Estimating Multivariate Models with brms

Paul Bürkner

2023-09-15

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

Group-Level Effects: 
~dam (Number of levels: 106) 
                                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(tarsus_Intercept)                     0.49      0.05     0.40     0.59 1.00      728
sd(back_Intercept)                       0.25      0.07     0.11     0.39 1.01      395
cor(tarsus_Intercept,back_Intercept)    -0.51      0.22    -0.93    -0.07 1.00      679
                                     Tail_ESS
sd(tarsus_Intercept)                     1088
sd(back_Intercept)                        801
cor(tarsus_Intercept,back_Intercept)      637

~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      812
sd(back_Intercept)                       0.35      0.06     0.24     0.47 1.01      491
cor(tarsus_Intercept,back_Intercept)     0.70      0.20     0.24     0.98 1.01      365
                                     Tail_ESS
sd(tarsus_Intercept)                     1088
sd(back_Intercept)                        937
cor(tarsus_Intercept,back_Intercept)      784

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     2006     1476
back_Intercept      -0.01      0.06    -0.14     0.11 1.00     2612     1575
tarsus_sexMale       0.77      0.06     0.66     0.88 1.00     3822     1525
tarsus_sexUNK        0.23      0.13    -0.01     0.47 1.00     3271     1582
tarsus_hatchdate    -0.05      0.06    -0.16     0.07 1.00     1447     1363
back_sexMale         0.01      0.07    -0.12     0.14 1.00     3510     1336
back_sexUNK          0.15      0.15    -0.14     0.44 1.00     4016     1625
back_hatchdate      -0.09      0.05    -0.19     0.02 1.00     2120     1358

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     2430     1623
sigma_back       0.90      0.02     0.86     0.95 1.00     2830     1553

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     3485     1684

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 0.4355111 0.02317053 0.3889295 0.4789890
R2back   0.1987557 0.02765139 0.1450809 0.2540395

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

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     0.40     0.59 1.00      946
sd(back_Intercept)                       0.25      0.08     0.09     0.39 1.01      288
cor(tarsus_Intercept,back_Intercept)    -0.50      0.24    -0.94    -0.02 1.01      647
                                     Tail_ESS
sd(tarsus_Intercept)                     1353
sd(back_Intercept)                        446
cor(tarsus_Intercept,back_Intercept)      732

~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     0.37 1.00      735
sd(back_Intercept)                       0.35      0.06     0.23     0.46 1.00      455
cor(tarsus_Intercept,back_Intercept)     0.67      0.21     0.18     0.98 1.01      281
                                     Tail_ESS
sd(tarsus_Intercept)                     1164
sd(back_Intercept)                       1268
cor(tarsus_Intercept,back_Intercept)      571

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     1631     1337
back_Intercept       0.00      0.05    -0.11     0.11 1.00     2899     1505
tarsus_sexMale       0.77      0.06     0.65     0.88 1.00     3834     1542
tarsus_sexUNK        0.23      0.12    -0.01     0.48 1.00     3774     1629
back_hatchdate      -0.09      0.05    -0.19     0.02 1.00     2619     1644

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     2256     1328
sigma_back       0.90      0.02     0.85     0.95 1.00     2150     1350

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     2645     1588

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  -2124.4 33.4
p_loo       174.6  7.2
looic      4248.7 66.9
------
Monte Carlo SE of elpd_loo is NA.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     812   98.1%   316       
 (0.5, 0.7]   (ok)        15    1.8%   85        
   (0.7, 1]   (bad)        1    0.1%   58        
   (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  -2124.4 33.6
p_loo       173.9  7.3
looic      4248.9 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%   111       
 (0.5, 0.7]   (ok)        14    1.7%   109       
   (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.

Model comparisons:
     elpd_diff se_diff
fit1  0.0       0.0   
fit2 -0.1       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.

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

Smooth Terms: 
                       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sds(back_shatchdate_1)     1.96      1.00     0.40     4.21 1.00      664      836

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     1083
sd(back_Intercept)                       0.24      0.07     0.11     0.37 1.00      409
cor(tarsus_Intercept,back_Intercept)    -0.51      0.22    -0.92    -0.09 1.00      681
                                     Tail_ESS
sd(tarsus_Intercept)                     1251
sd(back_Intercept)                        772
cor(tarsus_Intercept,back_Intercept)      891

~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     0.36 1.00      589
sd(back_Intercept)                       0.31      0.06     0.19     0.43 1.00      622
cor(tarsus_Intercept,back_Intercept)     0.65      0.21     0.17     0.97 1.01      325
                                     Tail_ESS
sd(tarsus_Intercept)                     1038
sd(back_Intercept)                       1045
cor(tarsus_Intercept,back_Intercept)      640

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    -0.29 1.00     1748     1778
back_Intercept           0.00      0.05    -0.10     0.10 1.00     2676     1542
tarsus_sexMale           0.77      0.06     0.66     0.89 1.00     4276     1578
tarsus_sexUNK            0.22      0.12    -0.02     0.45 1.00     4078     1615
sigma_tarsus_sexFem     -0.30      0.04    -0.38    -0.22 1.00     2645     1794
sigma_tarsus_sexMale    -0.24      0.04    -0.32    -0.16 1.00     2755     1745
sigma_tarsus_sexUNK     -0.39      0.13    -0.63    -0.15 1.00     3251     1694
back_shatchdate_1       -0.27      3.10    -5.79     6.57 1.00     1483     1603

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     3088     1128
alpha_tarsus    -1.21      0.44    -1.86     0.05 1.00     1701      865

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.