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.

```
<- brm(
fit1 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

```
<- 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 1.00 789
sd(back_Intercept) 0.24 0.08 0.09 0.39 1.01 286
cor(tarsus_Intercept,back_Intercept) -0.52 0.23 -0.95 -0.08 1.01 377
Tail_ESS
sd(tarsus_Intercept) 1353
sd(back_Intercept) 606
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) 0.27 0.05 0.16 0.38 1.00 664
sd(back_Intercept) 0.35 0.06 0.23 0.47 1.00 453
cor(tarsus_Intercept,back_Intercept) 0.68 0.21 0.20 0.98 1.00 316
Tail_ESS
sd(tarsus_Intercept) 1163
sd(back_Intercept) 952
cor(tarsus_Intercept,back_Intercept) 557
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 1356 1387
back_Intercept -0.01 0.06 -0.14 0.11 1.00 2247 1577
tarsus_sexMale 0.77 0.06 0.66 0.87 1.00 3631 1251
tarsus_sexUNK 0.23 0.13 -0.03 0.48 1.00 3904 1252
tarsus_hatchdate -0.04 0.06 -0.16 0.07 1.00 1221 1286
back_sexMale 0.01 0.07 -0.12 0.13 1.00 3669 1885
back_sexUNK 0.15 0.15 -0.13 0.44 1.00 3706 1602
back_hatchdate -0.09 0.05 -0.19 0.02 1.00 2152 1551
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 1928 1652
sigma_back 0.90 0.02 0.85 0.95 1.00 2607 1610
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 2597 1370
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 0.4341246 0.02329163 0.3857624 0.4776837
R2back 0.1977502 0.02836021 0.1408088 0.2523545
```

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, data = BTdata, chains = 2, cores = 2) fit2
```

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 1.00 800
sd(back_Intercept) 0.25 0.07 0.11 0.39 1.00 392
cor(tarsus_Intercept,back_Intercept) -0.49 0.22 -0.91 -0.06 1.00 576
Tail_ESS
sd(tarsus_Intercept) 1270
sd(back_Intercept) 589
cor(tarsus_Intercept,back_Intercept) 649
~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 1.00 642
sd(back_Intercept) 0.35 0.06 0.23 0.45 1.00 511
cor(tarsus_Intercept,back_Intercept) 0.70 0.20 0.24 0.98 1.00 229
Tail_ESS
sd(tarsus_Intercept) 937
sd(back_Intercept) 1098
cor(tarsus_Intercept,back_Intercept) 311
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
tarsus_Intercept -0.42 0.07 -0.55 -0.28 1.00 1248 1372
back_Intercept 0.00 0.05 -0.10 0.10 1.00 1552 1414
tarsus_sexMale 0.77 0.06 0.66 0.88 1.00 3231 1537
tarsus_sexUNK 0.23 0.13 -0.03 0.49 1.00 2914 1644
back_hatchdate -0.08 0.05 -0.19 0.02 1.00 2024 1543
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 2089 1584
sigma_back 0.90 0.02 0.85 0.95 1.00 2066 1532
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 2480 1767
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 -2127.2 33.5
p_loo 176.2 7.4
looic 4254.3 67.1
------
Monte Carlo SE of elpd_loo is 0.4.
Pareto k diagnostic values:
Count Pct. Min. n_eff
(-Inf, 0.5] (good) 814 98.3% 196
(0.5, 0.7] (ok) 14 1.7% 76
(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 -2124.6 33.6
p_loo 173.5 7.3
looic 4249.2 67.3
------
Monte Carlo SE of elpd_loo is NA.
Pareto k diagnostic values:
Count Pct. Min. n_eff
(-Inf, 0.5] (good) 813 98.2% 208
(0.5, 0.7] (ok) 14 1.7% 107
(0.7, 1] (bad) 1 0.1% 32
(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 -2.5 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) 2.10 1.11 0.34 4.66 1.00 563 515
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 1.00 476
sd(back_Intercept) 0.23 0.07 0.08 0.36 1.00 249
cor(tarsus_Intercept,back_Intercept) -0.54 0.24 -0.95 -0.05 1.00 458
Tail_ESS
sd(tarsus_Intercept) 1100
sd(back_Intercept) 302
cor(tarsus_Intercept,back_Intercept) 706
~fosternest (Number of levels: 104)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(tarsus_Intercept) 0.26 0.06 0.15 0.37 1.01 362
sd(back_Intercept) 0.32 0.06 0.21 0.43 1.00 478
cor(tarsus_Intercept,back_Intercept) 0.66 0.23 0.16 0.99 1.01 148
Tail_ESS
sd(tarsus_Intercept) 518
sd(back_Intercept) 696
cor(tarsus_Intercept,back_Intercept) 306
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 1.00 797 1161
back_Intercept 0.00 0.05 -0.10 0.11 1.00 1219 1284
tarsus_sexMale 0.77 0.06 0.66 0.88 1.00 2604 1442
tarsus_sexUNK 0.21 0.12 -0.02 0.44 1.00 2548 1583
sigma_tarsus_sexFem -0.30 0.04 -0.39 -0.21 1.00 2220 1266
sigma_tarsus_sexMale -0.24 0.04 -0.32 -0.16 1.00 1723 1329
sigma_tarsus_sexUNK -0.39 0.13 -0.63 -0.14 1.00 1636 1461
back_shatchdate_1 -0.08 3.24 -5.80 7.09 1.00 880 1003
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 2079 1550
alpha_tarsus -1.22 0.43 -1.85 0.10 1.01 922 445
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.