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1 Estimating and Testing Mediation David A. Kenny University of Connecticut

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2 Overview (click to go there) (click to go there) Introduction (click to go there) (click to go there) (click) (click) Baron & Kenny Steps (click) (click) (click) Power (click) (click) (click) Test of the Indirect Effect (click)(click) (click) Assumptions (click) (click) (click) Additional Variables (click) (click)

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Introduction 3

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4 Interest in Mediation Mentions of mediation or mediator in psychology abstracts: –1980: 36 –1990: 122 –2000: 339 –2010: 1,198 4

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5 Why the Interest in Mediation? Understand the mechanism –theoretical concerns –cost and efficiency concerns Find more proximal endpoints Understand why the intervention did not work –finding the missing link –compensatory processes

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6 The Beginning Model

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7 The Mediational Model

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8 The Four Paths X Y: path c X M: path a M Y (controlling for X): path b X Y (controlling for M): path c (standardized or unstandardized)

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Baron & Kenny Steps 9

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11 Baron and Kenny Steps Step 1: X Y (test path c) Step 2: X M (test path a) Step 3: M (and X) Y (test path b) Step 4: X (and M) Y (test path c ) Note that Steps 3 and 4 use the same regression equation.

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12 Total and Partial Mediation Total Mediation Meet steps 1, 2, and 3 and find that c equal zero. Partial Mediation Meet steps 1, 2, and 3 and find that c is smaller in absolute value than c.

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13 Example Dataset Morse et al. – J. of Community Psychology, 1994 –treatment housing contacts days of stable housing – persons randomly assigned to treatment groups. – 109 people

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14 Variables in the Example Treatment –1 = treated (intensive case management) –0 = treatment as usual Housing Contacts: number of contacts for the year after the intervention began Stable Housing – days per month with adequate housing (0 to 30) –measured for 9 months, one year after the intervention began

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15 Morse et al. Example Step 1: X Y c = 6.558, p =.009 Step 2: X M a = 1.831, p =.013 Step 3: M (and X) Y b = 1.398, p <.001 Step 4: X (and M) Y c = 3.998, p =.089

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16 Decomposition of Effects Total Effect = Direct Effect + Indirect Effect c = c + ab Note that ab = c - c This equality exactly holds for multiple regression, but not necessarily for other estimation methods. Example: = (1.831x 1.398) And 100(2.56/6.56) = 39% of the total effect is explained (ab/c or equivalently 1 - c/c).

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17 When Can We Conclude Complete Mediation? When c is not statistically different from zero, but c is? NOT REALLY Ideally when ab is substantial and c is small, not just statistically non-significant.

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18 Estimating the Total Effect (c) The total effect or c can be inferred from direct and indirect effect as c + ab. We need not perform Step 1 to estimate c. This can be useful in situations when c does not exactly equally c + ab.

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19 All Four Steps Essential? Steps 1 and 4 are not so key. Steps 2 and 3 are essential. Note that ab measures the indirect or the amount of the total effect that is mediated. Need a way of testing the null hypothesis that the indirect effect of ab is zero (discussed later). 19

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20 Inconsistent Mediation ab and c have a different sign X as a suppressor variable Example: Stress and Mood with Coping as a Mediator Consequences –Step 1 may fail –Percent mediated greater than 100% Do we have mediation? –Yes. There is an indirect effect (ab > 0). –No. There is no effect that is mediated.

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21 Presenting Mediation 21

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Power 22

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23 Power and the Test of c If c is zero, then c equals ab. If both a and b have a moderate effect size, then c has a smaller than small effect size (assuming c is zero). Thus, it is very possible to have tests of significance for a and b be statistically significant, but c is not. Note that for the example if a and b have moderate effect sizes (r =.3) and c = 0, the power of the test of c is only.15.

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24 Power and the Test of c When b = 0, c = c and in this case c might be statistically significant, yet c might not be significant even though. Why? There is multicollinearity between X and M due to path a. Note that as M becomes a more successful mediator and path a gets larger, multicollinearity becomes more of a issue in the testing of c (and b).

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25 Power and the Test of b Test of path a has more power than the test of path b (b =.3, standardized): a N (a is standardized; sample size needed for 80% power to reject the null hypothesis that b = 0) Conclusion: Power of the test of b declines as a increases.

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Test of the Indirect Effect 26

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27 Strategies to Test ab = 0 Test a and b separately Sobel test Bootstrapping

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28 Test a and b Separately Easy to do Works fairly well

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29 Sobel Test of Mediation Compute the square root of a 2 s b 2 + b 2 s a 2 which is denoted as s ab Note that s a and s b are the standard errors of a and b, respectively; t a = a/s a and t b = b/s b. Divide ab by s ab and treat that value as a Z. So if ab/s ab greater than 1.96 in absolute value, reject the null hypothesis that the indirect effect is zero.

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30 Example a = and b = s a = and s b = ab = 2.56; s ab = Sobel test Z is 2.213, p =.027 We conclude that the indirect effect is statistically different from zero. Website:

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31 The distribution of ab is highly skewed which lowers the power of the test. Large values of ab are more variable than small values (i.e., 0).

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32 Bootstrapping Nonparametric way of computing a sampling distribution. Re-sampling (with replacement) Many trials (computationally intensive) Correct for bias –Mean of the bootstrap estimate differs slightly from the estimate. Compute a confidence interval which is asymmetric. Slight changes because empirically derived.

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33 Results of Bootstrapping 95% Bias Corrected Confidence Interval: Lower Upper Note that the CI is asymmetric for an estimate of Also values differ to sampling error. (Done using the Hayes & Preacher macro from

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Assumptions 34

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35 Assumptions: Multiple Regression Linearity –Is a problem in the example; housing contacts has a quadratic effect on stable housing (p =.034). Normal Distribution of Errors –Interval level of measurement of M and Y Equal Error Variance Independence –No clustering X and M Do Not Interact to Cause Y 35

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36 No XM Interaction: Linear Mediation Called Moderation in Baron & Kenny Add XM (and possibly other interaction terms, e.g., X 2 M) when explaining Y. Not significant for the example (p =.476)

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37 Causal Assumptions Perfect Reliability –for M and X No Reverse Causal Effects –Y may not cause M – M and Y not cause X No Omitted Variables –all common causes of M and Y, X and M, and X and Y measured and controlled (Guaranteed if X is manipulated.) 37

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38 Basic Mediational Causal Model Note that U1 and U2 are theoretical variables and not errors from a regression equation.

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39 Unreliability Usually safe to assume that X is perfectly reliable. Measurement error in Y does not bias unstandardized regression coefficients. Measurement error in M is problematic.

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40 Unreliability in M

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41 Effect of Unreliability in M b is attenuated (closer to zero) c is inflated (given consistent mediation) –more as a increases –more as b increases –Note that the bigger the indirect, the greater the bias in c.

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42 What to Do about Unreliability in M? Improve the reliability Adjust estimates using Structural Equation Modeling

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43 Omitted Variables

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44 What is the Effect of Omitted Variables? Usually, but not always, the sign of ef is the same as b. –Inflating the estimate of b –Deflates the estimate of c (could produce inconsistent mediation).

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45 Effect of Vitamin A Supplements in Northern Sumatra 45

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46 Standard Results 46

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47 Results with an Omitted Variable 47

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48 What to Do about Omitted Variables? Do not omit them: Include them in the analysis as covariates. If there is good reason to believe that c = 0, they can be allowed for. Sometimes the omitted variable is shared method effects. If an issue, measure M and Y by different methods. 48

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49 Reverse Causation

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50 Effect of Reverse Causation Typically, b and g have the same sign, which likely makes the value of b inflated and the value of c deflated.

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51 What to Do about Reverse Causation? Longitudinal designs If c = 0, then the model can be estimated. Instrumental variable method.

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52 Timing of the Measurement of the Mediator Mediator should be measured after X but before Y. X might be measured at the same time as Y (e.g., number of treatment sessions), but it must be assumed that X has not changed since when it affected Y.

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53 Controlling for Prior Values Obtain baseline measures of M and Y. Control for baseline M and Y in the analysis.

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54 Mediation: The Full Model

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55 Mediation: The Full Model – X Manipulated

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Additional Variables 56

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57 Additional Variables Multiple Xs Multiple Ms Multiple Ys Covariates 57

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58 Multiple Xs Consider two Xs. –happens when X is categorical and there are more than two treatment groups Now two indirect effects a 1 b and a 2 b. Can combine the Xs using a formative variable. 58

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59 Multiple Mediators Consider two mediators, 1 and 2, and two indirect effects a 1 b 1 and a 2 b 2. Can test: –Are both different from zero? –Is each different from zero? –Is one larger than the other? 59

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60 Example Two mediators: housing contacts and entitlement contacts. Tests: –Is the sum different from zero? c – c CI: ( to ) Yes –Is each different from zero? Housing Contacts CI: ( to ) Yes Entitlement Contacts CI: ( to ) Yes –Is one larger than the other? H Contacts – E contacts CI: ( to ) No 60

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61 Dual Mediation: Special Example of Two Mediators X has two levels Each level is intervention Both equally effective Each works through a different mechanism (i.e., mediator). 61

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62 Dual Mediation with No Intervention Effect 62

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63 Mediation with No Intervention Effect Note that total effect of X on Y is.25 + (-.25) = 0! 63

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64 Causal Chains One mediator causes another X M 1 M 2 Y Indirect effect the product of three terms: ab 1 b 2 64

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65 Multiple Outcomes Consider two outcomes. Now two indirect effects ab 1 and ab 2. Consider combining outcome variables into a single variable, e.g., as a latent variables. 65

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66 Covariates Often there are variables in the analysis that need to be controlled: –Demographics –Baseline measures If a covariate interacts with X, it becomes a moderator. 66

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67 Topics Not Discussed Use of SEM simultaneous estimation latent variables (reflective and formative) instrumental variable estimation Mediators or outcomes that are categorical or counted Clustering and multilevel mediation Mediated moderation and moderated mediation 67

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68 Conclusion Mediational Analyses Are Very Simple Mediational Analyses Are Very Difficult –Difficulties are more in the conceptualization and measurement than in the statistical analysis.

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Additional Slides 69

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70 Mediation with Moderation Moderation –A causal effect varies as a function of some variable. –Housing intervention program is stronger from men than women. –Tested by an intervention X gender interaction. Combining the two –Moderated Mediation –Mediated Moderation 70

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71 Moderated Mediation Begin with a mediation analysis and estimate a, b, and c. Two paths that can be moderated –a path might vary due to moderator –b path might vary due to moderator –If one or both is moderated, the indirect effect or ab is moderated. Strategy –Conduct two moderator analyses: one on M and one on Y. –Measure how the indirect effect changes as a function of the moderator. –That change need not be linear. 71

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72 Example: Moderated Mediation We find that use of services mediates the effect of the intervention. Two paths that can be moderated –The effect of the intervention on services is stronger for men than women. –The effect of services on the days housed is stronger for men who receive the intervention. Note that moderated mediation can be re- expressed as moderated mediation. 72

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73 Mediated Moderation Begin with moderation. –Show that the moderator affects the X Y Introduce a mediator and replace X with the interaction of X with Moderator –repeat Baron and Kenny steps but X is replaced by an X by moderator interaction 73

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74 Example: Mediated Moderation Step 1: Intervention effects stronger for men than women. Step 2: Men who receive the intervention receive more services than do women. Step 3: Services increase the number of days housed. Step 4: The gender difference in intervention effect disappears when services is controlled. 74

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75 Graph of Mediation Copied from MacKinnon et al. (2007) 75

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76 DataToText Project Have the researcher tell DataToText what is the research question. DataToText performs the requisite analyses. DataToText gives the results from those analyses: computer output a written description 76

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77 Morse et al.: The effect of Treatment on Stable Housing is mediated by Housing Contacts. 77

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