Estimating Multivariate Models with brms

Paul Bürkner

2020-02-21

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")). Similarily, 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) 
Samples: 2 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup samples = 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     0.39     0.60 1.00      719
sd(back_Intercept)                       0.25      0.07     0.10     0.39 1.01      243
cor(tarsus_Intercept,back_Intercept)    -0.52      0.23    -0.95    -0.04 1.01      311
                                     Tail_ESS
sd(tarsus_Intercept)                     1066
sd(back_Intercept)                        537
cor(tarsus_Intercept,back_Intercept)      304

~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 1.01      487
sd(back_Intercept)                       0.35      0.06     0.23     0.46 1.01      357
cor(tarsus_Intercept,back_Intercept)     0.71      0.21     0.25     0.99 1.02      135
                                     Tail_ESS
sd(tarsus_Intercept)                      664
sd(back_Intercept)                        756
cor(tarsus_Intercept,back_Intercept)      328

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     1070     1390
back_Intercept      -0.01      0.07    -0.14     0.12 1.00     1495     1376
tarsus_sexMale       0.77      0.06     0.66     0.88 1.00     2241     1366
tarsus_sexUNK        0.23      0.13    -0.03     0.49 1.00     2491     1218
tarsus_hatchdate    -0.04      0.06    -0.15     0.07 1.00      725      903
back_sexMale         0.01      0.07    -0.13     0.14 1.00     3309     1434
back_sexUNK          0.14      0.15    -0.13     0.44 1.00     2010     1582
back_hatchdate      -0.09      0.05    -0.20     0.01 1.00     1046     1515

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     2177     1501
sigma_back       0.90      0.02     0.86     0.95 1.00     2103     1559

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     1954     1513

Samples were drawn 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 comparions. 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 0.4344939 0.02346996 0.3882848 0.4776280
R2back   0.1996353 0.02724460 0.1465235 0.2531614

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) 
Samples: 2 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup samples = 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 1.01      881
sd(back_Intercept)                       0.25      0.07     0.11     0.39 1.00      237
cor(tarsus_Intercept,back_Intercept)    -0.50      0.22    -0.94    -0.06 1.00      496
                                     Tail_ESS
sd(tarsus_Intercept)                     1060
sd(back_Intercept)                        823
cor(tarsus_Intercept,back_Intercept)      460

~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 1.00      747
sd(back_Intercept)                       0.35      0.06     0.23     0.46 1.00      437
cor(tarsus_Intercept,back_Intercept)     0.68      0.20     0.24     0.97 1.00      311
                                     Tail_ESS
sd(tarsus_Intercept)                     1023
sd(back_Intercept)                       1247
cor(tarsus_Intercept,back_Intercept)      737

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     1441     1365
back_Intercept       0.00      0.05    -0.10     0.10 1.00     1937     1650
tarsus_sexMale       0.77      0.06     0.66     0.88 1.01     3238     1729
tarsus_sexUNK        0.23      0.13    -0.03     0.47 1.00     2806     1794
back_hatchdate      -0.08      0.05    -0.18     0.01 1.00     2208     1659

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     1508
sigma_back       0.90      0.03     0.85     0.95 1.00     2744     1480

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     3066     1229

Samples were drawn 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  -2126.9 33.8
p_loo       176.8  7.7
looic      4253.8 67.5
------
Monte Carlo SE of elpd_loo is NA.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     812   98.1%   156       
 (0.5, 0.7]   (ok)        14    1.7%   96        
   (0.7, 1]   (bad)        2    0.2%   24        
   (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  -2123.9 33.6
p_loo       173.7  7.3
looic      4247.9 67.1
------
Monte Carlo SE of elpd_loo is NA.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     818   98.8%   123       
 (0.5, 0.7]   (ok)         9    1.1%   78        
   (0.7, 1]   (bad)        1    0.1%   185       
   (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 -3.0       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 males and femals 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) 
Samples: 2 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup samples = 2000

Smooth Terms: 
                       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sds(back_shatchdate_1)     2.56      1.49     0.43     5.86 1.01      262      341

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      572
sd(back_Intercept)                       0.23      0.07     0.09     0.36 1.00      289
cor(tarsus_Intercept,back_Intercept)    -0.54      0.23    -0.96    -0.08 1.01      378
                                     Tail_ESS
sd(tarsus_Intercept)                      878
sd(back_Intercept)                        441
cor(tarsus_Intercept,back_Intercept)      443

~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     0.37 1.00      384
sd(back_Intercept)                       0.31      0.06     0.20     0.42 1.00      494
cor(tarsus_Intercept,back_Intercept)     0.65      0.23     0.13     0.97 1.00      208
                                     Tail_ESS
sd(tarsus_Intercept)                      512
sd(back_Intercept)                        789
cor(tarsus_Intercept,back_Intercept)      570

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 1.01      639     1093
back_Intercept          -0.00      0.05    -0.10     0.10 1.00     1145     1426
tarsus_sexMale           0.77      0.06     0.66     0.89 1.00     1897     1409
tarsus_sexUNK            0.22      0.12    -0.02     0.46 1.00     1223     1038
sigma_tarsus_sexFem     -0.30      0.04    -0.38    -0.22 1.00     1890     1278
sigma_tarsus_sexMale    -0.25      0.04    -0.32    -0.17 1.00     1664     1464
sigma_tarsus_sexUNK     -0.39      0.13    -0.63    -0.13 1.00     1350     1353
back_shatchdate_1        0.35      3.66    -5.87     8.43 1.00      514      652

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     1665     1510
alpha_tarsus    -1.21      0.45    -1.89     0.13 1.00      660      278

Samples were drawn 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.