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.

```
fit1 <- brm(
mvbind(tarsus, back) ~ sex + hatchdate + (1|p|fosternest) + (1|q|dam),
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)
Samples: 2 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup samples = 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 1.00 677
sd(back_Intercept) 0.24 0.07 0.09 0.38 1.00 341
cor(tarsus_Intercept,back_Intercept) -0.53 0.22 -0.95 -0.08 1.00 704
Tail_ESS
sd(tarsus_Intercept) 1364
sd(back_Intercept) 529
cor(tarsus_Intercept,back_Intercept) 793
~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 1.00 877
sd(back_Intercept) 0.36 0.06 0.24 0.47 1.00 560
cor(tarsus_Intercept,back_Intercept) 0.70 0.20 0.23 0.98 1.00 229
Tail_ESS
sd(tarsus_Intercept) 1037
sd(back_Intercept) 1169
cor(tarsus_Intercept,back_Intercept) 527
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
tarsus_Intercept -0.40 0.07 -0.53 -0.27 1.00 1327 1521
back_Intercept -0.01 0.07 -0.14 0.11 1.00 2966 1525
tarsus_sexMale 0.77 0.06 0.66 0.88 1.00 3575 1405
tarsus_sexUNK 0.23 0.12 -0.01 0.47 1.00 3265 1630
tarsus_hatchdate -0.04 0.06 -0.16 0.07 1.00 1615 1428
back_sexMale 0.01 0.06 -0.12 0.13 1.00 4241 1438
back_sexUNK 0.15 0.15 -0.15 0.43 1.00 4219 1700
back_hatchdate -0.09 0.05 -0.19 0.01 1.00 2609 1812
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 2438 1493
sigma_back 0.90 0.02 0.86 0.95 1.00 2185 1411
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 3225 1668
Samples were drawn 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 0.4340881 0.02315548 0.3866977 0.4763724
R2back 0.1986577 0.02718904 0.1438330 0.2512083
```

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, 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)
Samples: 2 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup samples = 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 1.00 893
sd(back_Intercept) 0.25 0.08 0.09 0.39 1.02 283
cor(tarsus_Intercept,back_Intercept) -0.50 0.23 -0.91 -0.04 1.00 518
Tail_ESS
sd(tarsus_Intercept) 1504
sd(back_Intercept) 299
cor(tarsus_Intercept,back_Intercept) 978
~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 1.00 645
sd(back_Intercept) 0.35 0.06 0.23 0.46 1.01 357
cor(tarsus_Intercept,back_Intercept) 0.70 0.20 0.23 0.98 1.01 322
Tail_ESS
sd(tarsus_Intercept) 1191
sd(back_Intercept) 820
cor(tarsus_Intercept,back_Intercept) 667
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 1675 1631
back_Intercept 0.00 0.05 -0.11 0.11 1.00 2149 1683
tarsus_sexMale 0.77 0.06 0.66 0.88 1.00 4192 1468
tarsus_sexUNK 0.23 0.13 -0.02 0.48 1.00 3270 1713
back_hatchdate -0.09 0.06 -0.19 0.02 1.00 2203 1581
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 3098 1350
sigma_back 0.90 0.02 0.85 0.95 1.00 2762 1266
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 3023 1471
Samples were drawn 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 -2124.7 33.5
p_loo 173.9 7.3
looic 4249.4 67.1
------
Monte Carlo SE of elpd_loo is 0.4.
Pareto k diagnostic values:
Count Pct. Min. n_eff
(-Inf, 0.5] (good) 818 98.8% 296
(0.5, 0.7] (ok) 10 1.2% 61
(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.
Output of model 'fit2':
Computed from 2000 by 828 log-likelihood matrix
Estimate SE
elpd_loo -2124.0 33.6
p_loo 172.7 7.4
looic 4248.1 67.3
------
Monte Carlo SE of elpd_loo is NA.
Pareto k diagnostic values:
Count Pct. Min. n_eff
(-Inf, 0.5] (good) 811 97.9% 253
(0.5, 0.7] (ok) 16 1.9% 124
(0.7, 1] (bad) 1 0.1% 29
(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 -0.7 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)
Samples: 2 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup samples = 2000
Smooth Terms:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sds(back_shatchdate_1) 2.01 1.04 0.35 4.50 1.01 494 498
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 1.00 805
sd(back_Intercept) 0.25 0.07 0.10 0.38 1.01 396
cor(tarsus_Intercept,back_Intercept) -0.49 0.23 -0.91 -0.06 1.01 545
Tail_ESS
sd(tarsus_Intercept) 1371
sd(back_Intercept) 373
cor(tarsus_Intercept,back_Intercept) 634
~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 1.00 587
sd(back_Intercept) 0.31 0.06 0.19 0.43 1.00 574
cor(tarsus_Intercept,back_Intercept) 0.65 0.22 0.15 0.97 1.00 207
Tail_ESS
sd(tarsus_Intercept) 913
sd(back_Intercept) 910
cor(tarsus_Intercept,back_Intercept) 442
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 1063 1486
back_Intercept 0.00 0.05 -0.10 0.10 1.00 1780 1265
tarsus_sexMale 0.77 0.06 0.66 0.88 1.00 3454 1735
tarsus_sexUNK 0.21 0.12 -0.01 0.44 1.00 3114 1637
sigma_tarsus_sexFem -0.30 0.04 -0.39 -0.22 1.00 2386 1464
sigma_tarsus_sexMale -0.25 0.04 -0.33 -0.16 1.00 2066 1547
sigma_tarsus_sexUNK -0.39 0.13 -0.64 -0.13 1.00 2254 1427
back_shatchdate_1 -0.39 3.25 -6.20 6.63 1.00 853 1146
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 2551 1384
alpha_tarsus -1.20 0.45 -1.88 0.09 1.00 772 295
Samples were drawn 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.