Different posterior-based ways for describing the effect size?

Storytelling with data is extremely important. I am trying to more discover the flexibility of a posterior for presenting an effect size. What are the different possibilities for showing a posterior-based effect size (point estimate + credible intervals)?

I give few examples (group A and B weight difference) that I found out by myself.

Data, model and posterior

df = bind_rows(
  tibble(weight = rnorm(100, 80, 5), group = rep("A", length.out = T)),
  tibble(weight = rnorm(100, 60, 5), group = rep("B", length.out = T)),

fit = brm(weight ~ (1 | group), df)

posterior = posterior_samples(fit) %>% clean_names() %>% mutate(A = b_intercept + r_group_a_intercept, B = b_intercept + r_group_b_intercept) %>% select(A, B)

Option 1: showing difference in natural scale (group A - group B)

posterior %>% mutate(difference = A-B) %>% posterior_summary()

           Estimate Est.Error     Q2.5    Q97.5
A          79.00722 0.5201468 77.97750 80.01733
B          60.00173 0.5233391 58.97731 61.02931
difference 19.00550 0.7388529 17.55313 20.41760

Option 2: showing difference in times (group A/group B)

posterior %>% mutate(difference = A/B) %>% posterior_summary()

           Estimate  Est.Error      Q2.5     Q97.5
A          79.00722 0.52014681 77.977501 80.017334
B          60.00173 0.52333915 58.977314 61.029313
difference  1.31685 0.01441019  1.289337  1.344716

Option 3: showing difference as percentage ((A-B)*100)/A)

posterior %>% mutate(difference = ((A-B)*100)/A) %>% posterior_summary()

           Estimate Est.Error     Q2.5    Q97.5
A          79.00722 0.5201468 77.97750 80.01733
B          60.00173 0.5233391 58.97731 61.02931
difference 24.05209 0.8311302 22.44078 25.63488

Are these correct? What are other cool possibilities that I could use for showing the difference (effect size)?

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They appear “correct” in the sense of “implemented correctly”. Whether they are “correct” for answering a specific question depends heavily on the question you are asking, so I don’t think there is a general answer. I think most often difference in natural scale would be the easiest to interpret. It also matches the model closely - when one talks about relative multiplicative change, one usually wants to work with models whose responses are constrained to be positive (as otherwise relative difference is a weird measure), so I would expect those primarily with something like lognormal or Poisson model. But I can imagine scenarios where either Option 2 or Option 3 is most useful even with a normal model.

I’ll note that here you seem to be replicating the functionality of posterior_epred - that’s a good exercise to make sure you understand the model, just wanted to highlight that this functionality is already available in brms out of the box.

In more complex models, you often get a lot of valid prediction tasks for the same model (e.g. which covariates to keep fixed, which to change; which random effects to include; …) and the interesting part is thinking which prediction task (and thus which effect size) is the most relevant to your question.

Does that answer your question?

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