Question regarding the value of upper 95% CI over 100% generated from Proc CAUSALMED

[Rosy1]

Hi Everyone,

    I used Proc CAUSALMED to perform mediation analysis and found some values of Percentage Mediated Wald 95%

Confidence Limits are 100%. Does it make sense?

    P.S. I also noticed that SAS PROC  also showed similar results. Please see the link below. Page: 2127

Output 33.3.8: Percentage Due to Interaction -68.5853 167.6 -397.04 259.87

https://support.sas.com/documentation/onlinedoc/stat/143/causalmed.pdf

 Thank you,

Rosy

[MichaelL_SAS]

Good question.

The Wald Confidence Limits reporting a value above 100% is most likely due to high variability in the estimate, like in the documentation example you reference where the standard error estimate is 167.58. 

The estimate reported as the percentage due to interaction could be over 100%, but in that case the estimate is not interpretable. This point is touched on in the Counterfactual Framework for Defining Causal Mediation Effects section of the PROC CAUSALMED documentation but can be easy to miss since it involves a bit of notation and a few different equations. I'll provide more detail below, but basically the estimate for the percentage due to interaction might be over 100% if a portion of the four way decomposition of the total effect has the opposite sign of the total effect, and when that occurs the contributions in the decomposition are not interpretable. 

Specifically, the four way decomposition of the total effect (TE) says it is equal to the sum of the controlled directed effect (CDE), reference interaction (IRF), mediated interaction (IMD), and pure indirect effect (PIE). The portion attributed to interaction (PAI) is then defined to be PAI=IRF + IMD.

This leads to the three way decomposition, TE = PAI + CDE + PIE. So if the PIE and CDE have the opposite sign than than the TE, then the percentage due to interaction PAI/TE will be over 100%, again in this case it would not be interpreatable.  

[Rosy1 · Accepted solution]

Sounds great. Many thanks Michaell for your explanation. 

Rosy

[Rick_SAS]

I'll merely add that other statistical models can suffer from a similar situation. For example, in PROC FACTOR, you can estimate the squared correlations. Sometimes the estimates exceed 1 or might even be negative, which is of course impossible. This is known as a "Heywood condition." 

[Rosy1]

Super. Many thanks Rick.

Rosy