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) 0.48 0.05 0.39 0.58 1.00 908
sd(back_Intercept) 0.24 0.07 0.10 0.38 1.01 333
cor(tarsus_Intercept,back_Intercept) -0.52 0.23 -0.92 -0.06 1.00 499
Tail_ESS
sd(tarsus_Intercept) 1662
sd(back_Intercept) 571
cor(tarsus_Intercept,back_Intercept) 566
~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 1.00 753
sd(back_Intercept) 0.35 0.06 0.24 0.46 1.00 601
cor(tarsus_Intercept,back_Intercept) 0.69 0.20 0.22 0.98 1.03 257
Tail_ESS
sd(tarsus_Intercept) 1286
sd(back_Intercept) 1212
cor(tarsus_Intercept,back_Intercept) 582
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 -0.27 1.00 1646 1420
back_Intercept -0.01 0.07 -0.15 0.11 1.00 3194 1546
tarsus_sexMale 0.77 0.06 0.66 0.88 1.00 3994 1428
tarsus_sexUNK 0.23 0.13 -0.03 0.47 1.00 4522 1762
tarsus_hatchdate -0.04 0.06 -0.16 0.07 1.00 1756 1582
back_sexMale 0.01 0.07 -0.12 0.14 1.01 4563 1487
back_sexUNK 0.15 0.15 -0.16 0.44 1.00 3878 1375
back_hatchdate -0.09 0.05 -0.19 0.01 1.00 2719 1631
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 3043 1382
sigma_back 0.90 0.02 0.86 0.95 1.00 2481 1515
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 3473 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).
```

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 0.4349094 0.02250483 0.3884970 0.4760042
R2back 0.1981263 0.02842267 0.1431563 0.2548206
```

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 0.39 0.59 1.00 827
sd(back_Intercept) 0.25 0.07 0.10 0.38 1.01 307
cor(tarsus_Intercept,back_Intercept) -0.50 0.22 -0.92 -0.08 1.00 554
Tail_ESS
sd(tarsus_Intercept) 1196
sd(back_Intercept) 536
cor(tarsus_Intercept,back_Intercept) 695
~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 1.00 603
sd(back_Intercept) 0.35 0.06 0.23 0.47 1.00 349
cor(tarsus_Intercept,back_Intercept) 0.66 0.21 0.19 0.97 1.00 228
Tail_ESS
sd(tarsus_Intercept) 812
sd(back_Intercept) 954
cor(tarsus_Intercept,back_Intercept) 581
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 1334 1317
back_Intercept 0.00 0.06 -0.11 0.11 1.00 1769 1450
tarsus_sexMale 0.77 0.06 0.65 0.89 1.00 2775 1434
tarsus_sexUNK 0.22 0.13 -0.03 0.47 1.00 3137 1668
back_hatchdate -0.08 0.05 -0.18 0.02 1.00 1816 1315
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 1812 848
sigma_back 0.90 0.02 0.86 0.95 1.00 1938 1509
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 2589 1656
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 -2126.6 33.6
p_loo 176.5 7.4
looic 4253.2 67.2
------
Monte Carlo SE of elpd_loo is NA.
Pareto k diagnostic values:
Count Pct. Min. n_eff
(-Inf, 0.5] (good) 810 97.8% 206
(0.5, 0.7] (ok) 17 2.1% 78
(0.7, 1] (bad) 1 0.1% 65
(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 -2125.4 33.6
p_loo 174.6 7.4
looic 4250.7 67.2
------
Monte Carlo SE of elpd_loo is NA.
Pareto k diagnostic values:
Count Pct. Min. n_eff
(-Inf, 0.5] (good) 804 97.1% 180
(0.5, 0.7] (ok) 22 2.7% 98
(0.7, 1] (bad) 2 0.2% 45
(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 -1.3 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) 1.98 1.03 0.36 4.31 1.00 553 496
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 0.58 1.00 774
sd(back_Intercept) 0.23 0.07 0.10 0.37 1.01 256
cor(tarsus_Intercept,back_Intercept) -0.54 0.23 -0.96 -0.08 1.01 256
Tail_ESS
sd(tarsus_Intercept) 1167
sd(back_Intercept) 591
cor(tarsus_Intercept,back_Intercept) 218
~fosternest (Number of levels: 104)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(tarsus_Intercept) 0.26 0.06 0.14 0.37 1.01 374
sd(back_Intercept) 0.31 0.06 0.20 0.43 1.00 500
cor(tarsus_Intercept,back_Intercept) 0.65 0.22 0.17 0.97 1.01 271
Tail_ESS
sd(tarsus_Intercept) 717
sd(back_Intercept) 901
cor(tarsus_Intercept,back_Intercept) 486
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 -0.28 1.00 842 1422
back_Intercept 0.00 0.05 -0.10 0.10 1.00 1270 1572
tarsus_sexMale 0.77 0.05 0.66 0.87 1.00 3045 1141
tarsus_sexUNK 0.21 0.12 -0.02 0.44 1.00 2731 1746
sigma_tarsus_sexFem -0.30 0.04 -0.38 -0.22 1.00 2929 1561
sigma_tarsus_sexMale -0.24 0.04 -0.32 -0.17 1.00 2338 1622
sigma_tarsus_sexUNK -0.39 0.13 -0.64 -0.14 1.00 2202 1560
back_shatchdate_1 -0.16 3.18 -5.64 6.81 1.00 897 1036
Family Specific 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 1674 1801
alpha_tarsus -1.22 0.43 -1.87 0.05 1.00 1148 481
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.