I fit three Bayesian binary multilevel logistic regression models. Schematically they look like this:

model1 <- brm(DEP.VAR ~ IND.VAR1 + (1|WORD), data = data, family = Bernoulli, warmup = 4000, iter=20000)
model2 <- brm(DEP.VAR ~ IND.VAR1 + IND.VAR2 + (1|WORD), data = data, family = Bernoulli, warmup = 4000, iter=20000)
model3 <- brm(DEP.VAR ~ IND.VAR1 + IND.VAR2 + IND.VAR3 + (1|WORD), data = data, family = Bernoulli, warmup = 4000, iter=20000)

I calculated the ICC for each model with icc() from the package performance. The ICC is calculated with the latent variable method:

$\frac{\tau^2}{\tau^2+ \frac{\pi^2}{3}}$

$\tau^2$ is the variance of the distribution of the varying intercepts.

With each additional covariate, the ICC increases. Why? I don't understand why adding predictor variables should increase the amount of within-group homogeneity.

Does the ICC have to have a conditional interpretation when there are covariates? For instance, this blog says:

``Note though that when there are predictors in the model, the ICC should have a conditional interpretation: of the residual variation in outcomes that remains after accounting for the variables in the model, it is the proportion that is attributable to systematic differences between clusters.''

If this is true, why is the residual variation in outcomes (as measured by the ICC) increasing? I would think that the residual variation in outcomes would on the whole decrease as covariates were added. There would then be less residual variation, which I would have expected to lead to a lower ICC.