Bayesian lognormal model: how to correctly back-transform the estimates?

Question

I have a Bayesian model of the form:

$$ \begin{align} y & \sim logNormal(\mu, \sigma)\\ \mu_n & = \alpha + \beta_0 c_n + \beta_1 d_n + \beta_2 c_n d_n \end{align} $$ Where:

• $y$ is a variable measured in ms
• $c$ is a sum-contrast coded variable equal to 1 or -1.
• $d$ is a scaled and centered continuous variable

This omits assumed prior distributions, and slope and intercept adjustments, I believe they are not important with respect to the main point. Since the likelihood is logNormal the results are on the log scale.

I want to back-transform the estimates from the model and compute the effect sizes of different variables. I think I know how to do this for $c$:

$$ \mu|_{c=1} - \mu|_{c=-1} = \exp (\alpha + \beta_0 + \beta_1 d_n + \beta_2 d_n ) - \exp (\alpha - \beta_0 + \beta_1 d_n - \beta_2 d_n ) $$

For interaction this is slightly more complicated. Let the interaction effect be: $IE = \frac{\partial \mu}{\partial d}|_{c = 1} - \frac{\partial \mu}{\partial d}|_{c = -1}$. I remove the log by exponentiation and then I take the derivative which results in:

$$ \frac{\partial \mu}{\partial d} = \exp (\alpha + \beta_0 c_n + \beta_1 d_n + \beta_2 c_n d_n ) (\beta_1 + \beta_2 c_n) $$

So the interaction effect is the difference of these two terms:

$$ \begin{align} \frac{\partial \mu}{\partial d}|_{c = 1} = & \exp (\alpha + \beta_0 + \beta_1 d_n + \beta_2 d_n ) (\beta_1 + \beta_2) \\ \frac{\partial \mu}{\partial d}|_{c = -1} = & \exp (\alpha - \beta_0 + \beta_1 d_n - \beta_2 d_n ) (\beta_1 - \beta_2) \\ \end{align} $$

My questions are:

1. Are the derivations for the interaction effect and the effect of $c$ correct? If yes, this means that these effects are functions of $d$, but fitting the model returns the effects (on the log scale) as constants. Where does this difference come from?
2. How do I derive and interpret the effect for $d$? Ie. what are the meaningful points at which to evaluate the function $\mu$? Or maybe is there a better way of doing this?

Answer 1

The effect size of a log-normal model has not a univoque value on the same scale of y, because of the log-relation between $\mu$ and the position of the log-normal distribution (its median, its mean). When changing a covariate, the model predictions move linearly with the covariate on the log scale, not on the original scale, so on the log scale there is a constant effect size, while on the original scale there is not, it varies.

The effect sizes you already have are good as they are, you can't "back-transform" them without making a mess of them. Although you may make a simple exponentiation of the betas. This is called "time ratio" in AFT models, and expresses the proportion between the predicted value for two observations with unit variation in the corresponding covariate (the proportion, because the difference varies!).