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Mediation Models

Mediation Models. Laura Stapleton UMBC. Mediation Models. Tasha Beretvas University of Texas at Austin. Session outline. What is mediation? Basic single mediator model Short comment on causality Tests of the hypothesized mediation effect Mediation models for cluster randomized trials

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Mediation Models

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  1. Mediation Models Laura Stapleton UMBC

  2. Mediation Models Tasha Beretvas University of Texas at Austin

  3. Session outline • What is mediation? • Basic single mediator model • Short comment on causality • Tests of the hypothesized mediation effect • Mediation models for cluster randomized trials • Brief mention of advanced issues

  4. What is mediation? • A mediator explains how or why two variables are related. • In the context of interventions, a mediator explains how or why a Tx effect occurs • A mediator is thought of as the mechanism or processes through which a Tx influences an outcome (Barron & Kenny, 1986). • If X M and M  Y, then M is a mediator • X causes proximal variable, M, to vary which itself causes distal, variable,Y, to vary

  5. What is mediation? • Mediational process can be • Observed or latent • Internal or external • At the individual or cluster level • Based on multiple or sequential processes • Who cares?! • Mediation analyses can identify important processes/mechanisms underlying effective (or ineffective!) treatments thereby providing important focal points for future interventions.

  6. Mediation Examples • Bacterial exposure  Disease • Bacterial exposure  Infection  Disease • Stimulus  Response • Might work for simple organisms (amoebae!), however, for more complex creatures: • Stimulus  Organism  Response • Stimulus  Expectancy Response • Monkey and lettuce example • Maze-bright, maze-dull rats and maze performance example

  7. Mediation Examples Intervention  Outcome Intervention  Receptivity  Outcome Intervention  Tx Fidelity  Outcome Intervention  TchConfid Outcome Intervention  Soc Comp Achievement Intervention  Phon Aware  Reading Intervention Peer Affil  DelinqBeh

  8. Mediation  Moderation • A moderator explains when an effect occurs • Relationship between X and Ychanges for different values of M • More in later session of workshop…

  9. Basic (single-level) mediation model c Treatment Outcome Mediator a b Treatment Outcome c’ total effect = indirect effect + direct effect c= ab+ c’

  10. Causality concerns • Just because you estimate the model X M  Y does not mean that the relationships are causal • Unless you manipulate M, causal inferences are limited. • Mediation model differs from Mediation design

  11. Causality concerns – mediation model • Remember, if the mediator is not typically manipulated, causal interpretations are limited Z Mediator M a b  Treatment T Ok! Outcome Y • Possible misspecification • For now, be sure to substantively justify the causal direction and “assumeor hypothesize that M causes Y and assuming that, here’s the strength of that effect…” • In future research, manipulate mediator

  12. Tests of the hypothesized mediation effect Mediator M a b • The estimate of the indirect effect, ab, is based on the sample • To infer that a non-zero αβexists in the population, a test of the statistical significance of ab is needed • Several approaches have been suggested and differ in their ability to “see” a true effect (power) Treatment T Outcome Y c’

  13. Tests of the hypothesized mediation effect • Causal steps approach (Baron & Kenny) • Test of joint significance • z test of ab(with normal theory confidence interval) • Asymmetric confidence interval (Empirical M or distribution of the product) • Bootstrap resampling

  14. Causal steps approach • Step 1: test the effect of T on Y (path c) c Treatment Outcome • Step 2: test the effect of T on M (path a) Mediator a Treatment

  15. Causal steps approach • Step 3: test the effect of M on Y, controlling for T (path b) Mediator b Treatment Outcome c’ • Step 4: to decide on partial or complete mediation, test the effect of T on Y, controlling for M (path c’)

  16. Causal steps approach: performance • Step 1 may be non-significant when true mediation exists Mediator FdF +2 +3 What if… Treatment T Outcome Dep -6 Mediator FdF +2 +3 or… Treatment T Outcome Dep +3 -2 Mediator SS

  17. Causal steps approach: performance • Lacks power • Power is a function of the product of the power to test each of the three paths • Power discrepancy worsens for smaller n and smaller effects • Lower Type I error rate than expected • i.e., too conservative

  18. Test of joint significance • Very similar to causal steps approach Mediator a b Treatment Outcome c’ • 1st: test the effect of T on M (path a) • 2nd: test the effect of M on Y, controlling for T (path b) • If both significant, then infer significant mediation

  19. Test of joint significance: performance • Better power than causal steps approach • Type I error rate slightly lower than expected • Power nearly as good as newer methods in single- level models • Power lower than other methods in multilevel models • No confidence interval around the indirect effect is available

  20. z test of ab product • Calculate z = • Sobel’sseab= • Compare z test value to critical values from the standard normal distribution • Can also calculate confidence interval around ab CI =

  21. z test of ab product: performance • One of the least powerful approaches • Type I error rate much lower than expected .05. • Single-level models, it approaches the power of other methods when sample size are 500 or greater, or effect sizes are large • Multilevel models, it never reaches the levels of other models although it does get closer although still lower • Problem is that the ab product is not normally distributed, so critical values are inappropriate • How is the abproduct distributed?

  22. Sampled 1,000 a ~ N(0,1) and of b ~ N(0,1) Distribution of path a Distribution of path b Distribution of product of axb

  23. Empirical M-test (asymmetric CI) • Determines empirical (more leptokurtic) distribution of zof the abproduct (not assuming normality) • αβ=0: dist’n is leptokurtic and symmetric • αβ>0: dist’n is less leptokurtic and +ly skewed • αβ<0: dist’n is less leptokurtic and -ly skewed • Due to asymmetry, different upper and lower critical values needed to calculate asymmetric confidence intervals (CIs). • Meeker derived tables for various combinations of Za and Zbvalues (increments of 0.4) that could be used to calculate asymmetric CIs.

  24. Empirical M-test (asymmetric CI) MacKinnon et al created PRODCLIN that, given a, b, and their SEs, determines the distribution of ab and relevant critical values for calculating asymmetric CI. (MacKinnon & Fritz, 2007, 384-389). Confidence interval limits: If CI does not include zero, then significant

  25. Empirical M-test: performance • Good balance of power while maintaining nominal Type I error rate • Performed well in both single-level and multi-level tests of mediation • Only bootstrap resampling methods had (very slightly) better power than this method • PRODCLIN software is easy to use

  26. Bootstrap resampling methods • Determines empirical distribution of the ab product • Several variations • Parametric percentile • Non-parametric percentile • Bias-corrected versions of both • Can bootstrap cases or bootstrap residuals. • It is typical in multilevel designs to bootstrap residuals.

  27. Parametric percentile bootstrap • With original sample, run the analysis and obtain estimates of variance(s) of residuals • New residuals are then resampled from a distribution ~N(0,σ2) (thus, the “parametric”). • New values of M are created by using the fixed effects estimates from the original analysis, T and the resampled residual(s). • New values of Y are created using the fixed effects, and T and M values and residual(s). • Then, the analysis is run and the ab product is estimated

  28. Parametric percentile bootstrap • The process of resampling and estimating ab is repeated many times (commonly 1,000 times) • The ab estimates are then ordered • With 1,000 estimates, the 25th and the 975th are taken as the lower and upper limits of the 95% (empirically derived) CI. • Note that the CI limits may not be symmetric around the original ab estimate • If CI does not include zero, then significant mediation

  29. Non-parametric percentile bootstrap • The parametric bootstrap involves the assumption that the residuals are normally distributed • Instead, residuals can be resampled with replacement from the empirical distribution of actual residuals (saved from the original sample’s analysis) • The remaining process is the same as for the parametric version

  30. Bias-corrected bootstrap • With both the parametric and non-parametric bootstrap, the initial ab product may not be at the median of the bootstrap ab distribution • Thus, the initial ab estimate is biased • BC-bootstrap procedures “shift” the confidence interval to adjust for the difference in the initial estimate and the median

  31. Bootstrap resampling methods: performance • Resampling methods provide the most power and accurate Type I error rates of all methods • Parametric has best confidence interval coverage • BC-parametric had best power, especially with low effect sizes with normal and non-normally distributed residuals; Type I error rate was slightly high for multilevel analyses • Non-parametric had the most accurate Type I error rates; good overall power • BC Non-parametric had good power • But … complicated to program

  32.  Summary: tests of the hypothesized mediation effect • Causal steps approach • Test of joint significance • z test of ab • Empirical M • Bootstrap resampling  OK for single level…  Yes! Easy!  Yes! Not quite as easy… but does have the most power

  33. Example for today • Social-emotional curriculum = Tx • Child social competence = outcome • Randomly selected classrooms (one per school) • Why would Tx affect outcome? • Teacher attitude about importance? • Child understanding of others’ behaviors? • Puppet show down-time soothes child? • Researcher should think in advance of possible mediators to measure

  34. Mediation models for cluster randomized trials • Extend basic model to situations when treatment is administered at cluster level • Model depends on whether mediator is measured at cluster or individual level • Definition (Krull & MacKinnon, 2001) depends on level at which each variable is measured: T→ M →Y • Upper-level mediation [2→2→1] • Cross-level mediation [2→1→1] • Cross-level and upper-level mediation [2→(1 & 2) →1]

  35. Measured variable partitioning Cluster uoj • First, consider that any variable may be partitioned into individual level components and cluster level components Yij Individual rij Note: No intercepts depicted

  36. Mediation model possibilities Tx Cluster M Cluster Y Cluster Tx M Y Tx Individual M Individual Y Individual

  37. Data Example Context • Cluster randomized trial (hierarchical design) • 14 preschools: ½ treatment, ½ control • 6 kids per school (/classroom) • Socio-emotional curriculum • Outcome is child social competence behavior • Possible mediators: teacher attitude about importance of including this kind of training in classroom, child socio-emotional knowledge • Sample data are on handout

  38. Total effect of treatment Before we examine mediation, let’s examine the total effect of treatment on the outcome… Tx Cluster Y Cluster 01 Tx Y Y Cluster

  39. Total effect of treatment: FEResults Final estimation of fixed effects: ---------------------------------------------------------------------------- Standard Approx. Fixed Effect Coefficient Error T-ratio d.f. P-value ---------------------------------------------------------------------------- For INTRCPT1, B0 INTRCPT2, G00 34.357143 1.029102 33.386 12 0.000 T, G01 4.238095 1.455370 2.912 12 0.014 ---------------------------------------------------------------------------- c

  40. Upper-level mediation model (2→2→1) M Cluster 01 ’01 Tx Cluster Y Cluster ’02 Tx M Y Y Cluster

  41. Upper-level mediation model: Results To estimate the a path, I ran an OLS regression in SPSS using the Level 2 file What is the estimate of a and its SE?

  42. Upper-level mediation model: Results To estimate the b path, I ran a model in HLM Final estimation of fixed effects: ---------------------------------------------------------------------------- Standard Approx. Fixed Effect Coefficient Error T-ratio d.f. P-value ---------------------------------------------------------------------------- For INTRCPT1, B0 INTRCPT2, G00 34.640907 1.036530 33.420 11 0.000 M1, G01 0.794540 0.656229 1.211 11 0.252 T, G02 3.670567 1.502879 2.442 11 0.033 ---------------------------------------------------------------------------- What is the estimate of b and its SE? What is the estimate of c’ and its SE?

  43. Upper-level mediation model: Results M Cluster .714 .795 Tx Cluster Y Cluster 3.671 Tx M Y Y Cluster • Direct effect = 3.671 • Indirect effect = (.714)(.795) = .568 • Total effect = DE + IE = 3.671 + .568 = 4.239

  44. Upper-level mediation model: Results • Causal steps approach • Test of joint significance • z test of ab product • Empirical-M test • BC parametric bootstrap Step 1 significant, but not Steps 2 and 3 No. Neither path a nor path bare significant No. se=.68, z=.83, p=.41 95% CI = -.78 to 1.91 No. No. 95% CI = -.47 to 2.26 No. 95% CI= -.42 to 3.68

  45. Upper-level mediation model: Results • PRODCLINhttp://www.public.asu.edu/~davidpm/ripl/ Prodclin/

  46. Cross-level mediation model (2→1→1) Model A Model B Mediator CLUSTER γ01 Treatment CLUSTER Treatment CLUSTER Outcome CLUSTER γ’01 Mediator Mediator Treatment Treatment Outcome Mediator INDIVIDUAL Mediator INDIVIDUAL γ’10 Outcome INDIVIDUAL

  47. Cross-level mediation model: Results To estimate the a path: Final estimation of fixed effects: ---------------------------------------------------------------------------- Standard Approx. Fixed Effect Coefficient Error T-ratio d.f. P-value ---------------------------------------------------------------------------- For INTRCPT1, B0 INTRCPT2, G00 39.309524 0.845210 46.509 12 0.000 T, G01 2.642857 1.195308 2.211 12 0.047 ---------------------------------------------------------------------------- What is a and its SE?

  48. Cross-level mediation model: Results To estimate the b path: Final estimation of fixed effects: ---------------------------------------------------------------------------- Standard Approx. Fixed Effect Coefficient Error T-ratio d.f. P-value ---------------------------------------------------------------------------- For INTRCPT1, B0 INTRCPT2, G00 35.138955 0.941637 37.317 12 0.000 T, G01 2.674528 1.358185 1.969 12 0.072 For M2_GRAND slope, B1 INTRCPT2, G10 0.591620 0.142895 4.140 81 0.000 ---------------------------------------------------------------------------- What is b and its SE? And for c’?

  49. Cross-level mediation model: Results Model A Model B Mediator CLUSTER 2.643 Treatment CLUSTER Treatment CLUSTER Outcome CLUSTER 2.675 Mediator Mediator Treatment Treatment Outcome Mediator INDIVIDUAL Mediator INDIVIDUAL .592 Outcome INDIVIDUAL • Direct effect = 2.675 • Indirect effect = (2.643)(.592) = 1.564 • Total effect = 2.675 + 1.564 = 4.239

  50. Cross-level mediation model: Results • Causal steps approach • Test of joint significance • z test of ab product • Empirical-M test • BC parametric bootstrap Yes Steps 1, 2 and 3 significant Yes Paths a and bsignificant se=.802, z=1.95, p=.051 95% CI = -.01 to 3.13 No Yes 95% CI = .19 to 3.32 95% CI = .31 to 3.57 Yes

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