Estimating Multivariate Models with brms

Paul Bürkner

2023-09-25

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)                     0.48      0.05     0.39     0.58 1.00      908
sd(back_Intercept)                       0.24      0.07     0.10     0.38 1.01      333
cor(tarsus_Intercept,back_Intercept)    -0.52      0.23    -0.92    -0.06 1.00      499
                                     Tail_ESS
sd(tarsus_Intercept)                     1662
sd(back_Intercept)                        571
cor(tarsus_Intercept,back_Intercept)      566

~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      753
sd(back_Intercept)                       0.35      0.06     0.24     0.46 1.00      601
cor(tarsus_Intercept,back_Intercept)     0.69      0.20     0.22     0.98 1.03      257
                                     Tail_ESS
sd(tarsus_Intercept)                     1286
sd(back_Intercept)                       1212
cor(tarsus_Intercept,back_Intercept)      582

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     1646     1420
back_Intercept      -0.01      0.07    -0.15     0.11 1.00     3194     1546
tarsus_sexMale       0.77      0.06     0.66     0.88 1.00     3994     1428
tarsus_sexUNK        0.23      0.13    -0.03     0.47 1.00     4522     1762
tarsus_hatchdate    -0.04      0.06    -0.16     0.07 1.00     1756     1582
back_sexMale         0.01      0.07    -0.12     0.14 1.01     4563     1487
back_sexUNK          0.15      0.15    -0.16     0.44 1.00     3878     1375
back_hatchdate      -0.09      0.05    -0.19     0.01 1.00     2719     1631

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     3043     1382
sigma_back       0.90      0.02     0.86     0.95 1.00     2481     1515

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     3473     1379

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 0.4349094 0.02250483 0.3884970 0.4760042
R2back   0.1981263 0.02842267 0.1431563 0.2548206

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     0.39     0.59 1.00      827
sd(back_Intercept)                       0.25      0.07     0.10     0.38 1.01      307
cor(tarsus_Intercept,back_Intercept)    -0.50      0.22    -0.92    -0.08 1.00      554
                                     Tail_ESS
sd(tarsus_Intercept)                     1196
sd(back_Intercept)                        536
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)                     0.26      0.05     0.16     0.37 1.00      603
sd(back_Intercept)                       0.35      0.06     0.23     0.47 1.00      349
cor(tarsus_Intercept,back_Intercept)     0.66      0.21     0.19     0.97 1.00      228
                                     Tail_ESS
sd(tarsus_Intercept)                      812
sd(back_Intercept)                        954
cor(tarsus_Intercept,back_Intercept)      581

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     1334     1317
back_Intercept       0.00      0.06    -0.11     0.11 1.00     1769     1450
tarsus_sexMale       0.77      0.06     0.65     0.89 1.00     2775     1434
tarsus_sexUNK        0.22      0.13    -0.03     0.47 1.00     3137     1668
back_hatchdate      -0.08      0.05    -0.18     0.02 1.00     1816     1315

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     1812      848
sigma_back       0.90      0.02     0.86     0.95 1.00     1938     1509

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     2589     1656

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  -2126.6 33.6
p_loo       176.5  7.4
looic      4253.2 67.2
------
Monte Carlo SE of elpd_loo is NA.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     810   97.8%   206       
 (0.5, 0.7]   (ok)        17    2.1%   78        
   (0.7, 1]   (bad)        1    0.1%   65        
   (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  -2125.4 33.6
p_loo       174.6  7.4
looic      4250.7 67.2
------
Monte Carlo SE of elpd_loo is NA.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     804   97.1%   180       
 (0.5, 0.7]   (ok)        22    2.7%   98        
   (0.7, 1]   (bad)        2    0.2%   45        
   (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 -1.3       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)     1.98      1.03     0.36     4.31 1.00      553      496

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      774
sd(back_Intercept)                       0.23      0.07     0.10     0.37 1.01      256
cor(tarsus_Intercept,back_Intercept)    -0.54      0.23    -0.96    -0.08 1.01      256
                                     Tail_ESS
sd(tarsus_Intercept)                     1167
sd(back_Intercept)                        591
cor(tarsus_Intercept,back_Intercept)      218

~fosternest (Number of levels: 104) 
                                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(tarsus_Intercept)                     0.26      0.06     0.14     0.37 1.01      374
sd(back_Intercept)                       0.31      0.06     0.20     0.43 1.00      500
cor(tarsus_Intercept,back_Intercept)     0.65      0.22     0.17     0.97 1.01      271
                                     Tail_ESS
sd(tarsus_Intercept)                      717
sd(back_Intercept)                        901
cor(tarsus_Intercept,back_Intercept)      486

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      842     1422
back_Intercept           0.00      0.05    -0.10     0.10 1.00     1270     1572
tarsus_sexMale           0.77      0.05     0.66     0.87 1.00     3045     1141
tarsus_sexUNK            0.21      0.12    -0.02     0.44 1.00     2731     1746
sigma_tarsus_sexFem     -0.30      0.04    -0.38    -0.22 1.00     2929     1561
sigma_tarsus_sexMale    -0.24      0.04    -0.32    -0.17 1.00     2338     1622
sigma_tarsus_sexUNK     -0.39      0.13    -0.64    -0.14 1.00     2202     1560
back_shatchdate_1       -0.16      3.18    -5.64     6.81 1.00      897     1036

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     1674     1801
alpha_tarsus    -1.22      0.43    -1.87     0.05 1.00     1148      481

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.