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).
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
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
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) 0.48 0.05 0.40 0.58 1.00 1011
sd(back_Intercept) 0.25 0.08 0.10 0.40 1.01 301
cor(tarsus_Intercept,back_Intercept) -0.53 0.22 -0.94 -0.09 1.01 457
Tail_ESS
sd(tarsus_Intercept) 1276
sd(back_Intercept) 564
cor(tarsus_Intercept,back_Intercept) 405
~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 1.00 674
sd(back_Intercept) 0.35 0.06 0.23 0.47 1.01 480
cor(tarsus_Intercept,back_Intercept) 0.71 0.20 0.23 0.98 1.00 303
Tail_ESS
sd(tarsus_Intercept) 843
sd(back_Intercept) 873
cor(tarsus_Intercept,back_Intercept) 524
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 1586 1491
back_Intercept -0.01 0.06 -0.14 0.11 1.01 2513 1758
tarsus_sexMale 0.77 0.06 0.66 0.88 1.00 3288 1349
tarsus_sexUNK 0.23 0.13 -0.02 0.48 1.00 3744 1637
tarsus_hatchdate -0.04 0.06 -0.15 0.08 1.00 1783 1639
back_sexMale 0.01 0.07 -0.12 0.15 1.00 4621 1461
back_sexUNK 0.15 0.16 -0.16 0.46 1.00 3334 1238
back_hatchdate -0.09 0.05 -0.20 0.01 1.00 2092 1520
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 2903 1636
sigma_back 0.90 0.02 0.85 0.95 1.00 2743 1457
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 2931 1386
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 comparisons. Next, let’s take a look at some posterior-predictive checks, which give us a first impression of the model fit.
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.
Estimate Est.Error Q2.5 Q97.5
R2tarsus 0.4343804 0.02320407 0.3885923 0.4799999
R2back 0.1999615 0.02785385 0.1438462 0.2520180
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.
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.
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 0.40 0.58 1.00 793
sd(back_Intercept) 0.25 0.07 0.12 0.39 1.00 359
cor(tarsus_Intercept,back_Intercept) -0.50 0.22 -0.91 -0.06 1.01 537
Tail_ESS
sd(tarsus_Intercept) 1260
sd(back_Intercept) 1040
cor(tarsus_Intercept,back_Intercept) 1002
~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 1.00 771
sd(back_Intercept) 0.35 0.06 0.23 0.47 1.00 637
cor(tarsus_Intercept,back_Intercept) 0.69 0.20 0.23 0.98 1.00 348
Tail_ESS
sd(tarsus_Intercept) 1273
sd(back_Intercept) 918
cor(tarsus_Intercept,back_Intercept) 802
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 1504 1366
back_Intercept -0.00 0.05 -0.10 0.11 1.00 2334 1649
tarsus_sexMale 0.77 0.06 0.66 0.88 1.00 4323 1494
tarsus_sexUNK 0.23 0.13 -0.03 0.47 1.00 4023 1508
back_hatchdate -0.08 0.05 -0.19 0.02 1.00 2386 1440
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma_tarsus 0.75 0.02 0.72 0.80 1.00 3575 1662
sigma_back 0.90 0.02 0.85 0.95 1.00 2484 1306
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 2890 1541
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:
Output of model 'fit1':
Computed from 2000 by 828 log-likelihood matrix
Estimate SE
elpd_loo -2125.8 33.8
p_loo 175.9 7.6
looic 4251.6 67.6
------
Monte Carlo SE of elpd_loo is NA.
Pareto k diagnostic values:
Count Pct. Min. n_eff
(-Inf, 0.5] (good) 816 98.6% 309
(0.5, 0.7] (ok) 10 1.2% 142
(0.7, 1] (bad) 2 0.2% 17
(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 -2123.8 33.7
p_loo 173.7 7.3
looic 4247.7 67.4
------
Monte Carlo SE of elpd_loo is NA.
Pareto k diagnostic values:
Count Pct. Min. n_eff
(-Inf, 0.5] (good) 811 97.9% 313
(0.5, 0.7] (ok) 14 1.7% 199
(0.7, 1] (bad) 3 0.4% 18
(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 -1.9 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.
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) 2.04 1.07 0.29 4.44 1.00 326 375
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 1.00 537
sd(back_Intercept) 0.23 0.07 0.08 0.37 1.01 176
cor(tarsus_Intercept,back_Intercept) -0.51 0.23 -0.93 -0.04 1.00 321
Tail_ESS
sd(tarsus_Intercept) 940
sd(back_Intercept) 172
cor(tarsus_Intercept,back_Intercept) 307
~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 1.00 274
sd(back_Intercept) 0.31 0.06 0.20 0.43 1.00 388
cor(tarsus_Intercept,back_Intercept) 0.64 0.24 0.10 0.99 1.01 124
Tail_ESS
sd(tarsus_Intercept) 314
sd(back_Intercept) 891
cor(tarsus_Intercept,back_Intercept) 132
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 1.00 793 1121
back_Intercept 0.01 0.05 -0.10 0.10 1.00 1131 1247
tarsus_sexMale 0.77 0.06 0.65 0.89 1.00 2376 1489
tarsus_sexUNK 0.21 0.12 -0.03 0.45 1.00 1763 1675
sigma_tarsus_sexFem -0.30 0.04 -0.38 -0.22 1.00 1809 1424
sigma_tarsus_sexMale -0.24 0.04 -0.32 -0.16 1.00 2081 1516
sigma_tarsus_sexUNK -0.39 0.13 -0.64 -0.14 1.00 1429 1293
back_shatchdate_1 -0.11 3.39 -5.89 8.13 1.00 647 743
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma_back 0.90 0.03 0.86 0.95 1.00 1627 1245
alpha_tarsus -1.21 0.46 -1.89 0.20 1.00 666 327
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
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.
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.