# Estimating Multivariate Models with brms

## Introduction

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

## Basic Multivariate Models

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

fit1 <- add_criterion(fit1, "loo")
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)
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.

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 0.4340881 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.

## More Complex Multivariate Models

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.

fit2 <- add_criterion(fit2, "loo")
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)
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:

loo(fit1, fit2)
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,
)

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

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

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.

## References

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.