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

Omitted Variables 7 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 s O. 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 = s M r M, f = s Y r Y, and s O = 1

10 Adjust Estimates Let p = efs O 2 /[s M 2 (1 – r XM 2 )] 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 = New (Old) CI b: (0.466) to c′: (3.992) to ab: (2.566) 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 = r MY.X s M.X s Y.X /(s M s Y ) 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

fig 15 Specify e and f.

Fig with rs and sds 16 Specify r M and r Y.

17 Example Using SEM with Standardized e and f to.3 New (Old) CI b: (0.466) to c′: (3.992) to ab: (2.566) 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 . 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/  c′: c′ - ab(1 –  ab: ab/  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  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, to c′: ab: 3.208, to 6.246

25 SEM Strategy SEM approach (based on Williams and Hazer). Fix error variance in M to: s M 2 (1 –  )(1 – r XM 2 ). 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, to c′: 3.358, to ab: 3.256, 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  =.8. Revised estimates with CIs (old estimates in parentheses): b: (0.466) c′: (3.992) ab: (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 rs M /s Y 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