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

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?

• Would you please post the data (or a link to the data) before transform? May 17, 2019 at 12:20
• No you can't. Y could increase 10 units due to a change in X1 irrespective of X2, and Y could increase 10 units due to a change in X2 irrespective of X1. May 17, 2019 at 16:47

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.

• The response is not a time-to-event. Even if it was, a time-to-event model does not have the ability to attribute hazard differences to two or more covariates. May 17, 2019 at 16:48