How to interpret id-level beta parameters in lognormal model?

General
1

I’m trying to interpret the results of my lognormal regression model fitted with brms. I’ve looked at the excellent notes that folks have referred to in similar threads (e.g. [FAQ How do I interpret a regression model when some variables are log transformed?]) and think I understand them, but I just need to be sure that when transforming the axes for my plots I’m using the correct interpretation.
So my call the the model is;

lognorm_multi_elq3 <- brm(formula = sales ~ 1 + 
                            APD + 
                            (1  + APD | ID ), 
                          data = elq3, 
                          family = lognormal(),
                          prior = set_prior("normal(1,10)", class = "b"),
                          iter = 10000)

So I think I understand that the lognormal model is reporting the parameters on the log scale. So for the population-level intercept term, that means exp(Intercept) is roughly the value of sales when all independent variables are at 1 (not 0, because exp(0) = 1).

If I use the following transformation in the call to mcmc_areas()…

# plot the intercepts for a subsample of 10 ids 
mcmc_areas(
  model_object, 
  pars = vars(param_glue("r_ID[{id},{var}]",
                         var = c("Intercept"),
                         id = sample(data$ID, 10))),
  prob = 0.8, # 80% intervals
  prob_outer = 0.95, # 99%
  point_est = "median",
  transformations = function(x) (exp(x)* exp(fixef(model_object)[1]))
  ) + geom_vline(xintercept=exp(fixef(lognorm_multi_elq1)[1]), size=1.5, color="grey") + labs(x = "Sales")

So here exp(fixef(model_object)[1]) will give us the intercept (which is at index [1] in the output of fixef(model_object)) in the original scale (sales in $), and the transformation calculates id-level intercepts as a proportion of that population level intercept.

Now the situation with the beta (slope) parameters are a little more tricky to me. If exp(beta) of the population-level beta is the % increase for every unit increase in APD (or 1 SD increase if I’ve scaled and centered my independent variables), then what are the exp(beta_id) i.e. id-level betas? Are they an additional % increase per unit increase in APD, or are they also a multiplicative? Because of the way I’ve written my model formula above I feel like it’s an additional increase, example if exp(beta) for a population is 1.03 and exp(beta_id) for this ID is 1.03, the total increase in sales for a 1 SD increase in APD would be 6% for this ID. Is that correct?

2

I haven’t gotten a lot of traction on this thread, despite getting a few folks reading it, so perhaps I should rephrase my question.

For a given individual in my model, I have the population wide constant slope estimate, and an id-level varying slope estimate. I can use fixef and ranef() from brms to extract these estimates from my model object. Now, because my model is using the lognormal family, I need to use exp() to transform the estimates back, and doing so will give me the % increase in y for every 1 standard deviation increase in x.

Now if I wanted to get the TOTAL effect for this individual, would it be exp(fixed) * exp(random) or exp(fixed) + exp(random)?

3

Hi,
just a few thoughts:

exp(Intercept) is roughly the value of sales when all independent variables are at 0 - because all of those variables act before taking the exponent.

I think the best way to think about those models is to really internalize logarithms/exps. So when you add x on the log scale, it is the same as multiplying by exp(x) on the response scale. So if the population level effect of APD is fixed_APD and the varying effect of APD for ID 1 is beta_APD[i] then one unit increase in APD results in multiplying the outcome for ID 1 by exp(fixed_APD) * exp(beta_APD[i]) = exp( fixed_APD + beta_APD[i]). Does that clarify things?

4

Hi Martin,

Thanks so much for getting back to me on this. You’re right, thinking about the behavior of logarithms first before applying it to the model helped me a lot, and I see that exp(fixed_APD) * exp(beta_APD[i]) = exp( fixed_APD + beta_APD[i])

This has given me confidence that I’m applying the transformations in the right way, appreciate your help.

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