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 Fem
```

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

```
Family: MV(gaussian, gaussian)
Links: mu = identity; sigma = identity
mu = identity; sigma = 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
Group-Level Effects:
~dam (Number of levels: 106)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(tarsus_Intercept) 0.49 0.05 0.40 0.59 1.00 728
sd(back_Intercept) 0.25 0.07 0.11 0.39 1.01 395
cor(tarsus_Intercept,back_Intercept) -0.51 0.22 -0.93 -0.07 1.00 679
Tail_ESS
sd(tarsus_Intercept) 1088
sd(back_Intercept) 801
cor(tarsus_Intercept,back_Intercept) 637
~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 0.38 1.00 812
sd(back_Intercept) 0.35 0.06 0.24 0.47 1.01 491
cor(tarsus_Intercept,back_Intercept) 0.70 0.20 0.24 0.98 1.01 365
Tail_ESS
sd(tarsus_Intercept) 1088
sd(back_Intercept) 937
cor(tarsus_Intercept,back_Intercept) 784
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
tarsus_Intercept -0.41 0.07 -0.55 -0.27 1.00 2006 1476
back_Intercept -0.01 0.06 -0.14 0.11 1.00 2612 1575
tarsus_sexMale 0.77 0.06 0.66 0.88 1.00 3822 1525
tarsus_sexUNK 0.23 0.13 -0.01 0.47 1.00 3271 1582
tarsus_hatchdate -0.05 0.06 -0.16 0.07 1.00 1447 1363
back_sexMale 0.01 0.07 -0.12 0.14 1.00 3510 1336
back_sexUNK 0.15 0.15 -0.14 0.44 1.00 4016 1625
back_hatchdate -0.09 0.05 -0.19 0.02 1.00 2120 1358
Family Specific 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 2430 1623
sigma_back 0.90 0.02 0.86 0.95 1.00 2830 1553
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 3485 1684
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. Within dams, tarsus length and back color
seem to be negatively correlated, while within 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.

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.

```
Estimate Est.Error Q2.5 Q97.5
R2tarsus 0.4355111 0.02317053 0.3889295 0.4789890
R2back 0.1987557 0.02765139 0.1450809 0.2540395
```

Clearly, 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.

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.

```
Family: MV(gaussian, gaussian)
Links: mu = identity; sigma = identity
mu = identity; sigma = 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
Group-Level Effects:
~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 1.00 946
sd(back_Intercept) 0.25 0.08 0.09 0.39 1.01 288
cor(tarsus_Intercept,back_Intercept) -0.50 0.24 -0.94 -0.02 1.01 647
Tail_ESS
sd(tarsus_Intercept) 1353
sd(back_Intercept) 446
cor(tarsus_Intercept,back_Intercept) 732
~fosternest (Number of levels: 104)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(tarsus_Intercept) 0.27 0.05 0.16 0.37 1.00 735
sd(back_Intercept) 0.35 0.06 0.23 0.46 1.00 455
cor(tarsus_Intercept,back_Intercept) 0.67 0.21 0.18 0.98 1.01 281
Tail_ESS
sd(tarsus_Intercept) 1164
sd(back_Intercept) 1268
cor(tarsus_Intercept,back_Intercept) 571
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
tarsus_Intercept -0.41 0.07 -0.55 -0.27 1.00 1631 1337
back_Intercept 0.00 0.05 -0.11 0.11 1.00 2899 1505
tarsus_sexMale 0.77 0.06 0.65 0.88 1.00 3834 1542
tarsus_sexUNK 0.23 0.12 -0.01 0.48 1.00 3774 1629
back_hatchdate -0.09 0.05 -0.19 0.02 1.00 2619 1644
Family Specific 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 2256 1328
sigma_back 0.90 0.02 0.85 0.95 1.00 2150 1350
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 2645 1588
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:

```
Output of model 'fit1':
Computed from 2000 by 828 log-likelihood matrix
Estimate SE
elpd_loo -2124.4 33.4
p_loo 174.6 7.2
looic 4248.7 66.9
------
Monte Carlo SE of elpd_loo is NA.
Pareto k diagnostic values:
Count Pct. Min. n_eff
(-Inf, 0.5] (good) 812 98.1% 316
(0.5, 0.7] (ok) 15 1.8% 85
(0.7, 1] (bad) 1 0.1% 58
(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 -2124.4 33.6
p_loo 173.9 7.3
looic 4248.9 67.1
------
Monte Carlo SE of elpd_loo is 0.4.
Pareto k diagnostic values:
Count Pct. Min. n_eff
(-Inf, 0.5] (good) 814 98.3% 111
(0.5, 0.7] (ok) 14 1.7% 109
(0.7, 1] (bad) 0 0.0% <NA>
(1, Inf) (very bad) 0 0.0% <NA>
All Pareto k estimates are ok (k < 0.7).
See help('pareto-k-diagnostic') for details.
Model comparisons:
elpd_diff se_diff
fit1 0.0 0.0
fit2 -0.1 1.3
```

Apparently, 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.

```
Family: MV(skew_normal, gaussian)
Links: mu = identity; sigma = log; alpha = identity
mu = identity; sigma = 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
Smooth Terms:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sds(back_shatchdate_1) 1.96 1.00 0.40 4.21 1.00 664 836
Group-Level Effects:
~dam (Number of levels: 106)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(tarsus_Intercept) 0.47 0.05 0.38 0.57 1.00 1083
sd(back_Intercept) 0.24 0.07 0.11 0.37 1.00 409
cor(tarsus_Intercept,back_Intercept) -0.51 0.22 -0.92 -0.09 1.00 681
Tail_ESS
sd(tarsus_Intercept) 1251
sd(back_Intercept) 772
cor(tarsus_Intercept,back_Intercept) 891
~fosternest (Number of levels: 104)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(tarsus_Intercept) 0.26 0.06 0.15 0.36 1.00 589
sd(back_Intercept) 0.31 0.06 0.19 0.43 1.00 622
cor(tarsus_Intercept,back_Intercept) 0.65 0.21 0.17 0.97 1.01 325
Tail_ESS
sd(tarsus_Intercept) 1038
sd(back_Intercept) 1045
cor(tarsus_Intercept,back_Intercept) 640
Population-Level Effects:
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 1748 1778
back_Intercept 0.00 0.05 -0.10 0.10 1.00 2676 1542
tarsus_sexMale 0.77 0.06 0.66 0.89 1.00 4276 1578
tarsus_sexUNK 0.22 0.12 -0.02 0.45 1.00 4078 1615
sigma_tarsus_sexFem -0.30 0.04 -0.38 -0.22 1.00 2645 1794
sigma_tarsus_sexMale -0.24 0.04 -0.32 -0.16 1.00 2755 1745
sigma_tarsus_sexUNK -0.39 0.13 -0.63 -0.15 1.00 3251 1694
back_shatchdate_1 -0.27 3.10 -5.79 6.57 1.00 1483 1603
Family Specific 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 3088 1128
alpha_tarsus -1.21 0.44 -1.86 0.05 1.00 1701 865
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

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.

Hadfield JD, Nutall A, Osorio D, Owens IPF (2007). Testing the
phenotypic gambit: phenotypic, genetic and environmental correlations of
colour. *Journal of Evolutionary Biology*, 20(2), 549-557.