Interpreting results from categorical() with brm

#1

I’m running a multi-nominal model with categorical() in brm(). With three categories, I have three parameters mu_1, mu_2, and mu_3 in the output. How should I interpret these three parameters? Are exp(mu_1), exp(mu_2), and exp(mu_3) the odds ratios for those three categories? Or is exp(mu_1) associated with the base category?

• Operating System: Linux system (Fedora 25)
• brms Version: 2.1.5

Thanks,
Gang

#2

Could you show the full summary output? This makes it easier to put the parameters in context.

#3

The interest is the group-level effect for “corrZ” per “rsfm” level. Below is the model summary:

Family: categorical
Links: mu1 = logit; mu2 = logit; mu3 = logit
Formula: Y ~ corrZ + (1 + corrZ | rsfm)
Data: dat (Number of observations: 959066)
Samples: 4 chains, each with iter = 1000; warmup = 500; thin = 1; total post-warmup samples = 2000

Group-Level Effects:
~rsfm (Number of levels: 247)
Estimate Est.Error l-95% CI u-95% CI Eff.Sample  Rhat
sd(mu1_Intercept) 0.00 0.00 0.00 0.01 2000       1.0
sd(mu1_corrZ) 0.22 0.05 0.11 0.32 430            1.0
sd(mu2_Intercept) 0.00 0.00 0.00 0.01 2000       1.0
sd(mu2_corrZ) 0.05 0.03 0.00 0.13 838            1.0
sd(mu3_Intercept) 0.00 0.00 0.00 0.01 2000       1.0
sd(mu3_corrZ) 0.15 0.07 0.01 0.28 234            1.03
cor(mu1_Intercept,mu1_corrZ) -0.08 0.58 -0.95 0.94 27  1.10
cor(mu2_Intercept,mu2_corrZ) -0.01 0.58 -0.95 0.94 1205 1.0
cor(mu3_Intercept,mu3_corrZ) 0.04 0.57 -0.92 0.95 260   1.01

Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
mu1_Intercept -3.13 0.01 -3.14 -3.12 2000 1.00
mu2_Intercept -3.04 0.01 -3.05 -3.03 2000 1.00
mu3_Intercept -3.46 0.01 -3.47 -3.45 2000 1.00
mu1_corrZ -0.13 0.03 -0.19 -0.07 2000 1.00
mu2_corrZ -0.05 0.03 -0.10 -0.00 2000 1.00
mu3_corrZ 0.03 0.03 -0.04 0.09 2000 1.00

#4

Are you sure you only have three categories? Because, in brms, categorial() models are implemented using the first category as reference category, seeing mu1, mu2, mu3 indicates that you actually have four categories.

#5

You’re correct! I misspoke: there are indeed FOUR categories! So, exp(mu_1), exp(mu_2), and exp(mu_3) are the odds ratios relative to the base category?

#6

Indeed, with the first category (let’s call it category 0) as reference, we have:

P(y = k) / P(y = 0) = \exp(\eta_k)