Milin, P., & Hadžić, O. (2011). Moderating and Mediating Variables in Psychological Research. In M. Lovrić (Ed.) International Encyclopedia of Statistical Science (pp. 849-852). Berlin: Springer. Moderating and Mediating Variables in Psychological Research Petar MILIN Department of Psychology, University of Novi Sad, Serbia Olga HADŽIĆ Department of Mathematics and Informatics, University of Novi Sad, Serbia Moderating and mediating variables, or simply moderators and mediators, are related but distinct concepts in both general statistics and its application in psychology. A moderating variable is a variable that a ects the relationship between two other variables. This e ect is usually referred to as an interaction. Thesimplestcaseofaninteractioncanoccur in analysis of variance (ANOVA). For example, we tested whether there is a significant di erence in the level of anxiety (as measured with an appropriate standardized psychological test) between married and unmarried participants (i.e., variable marital status). The e ect was not statistically significant. However, when we enter the third variable gender (female/male), it appears that, on average, unmarried males are significantly more anxious than married males, while for females the e ect is the reverse. Figure 1 represents the results from two models described above. In the left-hand panel, we can see that, on average, there are no di erences between married and unmarried participants in the level of anxiety. From the right-hand panel, we can conclude that gender moderates the e ect of marital status on the level of anxiety: married males and unmarried females are significantly less anxious than the other two groups (unmarried males and married females). We can generalize the previous example to more complex models, with two independent variables having more than just two levels for comparison, or even with more than two independent variables. If all variables in the model are continuous variables, we would apply multiple regression analysis, but the phenomenon of a moderating e ect would remain the same, in essence. For example, we confirmed a positive relationship between the hours of learning and the result in an assessment test. Yet,music loudness during learning can moderate test results. We can imagine this as if a hand on the volume knob of an amplifier rotates clockwise and turns the volume up, students get all the worse results the longer they learn. Depending on the music volume level, the relationship between the hours of learning and the knowledge assessment changes continuously. This outcome is presented in Figure 2. On the left-hand side, we can observe a positive influence of the hours of learning on the results in the assessment test, while on the right-hand side, we can see how music loudness moderates this relationship. 1
anxiety 5 10 15 20 25 30 anxiety 5 10 15 20 25 30 male female married not married married not married marital status marital status Figure 1: The main e ect of one categorical variable on a continuous dependent variable (left-hand panel), and how it is moderated by the third categorical variable (right-hand panel). music loudness (decile values) test results 35 40 45 test results 35 40 45 6 7 8 9 hours of learning 6 7 8 9 hours of learning Figure 2: The main e ect of one continuous variable on another (left-hand panel), and how it is moderated by a third continuous variable (right-hand panel). Lines on the right panel represent decile values for the moderator variable. 2
Figure 3: Schematic representation of a complete mediation e ect (panel A, upper), and an incomplete mediation e ect (panel B, lower). The general linear form with one dependent, one independent, and one moderating variable is as follows: Y = 0 + 1 X 1 + 2 X 2 + 3 (X 1 X 2 )+", where 3 evaluates the interaction between X 1 and X 2. Mediating variables typically emerge in multiple regression analysis, where the influence of some independent variable (predictor) onthedependentvariable(criterion) isnotdirect, but mediated through the third variable. For example, the correlation between ageing and the number of work accidents in the car industry appears to be strong and negative. Nevertheless, the missing link in this picture is work experience: ita ectsinjuryrate, and is itself a ected by the age of worker. In regression modeling, one can distinguish between complete mediation and incomplete mediation. In practice, if the e ects of ageing on the number of work injuries would not di er statistically from zero when work experience is included in the model, then mediation is complete. Otherwise, if this e ect still exists (in the statistical sense), then mediation is incomplete. Complete and incomplete mediation are presented in Figure 3. In principle, a mediating variable flattens the e ect of an independent variable on the dependent variable. The opposite phenomenon would occur if the mediator variable would increase the e ect. This is called suppression. It is a controversial concept in statistical theory and practice, but contemporary applied approaches take a more neutral position, and consider that suppression may provide better insights into the relationships between relevant variables. The simplest case of linear regression with one dependent, one independent, and one mediating variable is defined by the following equations: Y = 0 + 1 X + " 1 M = 0 + 1 X + " 2 Y = 0 0 + 0 1X + 2 M + " 3, where of particular interest are 1, whichiscalledthetotal e ect, and 0 1,namedthe direct e ect. If suppression does not take place, which would occur if 0 1 > 1,thenwe 3
can continue the analysis with a standard regression model. First, we ascertain whether mediation is complete or incomplete, depending on whether the direct e ect drops to zero ( 1 0 0). The most important step in the analysis is the inference about the indirect e ect, or the amount of mediation. Itisdefinedasthereductioninthee ectoftheinitialvariable 0 on the model outcome ( 1 1 ). In simple hierarchical regression models, the di erence of the coe cients is exactly the same as the product of the e ect of the independent variable on the mediating variable multiplied by the e ect of the mediating variable on the dependent variable. In the general case, this equality only approximately holds. Mediation and moderation can co-occur in statistical models. This is often the case in psychology. Mediated moderation takes place when the independent variable is actually an interaction (X = X A X B ). Thus, the mediator acts between interacting variables (X A and X B )anddependentvariable(y ). For example, the e ect of interacting variable hours of learning and music loudness on the dependent variable result in an assessment test can be mediated by the importance of the test, as rated by the participants. Conversely, moderated mediation is realized in two forms: (a) the e ect of the independent variable on the mediator is a ected by a moderator ( 1 varies; as if the e ect of ageing on work experience is moderated by a particular personality trait, like H. J. Eysenck s Neuroticism), or (b) a moderator may interact with the mediating variable ( 2 varies; as if the work experience and the level of anxiety would interact and mediate between ageing and number of work accidents). If moderated mediation exists, inference about its type must be given. Finally, special attention is required in moderation and mediation analyses since both can be influenced by multicollinearity, whichmakesestimatesofregressioncoe cientsunstable. In addition, in an analysis with a moderating term i.e., an interaction e ect, the product of the variables can be strongly related to either the independent or the moderating variable, or both of them. If two variables are collinear, one of them can be centred to its mean. In this way, half of its value will become negative, and consequently, collinearity will decrease. Another possibility is to regress the independent variable with a moderator or mediator, and then to use the residuals or unexplained values, of the independent variable in the main analysis. Thus, the independent variable will be orthogonal to the moderating or mediating variable, with zero correlation, which will bring collinearity under control. However, in applying the previous two remedies, and others that are available, one must choose a conservative approach. The risk of emphasizing, or even inventing, what is not present in the data ought to be as little as possible. In any circumstances, the ultimate way of securing more reliable estimates is simply to obtain enough data. Acknowledgment: We would like to thank Professor David Kenny for reading a draft of this article, and providing us with comments and suggestions which resulted in many improvements. 4
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