Model Drawing

Warner, R. M. (2021). Applied sta s cs II: Mul variable and Mul variate Techniques (3rd ed.). Los Angeles, CA: Sage Publica ons. ISBN: 9781544398723.

CHAPTER 9 MEDIATION 9.1 DEFINITION OF MEDIATION Media on involves a set of causal hypotheses. An ini al causal variable X1 may influence an outcome variable Y through a media ng variable X2. (Some books and websites use different nota ons for the three variables; for example, on Kenny’s media on webpage, h p://www.davidakenny.net/cm/mediate.htm, the ini al causal variable is denoted X, the outcome as Y, and the media ng variable as M.) Media on occurs if the effect of X1 on Y is partly or en rely “transmi ed” by X2. A mediated causal model involves a causal sequence; first, X1 causes or influences X2; then, X2 causes or influences Y. X1 may have addi onal direct effects on Y that are not transmi ed by X2. A media on hypothesis can be represented by a diagram of a causal model. Note that the term causal is used because the path diagram represents hypotheses about possible causal influence; however, when data come from nonexperimental designs, we can only test whether a hypothesized causal model is consistent or inconsistent with a par cular causal model. That analysis falls short of proof that any specific causal model is correct. 9.1.1 Path Model Nota on Path model nota on was introduced earlier and it is briefly reviewed here. We begin with two variables (X and Y). Arrows are used to correspond to paths that represent different types of rela ons between variables. The absence of an arrow between X and Y corresponds to an assump on that these variables are not related in any way; they are not correlated or confounded, and they are not directly causally connected. A unidirec onal arrow corresponds to the hypothesis that one variable has a causal influence on the other—for example, X → Y corresponds to the hypothesis that X causes or influences Y; Y → X corresponds to the hypothesis that Y causes or influences X. A bidirec onal or double-headed arrow represents a noncausal associa on, such as correla on or confounding of variables that does not arise from any causal connec on between them. In path diagrams, these double-headed arrows may be shown as curved lines. If we consider only two variables, X and Y, there are four possible models: (a) X and Y are not related in any way (this is denoted in a path diagram by the absence of a path between X and Y), (b) X causes Y (X → Y), (c) Y causes X (Y → X), and (d) X and Y are correlated but not because of any causal influence (XY). When a third variable is added, the number of possible rela onships among the variables X1, X2, and Y increases substan ally. One theore cal model corresponds to X1 and X2 as correlated causes of Y. For this model, the appropriate analysis is a regression to predict Y from both X1 and X2. Another possible hypothesis is that X2 may be a moderator of the rela onship between X1 and Y; this is also described as an interac on between X2 and X1 as predictors of Y. Sta s cal significance and nature of interac on can be assessed using procedures described in Chapter 7, on modera on. 9.1.2 Circumstances in Which Media on May Be a Reasonable Hypothesis

Because a mediated causal model includes the hypothesis that X1 causes or influences X2 and the hypothesis that X2 causes or influences Y, it does not make sense to consider media on analysis in situa ons where one or both hypotheses would be nonsense. For X1 to be hypothesized as a cause of X2, X1 should occur before X2, and there should be a plausible mechanism through which X1 could influence X2. For example, suppose we are interested in a possible associa on between height and salary (a few studies suggest that taller people earn higher salaries). It is conceivable that height influences salary (perhaps employers have a bias that leads them to pay tall people more money). It is not conceivable that a person’s salary changes his or her height. 9.2 HYPOTHETICAL RESEARCH EXAMPLE This hypothe cal correla onal study examines associa ons among three variables as an illustra on of a media on hypothesis: X1, age; X2, body weight; and Y, systolic blood pressure (SBP). The data are in an SPSS file named ageweightbp.sav. Note that for research applica ons of media on analysis, much larger sample sizes should be used. For these variables, it is plausible to hypothesize the following causal connec ons. Blood pressure tends to increase as people age. As people age, body weight tends to increase (this could be due to lower metabolic rate, reduced ac vity level, or other factors). Other factors being equal, increased body weight makes the cardiovascular system work harder, and this can increase blood pressure. It is possible that at least part of the age-related increase in blood pressure might be mediated by age-related weight gain. Figure 9.1 is a path model that represents this media on hypothesis for this set of three variables. To es mate the strength of associa on that corresponds to each path in Figure 9.1, a series of three ordinary least squares (OLS) linear regression analyses can be run. Note that a variable is dependent if it has one or more unidirec onal arrows poin ng toward it. We run a regression analysis for each dependent variable (such as Y), using all variables that have unidirec onal arrows that point toward Y as predictors. For the model in Figure 9.1, the first regression predicts Y from X1 (blood pressure from age). The second regression predicts X2 from X1 (weight from age). The third regression predicts Y from both X1 and X2 (blood pressure predicted from both age and weight). 9.3 LIMITATIONS OF “CAUSAL” MODELS Path models similar to Figure 9.1 are called “causal” models because each unidirec onal arrow represents a hypothesis about a possible causal connec on between two variables. However, the data used to es mate the strength of rela onship for the paths are almost always from nonexperimental studies, and nonexperimental data cannot prove causal hypotheses. If the path coefficient between two variables such as X2 and Y (this coefficient is denoted b in Figure 9.1) is sta s cally significant and large enough in magnitude to indicate a change in the outcome variable that is clinically or prac cally important, this result is consistent with the possibility that X2 might cause Y, but it is not proof of a causal connec on. Numerous other situa ons could yield a large path coefficient between X2 and Y. For example, Y may cause X2; both Y and X2 may be caused by some third variable, X3; X2 and Y may actually be measures of

the same variable; the rela onship between X2 and Y may be mediated by other variables, X4, X5, and so on; or a large value for the b path coefficient may be due to sampling error. Descrip on Figure 9.1 Hypothe cal Media on Example: Effects of Age on Systolic Blood Pressure (SBP) 9.3.1 Reasons Why Some Path Coefficients May Be Not Sta s cally Significant If the path coefficient between two variables is not sta s cally significantly different from zero, there are also several possible reasons. If the b path coefficient in Figure 9.1 is close to zero, this could be because there is no causal or noncausal associa on between X2 and Y. However, a small path coefficient could also occur because of sampling error or because assump ons required for regression are severely violated. 9.3.2 Possible Interpreta ons for Sta s cally Significant Paths A large and sta s cally significant b path coefficient is consistent with the hypothesis that X2 causes Y, but it is not proof of that causal hypothesis. Replica on of results (such as values of a, b, and c′ path coefficients in Figure 9.1) across samples increases confidence that findings are not due to sampling error. For predictor variables and/or hypothesized media ng variables that can be experimentally manipulated, experimental studies can be done to provide stronger evidence whether associa ons between variables are causal (MacKinnon, 2008). By itself, a single media on analysis only provides preliminary nonexperimental evidence to evaluate whether the proposed causal model is plausible (i.e., consistent with the data). 9.4 QUESTIONS IN A MEDIATION ANALYSIS Researchers typically ask two ques ons in a media on analysis. The first ques on is whether there is a sta s cally significant mediated path from X1 to Y via X2 (and whether the part of the Y outcome variable score that is predictable from this path is large enough to be of prac cal importance). Recall from the discussion of the tracing rule in Chapter 4 that when a path from X to Y includes more than one arrow, the strength of the rela onship for this mul ple-step path is obtained by mul plying the coefficients for each included path. Thus, the strength of the mediated rela onship (the path from X1 to Y through X2 in Figure 9.1) is es mated by the product of the a × b (ab) coefficients. The null hypothesis of interest is H0: ab = 0. Note that the unstandardized regression coefficients are used for this significance test. Later sec ons in this chapter describe test sta s cs for this null hypothesis. If this mediated path is judged to be nonsignificant, the media on hypothesis is not supported, and the data analyst would need to consider other explana ons. If there is a significant mediated path (i.e., the ab product differs significantly from zero), then the second ques on in the media on analysis is whether there is also a significant direct path from X1 to Y; this path is denoted c′ in Figure 9.1. If c′ is not sta s cally significant (or too small to be of any prac cal importance), a possible inference is that the effect of X1 on Y is completely mediated by X2. If c′ is sta s cally significant and large enough to be of prac cal importance, a possible inference is that the influence of X1 on Y is only par ally mediated by X2 and that X1 has some addi onal effect on Y that is not mediated by X2. In the hypothe cal data used for the

example in this chapter (in the SPSS file ageweightbp.sav), we will see that the effects of age on blood pressure are only par ally mediated by body weight. Of course, it is possible that there could be addi onal mediators of the effect of age on blood pressure; for example, age-related changes in the condi on of arteries might also influence blood pressure. Models with mul ple media ng variables are discussed briefly later in the chapter. 9.5 ISSUES IN DESIGNING A MEDIATION ANALYSIS STUDY A media on analysis begins with a minimum of three variables. Every unidirec onal arrow that appears in Figure 9.1 represents a hypothesized causal connec on and must correspond to a plausible theore cal mechanism. A model such as age → body weight → blood pressure seems reasonable; processes that occur with advancing age, such as slowing metabolic rate, can lead to weight gain, and weight gain increases the demands on the cardiovascular system, which can cause an increase in blood pressure. However, it would be nonsense to propose a model of the following form: blood pressure → body weight → age, for example; there is no reasonable mechanism through which blood pressure could influence body weight, and weight cannot influence age in years. 9.5.1 Types of Variables in Media on Analysis Usually all three variables (X1, X2, and Y) in a media on analysis are quan ta ve. A dichotomous variable can be used as a predictor in regression (Chapter 6), and therefore it is acceptable to include an X1 variable that is dichotomous (e.g., treatment vs. control) as the ini al causal variable in a media on analysis; OLS regression methods can s ll be used in this situa on. However, both X2 and Y are dependent variables in media on analysis; if one or both of these variables are categorical, then logis c regression is needed to es mate regression coefficients, and this complicates the interpreta on of outcomes (see MacKinnon, 2008, Chapter 11). It is helpful if scores on the variables can be measured in meaningful units because this makes it easier to evaluate whether the strength of influence indicated by path coefficients is large enough to be clinically or prac cally significant. For example, suppose that we want to predict annual salary in dollars (Y) from years of educa on (X1). An unstandardized regression coefficient is easy to interpret. A student who is told that each addi onal year of educa on predicts a $50 increase in annual salary will understand that the effect is too weak to be of any prac cal value, while a student who is told that each addi onal year of educa on predicts a $5,000 increase in annual salary will understand that this is enough money to be worth the effort. O en, however, measures are given in arbitrary units (e.g., happiness rated on a scale from 1 = not happy at all to 7 = extremely happy). In this kind of situa on, it may be difficult to judge the prac cal significance of a half-point increase in happiness. As in other applica ons of regression, measurements of variables are assumed to be reliable and valid. If they are not, regression results can be misleading.

9.5.2 Temporal Precedence or Sequence of Variables in Media on Studies Hypothesized causes must occur earlier in me than hypothesized outcomes (temporal precedence, as discussed in Volume I, Chapter 2 [Warner, 2020]). It seems reasonable to hypothesize that “being abused as a child” might predict “becoming an abuser as an adult”; it would not make sense to suggest that being an abusive adult causes a person to have experiences of abuse in childhood. Some mes measurements of the three variables X1, X2, and Y are all obtained at the same me (e.g., in a one- me survey). If X1 is a retrospec ve report of experiencing abuse as a child, and Y is a report of current abusive behaviors, then the requirement for temporal precedence (X1 happened before Y) may be sa sfied. In some studies, measures are obtained at more than one point in me; in these situa ons, it would be preferable to measure X1 first, then X2, and then Y; this may help establish temporal precedence. When all three variables are measured at the same point in me and there is no logical reason to believe one of them occurs earlier in me than the others, it may not be possible to establish temporal precedence. 9.5.3 Time Lags Between Variables When measures are obtained at different points in me, it is important to consider the me lag between measures. If this me lag is too brief, the effects of X1 may not be apparent yet when Y is measured (e.g., if X1 is ini a on of treatment with either placebo or Prozac, a drug that typically does not have full an depressant effects un l about 6 weeks, and Y is a measure of depression and is measured one day a er X1, then the full effect of the drug will not be apparent). Conversely, if the me lag is too long, the effects of X1 may have worn off by the

me Y is measured. Suppose that X1 is receiving posi ve feedback from a rela onship partner and Y is rela onship sa sfac on, and Y is measured 2 months a er X1. The effects of the posi ve feedback (X1) may have dissipated over this period of me. The op mal me lag will vary depending on the variables involved; some X1 interven ons or measured variables may have immediate but not long-las ng effects, while others may require a substan al me before effects are apparent. 9.6 ASSUMPTIONS IN MEDIATION ANALYSIS AND PRELIMINARY DATA SCREENING Unless the types of variables involved require different es ma on methods (e.g., if a dependent variable is categorical, logis c regression methods are required), the coefficients (a, b, and c′) associated with the paths in Figure 9.1 can be es mated using OLS regression. All of the assump ons required for regression (see Volume I, Chapter 11 [Warner, 2020], and Chapter 5 in the present volume) are also required for media on analysis. Because preliminary data screening was presented in greater detail earlier, data-screening procedures are reviewed here only briefly. For each variable, histograms or other graphic methods can be used to assess whether scores on all quan ta ve variables are reasonably normally distributed, without extreme outliers. If the X1 variable is dichotomous, both groups should have a reasonably large number of cases. Sca erplots can be used to evaluate whether rela onships between each pair of variables appear to be linear (X1 with Y, X1 with X2, and X2 with Y) and to iden fy bivariate outliers.

Baron and Kenny (1986) suggested that a media on model should not be tested unless there is a significant rela onship between X1 and Y. In more recent treatments of media on, it has been pointed out that in situa ons where one of the path coefficients is nega ve, there can be significant mediated effects even when X1 and Y are not significantly correlated (Hayes, 2009). This can be understood as a form of suppression. If none of the pairs of variables in the model are significantly related to one another in bivariate analyses, however, there is not much point in tes ng mediated models. MacKinnon, Krull, and Lockwood (2000) explained that the pa erns of results obtained from analysis of models such as the models presented here for par al or complete media on cannot prove media on hypotheses. The pa erns of results that might be interpreted as evidence of media on are equally consistent with the outcomes expected for suppression and confounded predictor. Empirical results do not make it possible to dis nguish which of these explana ons is “be er.” 9.7 PATH COEFFICIENT ESTIMATION The most common way to obtain es mates of the path coefficients that appear in Figure 9.1 is to run the following series of regression analyses. These steps are similar to those recommended by Baron and Kenny (1986), except that, as suggested in recent treatments of media on (MacKinnon, 2008), a sta s cally significant outcome on the first step is not considered a requirement before going on to subsequent steps. Step 1: First, a regression is run to predict Y (SBP) from X1 (age). (SPSS procedures for this type of regression were provided in Volume I, Chapter 11 [Warner, 2020], and Chapter 4 in the present volume.) The raw or unstandardized regression coefficient from this regression corresponds to path c. This step is some mes omi ed; however, it provides informa on that can help evaluate how much controlling for the X2 media ng variable reduces the strength of associa on between X1 and Y. Figure 9.2 shows the regression coefficients part of the output. The unstandardized regression coefficient for the predic on of Y (BloodPressure—note that there is no space within the SPSS variable name) from X1 (age) is c = 2.862; this is sta s cally significant, t(28) = 6.631, p < .001. (The N for this data set is 30; therefore, the df for this t ra o is N – 2 = 28.) Thus, the overall effect of age on blood pressure is sta s cally significant. Step 2: Next a regression is performed to predict the media ng variable (X2, weight) from the causal variable (X1, age). The results of this regression provide the path coefficient for the path denoted a in Figure 9.1 and also the standard error of a (sa) and the t test for the sta s cal significance of the a path coefficient (ta). The coefficient table for this regression appears in Figure 9.3. For the hypothe cal data, the unstandardized a path coefficient was 1.432, with t(28) = 3.605, p = .001. Step 3: Finally, a regression is performed to predict the outcome variable Y (blood pressure) from both X1 (age) and X2 (weight). (Detailed examples of regression with two predictor variables appeared in Chapter 4.) This regression provides es mates of the unstandardized coefficients for path b (and sb and tb) and also path c′ (the direct or remaining effect of X1 on Y

when the media ng variable has been included in the analysis). See Figure 9.1 for the corresponding path diagram. From Figure 9.4, path b = .49, t(27) = 2.623, p = .014; path c′ = 2.161, t(27) = 4.551, p < .001. These unstandardized path coefficients are used to label the paths in a diagram of the causal model (top panel of Figure 9.5). These values are also used later to test the null hypothesis H0: ab = 0. In many research reports, par cularly when the units in which the variables are measured are not meaningful or not easy to interpret, researchers report the standardized path coefficients (these are called beta coefficients in the SPSS output); the bo om panel of Figure 9.5 shows the standardized path coefficients. Some mes the es mate of the c coefficient appears in parentheses, next to or below the c′ coefficient, in these diagrams. Descrip on Figure 9.2 Regression Coefficient to Predict Blood Pressure (Y) From Age (X1) Descrip on Figure 9.3 Regression Coefficient to Predict Weight (Media ng Variable X2) From Age (X1) In addi on to examining the path coefficients from these regressions, the data analyst should pay some a en on to how well the X1 and X2 variables predict Y. From Figure 9.4, R2 = .69, adjusted R2 = .667, and this is sta s cally significant, F(2, 27) = 30.039, p < .001. These two variables do a good job of predic ng variance in blood pressure. 9.8 CONCEPTUAL ISSUES: ASSESSMENT OF DIRECT VERSUS INDIRECT PATHS When a path that leads from a predictor variable X to a dependent variable Y involves other variables and mul ple arrows, the overall strength of the path is es mated by mul plying the coefficients for each leg of the path (as discussed in the introduc on to the tracing rule in Chapter 4). Descrip on Figure 9.4 Regression Coefficient to Predict Blood Pressure (Y) From Age (X1) and Media ng Variable Weight (X2) 9.8.1 The Mediated or Indirect Path: ab The strength of the indirect or mediated effect of age on blood pressure through weight is es mated by mul plying the ab path coefficients. In many applica ons, one or more of the variables are measured in arbitrary units (e.g., happiness may be rated on a scale from 1 to 7). In such situa ons, the unstandardized regression coefficients may not be very informa ve, and research reports o en focus on standardized coefficients.1 The standardized (β) coefficients for the paths in the age, weight, and blood pressure hypothe cal data appear in the bo om panel of Figure 9.5. Throughout the remainder of this sec on, all path coefficients are given in standardized (β-coefficient) form. When the path from X to Y has mul ple parts or arrows, the overall strength of the associa on for the en re path is es mated by mul plying the coefficients for each part of the path. Thus,

the unit-free index of strength of the mediated effect (the effect of age on blood pressure, through the media ng variable weight) is given by the product of the standardized es mates of the path coefficients, ab. For the standardized coefficients, this product = (.563 × .340) = .191. The strength of the direct or nonmediated path from age to SBP corresponds to c′; the standardized coefficient for this path is .590. In other words, for a 1-SD increase in zAge, we predict a .191 increase in zSBP through the media ng variable zWeight. In addi on, we predict a .590 increase in zSBP due to direct effects of zAge (effects that are not mediated by zWeight); this corresponds to the c′ path. The total effect of zAge on zSBP corresponds to path c, and the standardized coefficient for path c is .782 (the beta coefficient to predict zSBP from zAge in Figure 9.5). Figure 9.5 Path Coefficients for the Media on Analysis of Age, Weight, and SBP 9.8.2 Mediated and Direct Path as Par on of Total Effect The media on analysis has par oned the total effect of age on blood pressure (c = .782) into a direct effect (c′ = .590) and a mediated effect (ab = .191). (Both of these are given in terms of standardized or unit-free path coefficients.) It appears that media on through weight, while sta s cally significant, explains only a small part of the total effect of age on blood pressure in this hypothe cal example. Within rounding error, c = c′ + ab, that is, the total effect is the sum of the direct and mediated effects. These terms are addi ve when OLS regression is used to obtain es mates of coefficients; when other es ma on methods such as maximum likelihood are used (as in structural equa on modeling [SEM] programs), these equali es may not hold. Also note that if there are missing data, each regression must be performed on the same set of cases in order for this addi ve associa on to work. Note that even if the researcher prefers to label and discuss paths using standardized regression coefficients, informa on about the unstandardized coefficients is required to carry out addi onal sta s cal significance tests (to find out whether the product ab differs significantly from zero, for example). 9.8.3 Magnitude of Mediated Effect When variables are measured in meaningful units, it is helpful to think through the magnitude of the effects in real units, as discussed in this paragraph. (The discussion in this paragraph is helpful primarily in research situa ons in which units of measurement have some real-world prac cal interpreta on.) All of the path coefficients in the rest of this paragraph are unstandardized regression coefficients. From the first regression analysis, the c coefficient for the total effect of age on blood pressure was c = 2.862. In simple language, for each 1-year increase in age, we predict an increase in blood pressure of 2.862 mm Hg. On the basis of the t- test result in Figure 9.2, this is sta s cally significant. Taking into account that people in wealthy countries o en live to age 70 or older, this implies substan al age-related increases in blood pressure; for example, for a 30-year increase in age, we predict an increase of 28.62 mm Hg in blood pressure, and that is sufficiently large to be clinically important. This tells us that the total effect of age on SBP is reasonably large in terms of clinical or prac cal importance. From the second regression, we find that the effect of age on weight is a = 1.432; this is also sta s cally significant, on the basis of the t test in Figure 9.3. For a 1-year increase in age, we predict almost

1.5 lb in weight gain. Again, over a period of 10 years, this implies a sufficiently large increase in predicted body weight (about 14.32 lb) to be of clinical importance. The last regression (in Figure 9.4) provides informa on about two paths, b and c′. The b coefficient that represents the effect of weight on blood pressure was b = .49; this was sta s cally significant. For each 1-lb increase in body weight, we predict almost a half-point increase in blood pressure. If we take into account that people may gain 30 or 40 lb over the course of a life me, this would imply weight-related increases in blood pressure on the order of 15 or 20 mm Hg. This also seems large enough to be of clinical interest. The indirect effect of age on blood pressure is found by mul plying a × b, in this case, 1.432 × .49 = .701. For each 1-year increase in age, a .7 mm Hg increase in blood pressure is predicted through the effects of age on weight. Finally, the direct effect of age on blood pressure when the media ng variable weight is sta s cally controlled or taken into account is represented by c′ = 2.161. Over and above any weight-related increases in blood pressure, we predict about a 2.2-unit increase in blood pressure for each addi onal year of age. Of the total effect of age on blood pressure (a predicted 2.862 mm Hg increase in SBP for each 1-year increase in age), a rela vely small part is mediated by weight (.701), and the remainder is not mediated by weight (2.161). (Because these are hypothe cal data, this outcome does not accurately describe the importance of weight as a mediator in real-life situa ons.) The media on analysis par ons the total effect of age on blood pressure (c = 2.862) into a direct effect (c′ = 2.161) and a mediated effect (ab = .701). Within rounding error, c = c′ + ab, that is, the total effect c is the sum of the direct (c′) and mediated (ab) effects. 9.9 EVALUATING STATISTICAL SIGNIFICANCE Several methods to test the sta s cal significance of mediated models have been proposed. The four most widely used procedures are briefly discussed: Baron and Kenny’s (1986) causal-steps approach, joint significance tests for the a and b path coefficients, the Sobel test (Sobel, 1982) for H0: ab = 0, and the use of bootstrapping to obtain confidence intervals (CIs) for the ab product that represents the mediated or indirect effect. 9.9.1 Causal-Steps Approach Fritz and MacKinnon (2007) reviewed and evaluated numerous methods for tes ng whether media on is sta s cally significant. A subset of these methods is described here. Their review of media on studies conducted between 2000 and 2003 revealed that the most frequently reported method was the causal-steps approach described by Baron and Kenny (1986). In Baron and Kenny’s ini al descrip on of this approach, in order to conclude that media on may be present, several condi ons were required: first, a significant total rela onship between X1, the ini al cause, and Y, the final outcome variable (i.e., a significant path c); significant a and b paths; and a significant ab product using the Sobel test or a similar method, as described in Sec on 9.9.3. The decision of whether to call the outcome par al or complete media on then depends on whether the c′ path that represents the direct path from X1 to Y is sta s cally significant; if c′ is not sta s cally significant, the result may be interpreted as complete media on; if c′ is sta s cally significant, then only par al media on may be occurring. Kenny has also noted elsewhere (h p://www.davidakenny.net/cm/mediate.htm) that other factors, such as the sizes of coefficients and whether they are large enough to be of prac cal

significance, should also be considered and that, as with any other regression analysis, meaningful results can be obtained only from a correctly specified model. This approach is widely recognized, but it is not the most highly recommended procedure at present for two reasons. First, there are (rela vely rare) cases in which media on may occur even when the original X1, Y associa on is not significant. For example, if one of the paths in the media on model is nega ve, a form of suppression may occur such that posi ve direct and nega ve indirect effects tend to cancel each other out to yield a small and nonsignificant total effect. (If a is nega ve, while b and c′ are posi ve, then when we combine a nega ve ab product with a posi ve c′ coefficient to recons tute the total effect c, the total effect c can be quite small even if the separate posi ve direct path and nega ve indirect paths are quite large.) MacKinnon, Fairchild, and Fritz (2007) referred to this as inconsistent media on; the mediator acts as a suppressor variable. See Chapter 3 for further discussion and an example of inconsistent media on. Second, among the methods compared by Fritz and MacKinnon (2007), this approach had rela vely low sta s cal power. 9.9.2 Joint Significance Test Fritz and MacKinnon (2007) also discussed a joint significance test approach to tes ng the significance of media on. The data analyst simply asks whether the a and b coefficients that cons tute the mediated path are both sta s cally significant; the t tests from the regression results are used. (On his media on webpage at h p://www.davidakenny.net/cm/mediate.htm, Kenny suggests that if this approach is used, and if an overall risk for Type I error of .05 is desired, each test should use α = .025, two tailed, as the criterion for significance.) This approach is easy to implement and has moderately good sta s cal power compared with the other test procedures reviewed by Fritz and MacKinnon. However, it is not the most frequently reported method; journal reviewers may prefer be er known procedures. 9.9.3 Sobel Test of H0: ab = 0 Another method to assess the significance of media on is to examine the product of the a, b coefficients for the mediated path. (This is done as part of Baron and Kenny’s [1986] causal- steps approach.) The null hypothesis, in this case, is H0: ab = 0. To set up a z-test sta s c, an es mate of the standard error of this ab product (SEab) is needed. Sobel (1982) provided the following approximate es mate for SEab: Other

(9.1) where a and b are the raw (unstandardized) regression coefficients that represent the effect of X1 on X2 and the effect of X2 on Y, respec vely; sa is the standard error of the a regression coefficient;

sb is the standard error of the b regression coefficient. Using the standard error from Equa on 9.1 as the divisor, the following z ra o for the Sobel test can be set up to test the null hypothesis H0: ab = 0: Other

(9.2) The ab product is judged to be sta s cally significant if z is greater than +1.96 or less than – 1.96. This test is appropriate only for large sample sizes. The Sobel test is rela vely conserva ve, and among the procedures reviewed by Fritz and MacKinnon (2007), it had moderately good sta s cal power. It is some mes used in the context of Baron and Kenny’s (1986) causal-steps procedure and some mes reported without the other causal steps. The Sobel test can be done by hand; Preacher and Leonardelli (2008) provide an online calculator at h p://quantpsy.org/sobel/sobel.htm to compute this z test given either the unstandardized regression coefficients and their standard errors or the t ra os for the a and b path coefficients. Their program also provides z tests on the basis of alterna ve methods of es ma ng the standard error of ab suggested by the Aroian test (Aroian, 1947) and Goodman test (Goodman, 1960). The Sobel test was carried out for the hypothe cal data on age, weight, and blood pressure. (Note again that the N in this demonstra on data set is too small for the Sobel test to yield accurate results; these data are used only to illustrate the use of the techniques.) For these hypothe cal data, a = 1.432, b = .490, sa = .397, and sb = .187. These values were entered into the appropriate lines of the calculator provided at Preacher’s webpage; the results appear in Figure 9.6. Because z = 2.119, with p = .034, two tailed, the ab product that represents the effect of age on blood pressure mediated by weight can be judged sta s cally significant. Note that the z tests for the significance of ab assume that values of this ab product are normally distributed across samples from the same popula on; it has been demonstrated empirically that this assump on is incorrect for many values of a and b. Because of this, authori es on media on analysis (MacKinnon, Preacher, and their colleagues) now recommend bootstrapping methods to obtain CIs for es mates of ab. 9.9.4 Bootstrapped Confidence Interval for ab Bootstrapping has become widely used in situa ons where the analy c formula for the standard error of a sta s c is not known and/or there are viola ons of assump ons of normal distribu on shape (Iacobucci, 2008). A sample is drawn from the popula on (with replacement), and values of a, b, and ab are calculated for this sample. This process is repeated many mes (bootstrapping procedures typically allow users to request from 1,000 up to 5,000 different samples). The value of ab is tabulated across these samples; this provides an empirical sampling distribu on that can be used to derive a value for the standard error of ab. Results of such bootstrapping indicate that the distribu on of ab values is o en asymmetrical, and this

asymmetry should be taken into account when se ng up CI es mates of ab. This CI provides a basis for evalua on of the single es mate of ab obtained from analysis of the en re data set. Bootstrapped CIs do not require that the ab sta s c have a normal distribu on across samples. If this CI does not include zero, the analyst concludes that there is sta s cally significant media on. Some bootstrapping programs include addi onal refinements, such as bias correc on (see Fritz & MacKinnon, 2007). Most SEM programs, such as Amos, can provide bootstrapped CIs. A detailed example is presented in Chapter 15, on structural equa on modeling. Descrip on Figure 9.6 Sobel Test Results for H0: ab = 0, Using Calculator Provided by Preacher and Leonardelli at h p://quantpsy.org/sobel/sobel.htm 9.10 EFFECT SIZE INFORMATION Effect size informa on is usually given in unit-free form (Pearson’s r and r2 can both be interpreted as effect sizes). The raw or unstandardized path coefficients from media on analysis can be converted to standardized slopes; alterna vely, we can examine the correla on between X1 and X2 to obtain effect-size informa on for the a path, as well as the par al correla on between X2 and Y (controlling for X1) to obtain effect-size informa on for the b path. There are poten al problems with comparisons among standardized regression or path coefficients. For example, if the same media on analysis involving the same set of three variables is conducted in two different samples (e.g., a sample of women and a sample of men), these samples may have different standard devia ons on variables such as the predictor X1 and the outcome variable Y. Suppose that the male and female samples yield b and c′ coefficients that are very similar, sugges ng that the amount of change in Y as a func on of X1 is about the same across the two groups. When we convert raw-score slopes to standardized slopes, this may involve mul plying and dividing by different standard devia ons for men and women, and different standard devia ons within these groups could make it appear that the groups have different rela onships between variables (different standardized slopes but similar unstandardized slopes). Unfortunately, both raw score (b) and standardized (β) regression coefficients can be influenced by numerous sources of ar fact that may operate differently in different groups. Appendix 10C in Volume I (Warner, 2020) reviewed numerous factors that can ar factually influence the size of r (such as outliers, curvilinearity, different distribu on shapes for X and Y, unreliability of measurement of X and Y, etc.). Chapter 11 in Volume I demonstrated that β coefficients can be computed from bivariate correla ons and that b coefficients are rescaled versions of β. When Y is the outcome and X is the predictor, b = β × (SDY/SDX). Both b and β coefficients can be influenced by many of the same problems as correla ons. Therefore, if we try to compare regression coefficients across groups or samples, differences in regression coefficients across samples may be due partly to ar facts. Considerable cau on is required whether we want to compare standardized or unstandardized coefficients.

Despite concerns about poten al problems with standardized regression slopes (as discussed by Greenland et al., 1991), data analysts o en include standardized path coefficients in reports of media on analysis, par cularly when some or all variables are not measured in meaningful units. In repor ng results, authors should make it clear whether standardized or unstandardized path coefficients are reported. Given the difficul es just discussed, it is a good idea to include both types of path coefficients. 9.11 SAMPLE SIZE AND STATISTICAL POWER Assuming the hypothesis of primary interest is H0: ab = 0, how large does sample size need to be to have an adequate level of sta s cal power? Answers to ques ons about sample size depend on several pieces of informa on: the alpha level, desired level of power, the type of test procedure, and the popula on effect sizes for the strength of the associa on between X1 and X2, as well as X2 and Y. O en, informa on from past studies can help researchers make educated guesses about effect sizes for correla ons between variables. In the discussion that follows, α = .05 and desired power of .80 are assumed. We can use the correla on between X1 and X2 as an es mate of the effect-size index for a and the par al correla on between X2 and Y, controlling for X1, as an es mate of the effect size for b. On the basis of recommenda ons about verbal labels for effect size given by Cohen (1988), Fritz and MacKinnon (2007) designated a correla on of .14 as small, a correla on of .39 as medium, and a correla on of .59 as large. They reported sta s cal power for combina ons of small (S), medium (M), and large (L) effect sizes for the a and b paths. For example, if a researcher plans to use the Sobel test and expects that both the a and b paths correspond to medium effects, the minimum recommended sample size from Table 9.1 would be 90. A few cau ons are in order: Sample sizes from this table may not be adequate to guarantee significance, even if the researcher has not been overly op mis c about an cipated effect size. Even when the power table suggests that fewer than 100 cases might be adequate for sta s cal power for the test of H0: ab = 0, analysts should keep in mind that small samples lead to more sampling error in es mates of path coefficients. For most studies that test media on models, minimum sample sizes of 150 to 200 would be advisable if possible. 9.12 ADDITIONAL EXAMPLES OF MEDIATION MODELS Several varia ons of the basic media on model in Figure 9.1 are possible. For example, the effect of X1 on Y could be mediated by mul ple variables instead of just one (see Figure 9.7). Media on could involve a mul ple-step causal sequence. Media on and modera on can both occur together. The following sec ons provide a brief introduc on to each of these research situa ons; for more extensive discussion, see MacKinnon (2008). 9.12.1 Mul ple Media ng Variables In many situa ons, the effect of a causal variable X1 on an outcome Y might be mediated by more than one variable. Consider the effects of personality traits (such as extraversion and neuro cism) on happiness. Extraversion is moderately posi vely correlated with happiness. Tkach and Lyubomirsky (2006) suggested that the effects of trait extraversion on happiness may be at least par ally mediated by behaviors such as social ac vity. For example, people who

score high on extraversion tend to engage in more social ac vi es, and people who engage in more social ac vi es tend to be happier. They demonstrated that, in their sample, the effects of extraversion on happiness were par ally mediated by engaging in social ac vity, but there was s ll a significant direct effect of extraversion on happiness. Their media on analyses examined only one behavior at a me as a poten al mediator. Mul ple mediators can easily be examined using SEM programs such as Amos, discussed later in this chapter. Table 9.1 Empirical Es mates of Sample Size Needed for Power of .80 When Using α = .05 as the Criterion for Sta s cal Significance in Three Different Types of Media on Analysis

Note: These power es mates may be inaccurate when measures of variables are unreliable, assump ons of normality are violated, or categorical variables are used rather than quan ta ve variables. aSS indicates that both a and b are small effects, SM indicates that a is small and b is medium, and SL indicates that a is small and b is large. bJoint significance test: Requirement that the a and b coefficients each are sta s cally significant.

cA z test for H0: ab using a method to es mate SEab proposed by Sobel (1982). dWithout bias correc on. 9.12.2 Mul ple-Step Mediated Paths It is possible to examine a media on sequence that involves more than one intermediate step, as in the sequence X1 → X2 → X3 → Y. If only par al media on occurs, addi onal paths would need to be included in this type of model; for further discussion, see Taylor, MacKinnon, and Tein (2008). 9.12.3 Mediated Modera on and Moderated Media on It is possible for modera on (described in another chapter) to co-occur with media on in two different ways. Mediated modera on occurs when two ini al causal variables (let’s call these variables A and B) have an interac on (A × B), and the effects of this interac on involve a media ng variable. In this situa on, A, B, and the A × B interac on are included as ini al causal variables, and the media on analysis is conducted to assess the degree to which a poten al media ng variable explains the impact of the A × B interac on on the outcome variable. The PROCESS macros provided by Andrew Hayes (2017, 2019) are extremely useful for assessment of models with moderated media on, or mediated modera on. Descrip on Figure 9.7 Path Model for Mul ple Media ng Variables Showing Standardized Path Coefficients Moderated media on occurs when you have two different groups (e.g., men and women), and the strength or signs of the paths in a media on model for the same set of variables differ across these two groups. Many SEM programs, such as Amos, make it possible to compare path models across groups and to test hypotheses about whether one, or several, path coefficients differ between groups (e.g., men vs. women). Further discussion can be found in Edwards and Lambert (2007); Muller, Judd, and Yzerbyt (2005); and Preacher, Rucker, and Hayes (2007). Comparison of models across groups using the Amos SEM program was demonstrated by Byrne (2016). 9.13 NOTE ABOUT USE OF STRUCTURAL EQUATION MODELING PROGRAMS TO TEST MEDIATION MODELS SEM programs such as LISREL, EQS, Mplus, and Amos make it possible to test models that include mul ple-step paths (e.g., media on hypotheses) and to compare results across groups (to test modera on hypotheses). In addi on, SEM programs make it possible to include mul ple indicator variables for some or all of the constructs; in theory, this makes it possible to assess mul ple indicator measurement reliability. Most SEM programs now also provide bootstrapping; most analysts now view SEM programs as the preferred method for assessment of mediated models. More extensive discussion of other types of analyses that can be performed using SEM is beyond the scope of this book; for further informa on, see Byrne (2009) or Kline (2016).

There are two reasons why it is worthwhile to learn how to use Amos (or other SEM programs) to test mediated models. First, it is now generally agreed that bootstrapping is the preferred method to test the sta s cal significance of indirect effects in mediated models; bootstrapping may be more robust to viola ons of assump ons of normality. Second, once a student has learned to use Amos (or other SEM programs) to test simple media on models similar to the example in this chapter, the program can be used to add addi onal predictor and/or mediator variables, as shown in Figure 9.11. 9.14 RESULTS SECTION For the hypothe cal data in this chapter, a “Results” sec on could read as follows. Results presented here are based on the output from linear regression (Figures 9.2–9.4) and the Sobel test result in Figure 9.6. (Results would include slightly different numerical values if the Amos output is used.) Results A media on analysis was performed using Baron and Kenny’s (1986) causal-steps approach; in addi on, a bootstrapped CI for the ab indirect effect was obtained using procedures described by Preacher and Hayes (2008). The ini al causal variable was age, in years; the outcome variable was systolic blood pressure (SBP), in millimeters of mercury; and the proposed media ng variable was body weight, in pounds. [Note: The sample N, mean, standard devia on, minimum and maximum scores for each variable, and correla ons among all three variables would generally appear in earlier sec ons.] Refer to Figure 9.1 for the path diagram that corresponds to this media on hypothesis. Preliminary data screening suggested that there were no serious viola ons of assump ons of normality or linearity. All coefficients reported here are unstandardized, unless otherwise noted; α = .05, two tailed, is the criterion for sta s cal significance. The total effect of age on SBP was significant, c = 2.862, t(28) = 6.631, p < .001; each 1-year increase in age predicted approximately a 3-point increase in SBP. Age was significantly predic ve of the hypothesized media ng variable, weight; a = 1.432, t(28) = 3.605, p = .001. When controlling for age, weight was significantly predic ve of SBP, b = .490, t(27) = 2.623, p = .014. The es mated direct effect of age on SBP, controlling for weight, was c′ = 2.161, t(27) = 4.551, p < .001. SBP was predicted quite well from age and weight, with adjusted R2 = .667 and F(2, 27) = 30.039, p < .001. The indirect effect, ab, was .701. This was judged to be sta s cally significant using the Sobel test, z = 2.119, p = .034. [Note: The Sobel test should be used only with much larger sample sizes than the N of 30 for this hypothe cal data set.] Using the SPSS script for the indirect procedure (Preacher & Hayes, 2008), bootstrapping was performed; 5,000 samples were

requested; a bias-corrected and accelerated CI was created for ab. For this 95% CI, the lower limit was .0769 and the upper limit was 2.0792. Several criteria can be used to judge the significance of the indirect path. In this case, both the a and b coefficients were sta s cally significant, the Sobel test for the ab product was significant, and the bootstrapped CI for ab did not include zero. By all these criteria, the indirect effect of age on SBP through weight was sta s cally significant. The direct path from age to SBP (c′) was also sta s cally significant; therefore, the effects of age on SBP were only partly mediated by weight. The upper diagram in Figure 9.5 shows the unstandardized path coefficients for this media on analysis; the lower diagram shows the corresponding standardized path coefficients. Comparison of the coefficients for the direct versus indirect paths (c′ = 2.161 vs. ab = .701) suggests that a rela vely small part of the effect of age on SBP is mediated by weight. There may be other media ng variables through which age might influence SBP, such as other age- related disease processes. 9.15 SUMMARY This chapter demonstrates how to assess whether a proposed media ng variable (X2) may partly or completely mediate the effect of an ini al causal variable (X1) on an outcome variable (Y). The analysis par ons the total effect of X1 on Y into a direct effect, as well as an indirect effect through the X2 media ng variable. The path model represents causal hypotheses, but readers should remember that the analysis cannot prove causality if the data are collected in the context of a nonexperimental design. If controlling for X2 completely accounts for the correla on between X1 and Y, this could happen for reasons that have nothing to do with mediated causality; for example, this can occur when X1 and X2 are highly correlated with each other because they measure the same construct. A media on analysis should be undertaken only when there are good reasons to believe that X1 causes X2 and that X2 in turn causes Y. In addi on, it is highly desirable to collect data in a manner that ensures temporal precedence (i.e., X1 occurs first, X2 occurs second, and Y occurs third). These analyses can be done using OLS regression; however, use of SPSS scripts provided by Preacher and Hayes (2008) provides bootstrapped es mates of CIs, and most analysts now believe this provides be er informa on than sta s cal significance tests that assume normality. SEM programs provide even more flexibility for assessment of more complex models. If a media on analysis suggests that par al or complete media on may be present, addi onal research is needed to establish whether this is replicable and real. If it is possible to manipulate or block the effect of the proposed media ng variable experimentally, experimental work can provide stronger evidence of causality (MacKinnon, 2008). COMPREHENSION QUESTIONS

1. Suppose that a researcher first measures a Y outcome variable, then measures an X1 predictor and an X2 hypothesized media ng variable. Why would this not be a good way to collect data to test the hypothesis that the effects of X1 on Y may be mediated by X2?

2. Suppose a researcher wants to test a media on model that says that the effects of math ability (X1) on science achievement (Y) are mediated by sex (X2). Is this a reasonable media on hypothesis? Why or why not?

3. A researcher believes that the predic on of Y (job achievement) from X1 (need for power) is different for men versus women (X2). Would a media on analysis be appropriate? If not, what other analysis would be more appropriate in this situa on?

4. Refer to Figure 9.1. If a, b, and ab are all sta s cally significant (and large enough to be of prac cal or clinical importance), and c′ is not sta s cally significant and/or not large enough to be judged prac cally or clinically important, would you say that the effects of X1 on Y are par ally or completely mediated by X2?

5. What pa ern of outcomes would you expect to see for coefficient es mates in Figure 9.1; for example, which coefficients would need to be sta s cally significant and large enough to be of prac cal importance, for the interpreta on that X2 only partly mediates the effects of X1 on Y? Which coefficients (if any) should be not sta s cally significant if the effect of X1 on Y is only partly mediated by X2?

6. In Figure 9.1, suppose that you ini ally find that path c (the total effect of X1 on Y) is not sta s cally significant and too small to be of any prac cal or clinical importance. Does it follow that there cannot possibly be any indirect effects of X1 on Y that are sta s cally significant? Why or why not?

7. Using Figure 9.1 again, consider this equa on: c = (a × b) + c′. Which coefficients represent direct, indirect, and total effects of X1 on Y in this equa on?

8. A researcher believes that the a path in a mediated model (see Figure 9.1) corresponds to a medium unit-free effect size and the b path in a mediated model also corresponds to a medium unit-free effect size. If assump ons are met (e.g., scores on all variables are quan ta ve and normally distributed), and the researcher wants to have power of about .80, what sample size would be needed for the Sobel test (according to Table 9.1)?

9. Give an example of a three-variable study for which a media on analysis would make sense. Be sure to make it clear which variable is the proposed ini al predictor, mediator, and outcome.

10. Briefly comment on the difference between the use of a bootstrapped CI (for the unstandardized es mate of ab) versus the use of the Sobel test. What programs can be used to obtain the es mates for each case? Which approach is less dependent on assump ons of normality?

NOTES 1 For discussion of poten al problems with comparisons among standardized regression coefficients, see Greenland, Maclure, Schlesselman, Poole, and Morgenstern (1991). Despite the problems they and others have iden fied, research reports s ll commonly report standardized regression or path coefficients, par cularly in situa ons where variables have arbitrary units of measurement.