Understanding and Using Mediators and Moderators

Transcription

1 Soc Indic Res (2008) 87: DOI /s Understanding and Using Mediators and Moderators Amery D. Wu Æ Bruno D. Zumbo Accepted: 14 May 2007 / Published online: 6 June 2007 Ó Springer Science+Business Media B.V Abstract Mediation and moderation are two theories for refining and understanding a causal relationship. Empirical investigation of mediators and moderators requires an integrated research design rather than the data analyses driven approach often seen in the literature. This paper described the conceptual foundation, research design, data analysis, as well as inferences involved in a mediation and/or moderation investigation in both experimental and non-experimental (i.e., correlational) contexts. The essential distinctions between the investigation of mediators and moderators were summarized and juxtaposed in an example of a causal relationship between test difficulty and test anxiety. In addition, the more elaborate models, moderated mediation and mediated moderation, the use of structural equation models, and the problems with model misspecification were discussed conceptually. Keywords Mediator Moderator Moderated mediation Mediated moderation Cause and effect Structural equation model Experimental design The methodology of mediation and moderation is commonly used in social science, health, psychological, educational, and sociological research. The purpose of this paper is to provide an up-to-date review of the concepts, uses, and methodology of mediation and moderation. The examples, notation, and statistical demands are purposefully broad and easily accessible, involving several different disciplines, so as to capture as wide an audience of readers as possible. Likewise, we begin with conceptual matters and matters of research design so as to clear ground of many of these confusing foundational ideas before moving to the statistical ideas that build from these foundations. Mediation and moderation are theories for refining and understanding a causal relationship. They, in essence, are researchers hypotheses about how a cause leads to an A. D. Wu B. D. Zumbo (&) Department of ECPS, University of British Columbia, Scarfe Building, 2125 Main Mall, Vancouver, BC, Canada V6T 1Z4 bruno.zumbo@ubc.ca

2 368 A. D. Wu, B. D. Zumbo effect. The investigation of mediation and moderation effects demands an integrated research plan from articulating the theoretical rationale, choosing a research design, analyzing the data, to drawing conclusions. Unfortunately, mediation and moderation effects have been widely misunderstood and misused as data analytical tools and statistical hypothesis testing with little theoretical and methodological considerations. Failure to attend to the necessity of an integrated research plan may lead to spurious conclusions. The purpose of this paper is to shed light on the investigation of mediation and moderation effects by describing the conceptual framework, the design demand, the data analyses, as well as the inferences one can and cannot make regarding mediation and moderation effects. To do so, both classical and contemporary methodologies were described and compared. This paper is organized as follows. First, the general concept and meaning of mediation and moderation effects are introduced with an analogy, example, and graphical demonstration. Second, the issues about the design demand and data analysis for modeling mediation and moderation are discussed, in more detail, in the context of a hypothetical experiment. Third, the distinction between a mediator and moderator is contrasted and summarized with examples. Fourth, the basic concepts for more elaborate models (i.e., mediated moderation and moderated mediation) are introduced. Fifth, the use of structural equation modeling (SEM) in investigating mediation and moderation effects from correlational data is briefly described. In the last section, the problem surrounding model misspecification of mediation and moderation is discussed. 1 Conceptualization: What are Mediators and Moderators? The cause-and-effect relationship has been the pursuit of many scholars in the fields of behavioral science. Testing causal hypotheses not only verifies researchers substantive theories around a phenomenon but also answers practical questions about whether an intervention or treatment program has the expected effect. However, as the findings mature, researchers often go beyond the simple account of the bivariate cause-and-effect relationship, and attempt to understand what bridges the causal relationship and what alters the magnitude or direction of the causal relationship (Frazier et al. 2004; Rose et al. 2004). Mediators and moderators are two tools that engage with these puzzles. A mediator is a third variable that links a cause and an effect. A moderator is a third variable that modifies a causal effect. As we shall describe in detail below, mediation and moderation are causal models. Here, a causal model refers to a theoretical hypothesis about how changes in one variable results in changes in another. Testing a causal hypothesis entails investigating whether a causal inference such that X causes Y is viable. Wegener and Fabrigar (2000) explicitly stated that there are three types of common causal hypotheses: direct causal effect, mediated causal effect, and moderated causal effect. Even if the data does not permit a causal conclusion (e.g., cross-sectional or non-experimental data), mediation and moderation models are, by nature, causal models because the underlying theories suggest directional inferences that are intrinsically causal (Rose et al. 2004), as one will see in the forthcoming discussions. General speaking, mediators and moderators are third variables, whose purpose is to enhance a deeper and more refined understanding of a causal relationship between an independent variable and dependent variable. The causal nature of mediation and mod-

3 Mediators and Moderators 369 eration is often overlooked or simply misunderstood hence, leading to misapplication and misinterpretation in much of applied research (Frazier et al. 2004; MacKinnon et al. 2002; Rose et al. 2004). Methodologists often present mediation and moderation effects concurrently because, conventionally, they are two competing causal theories about the mechanism through which a third variable operates between a cause and an effect (Frazier et al. 2004; Rose et al. 2004). That is, third variables are often hypothesized to function either as a mediator or a moderator to explain a causal relationship. At this point, it is crucial to pinpoint a misconception that is widely suggested and accepted among researchers. That is, a variable is tested for a mediation effect, if the hypothesis is disconfirmed, then the same variable is tested for moderation effect, or vice versa. We contend that the same operationalized variable should not be tested for both mediation and moderation effects. The appropriate role a third variable plays should be determined primarily by the researchers substantive theory and appropriate operationalization, as we will explain in detail later. 1.1 Mediators Mediation is a causal model (Rose et al. 2004; Wengener and Fabrigar 2000) that explains the process of why and how a cause-and-effect happens (Baron and Kenny 1986; Frazier et al. 2004). Hence, a mediational analysis attempts to identify the intermediary process that leads from the independent variable to the dependent variable (Muller et al. 2005, p. 852). In other words, in a simple mediational model, the independent variable is presumed to cause the mediator, and in turn, the mediator causes the dependent variable. For this reason, a mediation effect is also termed an indirect effect, surrogate effect, intermediates effect, or intervening effect (MacKinnon et al. 2002). Collins et al. (1998) provided a vivid analogy for the mediation effect. They described the mediation process as a line of dominos and knocking over the first domino starts a sequence where the rest of the dominos are knocked over one after another (p. 297). They also provided an easily understood example for a mediation model where a drug abuse prevention program (i.e., independent variable; treatment or control) is hypothesized to affect a participant s resistance to drugs (i.e., mediator), and in turn resistance to drug use affects the outcome of a drug offer (i.e., dependent variable; acceptance or refusal). The theoretical conceptualization of mediation has long been articulated in psychology (e.g., MacCorquodale and Meehl 1948; Rozeboom 1956). Woodworth s (1928) work is one of the earliest to introduce the notion of mediation in his formulation of the Stimulus Organism Response (S O R) approach to psychology in contrast to the strictly Stimulus Response (S R) approach of the behaviorist. The Organism mediates the stimulus and the response, and is perceived as an active processor between a stimulus and response. However, empirically modeling mediation was not popularized until the 1980s when a group of social-cognitive, personality, and organizational researchers such as Baron and Kenny (1986), James and Brett (1984), as well as Judd and Kenny (1981) began to formalize the data analytical strategies. Figure 1 demonstrates a mediation model using path diagrams (Baron and Kenny 1986; Frazier et al. 2004). Path diagram A, at the top, shows that there is an overall causal effect denoted as c that leads directly from X (e.g., drug abuse prevention program) to Y (e.g., outcome of a drug offer). Path diagram B introduces a mediator denoted as Me (e.g., resistance to drug use) to explain the processing mechanism between the simple X Y causal relationship. In addition to the partial direct effect of X on Y, denoted as c 0, X also

4 370 A. D. Wu, B. D. Zumbo Fig. 1 Path diagram for mediation effect has an effect on the mediator, denoted as a, and in turn, the mediator has an effect on Y denoted as b. In essence, a mediator plays dual roles in a causal relationship. On one hand, a mediator is the dependent variable for X, and on the other hand, it acts like an independent variable for Y. 1.2 Moderators A moderation effect is a causal model that postulates when or for whom an independent variable most strongly (or weakly) causes a dependent variable (Baron and Kenny 1986; Frazier et al. 2004; Kraemer et al. 2002). In essence, a moderator modifies the strength or direction (i.e., positive or negative) of a causal relationship. A simple analogy for a moderator is a dimmer that adjusts the strength of a switch on the lighting. For example, a teacher researcher may not be merely interested in knowing whether a new instructional method leads to a better learning outcome. Additionally, he or she may wish to know if the new instruction method is equally effective for students with low and high parental involvement parental involvement being the moderator. Perhaps the moderation effect is more commonly known as the statistical term interaction effect where the strength or direction of an independent variable effect on the dependent variable depends on the level (e.g., male or female) or the value (e.g., attitude) of the other independent variable. In fact, the embryo of the moderation effect can at least be traced back to the notion of the interaction effect in the context of analysis of variance (Saunders 1956). However, it is important to point out the implicit distinction in the semantic use of moderation effect and interaction effect. Interaction analysis has been extensively applied to both correlational and experimental data, as a result, the term interaction effect seems tacitly accepted as modeling hypothesis that are not necessarily causal in nature (e.g., Chaplin 1991). In contrast, the term moderation effect has continuously been reserved for models that intend to make causal hypotheses. Namely, a moderation effect is a special case of an interaction effect, a causal interaction effect, which requires a causal theory and design behind the data. In other words, a moderation effect is certainly an interaction effect, but an interaction effect is not necessarily a moderation effect. Figure 2 demonstrates the moderation effect using a conceptual path diagram. The causal effect of X on Y, denoted as c, is dependent on the value or level of the moderator, Mo. The change of strength or direction of the casual effect is indicated by the gradient shading of the arrow going from X to Y rather than a solid line as in Fig. 1.

5 Mediators and Moderators 371 Fig. 2 Conceptual path diagram for moderation effect 2 Control over Causal Design Since both mediation and moderation are casual models, to further the understanding of mediation and moderation, it is necessary to set the stage by describing the design control for general causal models. There is a long history of philosophical and scientific debate about what constitutes causation and the methodology to demonstrate it (e.g., Holland 1986; Kenny 1979; Pearl 2000; Rogosa 1987; Rosenbaum 1984; Rubin 1974, 1986; Sobel 1995, 2005). This paper primarily takes a manipulationist view in which a cause is manipulated to show if the occurrence of the effect pattern is consistent with an experimenter s hypothesis (Holland 1986; Rubin 1974, 1986; Shadish et al. 2002). Having said that, this paper does not exclude alternative conceptualizations based on the correlation or probability (e.g., Wegener and Fabrigar 2000; Sobel 1995, 2005). The Cook and Campbell (1979, p. 2) definition of experiment is a test of a causal proposition. This definition does not restrict experiments only to designs with randomization, and mediation and moderation is not restricted only to those causal relationships that are established based on randomized experiments. Nonetheless, the power of an experiment in making causal claims does depend on how much control a researcher has in the design (i.e., the operations of the independent variable, Holland 1986). The more control one has over the experiment, the more power one has in making strict causal inferences. This notion is widely known as the internal validity of an experiment (Cook and Campbell 1979; Shadish et al. 2002). In the following discussion, we describe four levels of control in an experiment that are largely related to modeling mediation and moderation effects. To understand the four levels of control over an experimental design, consider four statements that claim: Test anxiety causes poorer math performance. 1. Both Linda and Helen did poorly in the math exam, and have high trait anxiety. 2. Linda and Helen did poorly in the math exam, their state anxiety level rose before the exam. 3. Linda practiced an effective anxiety reduction technique before the exam but Helen did not. Linda s performance was better than Helen s. Both have high trait anxiety. 4. Linda took the exam without practicing the effective anxiety reduction technique. Linda took the exam again (assuming Linda did not have any memory of the first exam), but this time she practiced the effective anxiety reduction technique before the exam. Linda s performance the second time was better than the first time. Linda has high trait anxiety. In the first statement, the independent variable, Linda and Helen s general anxiety level, is seen as a stable trait or innate attribute, and was observed (i.e., measured) by the experimenter. The statement provides no indication of when the attribute was observed. In the second statement, the observation of the independent variable preceded the observation

6 372 A. D. Wu, B. D. Zumbo of the dependent variable i.e., statement two clearly specifies that trait anxiety was observed before the math test. In the third statement, test anxiety was seen as something that could be manipulated before the math test by practicing or not practicing an effective anxiety reduction technique (i.e., treatment and control) instead of being of a stable trait of an individual. In the final statement, each of the manipulations was assigned to the same person, Linda, assuming she has no memory of the first exam; hence no exposure from the first exam would influence her performance in the second exam. It seems that as the statements moving from the first to the last one is gaining increasing inferential power to conclude that test anxiety causes poor math performance. Most readers would agree that statement four verifies a causal claim, if the assumption Linda has no memory of the first exam holds. However, in reality, it is impossible to erase Linda s first exam memory as if she could go back in time. Ideally, we would wish there were two identical Linda(s) who could take the exam simultaneously with only one practicing the anxiety reduction technique. This unrealistic requirement of two identical Linda(s) or erasing Linda s first exam memory is called the fundamental problem of causal inference (Holland 1986). At this point, it is appropriate to clarify the four levels of design control implied in each of the four statements: Observation: The independent variable is simply observed or measured, and is typically a stable trait and innate attribute of a person (i.e., Linda s general anxiety trait) Precedence: The observation or measurement of the independent variable precedes the observation of the dependent variable in time (i.e., measuring Linda s anxiety level before the test began). Manipulation: Each level (e.g., control and treatment) of the independent variable is assigned to different groups of participants (i.e., Linda practiced anxiety reduction technique, but Helen did not). Randomization: The participants are randomly assigned to each level of the independent variable. Random assignment provides an alternative solution to the fundamental problem of causal inference described earlier. If a large enough number of participants is randomly assigned to the control or treatment groups, it is expected that all the participants characteristics influencing causal relationship are homogeneous as if the two groups of participants represent the two identical Linda(s) Linda in the control group and Linda in the anxiety reduction group. Note that the four levels of control are hierarchical, that is, if a higher level of control is gained, a lower level of control is certainly gained. For example, if an experimenter has control over randomization, then s/he has control over manipulation, precedence, and observation. The debate over what level of control is required to make a causal claim is beyond the scope of this paper. Fortunately, most methodologists would agree that (1) level one control is not eligible for a causal inference, and, at best, a correlational inference can be made, (2) level four control is legitimate for a causal inference if done appropriately. If the situation allows, a researcher should always aim for level four controls for maximum power of causal inference. As for the other two levels, the legitimacy in making causal inference would depend upon the control of the other design aspects and perhaps, soundness of the researchers theoretical reasoning. Having described the basic design control for a causal design, we can move to review various methodological and data analytic strategies available today in studying mediation and moderation effects.

7 Mediators and Moderators Research Design and Data Analysis for Mediation Models Typically, mediating mechanisms are proposed only if a body of literature has tentatively documented a causal relationship between an independent variable and a dependent variable (Rose et al. 2004). A mediator is often a cognitive, affective, physiological, motivational state that functions as a person s psychological process after receiving a stimulus such as intervention treatments (Hoyle and Robinson 2003). In turn, the responsive changes in the mediator leads to the change in the outcome. A state is a temporary condition of mentality or mood, transitory level of arousal or drive, and currently evoked activities or processes (Messick 1989, p. 15). Theoretically and conceptually, a mediator should be a responsive variable that changes within a person. For this reason, psychological constructs that are believed to be relatively more stable such as personality traits (e.g., extraversion) or innate attributes (e.g., gender or ethnicity) are less likely to be a candidate for a mediator. Although several design frameworks for mediation effect have been proposed to date, the classic work by Kenny and colleagues (Baron and Kenny 1986; Judd and Kenny 1981; Kenny et al. 1998) is still the most prevalent approach, and is regarded as the default paradigm for modeling mediation (Spencer et al. 2005; Collins et al. 1998). Hence, our description will focus more on this line of conceptualization, which we refer to as the Kenny approach, and then briefly introduce other contemporary thinking cast as alternatives to the Kenny approach. 3.1 The Kenny Approach The basic formation of the Kenny approach is identical to that shown in Fig. 1. Note that in Kenny s original formulation, X is randomly manipulated to show its causal effect on Y, which is a measured variable. It is crucial to make clear that it is the random assignment of the independent variable that validates the causal inferences such that X causes Y, not the simple drawing of an arrow going from X towards Y in the path diagram. Nonetheless, the mediator is not manipulated; rather, it is an observed variable just like the dependent variable. The work by Kenny and his colleagues (Baron and Kenny 1986; Judd and Kenny 1981; Kenny et al. 1998) is summarized as a four-step data analytic method to establish a mediation effect. Please refer to Fig. 1 while reading the following. Step 1. Y ¼ i þ cx þ e ð1þ To show that the independent variable is related to the dependent variable, Y is predicted by X to estimate effect c in Fig. 1. This step establishes that there is an overall direct effect that may be mediated. Also, i denotes the regression intercept and e denotes the regression error. Step 2. Me ¼ i þ ax þ e ð2þ

8 374 A. D. Wu, B. D. Zumbo To show that the independent variable is correlated with the mediator, the mediator is treated as a dependent variable and is predicted by X to test effect a in Fig. 1. Step 3. Y ¼ i þ c 0 X þ bme þ e ð3þ To show that the mediator affects the dependent variable, Y is predicted by both X and Me to test effect b in Fig. 1. Note that it is insufficient just to regress the dependent variable only on the mediator; the mediator and the dependent variable can be related because they are both caused by the independent variable (i.e., X? Me and Y), which contradicts the specification of the mediation model (X? Me? Y). Hence, the independent variable must be controlled in establishing the unique effect of the mediator on the dependent variable in step 3. Step 4. Compare c in Step 1 with c 0 in Step 3. If the mediator completely mediates the X? Y relationship, the effect of c 0 in Eq. 3, which is the direct effect of X on Y controlling for the mediator, should turn to zero. The extent to which the mediator accounts for the overall direct X? Y relationship can be calculated by the simple subtraction c c 0 (Kenny and Judd 1984), which is the decrease from the overall direct effect c in path diagram A to the partial direct effect c 0 in path diagram B. Alternatively, the product of the two effects that sequentially connect the mediation paths a b indicates the mediation effect (Sobel 1982, 1988). Theoretically, c c 0 turn out to be identical to a b in the population within the ordinary least squares statistical framework (MacKinnon et al. 1995). If step 4 is met, the magnitude of the mediation effect c c 0, would equal to c 0=c, which is equal to the overall direct effect. In this case, the X? Y relationship is said to be completely mediated. That is to say, once the mediator is included to explain the variation in the dependent variable, the overall direct effect X? Y disappears. If c 0 departs from zero, the X? Y relationship is said to be partially mediated. In more recent thinking, Kenny et al. (1998) specified that only step 2 and 3 are essential conditions for establishing a mediation effect. Obviously, step 4 is necessary only if complete mediation is hypothesized, which is somewhat unlikely if only one mediator is modeled. As for step 1, many data analysts have argued against its necessity for the following reasons: (1) an overall effect is implied if step 2 and 3 are met, (2) it is possible that an overall effect would not be observed, if c 0 are opposite in sign to ab, a scenario that the mediator acts like a suppressor (Kraemer et al. 2001; MacKinnon et al. 2000), and (3) the overall effect will not be observed if multiple mediation effects are present and cancel each other out. Kenny also points out that meeting all four steps does not conclusively establish that the hypothesized mediation model has occurred because there are other alternative models that meet the above specification. For example, because the mediator is observed and not experimentally manipulated, a reverse hypothesis that the dependent variable causes the mediator is not ruled out. As emphasized by Kenny and his colleagues, this four-step procedure is not a direct statistical test of mediation effect; rather it uses data analysis as a tool to examine whether a mediation effect is in place (Kenny et al. 1998; Kraemer et al. 2002). The four steps are stated in terms of descriptive non-zero coefficients. Their rationale behind this warning is that a trivial coefficient can be statistically significant with a large sample size, and large

9 Mediators and Moderators 375 coefficients can be non-significant with a small sample size. Hence, the four-step approach is not intended to test the statistical significance of the mediation effect. Even though it is argued that the four-step method is not a statistical test, per se, the interest in testing the significance of the mediation effect has not been diminished in practice. In fact, it could be argued that this sort of statistical testing has increased in usage in the last decade. To date, the commonly accepted approach is to directly test the significance of the mediation effect, ab. However, the test of a mediation effect has been a continuous debate because there is no consensus about the best way to estimate the standard error. The most commonly applied significance test of mediation is the Sobel test (Sobel 1982, 1988; Holmbeck 2002), which directly tests the significance pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of ab against a normal Z distribution, with a standard error approximately equal to b 2 SEa 2 þ a2 SEb 2 (SE a and SE b are the standard errors of a and b). Preacher and Hayes (2004) provides the SAS and SPSS macros for using the Sobel test, which are downloadable at Unfortunately, the Sobel test has been shown to have low statistical power because the distribution of ab often departs from a normal distribution (Hoyle and Kenny 1999; MacKinnon et al. 2002, 1995). When the sample size is small, it is recommended to use other distributional alternatives (Hoyle and Kenny 1999). Two promising alternatives have been proposed. MacKinnon provides a Z 0 statistic against which ab is tested. The Z 0 table is derived from an empirical sampling distribution for a wide range values of a and b. On the basis of these empirical sampling distributions, critical values for different significance level were determined. Tables of these critical values can be found in electronic format at The other alternative is to bootstrap the standard error of ab. Mallinckrodt et al. (2006) and Shrout and Bolger (2002) provide step-by-step application procedures for bootstrapping. As noted earlier, in Kenny s formulation of mediation, the mediator is merely measured, despite the independent variable being randomly manipulated. This lends to the criticism that Kenny s mediational model cannot provide convincing evidence that the mediator actually causes Y. Many alternative interpretations can void the intended causal inference. For example, the correlation between Me and Y may be due to another third variable that has nothing to do with the independent variable. Neither is there clear evidence in making directional inferences going from Me to the dependent variable; it may well be equally likely that Y causes the mediator, which is contrary to the hypothesized direction. Obviously, there is a weak point at the second causal link in Kenny s mediation network, consequently, the strength in making mediational interpretation is considerably diminished by this. A mediation model has an intrinsic requirement for precedence control in the design (Frazier et al. 2004; Kraemer et al. 2002; Rose et al. 2004). That is, the independent variable precedes the mediator, which itself precedes the dependent variable. The operation of a mediator, either manipulated or observed, should carefully follow this temporal sequence. Apart from the temporal sequence, the timing of observation after a treatment is also critical. Observation should follow long enough after the manipulation for the causal effect to occur and soon enough that the causal effect has not started to dissipate. However, in Kenny s formulation, although precedence of the independent variable over the dependent variable is assured through manipulation, there is no articulation about the necessity in the precedence of the independent variable over the mediator, nor was the necessity in the precedence of the mediator over the dependent variable. The other drawback of the Kenny approach for modeling mediation is that it is data analytical in nature; as an unfortunate result, it can easily be mistakenly applied to any

10 376 A. D. Wu, B. D. Zumbo types of data including those collected with only observational control. In these cases, causal inference is far from conclusive and only a weak form of causality can be suggested. Unless the researcher has a strong empirically tested theory behind the causal claims or special design aspects are in place to rule out alternative inferences, the phrasing of interpretation should remain correlational. 3.2 Experimental-Causal-Chain Design As a caveat to the dominant approach by Kenny and his colleagues, Spencer et al. (2005), affirmed the distinction between theoretical mediation, which views mediation as a theoretical articulation, and statistical mediation, which applies to various design and data analytical procedures. They argued that Kenny s analytical approach is simply one of the methods to provide statistical evidence. They promoted the importance of more experimental controls to establish the mediation effect. In their view, The Kenny approach was referred to as measurement-of-mediation design, a design that only achieves the measurement level of control (i.e., observation) over the mediator when it plays a role as an independent variable. Spencer et al. (2005) promoted the notion of experimental-causal-chain design, where the mediator is, as usual, observed when it functions like a dependent variable for the manipulated independent variable. However, the mediator is manipulated subsequently to act like an independent variable for the outcome variable. In essence, a researcher conducts two separate manipulated experiments; one aims to establish the causal relationship of X on Me, and the other aims to establish the causal relationship of Me on Y. Figure 3 shows the path diagram of such mediational design. Using the drug abuse prevention program as an example, two separate experiments should be conducted. One shows that the drug abuse prevention program causes the changes in the resistance to drug use, and the other shows manipulation of the resistance to drug use causes the changes in the outcome of accepting or refusing a drug offer. Spencer et al. (2005) contended that such designs are underutilized in the social and behavioral sciences and should be given greater attention. In their view, this type of design provides strong evidence for the theoretically proposed psychological process even though it does not directly test the meditation effect statistically. In fact, they believe that this kind of design, because it utilizes the power of experiments to demonstrate causality, often does a better job of demonstrating the proposed psychological process than does the measurement-of-mediation design. The reason they make this claim is that by manipulating both the independent variable and the mediator, one can make strong inferences about the causal chain of events. They argued that such design is a more powerful way to examine psychological processes than the Kenny approach. The drawback of this approach is that it necessitates two separate experiments, which involves the manipulation of both the independent variable and the mediator. Manipulation Fig. 3 Experimental-causal-chain design for mediation

11 Mediators and Moderators 377 of both the independent variable and mediator may not be feasible because a mediator is often a psychological process that is unable to be manipulated. Also, the mediator must be first measured in the first experiment (i.e., Me dv in Fig. 3) and then manipulated in the second experiment (i.e., Me iv in Fig. 3); Operationally, the same psychological processes often do not allow both observation and manipulation. In addition, the underlying logic for this type of design is that as long as one can show if A then B and if B then C, then one can conclude that if A then C. This logic holds true only when B represents two identical events. In other words, for the experimental-causal-chain design to work, one must show that the mediator measured and the manipulated (i.e., Me dv and Me iv ) are, in fact, the same variable so that the two separate causal links can be connected to a single integrated mediational model. 3.3 Sequence-Stage-Chain-Reaction Design In addition to the limitations stated by Spencer et al. (2005), Kenny s four-step framework demands that the researcher s data meet the typical assumptions of ordinary least squares regression: linearity, normality, equal variances, and independence. The normality and equal variances assumptions will be violated if any of the dependent variables (i.e., Y or Me) in a mediational model are measured on a categorical scale. In this case, Kenny suggested the use of logistic regression for dichotomous dependent variables that follows the same four-step procedures. The one complication of logistic regression is computing the size of the mediation effect. In addition, the independence assumption may also be violated if the experiment involves a within-subject design. Collins et al. (1998) provided an alternative framework for mediation when the dependent variable is categorical and is obtained through within-subject design. Their approach emphasizes the intra-individual, time-ordered nature of mediation. They argued that even though the notion of mediation clearly indicates that a mediator occurs after what it mediates and before the outcome, mediation is frequently assessed in cross-sectional studies, where temporal patterns cannot be documented. They contended that mediation is a chain reaction. They described the mediation process as a line of dominos; the beginning independent variable affects a mediator that in turn affects the dependent variable. The design requirement is that the same individuals go through the temporal chain. No manipulation of the independent or the mediator is necessary, and the scale of measurement is categorical for all variables. Analogous to the Kenny approach, Collin et al. s (1998) approach is data analytical in nature; however, the statistical technique is based on probability. To demonstrate their approach, Collins et al. used the drug abuse prevention example mentioned earlier. The independent variable is the prevention program with two categories (i.e., treatment or control), the mediator is the resistance to drug use with two categories (i.e., above or below a threshold), and the dependent variable is the outcome of the drug offer with two categories (i.e., acceptance and refusal). They presented three conditions that are stage-sequenced for establishing a mediation effect. Condition 1. The independent variable affects the probability of the sequence: no mediator? positive mediator? positive outcome variable. The probability of undergoing the sequence from the positive mediator stage to the positive dependent stage, given that no mediation effect and outcome effect have occurred, is greater than in the treatment group than in the control group. It is expected that for those who have not acquired resistance skills and not used drugs, the probability of first acquiring

12 378 A. D. Wu, B. D. Zumbo the resistance skill and then refusing a drug offer is greater for the treatment group than for the control group. Condition 2. The independent variable affects the probability of a transition into the positive mediator stage. The probability of a transition into the positive mediator stage for those who have never acquired the resistance skills is greater for the treatment group than for the control group. That is, participation in the prevention program increases the probability of a transition into the above threshold resistance stage. Condition 3. The mediator affects the probability of transition into the positive outcome stage at every level of the independent variable. For those who have not used drugs, being in the positive mediator stage increases the probability of a transition to the positive outcome stage for both treatment and control groups. That is, having above-threshold resistance makes an individual more likely to refuse a drug offer regardless whether the individual is in the control or treatment group. Unfortunately, Collins et al. s (1998) introduction was mostly at the conceptual level and did not provide further details about how to conduct such probabilistic analyses, with which many applied researchers are less familiar compared to the ordinary least squares technique. Another design demand that makes it less feasible is the requirement that the same participants must be observed across a minimum of three time points. In addition, the minimum design control articulated by the authors is merely at the precedence level for all the variables, irrespective of their role in the mediation model (although higher level control is allowed). This may render the validity of causal claims less warranted, in spite of the use of a within-subject design across stages. For instance, if the independent variable is comprised of naturally occurring intact groups such as students in two different high schools in different neighborhoods, precedence control in design cannot rule out the possibility that it is actually the pre-existing difference in the students school or neighborhood characteristics that causes the participants to refuse or accept a drug offer. Table 1 summarizes the minimum design requirement of the three mediation frameworks introduced so far. One can see that the most flexible design requirement articulated by the authors is the Collins et al. s sequence-stage-chain reaction approach, and the most rigid is the experimental-causal-chain approach. However, the flexibility of the sequencestage-chain-reaction approach does not preclude its causal inferential power; conversely, it could hold tremendous power in making causal inference if the design allows the independent variable and the mediator to be manipulated or randomly assigned. Table 1 Comparisons for the design requirement among three approaches to mediation Mediation approach Minimum design requirement IV Me DV Metric of the mediator Measurement-of-mediation The Kenny Approach Experimental-causal-chain Spencer et al. (2005) Sequence-stage-chain-reaction Collins et al. (1998) M O O Quantity M M O iv: Category; dv: Quantity P P P Category Note. M indicates manipulation, O indicates observation, and P indicates precedence

13 Mediators and Moderators Research Design and Data Analysis for Moderation Models As described earlier, a moderator is a third variable that modifies the strength or direction of a causal relationship (Rose et al. 2004). A moderator is characterized as an innate attribute (i.e., gender or ethnicity), a relatively stable trait (i.e., personality types or disposition), or a relatively unchangeable background, environmental or contextual variable (i.e., parents education level or neighborhood). As Messick (1989, p. 15) stated a trait is a relatively stable characteristic of a person an attribute, enduring process, or disposition which is consistently manifested to some degree when relevant, despite considerable variation in the range of settings and circumstances. Such a variable can either be a person characteristic variable, which tackles questions like For whom the treatment works? or a situational variable, which tackles questions like Where or when does the treatment work? Contrary to a mediation effect, a moderation effect is often sought after when a hypothesized causal relationship is weak or not found empirically (Baron and Kenny 1986; Chaplin 1991; Frazier et al. 2004). The above statement does not imply that further investigation of a moderation effect is exempted if a causal effect is significant, nor that one will always find a moderation effect if an overall causal effect is not found. One of the reasons that a true causal effect is not found or unexpectedly weak may be that there is a hidden moderation effect. An overall causal effect may be non-significant because the causal effect is true only for a small group of the sample, but not for the rest. For instance, a new instructional method for mathematics that invokes high level reasoning ability may be beneficial to a small number of children with exceptional reasoning ability that is beyond the age norm but not for most of the children with age-appropriate reasoning ability. As a result, an overall effect may not be found if the new program is delivered to all students. A non-significant true causal effect could also happen if the causal relationship for one subgroup is positive but negative for the other. In this case, the opposite causal effects may cancel out the overall causal effect. For instance, a program intends to enhance young children s social behaviors by encouraging body contacts may have a positive social outcomes for girls but negative social outcomes for boys. Compared to mediation, the moderation effect is more familiar to many researchers, because it is, in convention, characterized statistically as an interaction effect. Interaction effects are extensively covered in textbooks for experimental design under the heading of, for example, two-way factorial ANOVA (Cohen et al. 2003; Frazier et al. 2004). In addition, the role of a moderator is easier to conceptualize than a mediator. As explained earlier, mediators have dual roles an outcome variable for the independent variable and an independent variable. A moderator, in contrast, takes a single role as an independent variable. However, the function of a moderator is somewhat different from that of a mediator when it functions as an independent variable. When a mediator is characterized as a cause for the dependent variable, the design would prefer, at least, a manipulation control to validate the causal role of the mediator. In contrast, a moderator s job is to explain the strength and direction of the causal effect of the focal independent variable (e.g., treatment) on the dependent variable. In this sense, a moderator is characterized more as an auxiliary variable to refine a hypothesized bivariate causal relationship, and less as a causal variable responsible for the outcome effect. The focal independent in a moderation model that is hypothesized to cause the change in the dependent variable is, at least, expected to be manipulated if not randomly assigned.

14 380 A. D. Wu, B. D. Zumbo In contrast, because a moderator is seen as an auxiliary variable to explain a causal relationship, the demand for design control in manipulation is less a concern for moderators. For this reason, moderators are typically observed rather than manipulated. It is important to note that a moderator does not change with the independent variable, nor should it correlate with the independent variable (Kraemer et al. 2002). Moderators occur prior to manipulation of an experiment (Kraemer et al. 2001, 2002; Rose et al. 2004). To show that a variable is a moderator of treatment, the variable must be a baseline or pretreatment characteristic, and is uncorrelated with the treatment (Kraemer et al. 2002). Of course, the observation of a moderator should also precede a dependent variable (Kramer et al. 2002, 2001). At least, the demand for precedence control must be met for a variable to be considered as a moderator. The measurement of the moderator can be on a continuous scale (e.g., value of selfconfidence) or a categorical scale (e.g., gender) (West et al. 1996). When the moderator is a categorical variable, the appropriate statistical technique is the familiar two-way factorial ANOVA, and the moderation effect is indicated by a significant interaction effect (Baron and Kenny 1986). The effect of an independent variable on the dependent variable for each category of the moderator is then calculated and tested for significance. When the moderator is measured on a quantitative scale, a regression analysis is often a more appropriate choice because it has superior statistical power than ANOVA where the continuous variable has to be collapsed into categories for the analysis to work (Cohen et al. 2003; Frazier et al. 2004; Jaccard et al. 1990; MacCallum et al. 2002; West et al. 1996). In cases of both categorical and continuous moderators, the moderation model can be written as a multiple regression such that Y ¼ i þ ax þ bmo þ cðx MoÞ ð4þ where i is the regression intercept, a is the partial regression coefficient for the focal independent variable X, b is the partial regression coefficient for the moderator, and c is the partial regression coefficient for the product term X*Mo, which is the moderation effect. Figure 4 illustrates a statistical path diagram [in contrast to Fig. 2 demonstrated in Baron and Kenny (1986), which is conceptual in nature]. In this diagram, the dependent variable Y is predicted by three variables: X, Mo, and X*Mo. Moderation is indicated by the significant effect of the product term X*Mo while X and Mo are controlled. The effect c of X*Mo represents the unique synergistic effect of the two variables working together, over and above their separate effects. Thus, two variables Fig. 4 Statistical path diagram for moderation effect

15 Mediators and Moderators 381 X and Mo are said to interact in accounting for the variance in Y; that is, over and above their separate effects, they have a joint effect. Although the statistical technique for interaction and moderation is identical as shown in Fig. 4, there is a subtle difference in the intention of interpretation. The general interpretation of an interaction regression is often that both X1 and X2 are seen as focal explanatory variables, and the interaction effect X1*X2 is seen as a non-linear effect created by the interaction between X1 and X2. In a moderation model, the focus of the intended interpretation is somewhat different. The focal variable X has a primary role as the cause for the dependent variable; the moderator, on the other hand, is seen as a third variable that alters the causal effect of the focal independent variable on the dependent variable. In this sense, a mediator has a secondary role in refining a causal effect. It is highly recommended that a continuous moderator be centered before creating a cross-product term and entering the regression analysis (Cohen et al. 2003; Frazier et al. 2004; Rose et al. 2004). The reason for centering is that, unless the moderator has a meaningful zero point, the interpretations of the main effects, a and b, are meaningless. Centering is accomplished by subtracting the sample mean from each individual s score on the continuous moderator. Centering produces two straightforward and meaningful interpretations of the main effects coefficients: (1) the effect of individual cause at the mean of the sample, and (2) average effect of each individual predictor across the range of the other variables. In addition, centering eliminates the problems of non-essential multicollinearity between the two independent variables, X and Mo, with the product term X*Mo (Cohen et al. 2003). Centering does not alter the significance of the moderation test, nor does it alter the value of the regression coefficient c. Note that centering a continuous dependent variable is unnecessary; in fact, keeping the metric of the dependent variable helps the interpretation consistent with the original metric of the data. Moderation effects are best detected when the causal effect is substantial (Chaplin 1991; Frazier et al. 2004; Jaccard et al. 1990). However, as mentioned before, the moderation effect is often examined when there are unexpectedly weak or no causal relations (Baron and Kenny 1986; Chaplin 1991). Aguinis et al. (2001) showed that the typical power of t- test for detecting a moderation effect ranges from.20 to.34, which is much lower than the recommended level of.80 (Cohen 1988). In addition, the effect size for moderation is generally small (Chaplin 1991). For this reason, it is crucial to conduct a power analysis and estimate the required sample size prior to the data collection for a true moderation effect to be detected (Frazier et al. 2004). To do so, the effect size of the moderation effect should be estimated. The effect size for moderation is often indexed as the change in R 2 value as a result of including the synergistic moderation effect X*Mo over and above the two main effects (Frazier et al. 2004). If a theory predicts that the X Y causal effect varies with the moderator, understanding and interpreting the causal effect would focus on how the moderator impacts the X Y causal relationship. This is achieved by plotting the regression of Y on X at each level of a categorical moderator called simple main effects (i.e., difference in group means) or at meaningful cut-off points (e.g., ± 1*SD) of a continuous moderator, called simple regression slopes (Cohen et al. 2003; Holmbeck 2002). Figure 5 demonstrates a hypothetical example for a moderation effect using the instructional method example described earlier. Parental involvement at home moderates the effect of a new instructional method on students learning outcomes. First the new instructional method should be shown to uncorrelated with parental involvement. It was observed that, at the baseline, children with high parental involvement performed 4 points better than children with medium and low parental involvement. In addition, the new instructional method is most effective among