann Jacoby Knowledge Media Research Center Schleichstr Tübingen Phone: Fax: j.jacoby@iwm-kmrc.de

2 Conditional Indirect Effects Among X, M, and Y 2 Abstract A particular model comprising three variables stipulates that an interaction of an independent variable X and a mediator M in predicting a dependent variable Y be considered as evidence for a case of conditional indirect effects. The notions of indirect effects and conditional effects are formally defined and it is then shown that the model with three variables that is claimed to constitute conditional indirect effects leads into logical contradiction. Further conceptual examination of the model also yields the recognition that the model is flawed. We therefore recommend not to use the model as a guide in data collection and not to test the model as it will necessarily be problematic, especially if data seems to corroborate it. We then provide two approaches to modify the model in order to solve its problems while preserving the psychological intuitions that the original model appears to capture. Finally, the problem arising in the three variable model in focus are discussed as specific instances of a more general problem in mediation analysis. (169 words) Keywords: Indirect effects, Conditional effects, Mediation, Moderation, Additivity

3 Conditional Indirect Effects Among X, M, and Y 3 Why an Interaction Term in a Three Variable Mediation Model Suggests That The Model is Problematic and How This Might Be Solved One important and popular endeavor of psychological research is to understand psychological processes. From an established and widely recognized perspective on this term, 'understanding a process' means that one is able to explain how an independent variable X causes an effect in a dependent variable Y (e.g., Hayes, 2009). Such explanation is meanwhile often achieved by conducting mediation analysis as proposed most prominently by Baron and Kenny (1986). Mediation analysis assesses the degree to which the total effect of X on Y can be statistically accounted for by an indirect pathway between these variables via a third variable M. If the statistical association between X and M is called path α, and the association of M and Y (while statistically controlling for X) is called path β, mediation is indicated if the product α β (i.e., the indirect effect) can be considered different from zero (Hayes, 2009). In contrast, an association between variables M and Y is said to be moderated if the X M interaction effect on Y is significant (Baron & Kenny, 1986). Also, then M can be said to have different conditional effects on Y depending on X; or X can be said to have different conditional effects on Y depending on M. While the distinction between moderation and mediation has shown to be very valuable and is now widely shared and understood, recent literature has made efforts to clarify how indirect effects and interaction effects can simultaneously hold in a data set and both can be tested within a single model (Fairchild & MacKinnon, 2009; Morgan-Lopez & MacKinnon, 2006). There are a number of different such models combining mediation and moderation (see e.g., Edwards & Lambert, 2007; Muller, Judd, & Yzerbyt, 2005; Preacher, Rucker & Hayes, 2007). In most of these combination models the variable that acts as a mediator and the variable that interacts with

4 Conditional Indirect Effects Among X, M, and Y 4 X in predicting Y are two different variables. However one particular model claims a combination of mediation and moderation where a variable M mediates the effect of X on Y in accordance with the criteria of Baron and Kenny (1986), but at the same time, the same variable M also interacts with X in predicting Y (see Figure 1, Preacher et al., 2007, Model 1; see also Brett & James, 1984; Judd & Kenny, 1981a; MacKinnon, 2008). This model is therefore special in that it seems to conflate the roles of a mediator and a moderator and this is the model we are concerned with in in this article. We will in the remainder of this discussion refer to this model as the model of conditional indirect effects among X, M, and Y. In the following, we derive formally for this particular model that there are no indirect effects of X on Y via M (i.e., there is no mediation, because the assumption of different indirect effects leads to mathematical contradiction). Then, we provide a more conceptual description of this model that shows how the contradiction that we formally derived can be understood in theoretical terms. This description will show that the model violates basic psychometric principles and is a priori misspecified. Finally, we propose approaches to remedy the problems of the model and then discuss an additional issue related to mediation analysis in general from the perspective of the current discussion. The Model is Recommended and Used The particular model under scrutiny here could be ignored as minor and inconsequential if it were not recommended, hypothesized and applied in research. However, methodological work has recommended this very model quite early (Brett & James, 1984; Judd & Kenny, 1981a). Preacher et al. (2007) incorporated it in a list of standard moderated mediation models under the name of 'Model 1' (p. 195) and provided a computational tool to test this model: the popular SPSS macro MODMED. More recently this model was again included in an extended and updated SPSS macro (PROCESS) by Hayes (2012a, 'Model 74' in Hayes, 2012b, p. 91).

5 Conditional Indirect Effects Among X, M, and Y 5 Kraemer, Kiernan, Essex and Kupfer (2008), while differing on the exact naming of the variable roles and interpretation, advocate and recommend the same mathematical model as useful and informative (Case 8 in Table 2, p. S106). MacKinnon (2008) discusses the model in detail ('Interaction between the mediator and the independent variable in the single mediator model', p. 279) and asserts that the interaction of X and M in the model 'may [...] reflect important mediational processes' (p. 54). These recommendations have lead to increasing use of the model in recent years, also in high ranking journals 1. Therefore, even though this model is not a prototype of conditional indirect effects and is indeed not the most hypothesized and used moderated mediation/mediated moderation models of those that have been described in the literature, it is being recommended in some cases quite strongly, see Kraemer et al. (2008) and also hypothesized and used. The increasing availability of easy-to-use computational tools for the analysis of processes and change might even further stimulate the use of this model in the future. Therefore, we find it sensible to publicly examine this model in more detail here and advise to use it only after substantial changes to its structure (see below). Formal Derivation of the Contradiction in the Model The model of conditional indirect effects among X, M, and Y holds that a) X predicts M, b) M predicts Y while X is statistically controlled for in the prediction of Y, and c) the association of M to Y is conditional on X (i.e., an X M product term also predicts Y, or X and M interact in predicting Y). The model of conditional indirect effects among X, M, and Y thus captures the notion that X exerts its effect on Y indirectly via M (hence: model of indirect effects). However, the degree to which this is the case, according to the model, is dependent on the value of X. Therefore, the model implies indirect effects of X on Y via M that are conditional on X. 2 These are conceptual statements about the model that are commonly made, albeit in different phrasing (e.g., MacKinnon, 2008; Preacher et al., 2007). The model is visualized in

6 Conditional Indirect Effects Among X, M, and Y 6 three different forms in Figure 1. These representations differ slightly in their details, but they are common ways to graph the hypothesized relationships in the model. They are equivalent regarding the proposition that they capture conditional indirect effects. In order to discuss the model in depth, we will first define the concepts of 'indirect effect' and 'conditional effect' formally. After that we derive from these definitions the notion of 'conditional indirect effects' in the model presently in focus as well as the contradiction in this model. Definition of an Indirect Effect The following definition is in line with Baron and Kenny (1986), Bullock et al. (2010), Hayes (2009), Jo (2008), Judd and Kenny (1981a), MacKinnon, Fairchild and Fritz (2007), Muller et al. (2005), Sobel (2008), Pearl (2001), Preacher and Hayes (2004), among others. Let the following set of equations hold among variables X, Y, and M: M =κ 0 +α X +ε 0 (1) Y =κ 1 +γ X +β M +ε 1 (2) Where κ 0 and κ 0 are constants and ε 0 and ε 1 are randomly distributed, unsystematic residuals with a mean equal to zero. Definition. X is said to have an indirect effect on Y through M in a model captured by Equations 11 and 12 if αβ 0, i.e., both of the parameters α (i.e., the effect of X on M) and β (i.e., the effect of M on Y) are simultaneously different from zero. Substituting Equation 11 into Equation 12 yields: Y =κ 1 +γ X +β(κ 0 +α X +ε 0 )+ε 1 =κ 1 +β κ 0 +αβ X +γ X +ε 1 +βε 0 (3) Defining ω= κ 1 +βκ 0 =const. 3 and ε= ε 1 +βε 0, Equation 13 reduces to Y =ω+(γ +αβ) X +ε (4)

7 Conditional Indirect Effects Among X, M, and Y 7 In Equation 14, the total effect of X on Y is decomposed into 1) γ, the direct effect of X on Y, and 2) αβ, the indirect effect of X on Y via M. Definition of a Conditional Effect The following definition is in line with Aiken and West (1991); Baron and Kenny (1986); Cohen, Cohen, West & Aiken (2003), Hayes and Matthes (2009); Jaccard and Turisi (2003); Preacher, Curran and Bauer (2006), among others. Let X, Y, and M be functionally related as in: Y =κ 2 +ν X +ξ M +ϑ XM +ε 2 (5) where κ 2 is a constant, and ε 2 is a randomly distributed, unsystematic disturbance with a mean equal to zero. Equation 1. Definition. M has conditional effects on Y (i.e., depending on the values of X), if ϑ 0 in Equation 1 can be rewritten to see how the effect of M on Y is conditional (i.e., depends on X) in the presence of the interaction term: Y =κ 2 +ν X +( ξ+ϑ X ) M +ε 2 (6) The effect of M on Y is quantified in the term ξ+ ϑx of that equation, the regression weight of M. Thus, the effect of M on Y is conditional on X in the sense that the term ξ+ ϑx takes on different values for different values of X. For different values of X= x i, x j, where x i x j, the effects of M on Y conditional on X can thus be determined by substituting the respective specific value of X into Equation 2: and y( x i )=κ 2 +ν x i +(ξ+ϑ x i ) M +ε 2 (7) y( x j )=κ 2 +ν x j +( ξ+ϑx j ) M +ε 2 (8)

8 Conditional Indirect Effects Among X, M, and Y 8 In Equations 3 and 4, the terms νx i and νx j are constants, and the terms ξ+ ϑx i and ξ+ ϑx j represent the slopes of M on Y at x i and x j, respectively (i.e., the conditional effects of M). Since x i x j, these conditional effects are also different. The model of conditional indirect effects among X, M, and Y We will now formally show that, even if one assumes that the relationships that are hypothesized in the model of conditional indirect effects among X, M, and Y have seemingly received empirical confirmation and the data and the model seem consistent, the fundamental statements of the model ('There are indirect effects of X on Y via M that are conditional on X'), along with the common and standard definitions of the notions of indirect effect and conditional effect, will lead into logical contradiction. For this demonstration we make two assumptions that are quite strong, and they are in favor of the model. Therefore, if our derivation below holds under ideal circumstances as those cast in these assumptions, the results of our derivation also hold under more unfavorable conditions. Assumption 1. 'Causal unambiguity assumption': The causal direction of an association can be ascertained in some way. While in reality this poses difficulties to analyses investigating processes (e.g., Bullock, Green, & Ha, 2010; Fiedler, Schott, & Meiser, 2011; Stone-Romero & Rosopa, 2008) we will preliminarily pretend here that this is not a problem. Assumption 2. 'Perfect data assumption': The model describes variables and their relationships in either the population or a very large sample of cases that accurately reflects the parameters of the model. With this assumption, we make it clear that neither statistical power, nor correlated unsystematic residuals, sampling bias, or other related problems arising in statistical inference are a cause for the contradiction to be derived. The contradiction that we focus on is

9 Conditional Indirect Effects Among X, M, and Y 9 independent of empirical data or the relationship of a particular data set and the model, but it is inherent in the conceptual model. The model of conditional indirect effects among variables X, Y, and M has been discussed by various authors under different names: James and Brett (1984, "moderated mediation", p. 310), Preacher et al. (2007, "Model 1", p. 195), MacKinnon (2008, "interaction between the mediator and the independent variable in the single mediator model", p. 279), among others. The MacArthur approach (Kraemer et al., 2008; see also Kraemer, Wilson, Fairburn, & Agras, 2002) also discusses the model and criticizes the use of the term 'moderation' in connection with the model. Kraemer et al. (2008) propose to refer to this model as a model of 'mediation' exclusively, but also recommend to regard it as conceptually meaningful. Presently, to avoid a debate about the implications of these labels (which are not at stake here), we name this model 'model of conditional indirect effects among X, M, and Y'. The presently discussed model of conditional indirect effects among X, M, and Y proposes that i. X has an indirect effect on Y through M, and ii. the effect of M on Y is conditional on X and therefore the indirect effect of X on Y through M is also conditional on X. In a model that is characterized by i), according to Equation 11, the component path α of an indirect effect should be estimated using the equation M =κ 0 +α X +ε 0 (9) Also, according to ii) and Equation 2, the conditional effects of M on Y depending on the value of X in this model should be Y =κ 2 + ν X + (ξ+ ϑ X )M + ε 2 (10)

10 Conditional Indirect Effects Among X, M, and Y 10 where the term ξ+ ϑx represents the conditional effect (or simple slope, see Aiken & West, 1991; Judd & McClelland, 1989) of M on Y, which evidently depends on the value of X. Analogous to Equation 14, the indirect effects of X on Y via M conditional on X should be captured by the product term α ξ+ ϑx. The component α should be estimated from Equation 5. This estimation of α requires X as a genuine variable, that is as a vector of values of all observations in the sample. 4 The component ξ+ ϑx (see Equation 6) however must be estimated using a particular value of X=x i. X is not a variable in this estimation, but is replaced by a single specific value since interest is in the effect of M on Y conditional on X, or at a specific value of X. The estimation of the term α ξ+ ϑx thus requires conflicting treatments of X: it would have to figure as a genuine variable (a vector) and as a single specific value (a scalar) at the same time. Fulfilling both of these requirements simultaneously is impossible, regardless of the individual value of X on which the indirect effect is to be conditioned. Therefore the expression α ξ+ ϑx cannot be assessed at any value X={x i, x j, x k,...}. Thus, no indirect effect of X on Y via M can be estimated, let alone conditional indirect effects. 5 In summary, the model of conditional indirect effects among X, Y, and M is logically incompatible with the widely recognized and accepted notion of an indirect effect and the notion that such indirect effects would be conditional on the value of X. Conceptual reformulation The foregoing formal demonstration clearly shows that the model is internally contradictory: The model claims conditional indirect effects. But if one applies the definitions of indirect effects and conditional effects and formally derives implications from these definitions this derivation yields that in fact there cannot be indirect effects, let alone conditional indirect effects in the model. But the conceptual idea of a state of psychological affairs corresponding to

11 Conditional Indirect Effects Among X, M, and Y 11 the model may still seem intuitively plausible on its surface: It may seem that even if the model is not one of conditional indirect effects, it could still be a valid model that can be tested using empirical data. We will therefore discuss the model in two ways which will show that, beyond the formal demonstration that the model cannot contain conditional indirect effects, the model contains a conceptual contradiction, which holds even, and especially, if data is available that appears to corroborate the model. These reformulations are different perspectives on the same contradiction inherent in the model, and help to understand why the model is implausible also on a conceptual, and not only on a mathematical level. The second of these reformulations will also lead over to the constructive third part of this article, where we propose ways to avoid the contradiction inherent in the model. First reformulation: Effects among different sample subgroups An indirect effect is present when the effects a) of X on M and b) of M on Y controlling for X are simultaneously present in a population or a sample of a population. Their simultaneous presence is represented by the product term αβ in Equation 14. In this indirect effect, both component paths can be viewed as shared variance between the respective variables: Path α indicates the amount of variance that M shares with X. Path β indicates the amount of variance that M shares with Y (after controlling for the variance that Y also shares with X). In the model of conditional indirect effects among X, M, and Y, it is proposed that indirect effects (i. e., products αβ) are different at different values of X because β is different at different values of X (the relationship between M and Y is conditional on X, i.e., the X M interaction is present). In more mathematical terms, β x i β x j for any x i x j. But the path α at specific values of X=x i is also different from α in the overall model: α x i is the effect of X on M conditional on X taking the specific value x i. x i is constant, it is the specific value of X

12 Conditional Indirect Effects Among X, M, and Y 12 conditional on which β must be considered. Since x i is a constant, the variable X does not vary and cannot have an effect on M at this individual value of X. Accordingly α is not defined at different values of X. Thus, at any specific value X=x i, α is undefined and therefore αβ is undefined. Consequently, independently of the estimate of a conditional path β, there cannot be an indirect effect among observations that are homogeneous with regard to X (i.e., at a specific value of X). Therefore, there can not be conditional indirect effects. In more abstract terms, the components of the product α β, α and β, of which the model holds that they are conditional on X, are effects in different subgroups of the sample. While conditional βs are effects among observations that are homogeneous in X, and thus among subgroups of a set of data, α is an effect across all observations in a set of data and therefore among observations that are heterogeneous in X. The product term that combines one effect from a systematic subgroup of a sample and one effect from the entire sample will yield incorrect estimates of an indirect effect both in the entire sample as well as within a particular subsample. Second reformulation: Independent variance components in M Consider the following example of an intended conditional indirect effects model among X, M, and Y: Suppose that a coaching program for high school students (treatment X, with values 1='participates in the program' and -1='does not participate in the program' or any other appropriate control group) successfully teaches self-management skills (M: amount of selfmanagement skills). The self-management skills available to children (measured at some point during or after the intervention program) are positively related to academic performance (Y: grade point average at some point in time after the program ended). So far, this example depicts a simple mediation: program participation leads to more self-management skills (X causes M, path α), self-management skills are associated with higher grade point average (M leads to Y) and, as a

13 Conditional Indirect Effects Among X, M, and Y 13 result, program participation leads to higher grades (X predicts Y), with this latter relationship being substantially reduced once self-management skills (M) are statistically controlled. However, suppose that, in addition to the mentioned relationships among the variables, for those students who participated in the program the relationship between self-management skills and academic performance is positive, but for students who have not participated in the program, there is a zero relationship. This appears to constitute a treatment (X) self-management skills (M) interaction effect on grade point average (Y). It thus seems that the degree to which program participation affects grades via self-management skills is different for the students who participated in the program than for those in the control group: β is different for the different groups defined by X, and therefore, according to the model, on the surface it appears that the product αβ should also be different between the two groups of X. But consider the conditional effects β X=1 and β X=-1, that is, the relationships between M and Y within the group with X=1 (the intervention group) and within the group X = -1 (the control group), respectively: As conditional effects (or simple slopes in a multiple regression analysis, Aiken & West, 1991) they capture variance that M and Y share given a specific value of X, respectively, which means that X is held constant for each conditional effect. Therefore shared variance between M and Y that is contained in the conditional effects is conditional on X (i.e., X = x i is a background condition for covariation between M and Y), but independent of X (i.e. within simple slopes values of M do not covary with X, and values of Y also do not covary with X, since X is constant). In the example, the variance in self-management skills between the experimental group and the control group is, by definition, shared with X (i.e., the group difference in the dependent variable is a function of the difference between X=1 and X=-1). The degree to which this

14 Conditional Indirect Effects Among X, M, and Y 14 variance that M shares with X is also shared with Y indicates the strength of the indirect effect and is mathematically represented by the product αβ. But the variance components in M within the experimental group on the one hand, and within the control group on the other, are not shared with the variable X. This within group variability in self-management skills that predicts academic performance as of the conditional effects implied by the interaction could, for example, originate in pre-study interindividual differences, or be the result of rearing practice in the home. This within-group variability will be comparable across conditions if participants have been properly randomized, and it is under these circumstances unrelated to the experimental groups defined by X. Within group variability it is by the definition of randomization independent of the experimental condition. Thus, with regard to the indirect effect of the treatment (X) on academic performance (Y) via self-management skills (M), the within group variability in the conditional effects is not pertinent. It is, to be clear, relevant as an independent, additional predictor of Y. And X also predicts the amount of shared variance between M on Y (i.e. as of the simple slopes). But the variance in the origin of these simple slopes, M, is unrelated to X, and therefore cannot possibly be relevant for the explanation of any effect of X. Thus the variance that M shares with Y as per the simple slopes (to a degree depending on X), is not relevant for the indirect effect of X on Y. Indirect effects via a third variable are assessed as the degree to which that third variable explains the relationship between an independent and a dependent variable. Surely, a variance component in the third variable M that is independent of the origin of the effect to be explained cannot help in explaining this effect. However, the variance of the X M interaction term is in fact independent of X and thus independent of the origin of a hypothesized indirect effect of X. Of course, regardless of their irrelevance for indirect effects, the conditional effects between M and Y posited in the model are clearly plausible and these conditional effects of M on

15 Conditional Indirect Effects Among X, M, and Y 15 Y are important when the prediction of Y is at the focus of an investigation. How then can these conditional effects be understood and incorporated in a model in a way that avoids the contradictions demonstrated earlier? As suggested above, the conditional effects between M and Y must be regarded as evidence of a variance component that M shares with Y, but not with X. These components are not pertinent for the question of whether M explains the effect of X on Y. In addition, in a model of simple mediation, there is a systematic variance component in M that is shared with both X and Y. This latter variance component is in fact a clear indication of an indirect effect of X on Y via M. Since the former of these systematic variance components in M is not shared with X, but the second component is shared with X, these components are necessarily independent. Thus, the model in focus here implies that the variable M is composed of at least two independent systematic variance components and a random residual comprising unsystematic variance portions from different sources (e.g., measurement error, unsystematic fluctuations due to instability etc.). The latter residuals capture random variance in the sense that they are independent of all other variables in the model. They can be ignored, as this neither biases estimation nor represent conceptually interesting variation in psychological constructs. The two former components of M, however, are systematic. They are related to other variables in the model in a regular fashion and cannot be ignored without introducing bias. But they are independent of each other. Collapsing two independent variance components into one single one-dimensional variable and using that one variable as a mediator in a mediation model is psychometrically problematic. Tolerance for imperfection? One may argue that it is unproblematic to knowingly use one single measurement of M as one one-dimensional variable when in fact it is composed of two independent variance portions. After all, as Box and Draper (1986) put it: "Essentially, all (mathematical) models are wrong, but

16 Conditional Indirect Effects Among X, M, and Y 16 some are useful" (p. 424). In addition, measurements in psychological research are rarely perfect; often, two different instruments yield measurements of the same construct that are not perfectly correlated; and composite scores that contain different facets of a construct are regularily and fruitfully used. However, we believe the present case to be different for three reasons: Model inherent flaws. The tolerance for imperfect models and imperfect measures that empirical psychologists must muster considering the insight formulated in the Box and Draper (1986) quote is a matter of empirical contigency. Mathematical models of a complex world that is impossible to perfectly measure, let alone fully explain, will necessarily yield a simplified (and thus not entirely accurate) picture of the world. Therefore a certain degree of unpredictability, uncertainty and ignorance will always be inherent in the enterprise of empirical research. These adversities, however, in principle, can be remedied by more precise measurements and more data. On the other hand, the fact that M in the present model must be composed of two different variance components is not an unfortunate nuisance arising from imperfect measurement nor similar contigent adversity that is typical in empirical research. It is an inherent error in the model that would be present even in a hypothetical world with perfect measurements, perfect samples, and full knowledge of all pertinent variables this is why the Causal unambiguity assumption and the Perfect data assumption that we made above are crucial. The fact of mathematical contradiction in the model is not unpredictable in fact, the variance components are systematic and thus predictable even in a very basic, statistical way. A researcher also need not be uncertain about this fact, as there is not much that is less uncertain than a formal demonstration as the one we have provided above. Instead, the model by itself regardless of the degree of precision, completeness, or robustness of the measures used in a design to test it, inherently clearly shows that M cannot contain only one systematic variance, but must be composed of several independent variance components. Box and Draper (1986) have characterized such model errors

17 Conditional Indirect Effects Among X, M, and Y 17 as systematic error (as opposed to random error) and advise that "... the ignoring of systematic error is not an innocuous approximation" (p. 425, emphasis in original). We consequently would find it odd to retreat to a position of uncertainty and ignorance in the face of such clear knowledge about the problems in hypothesizing M as a single one-dimensional variable before data collection that is based on the model of conditional indirect effects among X, M, and Y. Perfect independence. Secondly, the two systematic variance components in M in the model of conditional indirect effects among X, M, and Y do not represent slightly different aspects of the same thing, they are by design necessarily independent. There are thus no grounds to treat them as 'approximately the same' or 'related'. While, for example, verbal and spatial intelligence might be combined to obtain one rough measure of general intelligence to the degree that their variances overlap somewhat (albeit far from fully), which reflects that general intelligence influences both verbal and spatial abilities, the two variances in M in the present model do not overlap at all. Component scores as independent and dependent variables. Lastly, there are of course contexts and research areas where variables that are composed of different independent variance components are quite common and useful. Summary scores from items that are not necessarily intercorrelated or total scores from a symptom checklist of a multifaceted clinical disorder rightfully and correctly contain independent systematic variance components. Scales occasionally contain separable subscales. These are of special importance and value as dependent variables, capturing the often complex and multidimensional nature of phenomena studied in psychological research. As independent variables they allow for rich assessments and predictions as well as informed decision making. Thus, of course, not every variable that may contain more than one independent systematic variance component should be categorically dismissed.

18 Conditional Indirect Effects Among X, M, and Y 18 However, models implying indirect effects propose a double role for one one-dimensional variable M. In such models the variable M functions as both a dependent variable (in the component path α of an indirect path way from X to Y) and an independent variable (in the component path β). If both of these roles can be fulfilled simultaneously by a variable M, a model positing indirect effects via M can provide a psychological explanation and elucidate which processes are implicated in the generation of effects. However, if such a variable is composed of different independent variance components (and if X and M interact in the present model, this is necessarily the case), M cannot play this double role of a mediator without a considerable risk that one of the components acts as the dependent variable for the α path, and the other acts as the independent variable for the β path. If this occurs, the variable M (incorrectly considered to be one-dimensional) will not further our understanding of processes, but rather obscure these processes and impede efforts to better understand the complexities of the psychological mechanisms under investigation. While one-dimensional variables comprising distinct systematic variance components are useful as dependent or independent variables, their usage in the investigation of complex processes in both independent and dependent variable roles poses the risk of finding false representations of psychological processes. Therefore, if a researcher knows from her own model that the mediator candidate M has in fact two independent variance components (as in a model of conditional indirect effects among X, M, and Y), she should refrain from testing this model at face value. We recommend to modify the model using one of the strategies below. In summary, we have shown above that in the presently discussed model, M is correlated with X and also with Y, but the conditional effects of M on Y are conditional on X (i.e., their size can be predicted from the specific value of X), but not associated with differences in X (neither

19 Conditional Indirect Effects Among X, M, and Y 19 M, nor Y, are associated with X at a specific value of X). Therefore M must comprise (at least) two systematic, but independent variance components. This realization leads us to generally recommend against the use of the model of conditional indirect effects among X, M, and Y, but also points towards possible modifications of the model which eliminate the contradictions and actually sharpen the predictions a researcher wishes to make. Before we discuss these possible modifications, we shortly point to an aspect in traditional mediation analysis (Baron & Kenny, 1986) that has occasionally startled researchers (Kraemer et al., 2008, see also Collins, Graham & Flaherty, 1998), including ourselves. The current discussion however provides a perspective that elucidates this aspect. The omitted interaction term in the Baron and Kenny (1986) approach to mediation Baron and Kenny (1986, see also Judd & Kenny, 1981b) have popularized an approach to mediation analysis based on a number of regression equations. In one of these equations (the third equation discussed on p. 1177), the independent variable X and the mediator candidate M both predict the dependent variable Y. However, the interaction term of these two predictors (i.e., their product) is omitted from the regression model in Baron and Kenny's (1986) original formulation. Given that interaction terms in multiple regression analysis are very well understood, can be easily integrated into a regression model and in principle should be included at least to statistically refute that they are substantial (see e.g., Aiken & West, 1991; Jaccard & Turisi, 2003, Preacher et al., 2006) its omission by Baron and Kenny (1986) may seem curious at first glance. In fact, Kraemer at al. (2008) have pointed out that ignoring the X M interaction term in the Baron and Kenny (1986) mediation analysis strategy is not the same as testing it: "The Baron & Kenny approach assumes that the interaction between M and T [the independent variable X, JJ&KS] is zero in the population for mediation and, thus,

20 Conditional Indirect Effects Among X, M, and Y 20 does not include the interaction in the linear model. However, assuming the interaction is zero does not make it so." (p. S103) Kraemer et al. (2008) therefore call for including the interaction term in mediation analysis to avoid model mis-specification by omission. This thusly extended model with the interaction term, then, is in fact statistically equivalent to the model of conditional indirect effects among X, M, and Y. However, the present discussion of this model in which X and M do interact in predicting Y shows that this interaction term cannot be empirically tested while preserving the conceptual and mathematical integrity of the model. The model of conditional indirect effects among X, M, and Y (which includes the X M interaction term) is mis-specified as it posits one variable M when this variable necessarily contains two independent components. Thus, Baron and Kenny (1986) did not simply make an error of omission as Kraemer et al. (2008) seem to suggest. Instead, according to the present analysis, including the interaction term should be considered as a deliberate model mis-specification by commission. In the present context, therefore, the assumption of a zero interaction in the traditional mediation model popularized by Baron and Kenny (1986) is not an empirical hypothesis or an unheeding conjecture about an empirical state of affairs. It is an assumption in the sense of the word in formal logic. This assumption is a pre-condition that must be met in order for the central steps of the analysis strategy to be sound at all (see also MacKinnon, 2008). This precondition should be distinguished from the common stipulation that a model to be estimated be the correct model (e.g., MacKinnon, 2008). In the latter cases, researchers very often cannot know whether the model they test is correct, as the answer to that question is empirically contingent: It may turn out later that a model should be expanded to contain a particular interaction effect in order to be a better approximation of the actual state of affairs. However, the case of the presently examined model of conditional indirect effects among X, M, and Y is fundamentally different. Our analysis

21 Conditional Indirect Effects Among X, M, and Y 21 has shown that the X M interaction term will yield logical contradiction in a model casting indirect effects through M, precisely when the hypothesized X M interaction is clearly present in a population of observations and found in a sample. Thus, if the X M interaction is tested and found in empirical data, the addition of the interaction term to the model does not make the model more accurate or legitimate, but it in fact alerts to the fact that the model is problematic and any traditional mediation analysis with a one-dimensional mediator candidate M cannot be sensibly conducted. This fact is discernible beforehand, it is not contingent upon empirical data, but logically necessary. Baron and Kenny (1986) appear to be aware of this. In their discussion they probably did not omit the interaction term out of negligence or ignorance toward the fact that "assuming the interaction to be zero does not make it so" (Kraemer at al., 2008, p. S103). It is actually sensible to omit the interaction term because if there is an interaction of X and M, the traditional mediation analysis strategy that Baron and Kenny (1986) propose would be logically flawed. We are not aware of an alternative generic model of indirect effects and a commensurate design along with an analysis strategy that avoids the problematic aspects of the interaction term in the model of conditional indirect effects among X, M, and Y discussed here 6. Baron and Kenny (1986) correctly restricted their discussion to exactly the cases in which X and M do not interact. This restriction avoids the logical contradiction derived above. Baron and Kenny (1986) do, on the other hand, propose a framework to combine mediation and moderation in one model (Baron & Kenny, 1986, p. 1179). Other than in the presently focused case this framework explicitly introduces a fourth variable one that interacts with X in predicting Y, but X and M do not interact in this framework. This introduction of a fourth variable is consistent with the strategy to extend the model that we propose below. Subsequent indepth treatments of the combination of mediation and moderation in the literature have also discussed such combination in models comprising four variables instead of three (see e.g.,

22 Conditional Indirect Effects Among X, M, and Y 22 Edwards & Lambert, 2007; Iacobucci, 2007; Morgan-Lopez & MacKinnon, 2006; Muller et al., 2005). On the other hand, discussions of three variable mediation models that do contain the X M interaction term (e.g., James & Brett, 1984; Judd & Kenny, 1981a; Kraemer et al., 2008; MacKinnon, 2008; Preacher et al., 2007, Model 1) have, to our knowledge, so far not provided an account of how the logical contradictions in the model that we have formally and conceptually derived here could be resolved. Finding an unproblematic model If a researcher posits a model of conditional indirect effects among X, M, and Y, she will thus face a dilemma. If, on one hand, collected data will prove the X M interaction term not to be predictive of Y, then there are no conditional effects, and the interaction term can be omitted from the model. If, on the other hand, the data shows the interaction term to be predictive of Y, the resulting model will be logically contradictory and of limited use. Two approaches are available to resolve this dilemma. They consist in adjusting the model so that empirical observation can either turn out to be consistent with a particular, unproblematic model or to be inconsistent with that model. The first approach is based on hypothesizing nonlinear relationships among variables instead of only linear ones and their linear interactions. We will discuss this approach only briefly and refer to existing literature for relevant further elaboration. The second approach requires hypothesizing more than three variables in the model, and leaving the linear relationships intact. The latter approach requires an extended design (i.e., one more variable) whereas the former does not. Therefore, the decision whether an intended model of conditional indirect effects among X, M, and Y is modified in terms of the stipulated nature of the relationships or in terms of the number of variables required to test it, should be made before data collection.

23 Conditional Indirect Effects Among X, M, and Y 23 The possibly non-linear nature of the hypothesized relationships between variables A number of articles in the literature on interaction effects have alerted to the phenomenon that non-linear relationships can be mistaken for effects of product terms of two linear predictors that are substantially correlated (i.e., for interaction effects, Lubinski and Humphreys, 1990). Busemeyer and Jones have referred to such seeming interaction effects as "spurious" (p. 553), Ganzach (1997) has characterized them as "misleading" (p. 236), and Jaccard and Turrisi (2003) have gone as far as proposing that such seeming interaction effects that in fact arise from nonlinear relationships are invalid: "Data may be the result of a generating process that derives from a curvilinear relationship [ ], but when an interaction model is fit to the data [ ], a false interaction results." (p. 83). Consequently Lubinski and Humphreys (1990), for example, have recommended modeling non-linear relationships concurrently with product term predictors, and this recommendation is echoed in discussions of simple mediation models (Imai, Keele, & Tingley, 2010; Kraemer et al., 2008; Hayes & Preacher, 2010). A number of different non-linear bivariate relationships may be hypothesized (e.g., quadratic or exponential/logarithmic) that will produce data patterns that can appear to be, but in fact are not genuine interaction effects. We discuss in Appendix B how a quadratic relationship between variables can be mistaken for an interaction effect along with two ways how quadratic non-linear indirect effects can produce data patterns that are practically indistinguishable from those stipulated in the model of conditional indirect effects among X, M, and Y (see also Figure 3). Hypothesizing quadratic relationships along with linear ones among the variables X, M, and Y will not lead to the contradictions that the X M interaction in the model of conditional indirect effects among X, M, and Y raises. Non-linear relationships as component paths of indirect effects between X and Y via M can be tested and estimated conveniently (Hayes & Preacher, 2010).

24 Conditional Indirect Effects Among X, M, and Y 24 However, as a cautionary note, we add that while the same variables would be collected in testing a model with non-linear relationships as those in the original model of conditional indirect effects among X, M, and Y, testing non-linear relationships requires at least three levels of the predicting variable in such a relationship within a mediation model. That means that either X must have three (equidistant) levels if a non-linear relationship is hypothesized between X and M; or M must have at least three levels if a non-linear relationship is hypothesized between M and Y. This should be ensured before data is collected so as to avoid not being able to test a non-linear relationship with data that has been collected for a dichotomous predictor. Adding a fourth variable to the model If recasting a model that seems to be one of conditional indirect effects among X, M, and Y as comprising non-linear relationships is not viable, the number of variables in the model should be increased. This option has been implied by authors who have always discussed models combining moderation and mediation with a minimum of four variables (e.g., Edwards & Lambert, 2007; Muller et al., 2005). Most other models of moderated mediation or mediated moderation (e.g., Models 2-5 in Preacher et al., 2007) incorporate a fourth variable in the model in this way. The model of conditional indirect effects among X, M, and Y posits a one-dimensional variable M, consisting of systematic variance representing one psychological construct and an unsystematic residual. However, as we have shown above, the X M interaction logically entails that M must contain not only one, but at least two independent systematic variance components. Two independent systematic variance components in one variable representing one psychological construct is psychometrically problematic. Therefore, the second strategy to resolve this contradiction consists in modifying the model so that it becomes consistent with the basic notion of one variable consisting of one systematic component and an unsystematic residual. This

25 Conditional Indirect Effects Among X, M, and Y 25 strategy follows naturally from the recognition of two independent variance components: it consists in introducing a fourth variable that captures the second systematic variance component of M (or X if this variable is measured, see below). Consider the consequence of the derivation we have made earlier: The variable M in the focused model cannot be composed only of one systematic variance component and an unsystematic residual. Instead the variable must contain two independent components and the residual. We have shown above that one of the systematic components is shared with X and Y, and the other systematic component is shared only with Y in the form of the conditional effects on Y (but not with X). These two components can and should be modeled as two different variables. Otherwise, the separation into a) the component that is pertinent to the indirect effect of X on Y via M and b) the component that is relevant in the prediction of Y only, but not for the indirect effects originating in X, is not possible. Then the hypotheses inherent in the original model cannot adequately be reflected and parameter estimates will be incorrect. Thus the model can be extended by identifying two different variables M 1 and M 2 that were collapsed in the original variable M. Separating variance components of M. The modified model with M 1 and M 2 stipulates that X has effects on M 1 and Y (paths α and γ, respectively), and that M 1 has an effect on Y (path β). In addition, the variable M 2 may or may not have an independent effect by itself on Y (path ζ), but will definitely interact with X in predicting Y (path θ). This is not a model of moderated mediation or mediated moderation as defined by Muller et al. (2005), but one in which M 1 can be considered a mediator, and M 2 interacts with X in independently influencing Y (see also Model E: Direct Effect Moderation Model in Edwards & Lambert, 2007, p. 4). The regression equations defining this modified model are:

26 Conditional Indirect Effects Among X, M, and Y 26 M 1 =κ M +α X +ε M (11) Y =κ Y + β M 1 + γ X + ζ M 2 + θ XM 2 + ε Y (12) Panel a) in Figure 2 visualizes this model, emphasizing that the influence of M 2 on Y is conditional on X. Panel b) is mathematically and statistically equivalent to a), but emphasizes that the relationship between X and Y is conditional on M 2. The possibility to depict the model in these two different ways, a) and b), originates from the symmetrical nature of the interaction term (i.e., commutativity). M 2 is independent of X and M 1. We can translate this modification into the example of a training program to increase selfmanagement skills and thereby academic performance that we have described earlier. We have noted that the differences in self-management skills that are related to academic performance by different conditional effects are not a result of the training program. Rather, they must result from influences extraneous to the variables in the model. These differences can be measured before the intervention, even with the same measurement instrument that is used to measure differences after the beginning of the program (i.e., the differences constituting M 1 ). They thus provide a baseline measurement that can be modeled as M 2. Since this variable would be used to predict Y simultaneously with the measurement after the beginning of the program (and therefore these measurements are statistically controlled for each other), a researcher can be certain that only those variance components in M 1 that are uniquely due to the intervention but not to previous differences will be represented by path β. Any other pre-existing differences in self-management skills are statistically controlled for by M 2. On the other hand, if, as prescribed by evaluation research standards, participants have been assigned randomly to the training vs. control group, differences in M 2 will also uniquely predict differences in Y that are independent of the intervention program.