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 1.00 752
sd(back_Intercept) 0.25 0.07 0.11 0.39 1.01 409
cor(tarsus_Intercept,back_Intercept) -0.51 0.22 -0.92 -0.09 1.00 582
Tail_ESS
sd(tarsus_Intercept) 1390
sd(back_Intercept) 717
cor(tarsus_Intercept,back_Intercept) 687
~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 1.00 811
sd(back_Intercept) 0.35 0.06 0.23 0.47 1.00 550
cor(tarsus_Intercept,back_Intercept) 0.70 0.20 0.22 0.98 1.00 302
Tail_ESS
sd(tarsus_Intercept) 1259
sd(back_Intercept) 864
cor(tarsus_Intercept,back_Intercept) 536
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 1619 1338
back_Intercept -0.02 0.07 -0.14 0.11 1.00 2007 1530
tarsus_sexMale 0.77 0.06 0.66 0.88 1.00 3690 1352
tarsus_sexUNK 0.23 0.13 -0.02 0.48 1.00 3367 1422
tarsus_hatchdate -0.04 0.06 -0.16 0.07 1.00 1338 1167
back_sexMale 0.01 0.07 -0.12 0.14 1.00 3717 1633
back_sexUNK 0.15 0.15 -0.15 0.44 1.00 2925 1403
back_hatchdate -0.09 0.05 -0.19 0.01 1.00 2250 1685
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 2254 1606
sigma_back 0.90 0.02 0.85 0.95 1.00 2029 1433
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 3155 1455
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 0.4336721 0.02428850 0.3824333 0.4793242
R2back 0.2001391 0.02821679 0.1442740 0.2539608
```

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 0.59 1.00 854
sd(back_Intercept) 0.25 0.07 0.10 0.38 1.00 358
cor(tarsus_Intercept,back_Intercept) -0.49 0.23 -0.92 -0.05 1.00 596
Tail_ESS
sd(tarsus_Intercept) 1322
sd(back_Intercept) 786
cor(tarsus_Intercept,back_Intercept) 666
~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 1.00 818
sd(back_Intercept) 0.35 0.06 0.23 0.47 1.00 506
cor(tarsus_Intercept,back_Intercept) 0.66 0.21 0.19 0.98 1.00 271
Tail_ESS
sd(tarsus_Intercept) 1430
sd(back_Intercept) 907
cor(tarsus_Intercept,back_Intercept) 627
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 1553 1757
back_Intercept 0.00 0.05 -0.10 0.11 1.00 2218 1187
tarsus_sexMale 0.77 0.06 0.66 0.88 1.00 4317 1368
tarsus_sexUNK 0.22 0.13 -0.02 0.48 1.00 3217 1389
back_hatchdate -0.08 0.05 -0.18 0.02 1.00 2449 1501
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma_tarsus 0.75 0.02 0.72 0.80 1.00 2408 1344
sigma_back 0.90 0.02 0.86 0.95 1.00 2015 1364
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 2619 1499
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 -2127.7 33.7
p_loo 177.7 7.5
looic 4255.3 67.4
------
Monte Carlo SE of elpd_loo is NA.
Pareto k diagnostic values:
Count Pct. Min. n_eff
(-Inf, 0.5] (good) 815 98.4% 279
(0.5, 0.7] (ok) 10 1.2% 260
(0.7, 1] (bad) 3 0.4% 49
(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 -2124.6 33.7
p_loo 174.7 7.5
looic 4249.2 67.4
------
Monte Carlo SE of elpd_loo is NA.
Pareto k diagnostic values:
Count Pct. Min. n_eff
(-Inf, 0.5] (good) 811 97.9% 419
(0.5, 0.7] (ok) 15 1.8% 90
(0.7, 1] (bad) 2 0.2% 67
(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 -3.1 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)
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) 2.14 1.07 0.43 4.65 1.00 416 422
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 1.00 546
sd(back_Intercept) 0.24 0.07 0.09 0.38 1.00 260
cor(tarsus_Intercept,back_Intercept) -0.52 0.23 -0.94 -0.07 1.00 485
Tail_ESS
sd(tarsus_Intercept) 1119
sd(back_Intercept) 465
cor(tarsus_Intercept,back_Intercept) 726
~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 1.00 325
sd(back_Intercept) 0.31 0.06 0.19 0.43 1.00 348
cor(tarsus_Intercept,back_Intercept) 0.68 0.22 0.17 0.98 1.00 191
Tail_ESS
sd(tarsus_Intercept) 669
sd(back_Intercept) 667
cor(tarsus_Intercept,back_Intercept) 437
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
tarsus_Intercept -0.41 0.06 -0.54 -0.29 1.00 962 1168
back_Intercept 0.00 0.05 -0.10 0.10 1.00 965 991
tarsus_sexMale 0.77 0.06 0.66 0.89 1.00 2609 1639
tarsus_sexUNK 0.21 0.12 -0.02 0.45 1.00 1760 1531
sigma_tarsus_sexFem -0.30 0.04 -0.38 -0.22 1.00 2163 1528
sigma_tarsus_sexMale -0.25 0.04 -0.33 -0.16 1.00 2409 1313
sigma_tarsus_sexUNK -0.39 0.13 -0.64 -0.13 1.00 1958 1613
back_shatchdate_1 -0.37 3.45 -6.59 7.23 1.00 721 1177
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 2008 1253
alpha_tarsus -1.22 0.43 -1.91 0.00 1.00 795 346
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.