Estimating Multivariate Models with brms
Paul Bürkner
2026-07-22
Source:vignettes/brms_multivariate.Rmd
brms_multivariate.RmdIntroduction
In the present vignette, we want to discuss how to specify
multivariate multilevel models using brms. We call a
model multivariate if it contains multiple response variables,
each being predicted by its own set of predictors. Consider an example
from biology. Hadfield, Nutall, Osorio, and Owens (2007) analyzed data
of the Eurasian blue tit (https://en.wikipedia.org/wiki/Eurasian_blue_tit). They
predicted the tarsus length as well as the
back color of chicks. Half of the brood were put into
another fosternest, while the other half stayed in the
fosternest of their own dam. This allows to separate
genetic from environmental factors. Additionally, we have information
about the hatchdate and sex of the chicks (the
latter being known for 94% of the animals).
tarsus back animal dam fosternest hatchdate sex
1 -1.89229718 1.1464212 R187142 R187557 F2102 -0.6874021 Fem
2 1.13610981 -0.7596521 R187154 R187559 F1902 -0.6874021 Male
3 0.98468946 0.1449373 R187341 R187568 A602 -0.4279814 Male
4 0.37900806 0.2555847 R046169 R187518 A1302 -1.4656641 Male
5 -0.07525299 -0.3006992 R046161 R187528 A2602 -1.4656641 Fem
6 -1.13519543 1.5577219 R187409 R187945 C2302 0.3502805 FemBasic Multivariate Models
We begin with a relatively simple multivariate normal model.
bform1 <-
bf(mvbind(tarsus, back) ~ sex + hatchdate + (1|p|fosternest) + (1|q|dam)) +
set_rescor(TRUE)
fit1 <- brm(bform1, data = BTdata, chains = 2, cores = 2)As can be seen in the model code, we have used mvbind
notation to tell brms that both tarsus and
back are separate response variables. The term
(1|p|fosternest) indicates a varying intercept over
fosternest. By writing |p| in between we
indicate that all varying effects of fosternest should be
modeled as correlated. This makes sense since we actually have two model
parts, one for tarsus and one for back. The
indicator p is arbitrary and can be replaced by other
symbols that comes into your mind (for details about the multilevel
syntax of brms, see help("brmsformula")
and vignette("brms_multilevel")). Similarly, the term
(1|q|dam) indicates correlated varying effects of the
genetic mother of the chicks. Alternatively, we could have also modeled
the genetic similarities through pedigrees and corresponding relatedness
matrices, but this is not the focus of this vignette (please see
vignette("brms_phylogenetics")). The model results are
readily summarized via
fit1 <- add_criterion(fit1, "loo")
summary(fit1) Family: MV(gaussian, gaussian)
Links: mu = identity
mu = identity
Formula: tarsus ~ sex + hatchdate + (1 | p | fosternest) + (1 | q | dam)
back ~ sex + hatchdate + (1 | p | fosternest) + (1 | q | dam)
Data: BTdata (Number of observations: 828)
Draws: 2 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 2000
Multilevel Hyperparameters:
~dam (Number of levels: 106)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(tarsus_Intercept) 0.48 0.05 0.39 0.59 1.00 876
sd(back_Intercept) 0.25 0.07 0.11 0.39 1.00 467
cor(tarsus_Intercept,back_Intercept) -0.50 0.21 -0.89 -0.08 1.00 619
Tail_ESS
sd(tarsus_Intercept) 1292
sd(back_Intercept) 780
cor(tarsus_Intercept,back_Intercept) 859
~fosternest (Number of levels: 104)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(tarsus_Intercept) 0.27 0.06 0.16 0.38 1.00 777
sd(back_Intercept) 0.35 0.06 0.23 0.46 1.00 745
cor(tarsus_Intercept,back_Intercept) 0.70 0.21 0.23 0.99 1.00 357
Tail_ESS
sd(tarsus_Intercept) 1191
sd(back_Intercept) 1266
cor(tarsus_Intercept,back_Intercept) 748
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
tarsus_Intercept -0.41 0.07 -0.54 -0.27 1.00 1725 1445
back_Intercept -0.01 0.06 -0.14 0.11 1.00 2473 1699
tarsus_sexMale 0.77 0.06 0.65 0.89 1.00 3977 1181
tarsus_sexUNK 0.22 0.13 -0.03 0.48 1.00 3494 1640
tarsus_hatchdate -0.04 0.06 -0.16 0.07 1.00 1775 1335
back_sexMale 0.01 0.07 -0.13 0.14 1.00 3679 1596
back_sexUNK 0.15 0.15 -0.15 0.45 1.00 3324 1327
back_hatchdate -0.09 0.05 -0.19 0.01 1.00 2687 1584
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma_tarsus 0.76 0.02 0.72 0.80 1.00 2245 1562
sigma_back 0.90 0.02 0.85 0.95 1.00 2650 1439
Residual Correlations:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
rescor(tarsus,back) -0.05 0.04 -0.13 0.02 1.00 2963 1258
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).The summary output of multivariate models closely resembles those of
univariate models, except that the parameters now have the corresponding
response variable as prefix. Across dams, tarsus length and back color
seem to be negatively correlated, while across fosternests the opposite
is true. This indicates differential effects of genetic and
environmental factors on these two characteristics. Further, the small
residual correlation rescor(tarsus, back) on the bottom of
the output indicates that there is little unmodeled dependency between
tarsus length and back color. Although not necessary at this point, we
have already computed and stored the LOO information criterion of
fit1, which we will use for model comparisons. Next, let’s
take a look at some posterior-predictive checks, which give us a first
impression of the model fit.
pp_check(fit1, resp = "tarsus")
pp_check(fit1, resp = "back")
This looks pretty solid, but we notice a slight unmodeled left
skewness in the distribution of tarsus. We will come back
to this later on. Next, we want to investigate how much variation in the
response variables can be explained by our model and we use a Bayesian
generalization of the \(R^2\)
coefficient.
bayes_R2(fit1) Estimate Est.Error Q2.5 Q97.5
R2tarsus 0.4335855 0.02592390 0.3822516 0.4852822
R2back 0.1989664 0.02876377 0.1432445 0.2543291Clearly, there is much variation in both animal characteristics that we can not explain, but apparently we can explain more of the variation in tarsus length than in back color.
More Complex Multivariate Models
Now, suppose we only want to control for sex in
tarsus but not in back and vice versa for
hatchdate. Not that this is particular reasonable for the
present example, but it allows us to illustrate how to specify different
formulas for different response variables. We can no longer use
mvbind syntax and so we have to use a more verbose
approach:
bf_tarsus <- bf(tarsus ~ sex + (1|p|fosternest) + (1|q|dam))
bf_back <- bf(back ~ hatchdate + (1|p|fosternest) + (1|q|dam))
fit2 <- brm(bf_tarsus + bf_back + set_rescor(TRUE),
data = BTdata, chains = 2, cores = 2)Note that we have literally added the two model parts via
the + operator, which is in this case equivalent to writing
mvbf(bf_tarsus, bf_back). See
help("brmsformula") and help("mvbrmsformula")
for more details about this syntax. Again, we summarize the model
first.
fit2 <- add_criterion(fit2, "loo")
summary(fit2) Family: MV(gaussian, gaussian)
Links: mu = identity
mu = identity
Formula: tarsus ~ sex + (1 | p | fosternest) + (1 | q | dam)
back ~ hatchdate + (1 | p | fosternest) + (1 | q | dam)
Data: BTdata (Number of observations: 828)
Draws: 2 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 2000
Multilevel Hyperparameters:
~dam (Number of levels: 106)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(tarsus_Intercept) 0.48 0.05 0.39 0.58 1.00 909
sd(back_Intercept) 0.25 0.08 0.10 0.39 1.01 306
cor(tarsus_Intercept,back_Intercept) -0.50 0.22 -0.93 -0.06 1.01 505
Tail_ESS
sd(tarsus_Intercept) 1276
sd(back_Intercept) 748
cor(tarsus_Intercept,back_Intercept) 627
~fosternest (Number of levels: 104)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(tarsus_Intercept) 0.27 0.06 0.16 0.38 1.00 790
sd(back_Intercept) 0.35 0.06 0.23 0.46 1.01 504
cor(tarsus_Intercept,back_Intercept) 0.67 0.21 0.20 0.98 1.00 354
Tail_ESS
sd(tarsus_Intercept) 1158
sd(back_Intercept) 888
cor(tarsus_Intercept,back_Intercept) 853
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
tarsus_Intercept -0.41 0.07 -0.54 -0.27 1.00 1867 1526
back_Intercept 0.00 0.05 -0.11 0.11 1.00 2666 1360
tarsus_sexMale 0.77 0.05 0.66 0.87 1.00 3835 1534
tarsus_sexUNK 0.22 0.12 -0.01 0.45 1.00 3669 1657
back_hatchdate -0.08 0.05 -0.19 0.02 1.00 2811 1727
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma_tarsus 0.76 0.02 0.72 0.80 1.00 2035 1383
sigma_back 0.90 0.02 0.85 0.95 1.00 2364 1198
Residual Correlations:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
rescor(tarsus,back) -0.05 0.04 -0.12 0.02 1.00 2964 1379
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Let’s find out, how model fit changed due to excluding certain effects from the initial model:
loo(fit1, fit2)Output of model 'fit1':
Computed from 2000 by 828 log-likelihood matrix.
Estimate SE
elpd_loo -2126.3 33.6
p_loo 176.1 7.4
looic 4252.6 67.3
------
MCSE of elpd_loo is NA.
MCSE and ESS estimates assume MCMC draws (r_eff in [0.5, 1.6]).
Pareto k diagnostic values:
Count Pct. Min. ESS
(-Inf, 0.7] (good) 826 99.8% 92
(0.7, 1] (bad) 2 0.2% <NA>
(1, Inf) (very bad) 0 0.0% <NA>
See help('pareto-k-diagnostic') for details.
Output of model 'fit2':
Computed from 2000 by 828 log-likelihood matrix.
Estimate SE
elpd_loo -2125.1 33.5
p_loo 174.4 7.3
looic 4250.1 67.1
------
MCSE of elpd_loo is 0.4.
MCSE and ESS estimates assume MCMC draws (r_eff in [0.4, 1.7]).
All Pareto k estimates are good (k < 0.7).
See help('pareto-k-diagnostic') for details.
Model comparisons:
model elpd_diff se_diff p_worse diag_diff diag_elpd
fit2 0.0 0.0 NA
fit1 -1.2 1.3 0.83 |elpd_diff| < 4 2 k_psis > 0.7Apparently, there is no noteworthy difference in the model fit.
Accordingly, we do not really need to model sex and
hatchdate for both response variables, but there is also no
harm in including them (so I would probably just include them).
To give you a glimpse of the capabilities of brms’
multivariate syntax, we change our model in various directions at the
same time. Remember the slight left skewness of tarsus,
which we will now model by using the skew_normal family
instead of the gaussian family. Since we do not have a
multivariate normal (or student-t) model, anymore, estimating residual
correlations is no longer possible. We make this explicit using the
set_rescor function. Further, we investigate if the
relationship of back and hatchdate is really
linear as previously assumed by fitting a non-linear spline of
hatchdate. On top of it, we model separate residual
variances of tarsus for male and female chicks.
bf_tarsus <- bf(tarsus ~ sex + (1|p|fosternest) + (1|q|dam)) +
lf(sigma ~ 0 + sex) + skew_normal()
bf_back <- bf(back ~ s(hatchdate) + (1|p|fosternest) + (1|q|dam)) +
gaussian()
fit3 <- brm(
bf_tarsus + bf_back + set_rescor(FALSE),
data = BTdata, chains = 2, cores = 2,
control = list(adapt_delta = 0.95)
)Again, we summarize the model and look at some posterior-predictive checks.
fit3 <- add_criterion(fit3, "loo")
summary(fit3) Family: MV(skew_normal, gaussian)
Links: mu = identity; sigma = log
mu = identity
Formula: tarsus ~ sex + (1 | p | fosternest) + (1 | q | dam)
sigma ~ 0 + sex
back ~ s(hatchdate) + (1 | p | fosternest) + (1 | q | dam)
Data: BTdata (Number of observations: 828)
Draws: 2 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 2000
Smoothing Spline Hyperparameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sds(back_shatchdate_1) 1.89 1.06 0.14 4.45 1.01 329 222
Multilevel Hyperparameters:
~dam (Number of levels: 106)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(tarsus_Intercept) 0.47 0.05 0.39 0.57 1.01 645
sd(back_Intercept) 0.24 0.07 0.09 0.37 1.01 229
cor(tarsus_Intercept,back_Intercept) -0.51 0.23 -0.94 -0.08 1.00 266
Tail_ESS
sd(tarsus_Intercept) 1184
sd(back_Intercept) 506
cor(tarsus_Intercept,back_Intercept) 388
~fosternest (Number of levels: 104)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(tarsus_Intercept) 0.26 0.05 0.15 0.37 1.01 457
sd(back_Intercept) 0.31 0.06 0.20 0.43 1.00 321
cor(tarsus_Intercept,back_Intercept) 0.67 0.21 0.16 0.97 1.02 143
Tail_ESS
sd(tarsus_Intercept) 767
sd(back_Intercept) 776
cor(tarsus_Intercept,back_Intercept) 338
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
tarsus_Intercept -0.41 0.07 -0.54 -0.29 1.00 796 1134
back_Intercept -0.00 0.05 -0.10 0.11 1.00 942 1267
tarsus_sexMale 0.77 0.06 0.66 0.88 1.00 1825 1505
tarsus_sexUNK 0.22 0.12 -0.03 0.44 1.00 1272 1512
sigma_tarsus_sexFem -0.30 0.04 -0.38 -0.22 1.00 1892 1479
sigma_tarsus_sexMale -0.24 0.04 -0.32 -0.16 1.00 1277 1480
sigma_tarsus_sexUNK -0.39 0.13 -0.64 -0.14 1.00 1736 1576
back_shatchdate_1 -0.20 3.05 -5.57 7.05 1.00 581 737
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma_back 0.90 0.02 0.86 0.95 1.00 1891 1605
alpha_tarsus -1.23 0.43 -1.87 0.04 1.00 1060 588
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).We see that the (log) residual standard deviation of
tarsus is somewhat larger for chicks whose sex could not be
identified as compared to male or female chicks. Further, we see from
the negative alpha (skewness) parameter of
tarsus that the residuals are indeed slightly left-skewed.
Lastly, running
conditional_effects(fit3, "hatchdate", resp = "back")
reveals a non-linear relationship of hatchdate on the
back color, which seems to change in waves over the course
of the hatch dates.
There are many more modeling options for multivariate models, which
are not discussed in this vignette. Examples include autocorrelation
structures, Gaussian processes, or explicit non-linear predictors (e.g.,
see help("brmsformula") or
vignette("brms_multilevel")). In fact, nearly all the
flexibility of univariate models is retained in multivariate models.