Calculating Confidence Interval with log-transformation I fitted a linear regression with a 3rd order interaction and adjustment for confounding variable which looks like this: 
$$
log(Y) = β_1S + β_2T + β_3M + β_4ST + β_5SM + β_6TM + β_7STM + β_XX
$$
With $Y$ my outcome, $S$, $T$ and $M$ my interesting variables and $X$ my vector of confounding variables.
I have got MLE and standard deviation for every β.
By transformation, I can get back to Y like this: 
$$
Y={e^{β_1 } }^S×{e^{β_2 } }^T×{e^{β_3 } }^M×{e^{β_4 } }^{ST}×{e^{β_5 } }^{SM}×{e^{β_6 } }^{TM}×{e^{β_7 } }^{STM}×e^{β_XX}
$$ 
This form is quite harder to interpret than the additive form for non-stat experts (which my study is aiming at).
In order to simplify interpretations, I thought about computing the increase of $Y$ depending only on $S$, $T$ and $M$, which I'd think would take this form (since $β_XX$ contains intercept and confounding variables):
$$
\begin{align}
& Y'= Y × coef _{(S, T, M)} \\
& coef = {e^{β_1 } }^S×{e^{β_2 } }^T×{e^{β_3 } }^M×{e^{β_4 } }^{ST}×{e^{β_5 } }^{SM}×{e^{β_6 } }^{TM}×{e^{β_7 } }^{STM}
\end{align}
$$ 
This coefficient is easy to compute and to interpret as the effect of $S$, $T$ and $M$ on $Y$ for anyone, but 


*

*Is it valid ? Else, is it possible to compute something similar ?

*How to compute its confidence interval ? (wouldn't it even be a prediction interval in this case ?)


Feel free to ask for more details.
 A: It is valid.
To computer the CI for $coef$, the covariances between $\hat\beta_i$ and $\hat\beta)j$ are needed.
We have $$Var(S\hat\beta_1 + T\hat\beta_2 + ...+ STM\hat\beta_7) = S^2Var(\hat\beta_1) + ...+ (STM)^2Var(\hat\beta_7) + 2STCov(\hat\beta_1,\hat\beta_2) + ... + 2ST^2M^2Cov(\hat\beta_6,\hat\beta_7)$$
Then get the CI of $S\hat\beta_1 + T\hat\beta_2 + ...+ STM\hat\beta_7$ by $S\hat\beta_1 + T\hat\beta_2 + ...+ STM\hat\beta_7 +/- Z_{1-\alpha/2}\sqrt{Var(S\hat\beta_1 + T\hat\beta_2 + ...+ STM\hat\beta_7)}$
Then Exponential of them are the CI of $coef$.
If you are familiar with matrix, it is simpler in matrix in writing and in calculation.
Let $\hat β_1S + \hat β_2T + \hat β_3M + \hat β_4ST + \hat β_5SM + \hat β_6TM + \hat β_7STM = (S, T, M, ST, SM, TM, STM)(\hat β_1,\hat β_2,\hat β_3,\hat β_4,\hat β_5,\hat β_6,\hat β_7)' = A\hat\beta$
Then $Var(A\hat\beta) = ACov(\hat\beta)A'$, where $Cov(\hat\beta)$ is variance-covariance matrix of $\hat\beta$ and should be able to get its estimate from the model fitting software.
It seems Excel can perform this kind of matrix operation. 
