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
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 1.00 830
sd(back_Intercept) 0.25 0.08 0.10 0.39 1.01 328
cor(tarsus_Intercept,back_Intercept) -0.50 0.22 -0.92 -0.07 1.01 496
Tail_ESS
sd(tarsus_Intercept) 1315
sd(back_Intercept) 571
cor(tarsus_Intercept,back_Intercept) 579
~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 1.00 698
sd(back_Intercept) 0.35 0.06 0.23 0.47 1.00 623
cor(tarsus_Intercept,back_Intercept) 0.67 0.21 0.17 0.98 1.00 243
Tail_ESS
sd(tarsus_Intercept) 1107
sd(back_Intercept) 995
cor(tarsus_Intercept,back_Intercept) 573
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 1944 1636
back_Intercept -0.01 0.07 -0.14 0.12 1.00 2739 1810
tarsus_sexMale 0.77 0.06 0.66 0.89 1.00 4046 1594
tarsus_sexUNK 0.23 0.13 -0.02 0.49 1.00 3819 1484
tarsus_hatchdate -0.04 0.05 -0.15 0.06 1.00 2132 1791
back_sexMale 0.01 0.07 -0.12 0.14 1.00 3858 1560
back_sexUNK 0.15 0.15 -0.15 0.44 1.00 4109 1626
back_hatchdate -0.09 0.05 -0.19 0.01 1.00 2653 1415
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma_tarsus 0.75 0.02 0.72 0.79 1.00 2319 1487
sigma_back 0.90 0.02 0.85 0.95 1.01 2277 1274
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 3386 1423
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.

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 0.4358746 0.02306086 0.3880626 0.4780669
R2back 0.1991446 0.02717648 0.1484433 0.2526186
```

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
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 1.00 895
sd(back_Intercept) 0.26 0.07 0.11 0.41 1.01 347
cor(tarsus_Intercept,back_Intercept) -0.47 0.22 -0.90 -0.02 1.00 524
Tail_ESS
sd(tarsus_Intercept) 1482
sd(back_Intercept) 791
cor(tarsus_Intercept,back_Intercept) 791
~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 0.38 1.00 656
sd(back_Intercept) 0.34 0.06 0.22 0.46 1.00 510
cor(tarsus_Intercept,back_Intercept) 0.67 0.21 0.19 0.98 1.01 225
Tail_ESS
sd(tarsus_Intercept) 1186
sd(back_Intercept) 889
cor(tarsus_Intercept,back_Intercept) 391
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 1492 1513
back_Intercept 0.00 0.05 -0.10 0.11 1.00 1969 1556
tarsus_sexMale 0.77 0.06 0.65 0.89 1.00 4823 1363
tarsus_sexUNK 0.23 0.13 -0.02 0.47 1.00 3472 1378
back_hatchdate -0.08 0.05 -0.19 0.02 1.00 2182 1567
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 2609 1382
sigma_back 0.90 0.02 0.86 0.95 1.00 2688 1222
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 0.03 1.00 2870 1153
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 -2125.5 33.8
p_loo 176.6 7.6
looic 4251.0 67.6
------
MCSE of elpd_loo is NA.
MCSE and ESS estimates assume MCMC draws (r_eff in [0.5, 1.7]).
Pareto k diagnostic values:
Count Pct. Min. ESS
(-Inf, 0.7] (good) 824 99.5% 267
(0.7, 1] (bad) 4 0.5% <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 -2125.3 33.6
p_loo 175.8 7.5
looic 4250.6 67.2
------
MCSE of elpd_loo is NA.
MCSE and ESS estimates assume MCMC draws (r_eff in [0.4, 1.9]).
Pareto k diagnostic values:
Count Pct. Min. ESS
(-Inf, 0.7] (good) 826 99.8% 109
(0.7, 1] (bad) 2 0.2% <NA>
(1, Inf) (very bad) 0 0.0% <NA>
See help('pareto-k-diagnostic') for details.
Model comparisons:
elpd_diff se_diff
fit2 0.0 0.0
fit1 -0.2 1.4
```

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
Smoothing Spline Hyperparameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sds(back_shatchdate_1) 2.07 1.04 0.38 4.48 1.00 481 527
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 0.38 0.58 1.00 550
sd(back_Intercept) 0.23 0.07 0.09 0.37 1.02 220
cor(tarsus_Intercept,back_Intercept) -0.51 0.24 -0.95 -0.04 1.01 361
Tail_ESS
sd(tarsus_Intercept) 835
sd(back_Intercept) 615
cor(tarsus_Intercept,back_Intercept) 368
~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 0.38 1.01 433
sd(back_Intercept) 0.31 0.06 0.20 0.42 1.00 473
cor(tarsus_Intercept,back_Intercept) 0.62 0.22 0.14 0.96 1.00 286
Tail_ESS
sd(tarsus_Intercept) 698
sd(back_Intercept) 690
cor(tarsus_Intercept,back_Intercept) 585
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 1.00 873 1295
back_Intercept 0.00 0.05 -0.10 0.10 1.00 1140 1341
tarsus_sexMale 0.77 0.06 0.66 0.88 1.00 2629 1726
tarsus_sexUNK 0.21 0.12 -0.02 0.45 1.00 1969 1389
sigma_tarsus_sexFem -0.30 0.04 -0.38 -0.22 1.00 1761 1672
sigma_tarsus_sexMale -0.25 0.04 -0.32 -0.16 1.00 1810 1535
sigma_tarsus_sexUNK -0.40 0.13 -0.64 -0.13 1.00 1718 1476
back_shatchdate_1 -0.15 3.34 -6.30 7.17 1.00 798 746
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 1768 1355
alpha_tarsus -1.23 0.42 -1.86 -0.03 1.00 1356 559
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.