Estimating Multivariate Models with brms

Paul Bürkner

2022-09-19

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      778
sd(back_Intercept)                       0.25      0.07     0.10     0.39 1.01      321
cor(tarsus_Intercept,back_Intercept)    -0.53      0.22    -0.94    -0.09 1.01      402
                                     Tail_ESS
sd(tarsus_Intercept)                     1083
sd(back_Intercept)                        798
cor(tarsus_Intercept,back_Intercept)      580

~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     0.37 1.00      660
sd(back_Intercept)                       0.35      0.06     0.23     0.47 1.00      663
cor(tarsus_Intercept,back_Intercept)     0.71      0.20     0.22     0.99 1.01      260
                                     Tail_ESS
sd(tarsus_Intercept)                     1054
sd(back_Intercept)                        754
cor(tarsus_Intercept,back_Intercept)      502

Population-Level Effects: 
                 Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
tarsus_Intercept    -0.40      0.07    -0.55    -0.27 1.00     1156     1279
back_Intercept      -0.01      0.06    -0.14     0.11 1.00     2090     1653
tarsus_sexMale       0.77      0.06     0.66     0.88 1.00     4043     1646
tarsus_sexUNK        0.23      0.12    -0.01     0.48 1.00     3590     1358
tarsus_hatchdate    -0.04      0.06    -0.15     0.07 1.00      975     1300
back_sexMale         0.01      0.07    -0.13     0.14 1.00     3523     1648
back_sexUNK          0.15      0.15    -0.14     0.46 1.00     2903     1438
back_hatchdate      -0.09      0.05    -0.20     0.01 1.00     1980     1679

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     2452     1760
sigma_back       0.90      0.02     0.85     0.95 1.00     2373     1434

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     2367     1433

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 0.4335935 0.02434887 0.3824507 0.4776490
R2back   0.1994719 0.02749108 0.1432571 0.2502431

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      925
sd(back_Intercept)                       0.25      0.07     0.09     0.38 1.00      416
cor(tarsus_Intercept,back_Intercept)    -0.50      0.22    -0.92    -0.08 1.00      729
                                     Tail_ESS
sd(tarsus_Intercept)                     1144
sd(back_Intercept)                        581
cor(tarsus_Intercept,back_Intercept)      744

~fosternest (Number of levels: 104) 
                                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(tarsus_Intercept)                     0.27      0.06     0.16     0.38 1.00      531
sd(back_Intercept)                       0.35      0.06     0.23     0.47 1.00      602
cor(tarsus_Intercept,back_Intercept)     0.69      0.20     0.24     0.98 1.01      289
                                     Tail_ESS
sd(tarsus_Intercept)                     1139
sd(back_Intercept)                       1090
cor(tarsus_Intercept,back_Intercept)      625

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    -0.28 1.00     1498     1507
back_Intercept      -0.00      0.05    -0.11     0.11 1.00     2394     1500
tarsus_sexMale       0.77      0.06     0.66     0.87 1.00     4223     1612
tarsus_sexUNK        0.23      0.13    -0.03     0.48 1.00     4999     1689
back_hatchdate      -0.09      0.05    -0.19     0.01 1.00     2449     1561

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     3084     1630
sigma_back       0.90      0.02     0.85     0.95 1.00     2558     1630

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     2745     1506

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  -2126.5 33.6
p_loo       175.9  7.4
looic      4253.0 67.1
------
Monte Carlo SE of elpd_loo is NA.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     815   98.4%   271       
 (0.5, 0.7]   (ok)        10    1.2%   224       
   (0.7, 1]   (bad)        3    0.4%   39        
   (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  -2125.1 33.8
p_loo       175.2  7.5
looic      4250.3 67.6
------
Monte Carlo SE of elpd_loo is NA.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     802   96.9%   215       
 (0.5, 0.7]   (ok)        24    2.9%   103       
   (0.7, 1]   (bad)        2    0.2%   21        
   (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 -1.4       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

Smooth Terms: 
                       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sds(back_shatchdate_1)     1.95      1.08     0.21     4.44 1.01      363      270

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      487
sd(back_Intercept)                       0.24      0.07     0.10     0.37 1.01      333
cor(tarsus_Intercept,back_Intercept)    -0.52      0.23    -0.92    -0.07 1.01      424
                                     Tail_ESS
sd(tarsus_Intercept)                     1102
sd(back_Intercept)                        477
cor(tarsus_Intercept,back_Intercept)      542

~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     0.36 1.00      429
sd(back_Intercept)                       0.31      0.06     0.20     0.43 1.01      364
cor(tarsus_Intercept,back_Intercept)     0.65      0.22     0.15     0.98 1.00      192
                                     Tail_ESS
sd(tarsus_Intercept)                      849
sd(back_Intercept)                        796
cor(tarsus_Intercept,back_Intercept)      382

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    -0.28 1.00      657      992
back_Intercept           0.00      0.05    -0.10     0.10 1.00      748      948
tarsus_sexMale           0.77      0.06     0.66     0.88 1.00     2156     1664
tarsus_sexUNK            0.22      0.12    -0.02     0.44 1.00     1260     1540
sigma_tarsus_sexFem     -0.30      0.04    -0.38    -0.22 1.00     1926     1528
sigma_tarsus_sexMale    -0.24      0.04    -0.32    -0.17 1.00     1672     1556
sigma_tarsus_sexUNK     -0.39      0.13    -0.64    -0.15 1.00     1434     1437
back_shatchdate_1       -0.06      3.30    -5.67     7.56 1.00      635      710

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     1581     1630
alpha_tarsus    -1.26      0.37    -1.88    -0.33 1.00     1306      810

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.