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How to measure the percentage of the total effect that occurs through the indirect effects (i.e.,5.1%, 18.8%) and direct effect (i.e., 76.0%)?

For below figure, enter image description here

In this figure, HFT is the independent variable, Liquidity comovement is the dependent variable, Liquidity and Volitality are the mediator variables.

According to my understanding,

Liquidity comovement=β5×HFT+β3×Liquidity+β4×IVoL+e

Liquidity comovement=β5×HFT+β3(β1×HFT+e)+β4(β2×HFT+e)+e

Liquidity comovement=β5×HFT+β1β3×HFT+β2β4×HFT+e

Liquidity comovement=(β5+β1β3+β2β4)HFT+e

and thus, I suppose that

Indirect effect of Liquidity= β1β3/(β5+β1β3+β2β4)=(3.26×213.07)/(10.27+3.26×213.07+1431×1.78)=21.36%

Indirect effect of Volatility= β2β4/(β5+β1β3+β2β4)=(1431×1.78)/(10.27+3.26×213.07+1431×1.78)=78.33%

Direct effect of HFT= β5/(β5+β1β3+β2β4)= 0.387/(10.27+3.26×213.07+1431×1.78)=0.32%

However, my results are different from the percentage in the Figure. May I ask where I was wrong and how to get this percentage？Any help is appreciated.

Source of the Figure: https://www.sciencedirect.com/science/article/pii/S0304405X19300832?casa_token=rZ9cPnhMxc8AAAAA:_UXLAjwRqgpzExN8Y7pjkVn5BYKw03HRWZrIjjF739HGA5lCyNEpidv3Z1UQllL0v2aA1QK6xN8

kjetil b halvorsen
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Alex
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  • It looks that the percentage of the direct effect is (imprecisely) derived by dividing it by the total effect. But I can’t see a way to get the other percentages from just the coefficients on the diagram. What’s your guess? – kk68 Jun 27 '23 at 21:07
  • Are all the variables included in the model continuous? – kk68 Jun 28 '23 at 09:55

1 Answers1

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I write in the "answer" section because there is little space in the comments.

In the data you provided, the percentage of the direct effect (76%) seems to be derived by dividing the Direct Effect coefficient (10.27) by the Total Effect (13.48). The result might be approximate because the authors report only rounded decimals.

10.27/13.48 = 0.7618694

Also, the ratio between the percentages of the two Relative Indirect Effects is approximately equal to the ratio between the two indirect effects calculated in the standard way, meaning by multiplying the coefficients of the path a (IV to the mediator) and b (mediator to DV):

(1431*1.78) / (3.26*213.07)  ≈  18.8 / 5.1 
3.667074 ≈ 3.686275

However, as you have understood, there is no way that, in this case, the following equation holds true: Total Effect = Indirect Effects + Direct Effect.

A working hypothesis that might put you on the right track to solve the puzzle you propose is that it is not always the case in mediation models that this equation holds true. For example, as reported by @user1205901 - Слава Україні in this post:

David Kenny writes on his website that: The equation of c = c' + ab exactly holds when a) multiple regression (or structural equation modeling without latent variables) is used, b) the same cases are used in all the analyses, c) and the same covariates are in all the equations. However, the two are only approximately equal for multilevel models, logistic analysis, and structural equation modeling with latent variables.

By looking at the equations at the bottom of the diagram, for example, it seems that the last equation includes only three control variables instead of the four that are used in the other models. In this case, we can't check the c) point mentioned by Kelly. From the information you provide, we can't even tell if the b) condition is respected, as it is reported the total number of cases in the dataset but the actual sample could vary depending on the variables included in the model.

kk68
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