Estimating Multivariate Models with brms

Paul Bürkner

2024-03-19

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

Multilevel Hyperparameters:
~dam (Number of levels: 106) 
                                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
sd(tarsus_Intercept)                     0.49      0.05     0.39     0.59 1.00      990
sd(back_Intercept)                       0.25      0.07     0.10     0.39 1.01      520
cor(tarsus_Intercept,back_Intercept)    -0.51      0.21    -0.91    -0.08 1.00      696
                                     Tail_ESS
sd(tarsus_Intercept)                     1327
sd(back_Intercept)                        683
cor(tarsus_Intercept,back_Intercept)      876

~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 1.01      842
sd(back_Intercept)                       0.35      0.06     0.24     0.46 1.00      655
cor(tarsus_Intercept,back_Intercept)     0.71      0.20     0.25     0.99 1.00      271
                                     Tail_ESS
sd(tarsus_Intercept)                     1128
sd(back_Intercept)                       1256
cor(tarsus_Intercept,back_Intercept)      505

Regression Coefficients:
                 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     1762     1720
back_Intercept      -0.01      0.06    -0.13     0.11 1.00     3093     1676
tarsus_sexMale       0.77      0.06     0.66     0.88 1.00     3676     1450
tarsus_sexUNK        0.23      0.13    -0.03     0.49 1.00     4148     1472
tarsus_hatchdate    -0.04      0.06    -0.16     0.07 1.00     1493     1599
back_sexMale         0.01      0.07    -0.12     0.14 1.00     3994     1662
back_sexUNK          0.15      0.15    -0.15     0.45 1.00     3393     1772
back_hatchdate      -0.09      0.05    -0.19     0.01 1.00     2315     1746

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     2590     1583
sigma_back       0.90      0.03     0.86     0.95 1.00     3066     1576

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     2367     1491

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 0.4349759 0.02335079 0.3852365 0.4780449
R2back   0.1999334 0.02743501 0.1444006 0.2507715

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      623
sd(back_Intercept)                       0.25      0.07     0.11     0.39 1.01      339
cor(tarsus_Intercept,back_Intercept)    -0.49      0.23    -0.94    -0.04 1.01      343
                                     Tail_ESS
sd(tarsus_Intercept)                     1065
sd(back_Intercept)                        856
cor(tarsus_Intercept,back_Intercept)      439

~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 1.01      603
sd(back_Intercept)                       0.34      0.06     0.22     0.46 1.01      475
cor(tarsus_Intercept,back_Intercept)     0.66      0.21     0.21     0.98 1.01      205
                                     Tail_ESS
sd(tarsus_Intercept)                     1019
sd(back_Intercept)                       1064
cor(tarsus_Intercept,back_Intercept)      313

Regression Coefficients:
                 Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
tarsus_Intercept    -0.41      0.07    -0.54    -0.28 1.00      982     1116
back_Intercept       0.00      0.05    -0.10     0.10 1.00     1364     1291
tarsus_sexMale       0.77      0.06     0.66     0.88 1.00     2445     1291
tarsus_sexUNK        0.23      0.13    -0.03     0.48 1.00     2276     1607
back_hatchdate      -0.08      0.05    -0.19     0.02 1.00     1263     1527

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     1862     1583
sigma_back       0.90      0.03     0.85     0.95 1.00     2497     1220

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     1939     1486

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  -2125.3 33.5
p_loo       175.5  7.3
looic      4250.6 67.0
------
MCSE of elpd_loo is NA.
MCSE and ESS estimates assume MCMC draws (r_eff in [0.4, 2.0]).

Pareto k diagnostic values:
                         Count Pct.    Min. ESS
(-Inf, 0.7]   (good)     827   99.9%   102     
   (0.7, 1]   (bad)        1    0.1%   <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  -2125.9 33.7
p_loo       175.6  7.5
looic      4251.8 67.5
------
MCSE of elpd_loo is NA.
MCSE and ESS estimates assume MCMC draws (r_eff in [0.4, 1.7]).

Pareto k diagnostic values:
                         Count Pct.    Min. ESS
(-Inf, 0.7]   (good)     824   99.5%   135     
   (0.7, 1]   (bad)        4    0.5%   <NA>    
   (1, Inf)   (very bad)   0    0.0%   <NA>    
See help('pareto-k-diagnostic') for details.

Model comparisons:
     elpd_diff se_diff
fit1  0.0       0.0   
fit2 -0.6       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

Smoothing Spline Hyperparameters:
                       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sds(back_shatchdate_1)     1.94      1.06     0.21     4.40 1.00      359      402

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      628
sd(back_Intercept)                       0.23      0.07     0.09     0.38 1.01      312
cor(tarsus_Intercept,back_Intercept)    -0.54      0.22    -0.95    -0.09 1.01      414
                                     Tail_ESS
sd(tarsus_Intercept)                     1053
sd(back_Intercept)                        666
cor(tarsus_Intercept,back_Intercept)      646

~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      554
sd(back_Intercept)                       0.32      0.06     0.20     0.43 1.00      515
cor(tarsus_Intercept,back_Intercept)     0.67      0.22     0.16     0.98 1.04      147
                                     Tail_ESS
sd(tarsus_Intercept)                      961
sd(back_Intercept)                        879
cor(tarsus_Intercept,back_Intercept)      253

Regression Coefficients:
                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
tarsus_Intercept        -0.41      0.07    -0.54    -0.28 1.00      882     1217
back_Intercept           0.00      0.05    -0.10     0.11 1.00     1234     1355
tarsus_sexMale           0.77      0.06     0.66     0.88 1.00     2952     1532
tarsus_sexUNK            0.22      0.12    -0.02     0.44 1.00     2403     1594
sigma_tarsus_sexFem     -0.30      0.04    -0.38    -0.22 1.00     2462     1539
sigma_tarsus_sexMale    -0.25      0.04    -0.33    -0.16 1.00     2611     1479
sigma_tarsus_sexUNK     -0.40      0.13    -0.65    -0.15 1.00     1906     1635
back_shatchdate_1       -0.24      3.07    -6.02     6.72 1.00     1074     1215

Further Distributional Parameters:
             Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma_back       0.90      0.03     0.85     0.95 1.01     2505     1438
alpha_tarsus    -1.22      0.45    -1.87     0.12 1.00     1297      446

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.