Interpret effect of covariates in linear model with log-transformed response variable

Question

I am having difficulty interpreting the effects of the covariates of a linear model with log-transformed response for two specific time points.

This is the model: $log(Y_t) = \beta_0 + \beta_1 * X_{1t} + \beta_2 * X_{2t}$

Let's say I have $Y_t = 100$ and $Y_{t+1} = 110$. Now I want to explain this increase in $Y$ from $t$ to $t+1$ in terms of the explanatory variables. Is it possible to split this $10$ units increase in $Y$ between $X_1$ and $X_2$, e.g. $Y$ increased by $7$ units due to $X_1$ and by $3$ units due to $X_2$?

How could I mathematically split the increase in $Y$ between the covariates for two specific time points?

Answer 1

This is known as an Accelerated Failure Time model.

https://en.wikipedia.org/wiki/Accelerated_failure_time_model

From wiki:

The interpretation of ${\displaystyle \theta }$ in accelerated failure time models is straightforward: ${\displaystyle \theta =2}$ means that everything in the relevant life history of an individual happens twice as fast. For example, if the model concerns the development of a tumor, it means that all of the pre-stages progress twice as fast as for the unexposed individual, implying that the expected time until a clinical disease is 0.5 of the baseline time. However, this does not mean that the hazard function ${\displaystyle \lambda (t|\theta )}$ is always twice as high - that would be the proportional hazards model.

A textbook reference on this sort of models is

Kalbfleisch & Prentice (2002). The Statistical Analysis of Failure Time Data (2nd ed.). Hoboken, NJ: Wiley Series in Probability and Statistics.