Estimating Multivariate Models with brms
Paul Bürkner
2026-09-07
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 0.40 0.59 1.00 517
sd(back_Intercept) 0.24 0.07 0.09 0.39 1.00 276
cor(tarsus_Intercept,back_Intercept) -0.53 0.23 -0.96 -0.07 1.01 422
Tail_ESS
sd(tarsus_Intercept) 847
sd(back_Intercept) 487
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 0.05 0.17 0.37 1.00 546
sd(back_Intercept) 0.35 0.06 0.23 0.47 1.00 448
cor(tarsus_Intercept,back_Intercept) 0.69 0.21 0.20 0.99 1.02 155
Tail_ESS
sd(tarsus_Intercept) 829
sd(back_Intercept) 952
cor(tarsus_Intercept,back_Intercept) 365
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
tarsus_Intercept -0.40 0.07 -0.54 -0.27 1.01 972 1235
back_Intercept -0.01 0.07 -0.14 0.12 1.00 1566 1387
tarsus_sexMale 0.77 0.06 0.66 0.88 1.00 2792 1665
tarsus_sexUNK 0.23 0.13 -0.01 0.49 1.00 1777 1432
tarsus_hatchdate -0.04 0.06 -0.15 0.08 1.00 760 961
back_sexMale 0.01 0.07 -0.12 0.14 1.00 3337 1511
back_sexUNK 0.15 0.15 -0.14 0.45 1.00 2873 1712
back_hatchdate -0.09 0.05 -0.19 0.01 1.00 1341 1435
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 1788 1339
sigma_back 0.90 0.02 0.86 0.95 1.00 1701 1343
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 1954 1506
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 0.4334649 0.02608363 0.3816716 0.4833573
R2back 0.1982176 0.02855200 0.1410045 0.2545034Clearly, 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 0.59 1.00 908
sd(back_Intercept) 0.25 0.07 0.10 0.39 1.00 334
cor(tarsus_Intercept,back_Intercept) -0.49 0.22 -0.93 -0.06 1.00 509
Tail_ESS
sd(tarsus_Intercept) 1238
sd(back_Intercept) 545
cor(tarsus_Intercept,back_Intercept) 577
~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 1.00 512
sd(back_Intercept) 0.35 0.06 0.22 0.47 1.00 456
cor(tarsus_Intercept,back_Intercept) 0.70 0.21 0.20 0.99 1.00 307
Tail_ESS
sd(tarsus_Intercept) 812
sd(back_Intercept) 984
cor(tarsus_Intercept,back_Intercept) 565
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 1259 1202
back_Intercept -0.00 0.05 -0.10 0.11 1.00 1953 1686
tarsus_sexMale 0.77 0.06 0.66 0.88 1.00 2895 1599
tarsus_sexUNK 0.22 0.12 -0.03 0.46 1.00 2885 1849
back_hatchdate -0.09 0.05 -0.18 0.01 1.00 1761 1371
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 2141 1524
sigma_back 0.90 0.02 0.85 0.95 1.00 2679 1485
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 2134 1451
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 -2126.9 33.9
p_loo 176.6 7.6
looic 4253.8 67.7
------
MCSE of elpd_loo is NA.
MCSE and ESS estimates assume MCMC draws (r_eff in [0.4, 2.2]).
Pareto k diagnostic values:
Count Pct. Min. ESS
(-Inf, 0.7] (good) 826 99.8% 223
(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 -2124.2 33.6
p_loo 172.7 7.3
looic 4248.4 67.2
------
MCSE of elpd_loo is NA.
MCSE and ESS estimates assume MCMC draws (r_eff in [0.5, 1.8]).
Pareto k diagnostic values:
Count Pct. Min. ESS
(-Inf, 0.7] (good) 827 99.9% 91
(0.7, 1] (bad) 1 0.1% <NA>
(1, Inf) (very bad) 0 0.0% <NA>
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 1 k_psis > 0.7
fit1 -2.7 1.3 0.98 |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) 1.90 1.05 0.16 4.30 1.00 460 329
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 0.58 1.00 778
sd(back_Intercept) 0.24 0.07 0.10 0.37 1.00 254
cor(tarsus_Intercept,back_Intercept) -0.52 0.22 -0.93 -0.07 1.00 326
Tail_ESS
sd(tarsus_Intercept) 1399
sd(back_Intercept) 707
cor(tarsus_Intercept,back_Intercept) 671
~fosternest (Number of levels: 104)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(tarsus_Intercept) 0.26 0.05 0.16 0.37 1.00 600
sd(back_Intercept) 0.31 0.06 0.20 0.44 1.00 342
cor(tarsus_Intercept,back_Intercept) 0.65 0.22 0.18 0.98 1.00 243
Tail_ESS
sd(tarsus_Intercept) 617
sd(back_Intercept) 985
cor(tarsus_Intercept,back_Intercept) 513
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 1071 1363
back_Intercept 0.00 0.05 -0.09 0.10 1.00 1644 1643
tarsus_sexMale 0.77 0.06 0.66 0.88 1.00 3127 1639
tarsus_sexUNK 0.21 0.12 -0.02 0.44 1.00 2266 1359
sigma_tarsus_sexFem -0.30 0.04 -0.38 -0.22 1.00 2544 1761
sigma_tarsus_sexMale -0.25 0.04 -0.32 -0.17 1.00 1949 1425
sigma_tarsus_sexUNK -0.40 0.13 -0.63 -0.15 1.00 2021 1520
back_shatchdate_1 -0.22 3.11 -5.87 7.21 1.00 1004 811
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma_back 0.90 0.02 0.85 0.95 1.00 2609 1146
alpha_tarsus -1.22 0.42 -1.88 0.00 1.00 1133 527
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.