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Mediation: Sensitivity Analysis

Mediation: Sensitivity Analysis. David A. Kenny. You Should Know. Assumptions Detailed Example Solutions to Assumption Violation. Sensitivity Analysis. What if? Involves a mixture of knowledge and guesswork. Examining “worst case” scenarios. Causal Assumptions.

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Mediation: Sensitivity Analysis

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  1. Mediation: Sensitivity Analysis David A. Kenny

  2. You Should Know Assumptions Detailed Example Solutions to Assumption Violation

  3. Sensitivity Analysis What if? Involves a mixture of knowledge and guesswork. Examining “worst case” scenarios.

  4. Causal Assumptions (Guaranteed if X is manipulated.) Perfect Reliability for M and X No Reverse Causal Effects Y may not cause M M and Y not cause X No Omitted Variables (Confounders) all common causes of M and Y, X and M, and X and Y measured and controlled 4

  5. Three Sources of Specification Error Omitted variables Measurement error Reverse causation Will assume that X is manipulated.

  6. Example

  7. Omitted Variables e f

  8. Strategy Estimate a, b, and c′ ignoring the omitted variable. Adjust those estimates: Specify the units of O (the omitted variable) and so fix sO. Specify e and f. Using these values, adjust the estimates and their confidence intervals (CIs).

  9. How to Pick e and f Think about how big each is using the effects of other variables. Especially important is to decide if ef is positive or negative. Convert from r; pick small, medium, or large for r: e = sMrM,f = sYrY, and sO = 1

  10. Adjust Estimates Let p = efsO2/[sM2(1 – rXM2)] New values: b: b - p c′: c′ + ap ab: a(b - p) Note that the total effect does not change.

  11. Example Not so clear what e and f would be. It might be plausible that ef is negative if the omitted variable is baseline housing: e would be negative f would be positive Will assume that standardized e and f are moderate in size or .3.

  12. Example Setting Standardized e and f to .3 p = 0.10842 New (Old) CI b: 0.358 (0.466) 0.159 to 0.557 c′: 4.589 (3.992) -0.034 to 9.212 ab: 1.970 (2.566) 0.019 to 4.400

  13. “Failsafe” e and f Values Find the value of ef that will make b = 0 and so ab = 0. Standardized ef = rMY.XsM.XsY.X/(sMsY) See if that is a plausible value. Example: Standardized e and f, assuming e = f, would have equal .62 which would seem to be implausibly large.

  14. SEM Approach Less computation All done in one step

  15. fig Specify e and f.

  16. Fig with rs and sds Specify rM and rY.

  17. Example Using SEM with Standardized e and f to .3 New (Old) CI b: 0.358 (0.466) 0.161 to 0.555 c′: 4.589 (3.992) 0.035 to 8.970 ab: 1.970 (2.566) 0.504 to 3.988

  18. Omitted Variable When X Not Manipulated Single omitted variable can ordinarily explain the covariation between X, M, and Y without having a, b, or c′ (Brewer, Crano, & Campbell, 1970). Estimate a single latent variable with X, M, and Y loading on that variable. The one non-trivial exception is when there is inconsistent mediation. Also with complete the loading of the mediator is one.

  19. Unreliability in M

  20. Theoretical Approach Pick a measure of reliability or a. Using that measure of reliability, re-compute b, c′, and ab.

  21. Picking a Reliability Can use an empirical estimate such as Cronbach’s alpha; such measures are likely to be somewhat optimistic. Can just guess; .8 not a bad starting point. “Hard” measures have much lower reliability than might be thought.

  22. Adjust Estimates New values: b: b/a c′: c′ - ab(1 – a)/a ab: ab/a Can adjust confidence intervals for b and ab, but not c′.

  23. “Failsafe” Reliability Note unreliability can only make the indirect effect larger not smaller. What value of a yields a zero value of c′? It is ab/(c′ + ab) (only compute if there is consistent mediation) Note that it is the “old” indirect effect divided by the “old” total effect.

  24. Example Reliability set at .8. Revised estimates with CIs: b: 0.583, 0.334 to 0.832 c′: 3.350 ab: 3.208, 0.770 to 6.246

  25. SEM Strategy SEM approach (based on Williams and Hazer). Fix error variance in M to: sM2(1 – a)(1 – rXM2). With this approach, we get p values and CIs for all relevant parameters.

  26. Example Using SEM Reliability set at .8. Revised estimates with CIs: b: 0.582, 0.346 to 0.838 c′: 3.358, -1.271 to 7.987 ab: 3.256, 0.661 to 6.548

  27. Can Combine Omitted Variable with Measurement Error in M These two sorts of bias can pretty much cancel each other out if for the omitted variable b and ef are the same sign.

  28. Example Standardized e and f at .3 with a = .8. Revised estimates with CIs (old estimates in parentheses): b: 0.446 (0.466) c′: 4.103 (3.992) ab: 2.455 (2.566)

  29. Reverse Causation

  30. Effects Like other forms of specification error, b and c′ are affected. Additionally, path a is also biased. Will only use the SEM approach.

  31. Strategy Pick path g. Determine its sign. Pick a small, medium, or large value and then compute rsM/sY for g. Fix the path from Y to M to that value.

  32. Extensions Could combine all three sources of specification error in one model. Have assumed that X is manipulated. If not there are many other sources of specification error and many other possible sensitivity analyses.

  33. Thank You

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