# how to interpret coefficients in Mediation Analysis with logged variables

#### vail

Hello there!
I really hope u can help me because i've tried to find a solution to my problem for ages now..

i'm performing a mediation analysis with two regressions to get the coefficients a b and c. i know the interpretation of indirect effect (a*b) in mediation analysis when there is no transformed variable:

the amount by which Y is expected to change indirectly through M per a unit change in X

but i transformed some of my variables. now i m not sure what it means. generally in a regression analysis you can interpret the coefficients as elasticity when the iv and the dv is log transformed...

can anybody help me in figuring out how to interpret the results / coefficients in a mediation analysis logged iv, mediatorvariable and dv??

i couldnt even find another study that did this and interpreted the values...
i would be very grateful for any hints...

im guessing one can interpret the results as elasticities as well. but im not quite sure...

#### chiro

MHF Helper
Hey vail.

Can you please tell us what your full regression model is, the structure of your observations (response variable and independent variables), what transformation you are using, what hypothesis tests or inference tests you are using and ultimately what questions you are trying to answer.

#### vail

hey chiro
thanks for your reply. im not that good at english so i tried to keep it to the basics. i thought it was a quite general question so one might not need all the specific information.
because im just looking for the general interpretation of a cofficient in a mediator model with logged variables (independent variable, response variable and mediator variable are all in natural logarihms)

what im guessing is that the interpretion is
the amount (%) by which Y is expected to change indirectly through M per a 1% change in X

im sorry its just very complicated to describe the whole thing and i wonder how it can help with the question at all. im very grateful though that u responded. Do u really think it would help if i list all variables and goals ? Because if it's necessary i could try but i dont know.. i m not looking for a specific answer for my case

another question: one cant change posts in here? is that correct? why is that #### chiro

MHF Helper
Can you write your model Y as a function of M and X? Include any transformations and assumptions you are using.

#### vail

well i hope this is what u meant... :/

ln(Y)=c1*x1+c2*ln(x2)+....+b*ln(M)+const1 + eY
ln(M)=a1*x1+a2*ln(x2)+...+const2 + eM

const for intercepts
e for erorrs in the estimation
a from the regression M on X1...XJ
b and c from the Regression Y on M and X...XJ

some of the independent variables are logged some arent
mediator M is logged
some of the untransformed variables are binary variables

#### chiro

MHF Helper
The last thing I need is your hypothesis tests for the parameters that you are making inferences on.

One example combination is H0: a1 = 0, H1: a1 != 0. Another is H0: a1 = 0, H1: a1 > 0.

You should note that under large sample sizes these variables will have normal distributions which will help you derive the power sizes using standard formulas for the PDF of a Normal.

The normality condition (also known as the Central Limit Theorem) is the foundation of statistics and will make the power calculations easy.

If you are testing a function of multiple parameters in each hypothesis (like a contrast) then you need to consider co-variance terms as well if they are correlated.

#### vail

Thank you for your respond but I'm sorry I'm very confused now.

i do have hypotheses of both kinds H0: a1 = 0, H1: a1 != 0. and H0: a1 = 0, H1: a1 > 0

the size of my sample is 150

but i m not really looking for the power size :/ .. i m desperately trying to find the right interpretation of the indirect effect a*b .. since i can not say that "y" is increasing several units

like.. when i have a*b=.123 and the independent variable is logged and the mediator and the dependent variable also. can i say that y increases .123 % when x increases 1% (through the effect on m)
or isn't this the right way to interpret the coefficients?

when the independent variable isnt logged but the mediator is and the dependent variable is. can i say that y increases (e^(a*b)-1) % when x increases 1%?

do u know how this interpretation stuff works in mediation analysis with transformed variables?

#### chiro

MHF Helper
The interpretation of the interaction is basically the same as the other effects - an increase in the term gives a corresponding increase (or decrease) in the response.

In terms of actual context for your process that is a bit different.

The increase itself is not that simple when you have multiple factors. You could "fix" the other variables as a constant but I think you would be better off looking at the relative increase or decrease as opposed to the normal type.

The relative increase or decrease is basically coefficient/total where total = sum up all coefficients. You can take absolute values as well to get indicators of the "weight" of the coefficient.

In terms of transforming log(y) to y then since you are using MLE estimates you can also do an exponential to the RHS and look at the coefficients in terms of how they are transformed. Just note that you won't have a linear model anymore - even though with an MLE you can use the transformed functions to get said estimates.

What I mean by the above is that if we have an MLE estimate x and a function f then the estimate for the random variable f(X) will be f(x) where x is the estimate. This is a theory in statistics.

Finally I think you should consider what the interaction says. Positive interactions mean two positives or two negatives increase and a negative means an increase happens when one is opposite sign of the other.

Also if you are trying to make sure that both processes are independent of each other and contribute to the response then you will need to change your process until that term is statistically insignificant.

The interaction means that there is a dependency in how both are used to explain your response. This means that increasing one or the other will increase or decrease (with respect to the population response mean) depending on how you increase or decrease the other. A single unit increase of either corresponds to an increase or decrease represented by the coefficient - and I think this is what you were trying to get at.

This is the same for other factors. A unit increase in that factor has the same interpretation.

In this case it means a unit increase in factor1 depends on coefficient1 + coefficient3 and an increase in factor2 is coefficient2 + coefficient3. The last coefficient assumes the other factor is non-zero.

#### vail

firs i want to thank u for ur detailed respond
but since im not that good at statistics and english i m not sure if i understand. it seems so much more complicated than i thought it was

im not using interaction terms.. not sure if u just call it that or if this is relevant.. i just have one mediator.

i did to regressions. one with all the ivs and the mediator (to get the direct effect b and c ) to the dv and one with the mediatorvariable as dependent variable to get the effect on the mediator ( a)
so i presume the other ivs are constant when i raise x für one unit / percent

when u say i can use relative increase are u saying that this is right? :
indirect: a 1 % increase of x leads to a (effect of a*b) % increase in y through m
direct: a 1 % i increase of x leads to a (effect of c) % increase in y ------ mediator and all other ivs are constant
total: a 1 % i increase of x leads to a (effect of a*b+c) % increase in y ---- mediator not constant but all other ivs

#### vail

i was quite sure that the direct and total effect can be interpreted this way because in these to just come from a regression with logged variables and the cofficients in these normally can be interpreted as elasticities.. as far as i know