Estimating Multivariate Models with brms

Paul Bürkner

2024-09-23

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.

bform1 <- 
  bf(mvbind(tarsus, back) ~ sex + hatchdate + (1|p|fosternest) + (1|q|dam)) +
  set_rescor(TRUE)

fit1 <- brm(bform1, 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) 
  Draws: 2 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup draws = 2000

Multilevel Hyperparameters:
~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      830
sd(back_Intercept)                       0.25      0.08     0.10     0.39 1.01      328
cor(tarsus_Intercept,back_Intercept)    -0.50      0.22    -0.92    -0.07 1.01      496
                                     Tail_ESS
sd(tarsus_Intercept)                     1315
sd(back_Intercept)                        571
cor(tarsus_Intercept,back_Intercept)      579

~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      698
sd(back_Intercept)                       0.35      0.06     0.23     0.47 1.00      623
cor(tarsus_Intercept,back_Intercept)     0.67      0.21     0.17     0.98 1.00      243
                                     Tail_ESS
sd(tarsus_Intercept)                     1107
sd(back_Intercept)                        995
cor(tarsus_Intercept,back_Intercept)      573

Regression Coefficients:
                 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     1944     1636
back_Intercept      -0.01      0.07    -0.14     0.12 1.00     2739     1810
tarsus_sexMale       0.77      0.06     0.66     0.89 1.00     4046     1594
tarsus_sexUNK        0.23      0.13    -0.02     0.49 1.00     3819     1484
tarsus_hatchdate    -0.04      0.05    -0.15     0.06 1.00     2132     1791
back_sexMale         0.01      0.07    -0.12     0.14 1.00     3858     1560
back_sexUNK          0.15      0.15    -0.15     0.44 1.00     4109     1626
back_hatchdate      -0.09      0.05    -0.19     0.01 1.00     2653     1415

Further Distributional Parameters:
             Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma_tarsus     0.75      0.02     0.72     0.79 1.00     2319     1487
sigma_back       0.90      0.02     0.85     0.95 1.01     2277     1274

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     3386     1423

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. Across dams, tarsus length and back color seem to be negatively correlated, while across 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 0.4358746 0.02306086 0.3880626 0.4780669
R2back   0.1991446 0.02717648 0.1484433 0.2526186

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 + set_rescor(TRUE), 
            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) 
  Draws: 2 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup draws = 2000

Multilevel Hyperparameters:
~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      895
sd(back_Intercept)                       0.26      0.07     0.11     0.41 1.01      347
cor(tarsus_Intercept,back_Intercept)    -0.47      0.22    -0.90    -0.02 1.00      524
                                     Tail_ESS
sd(tarsus_Intercept)                     1482
sd(back_Intercept)                        791
cor(tarsus_Intercept,back_Intercept)      791

~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      656
sd(back_Intercept)                       0.34      0.06     0.22     0.46 1.00      510
cor(tarsus_Intercept,back_Intercept)     0.67      0.21     0.19     0.98 1.01      225
                                     Tail_ESS
sd(tarsus_Intercept)                     1186
sd(back_Intercept)                        889
cor(tarsus_Intercept,back_Intercept)      391

Regression Coefficients:
                 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     1492     1513
back_Intercept       0.00      0.05    -0.10     0.11 1.00     1969     1556
tarsus_sexMale       0.77      0.06     0.65     0.89 1.00     4823     1363
tarsus_sexUNK        0.23      0.13    -0.02     0.47 1.00     3472     1378
back_hatchdate      -0.08      0.05    -0.19     0.02 1.00     2182     1567

Further Distributional 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     2609     1382
sigma_back       0.90      0.02     0.86     0.95 1.00     2688     1222

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     0.03 1.00     2870     1153

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  -2125.5 33.8
p_loo       176.6  7.6
looic      4251.0 67.6
------
MCSE of elpd_loo is NA.
MCSE and ESS estimates assume MCMC draws (r_eff in [0.5, 1.7]).

Pareto k diagnostic values:
                         Count Pct.    Min. ESS
(-Inf, 0.7]   (good)     824   99.5%   267     
   (0.7, 1]   (bad)        4    0.5%   <NA>    
   (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  -2125.3 33.6
p_loo       175.8  7.5
looic      4250.6 67.2
------
MCSE of elpd_loo is NA.
MCSE and ESS estimates assume MCMC draws (r_eff in [0.4, 1.9]).

Pareto k diagnostic values:
                         Count Pct.    Min. ESS
(-Inf, 0.7]   (good)     826   99.8%   109     
   (0.7, 1]   (bad)        2    0.2%   <NA>    
   (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 -0.2       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 <- 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.

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) 
  Draws: 2 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup draws = 2000

Smoothing Spline Hyperparameters:
                       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sds(back_shatchdate_1)     2.07      1.04     0.38     4.48 1.00      481      527

Multilevel Hyperparameters:
~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      550
sd(back_Intercept)                       0.23      0.07     0.09     0.37 1.02      220
cor(tarsus_Intercept,back_Intercept)    -0.51      0.24    -0.95    -0.04 1.01      361
                                     Tail_ESS
sd(tarsus_Intercept)                      835
sd(back_Intercept)                        615
cor(tarsus_Intercept,back_Intercept)      368

~fosternest (Number of levels: 104) 
                                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(tarsus_Intercept)                     0.26      0.05     0.15     0.38 1.01      433
sd(back_Intercept)                       0.31      0.06     0.20     0.42 1.00      473
cor(tarsus_Intercept,back_Intercept)     0.62      0.22     0.14     0.96 1.00      286
                                     Tail_ESS
sd(tarsus_Intercept)                      698
sd(back_Intercept)                        690
cor(tarsus_Intercept,back_Intercept)      585

Regression Coefficients:
                     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      873     1295
back_Intercept           0.00      0.05    -0.10     0.10 1.00     1140     1341
tarsus_sexMale           0.77      0.06     0.66     0.88 1.00     2629     1726
tarsus_sexUNK            0.21      0.12    -0.02     0.45 1.00     1969     1389
sigma_tarsus_sexFem     -0.30      0.04    -0.38    -0.22 1.00     1761     1672
sigma_tarsus_sexMale    -0.25      0.04    -0.32    -0.16 1.00     1810     1535
sigma_tarsus_sexUNK     -0.40      0.13    -0.64    -0.13 1.00     1718     1476
back_shatchdate_1       -0.15      3.34    -6.30     7.17 1.00      798      746

Further Distributional 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     1768     1355
alpha_tarsus    -1.23      0.42    -1.86    -0.03 1.00     1356      559

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.

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.