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).

```
data("BTdata", package = "MCMCglmm")
head(BTdata)
```

```
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)
<- brm(bform1, data = BTdata, chains = 2, cores = 2) fit1
```

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

```
<- add_criterion(fit1, "loo")
fit1 summary(fit1)
```

```
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 1.00 778
sd(back_Intercept) 0.25 0.07 0.10 0.39 1.01 321
cor(tarsus_Intercept,back_Intercept) -0.53 0.22 -0.94 -0.09 1.01 402
Tail_ESS
sd(tarsus_Intercept) 1083
sd(back_Intercept) 798
cor(tarsus_Intercept,back_Intercept) 580
~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 1.00 660
sd(back_Intercept) 0.35 0.06 0.23 0.47 1.00 663
cor(tarsus_Intercept,back_Intercept) 0.71 0.20 0.22 0.99 1.01 260
Tail_ESS
sd(tarsus_Intercept) 1054
sd(back_Intercept) 754
cor(tarsus_Intercept,back_Intercept) 502
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
tarsus_Intercept -0.40 0.07 -0.55 -0.27 1.00 1156 1279
back_Intercept -0.01 0.06 -0.14 0.11 1.00 2090 1653
tarsus_sexMale 0.77 0.06 0.66 0.88 1.00 4043 1646
tarsus_sexUNK 0.23 0.12 -0.01 0.48 1.00 3590 1358
tarsus_hatchdate -0.04 0.06 -0.15 0.07 1.00 975 1300
back_sexMale 0.01 0.07 -0.13 0.14 1.00 3523 1648
back_sexUNK 0.15 0.15 -0.14 0.46 1.00 2903 1438
back_hatchdate -0.09 0.05 -0.20 0.01 1.00 1980 1679
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 2452 1760
sigma_back 0.90 0.02 0.85 0.95 1.00 2373 1434
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 2367 1433
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.

`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 0.4335935 0.02434887 0.3824507 0.4776490
R2back 0.1994719 0.02749108 0.1432571 0.2502431
```

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 ~ sex + (1|p|fosternest) + (1|q|dam))
bf_tarsus <- bf(back ~ hatchdate + (1|p|fosternest) + (1|q|dam))
bf_back <- brm(bf_tarsus + bf_back + set_rescor(TRUE),
fit2 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.

```
<- add_criterion(fit2, "loo")
fit2 summary(fit2)
```

```
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 1.00 925
sd(back_Intercept) 0.25 0.07 0.09 0.38 1.00 416
cor(tarsus_Intercept,back_Intercept) -0.50 0.22 -0.92 -0.08 1.00 729
Tail_ESS
sd(tarsus_Intercept) 1144
sd(back_Intercept) 581
cor(tarsus_Intercept,back_Intercept) 744
~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 1.00 531
sd(back_Intercept) 0.35 0.06 0.23 0.47 1.00 602
cor(tarsus_Intercept,back_Intercept) 0.69 0.20 0.24 0.98 1.01 289
Tail_ESS
sd(tarsus_Intercept) 1139
sd(back_Intercept) 1090
cor(tarsus_Intercept,back_Intercept) 625
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 -0.28 1.00 1498 1507
back_Intercept -0.00 0.05 -0.11 0.11 1.00 2394 1500
tarsus_sexMale 0.77 0.06 0.66 0.87 1.00 4223 1612
tarsus_sexUNK 0.23 0.13 -0.03 0.48 1.00 4999 1689
back_hatchdate -0.09 0.05 -0.19 0.01 1.00 2449 1561
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 3084 1630
sigma_back 0.90 0.02 0.85 0.95 1.00 2558 1630
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 2745 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).
```

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 -2126.5 33.6
p_loo 175.9 7.4
looic 4253.0 67.1
------
Monte Carlo SE of elpd_loo is NA.
Pareto k diagnostic values:
Count Pct. Min. n_eff
(-Inf, 0.5] (good) 815 98.4% 271
(0.5, 0.7] (ok) 10 1.2% 224
(0.7, 1] (bad) 3 0.4% 39
(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 33.8
p_loo 175.2 7.5
looic 4250.3 67.6
------
Monte Carlo SE of elpd_loo is NA.
Pareto k diagnostic values:
Count Pct. Min. n_eff
(-Inf, 0.5] (good) 802 96.9% 215
(0.5, 0.7] (ok) 24 2.9% 103
(0.7, 1] (bad) 2 0.2% 21
(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 -1.4 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 ~ sex + (1|p|fosternest) + (1|q|dam)) +
bf_tarsus lf(sigma ~ 0 + sex) + skew_normal()
<- bf(back ~ s(hatchdate) + (1|p|fosternest) + (1|q|dam)) +
bf_back gaussian()
<- brm(
fit3 + bf_back + set_rescor(FALSE),
bf_tarsus data = BTdata, chains = 2, cores = 2,
control = list(adapt_delta = 0.95)
)
```

Again, we summarize the model and look at some posterior-predictive checks.

```
<- add_criterion(fit3, "loo")
fit3 summary(fit3)
```

```
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) 1.95 1.08 0.21 4.44 1.01 363 270
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 1.00 487
sd(back_Intercept) 0.24 0.07 0.10 0.37 1.01 333
cor(tarsus_Intercept,back_Intercept) -0.52 0.23 -0.92 -0.07 1.01 424
Tail_ESS
sd(tarsus_Intercept) 1102
sd(back_Intercept) 477
cor(tarsus_Intercept,back_Intercept) 542
~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 0.36 1.00 429
sd(back_Intercept) 0.31 0.06 0.20 0.43 1.01 364
cor(tarsus_Intercept,back_Intercept) 0.65 0.22 0.15 0.98 1.00 192
Tail_ESS
sd(tarsus_Intercept) 849
sd(back_Intercept) 796
cor(tarsus_Intercept,back_Intercept) 382
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 -0.28 1.00 657 992
back_Intercept 0.00 0.05 -0.10 0.10 1.00 748 948
tarsus_sexMale 0.77 0.06 0.66 0.88 1.00 2156 1664
tarsus_sexUNK 0.22 0.12 -0.02 0.44 1.00 1260 1540
sigma_tarsus_sexFem -0.30 0.04 -0.38 -0.22 1.00 1926 1528
sigma_tarsus_sexMale -0.24 0.04 -0.32 -0.17 1.00 1672 1556
sigma_tarsus_sexUNK -0.39 0.13 -0.64 -0.15 1.00 1434 1437
back_shatchdate_1 -0.06 3.30 -5.67 7.56 1.00 635 710
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 1581 1630
alpha_tarsus -1.26 0.37 -1.88 -0.33 1.00 1306 810
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.

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.