Estimating Multivariate Models with brms

Paul Bürkner

2021-08-18

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) 
  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      923
sd(back_Intercept)                       0.25      0.08     0.10     0.39 1.01      358
cor(tarsus_Intercept,back_Intercept)    -0.53      0.22    -0.94    -0.08 1.00      589
                                     Tail_ESS
sd(tarsus_Intercept)                     1249
sd(back_Intercept)                        514
cor(tarsus_Intercept,back_Intercept)      696

~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      701
sd(back_Intercept)                       0.35      0.06     0.23     0.46 1.00      591
cor(tarsus_Intercept,back_Intercept)     0.69      0.21     0.23     0.98 1.00      239
                                     Tail_ESS
sd(tarsus_Intercept)                     1086
sd(back_Intercept)                        972
cor(tarsus_Intercept,back_Intercept)      647

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     1685     1467
back_Intercept      -0.01      0.07    -0.15     0.12 1.00     1993     1351
tarsus_sexMale       0.77      0.06     0.66     0.88 1.00     3497     1384
tarsus_sexUNK        0.23      0.13    -0.03     0.48 1.00     3683     1508
tarsus_hatchdate    -0.04      0.06    -0.17     0.08 1.00     1351     1415
back_sexMale         0.01      0.07    -0.12     0.14 1.00     3145     1358
back_sexUNK          0.15      0.15    -0.14     0.43 1.00     2773     1707
back_hatchdate      -0.09      0.05    -0.19     0.01 1.00     2064     1779

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     2749     1388
sigma_back       0.90      0.02     0.85     0.95 1.00     2144     1665

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     2003     1639

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 0.4343672 0.02389080 0.3873917 0.4808309
R2back   0.1985510 0.02812847 0.1428650 0.2538008

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) 
  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      836
sd(back_Intercept)                       0.25      0.08     0.09     0.39 1.01      381
cor(tarsus_Intercept,back_Intercept)    -0.51      0.23    -0.92    -0.05 1.00      572
                                     Tail_ESS
sd(tarsus_Intercept)                      978
sd(back_Intercept)                        701
cor(tarsus_Intercept,back_Intercept)      798

~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      696
sd(back_Intercept)                       0.35      0.06     0.23     0.46 1.00      598
cor(tarsus_Intercept,back_Intercept)     0.67      0.22     0.17     0.98 1.01      340
                                     Tail_ESS
sd(tarsus_Intercept)                     1097
sd(back_Intercept)                        944
cor(tarsus_Intercept,back_Intercept)      665

Population-Level Effects: 
                 Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
tarsus_Intercept    -0.41      0.07    -0.56    -0.27 1.00     1930     1525
back_Intercept       0.00      0.05    -0.10     0.11 1.00     2430     1192
tarsus_sexMale       0.77      0.06     0.65     0.89 1.01     4417     1154
tarsus_sexUNK        0.22      0.13    -0.03     0.48 1.00     3913     1585
back_hatchdate      -0.08      0.05    -0.19     0.02 1.00     3031     1601

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     0.79 1.00     2059     1691
sigma_back       0.90      0.02     0.85     0.95 1.00     3006     1203

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     4169     1576

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  -2126.8 33.6
p_loo       176.3  7.3
looic      4253.5 67.1
------
Monte Carlo SE of elpd_loo is NA.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     816   98.6%   293       
 (0.5, 0.7]   (ok)        11    1.3%   88        
   (0.7, 1]   (bad)        1    0.1%   36        
   (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  -2125.0 33.5
p_loo       173.8  7.2
looic      4250.0 67.0
------
Monte Carlo SE of elpd_loo is NA.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     817   98.7%   273       
 (0.5, 0.7]   (ok)        10    1.2%   65        
   (0.7, 1]   (bad)        1    0.1%   146       
   (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 -1.8       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)     2.12      1.10     0.29     4.79 1.00      359      337

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      630
sd(back_Intercept)                       0.24      0.07     0.11     0.38 1.00      301
cor(tarsus_Intercept,back_Intercept)    -0.53      0.22    -0.93    -0.08 1.00      365
                                     Tail_ESS
sd(tarsus_Intercept)                     1134
sd(back_Intercept)                        738
cor(tarsus_Intercept,back_Intercept)      493

~fosternest (Number of levels: 104) 
                                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(tarsus_Intercept)                     0.26      0.06     0.16     0.38 1.00      467
sd(back_Intercept)                       0.31      0.06     0.19     0.42 1.00      445
cor(tarsus_Intercept,back_Intercept)     0.65      0.23     0.13     0.98 1.00      195
                                     Tail_ESS
sd(tarsus_Intercept)                      522
sd(back_Intercept)                        744
cor(tarsus_Intercept,back_Intercept)      416

Population-Level Effects: 
                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
tarsus_Intercept        -0.42      0.07    -0.56    -0.28 1.00      968     1320
back_Intercept           0.00      0.05    -0.09     0.10 1.00      987     1392
tarsus_sexMale           0.77      0.06     0.65     0.89 1.00     2467     1411
tarsus_sexUNK            0.22      0.12    -0.03     0.46 1.00     2132     1398
sigma_tarsus_sexFem     -0.30      0.04    -0.39    -0.22 1.00     2010     1638
sigma_tarsus_sexMale    -0.25      0.04    -0.32    -0.17 1.00     2287     1716
sigma_tarsus_sexUNK     -0.39      0.12    -0.62    -0.14 1.00     1345     1637
back_shatchdate_1       -0.18      3.27    -5.72     7.18 1.00      939      972

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     1873     1453
alpha_tarsus    -1.24      0.39    -1.89    -0.27 1.00     1472      737

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.