A simple guide to Mediation

Lyytinen & GaskinMediation and Multi-group Analyses1MediationIn an intervening variable model, variable X, is postulated to exert an effect on an outcome variable, Y, through one or more intervening variables called mediators (M)mediational models advance an X M Y causal sequence, and seek to illustrate the mechanisms through which X and Y are related. (Mathieu & Taylor)

XYM2Mediation refers to causal relationships somewhat more complex than the simple x predicts y relationship. When hypothesizing and testing for mediation, a mediating variable, m, (click) plays some role in the relationship between x and y. Thus we are hypothesizing that the relationship between x and y is somewhat more complex than a simple direct effect from IV to DV. More complex mediation can also take place, for example, with two mediating variables, but testing more complex mediation is not only beyond the scope of a simple guide to mediation, there is also not currently an accepted or agreed upon method for testing these types of mediation. So, were going to keep it simple.2

Why Mediation?Seeking a more accurate explanation of the causal effect the antecedent (predictor) has on the DV (criterion , outcome) focus on mechanisms that make causal chain possibleMissing variables in the causal chainIntelligence PerformanceIntelligence Work Effectiveness Performance3So why even consider mediation? Why not just predict and test direct effects? Often the reason is because we believe that some x really does have an effect on some y, but we want to ensure that there is not a more accurate explanation for this cause and effect relationship. For example, lets say we want to predict what causes people to either (Click) use an umbrella or not. We may say that (click) rain causes this effect. However, this does not explain why many people do not use an umbrella when it rains. Thus we can explain more of the observed behavior in our sample if we include a mediator called, (click) Desire to stay dry. Rain causes those who wish to remain dry to use an umbrella when it rains, and those who do not have this desire do not use an umbrella. So in this example desire to stay dry mediates the effect rain has on using an umbrella. Without the mediator, we can only explain the behavior of those who use umbrellas in the rain; whereas, with the mediator, we can also explain the behavior of those who do not use umbrellas.3Conditions for mediation4(1) justify the causal order of variables including temporal precedence; (2) reasonably exclude the influence of outside factors; (3) demonstrate acceptable construct validity of their measures; (4) articulate, a priori, the nature of the intervening effects that they anticipate; and (5) obtain a pattern of effects that are consistent with their anticipated relationships while also disconfirming alternative hypotheses through statistical tests.Conditions for mediation5Inferences of mediation are founded first and foremost in terms of theory, research design, and the construct validity of measures employed, and second in terms of statistical evidence of relationships.Mediation analysis requires:1) inferences concerning mediational X MY relationships hinge on the validity of the assertion that the relationships depicted unfold in that sequence (Stone-Romero & Rosopa, 2004). As with SEM, multiple qualitatively different models can be fit equally well to the same covariance matrix. Using the exact same data, one could as easily confirm a YMX mediational chain as one can an XMY sequence (MacCallum, Wegener, Uchino, & Fabrigar, 1993).Conditions for mediation62) experimental designs is to isolate and test, as best as possible, XY relationships from competing sources of influence. In mediational designs, however, this focus is extended to a three phase XMY causal sequence requiring random assignments to both X and M and related treatmentsBecause researchers may not be able to randomly assign participants to conditions, the causal sequence of XMY is vulnerable to any selection related threats to internal validity (Cook & Campbell, 1979; Shadish et al., 2002). To the extent that individuals status on a mediator or criterion variable may alter their likelihood of experiencing a treatment, the implied causal sequence may also be compromised. For example, consider a typical: trainingself-efficacyperformance, mediational chain. If participation in training is voluntary, and more efficacious people are more likely to seek training, then the true sequence of events may well be self-efficacytrainingperformance. If higher performing employees develop greater self-efficacy (Bandura, 1986), then the sequence could actually be performanceefficacytraining. If efficacy and performance levels remain fairly stable over time, one could easily misconstrue and find substantial support for the trainingefficacyperformance sequence when the very reverse is actually occurring. (Mathieu and Taylor 2006)Conditions of mediation7It is a hallmark of good theories that they articulate the how and why variables are ordered in a particular way (e.g., Sutton & Staw, 1995; Whetten, 1989). This is perhaps the only basis for advancing a particular causal order in non-experimental studies with simultaneous measurement of the antecedent, mediator, and criterion variables (i.e., classic cross-sectional designs).Implicitly, mediational designs advance a time-based model of events whereby X occurs before M which in turn occurs before Y. It is the temporal relationships of the underlying phenomena that are at issue, not necessarily the timing of measurementsIn other words, in mediation analyses, omitted variables represent a significant threat to validity of the XM relationship if they are related both to the antecedent and to the mediator, and have a unique influence on the mediator. Likewise omitted variables (and related paths) may lead to conclude falsely that no direct effect XY exists, while in fact it holds in the populationImportance of theory Cause and effectPerformanceSelf-efficacyTrainingSelf-efficacyTrainingPerformanceTrainingSelf-efficacyPerformanceTrainingPerformanceSelf-efficacy8Just as with other hypotheses, mediation suggests a cause and effect relationship between variables. However, remember from previous lessons that causation is not implied by correlation, and statistically, the IVs, DVs, and mediating variables will likely be correlated statistically. Thus it is critical to develop mediation models based on theory first, then to test and support the theoretical model through statistics. To illustrate this point, consider the relationship shown. Self efficacy mediates the effect training has on performance. In other words, training effects performance through Self-efficacy. That makes sense.Now consider this alternative model (click). Performance mediates the effect Training has on Self-efficacy. Well, that also makes sense, and it is supported by a statistical test. Now consider these other models (click) (click), they also make sense in certain contexts, and they are also supported by statistical tests. So which model do we choose? Most likely, we have a certain theory and model in mind prior to doing the statistical tests. Our advice to you is to let this a priori theory guide your model development, rather than first seeing what the statistics say, then developing a model. See Mathieu and Taylor 2006 for a more in depth discussion of this specific example.8Types of MediationXMYIndirect EffectMXYPartial MediationXYMFull MediationSignificant PathInsignificant Path9Now, there are three main types of mediation. The first is called indirect effect. Indirect effects predict no direct effect from x to y while leaving the mediator out of the model. However, x has a direct effect on the mediator, and the mediator has a direct effect on y. Thus, x is said to have an indirect effect on y. This hypothesis can only be supported if the direct effect of x on y is insignificant, prior to testing for indirect effects. (Click) The next type of mediation is called Partial Mediation. Partial mediation predicts (click) significant direct (click) and indirect effects from x to y. Thus the unmediated relationship is significant, as well as the x to mediator and mediator to y relationships. In order to avoid concluding that a partially mediated effect is significant, when in fact, only the three direct effects are individually significant, a significance test for mediation must be performed. This will be explained more fully on the next slide. (click) The third type of mediation is called Full Mediation. Full mediation predicts that the direct effect of x on y will be significant, only if the mediator is absent. (click) When the mediator is present, this direct effect becomes (click) insignificant (click), while the indirect effect is significant. Lastly, if the x to mediator, and/or the mediator to y relationships are insignificant, no mediation is taking place.9More complex mediation structures10XM1YChain ModelM2M3XM1YM2M3Parallel ModelHypothesizing MediationAll types of mediation need to be explicitly and with good theoretical reasons and logic hypothesized before testing themIndirect EffectYou still need to assume and test that X has an indirect effect on Y, though there is no effect in path XYX has an indirect, positive effect on Y, through M.Partial or FullM partially/fully mediates the effect of X on Y.The effect of X on Y is partially/fully mediated by M. The effect of X on Y is partially/fully mediated by M1, M2, & M3. 11Statistical evidence of relationships.12Each type of mediation needs to be backed by appropriate statistical analysis Sometimes the analysis can be based on OLS, but in most cases it needs to be backed by SEM based path analysisThere are four types of analyses to detect presence of mediation relationshipsCausal steps approach (Baron-Kenny 1986) (tests for significance of different paths)Difference in coefficients (evaluates the changes in betas/coefficients and their significance when new paths are added to the model)Product of effect approach (tests for indirect effects a*b- this always needs to be tested or evaluated using bootstrapping)Sometimes evaluating differences in R squares

Statistical evidence of relationships13Convergent validity is critical for mediation tests as this forms the basis for reliability especially poor reliability of mediator as to the extent that a mediator is measured with less than perfect reliability, the MY relationship would likely be underestimated, whereas the XY would likely be overestimated when the antecedent and mediator are considered simultaneously (see Baron & Kenny 1986)Discriminant validity must be gauged in the context of the larger nomological network within which the relationships being considered are believed to reside. Discriminant validity does not imply that measures of different constructs are uncorrelated the issue is whether measures of different variables are so highly correlated as to raise questions about whether they are assessing different constructs. It is incumbent on researchers to demonstrate that their measures of X, M, and Y evidence acceptable discriminant validity before any mediational tests are justified.

Statistical evidence of relationships14Statistical evidence of the relationships15In simple partial mediation mx is the coefficient for X for predicting M, and ym.x and yx.m are the coefficients predicting Y from both M and X, respectively. Here yx.m is the direct effect of X, whereas the product mx*ym quantifies the indirect effect of X on Y through M. If all variables are observed then yx = yx.m + mx*ym or mx*ym = yx - yx.m Indirect effect is the amount by which two cases who differ by one unit of X are expected to differ on Y through Xs effects on M, which in turn affects YDirect effect part of the effect of X on Y that is independent of the pathway through MSimilar logic can be applied to more complex situationsWhat would be the paths here?16

Statistical analysis17The testing of the existence of the mediational effect depends on the type of indirect effectThe lack of direct effect XY (yx is either zero or not significant) is not a demonstration of the lack of mediated effectTherefore three different situations prevail (in this order)The presence of a indirect effect (mx*ym is significant)The presence of full mediation (yx is significant but yx.m is not)The presence of partial mediation (yx is significant and yx.m is non zero and significant)

Testing for indirect effect18

Testing for full mediation19

Testing for partial mediation20

Observations of statistical analysis21The key is to test for the presence of a significant indirect effect just demonstrating the significant of paths yx, yx.m,mx.y, and mx is not enoughOne reason is that Type I testing of statistical significance of paths is not based on inferences on indirect effects as products of effects and their quantitiesCan be done either using Sobel test (see e.g. www.quantpsy.org) or bootstrappingSobel tests assumes normality of product terms and relatively large sample sizes (>200)Lacks power with small sample sizes or if the distribution is not normal

Bootstrapping22Bootstrapping (available in most statistical packages, or there is additional code to accomplish it for most software packages)Samples the distribution of the indirect effect by treating the obtained sample of size n as a representation of the population as a minitiature and then resampling randomly the sample with replacement so that a sample size n is built by sampling cases from the original sample by allowing any case once drawn to be thrown back to be redrawn as the resample of size n is constructedmx and ym and their product is estimated for each sample recordedThe process is repeated for k times where k is large (>1000)Hence we have k estimates of the indirect effect and the distribution functions as an empirical approximation of the sampling distribution of the indirect effect when taking the sample of size n from the original populationSpecific upper and lower bound for confidence intervals are established to find ith lowest and jst largest value in the ordered rank of value estimates to reject the null hypothesis that the indirect effect is zero with e.g. 95 level of confidenceObservations of statistical analysis23In full and partial mediation bivariate XY (assessed via correlation rYX or coefficient yx) must be nonzero in the population if the effects of X on Y are mediated by MHence establishing a significant bivariate is conditional on sample sizeFor example Assume that N=100 and sample correlations rXM=.30 and rMY =.30 and both would be significant at p