Please also provide the following information in addition to your question:

- Operating System: Win 10
- brms Version: 2.6.0

Hi, I have a question how to interprete / deal with monotonic effects in models, e.g. with count-outcome.

There’s a paper describing how to model these effects with brms, using a linear model. For the linear model, the parameter *b* of the monotonic effect indicates direction and size (range between lowest and highest category of the ordinal predictor), while the simplex parameters indicate the “normalized” distance between each category.

How would this be interpreted, e.g., in a count-model? Especially if I would like to report incident rate ratios, I would usually exponentiate the “point estimate” (e.g. posterior median). Can I also exponentiate the parameter *b* of the monotonic effect, and does it then tell me the range of the “ratio”?

Example:

```
library(brms)
income_options <- c("below_20", "20_to_40", "40_to_100", "greater_100")
income <- factor(sample(income_options, 100, TRUE), levels = income_options, ordered = TRUE)
mean_ls <- c(30, 60, 70, 75)
ls <- mean_ls[income] + rnorm(100, sd = 7)
dat <- data.frame(income, ls, count = rpois(100, 2))
m1 <- brm(ls ~ mo(income), data = dat)
m2 <- brm(count ~ mo(income), data = dat, family = poisson)
summary(m1)
#> Population-Level Effects:
#> Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
#> Intercept 29.97 1.79 26.55 33.40 2686 1.00
#> moincome 46.18 2.47 41.27 50.87 2591 1.00
#>
#> Simplex Parameters:
#> Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
#> moincome1[1] 0.63 0.04 0.55 0.71 3690 1.00
#> moincome1[2] 0.20 0.05 0.11 0.29 3897 1.00
#> moincome1[3] 0.17 0.04 0.08 0.25 2750 1.00
summary(m2)
#> Population-Level Effects:
#> Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
#> Intercept 0.69 0.14 0.40 0.94 1516 1.00
#> moincome 0.12 0.20 -0.26 0.53 1488 1.00
#>
#> Simplex Parameters:
#> Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
#> moincome1[1] 0.35 0.23 0.01 0.84 2806 1.00
#> moincome1[2] 0.31 0.22 0.01 0.81 2707 1.00
#> moincome1[3] 0.34 0.23 0.01 0.85 2870 1.00
```

For the second model, the posterior mean for `moincome`

is 0.12, which is on the log-scale. `exp(0.12)`

~ 1.13, so the IRR of `moincome`

would be 1.13. Can I conclude that the ratio for the monotonic effect has a range up to 1.13 times higher for the highest category?

This is how the marginal effects plot looks like, which automatically transforms the predictions into the scale of the outcome:

```
marginal_effects(m2, "income")
```

So, how would I interprete the (exponentiated) coefficient of **moincome** for the poisson-model? Or does it only make sense to interprete these coefficients by looking at marginal effects plots?