Author: Paul Bürkner
Date: 2018-10-23

# Estimating Multivariate Models with brms

#### Paul Bürkner

#### 2018-10-23

## 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(
      cbind(tarsus, back) ~ sex + hatchdate + (1|p|fosternest) + (1|q|dam),
      data = BTdata, chains = 2, cores = 2
    )

    In file included from C:/Users/paulb/Documents/R/win-library/3.5/BH/include/boost/config.hpp:39:0,
                     from C:/Users/paulb/Documents/R/win-library/3.5/BH/include/boost/math/tools/config.hpp:13,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math/rev/core/var.hpp:7,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math/rev/core/gevv_vvv_vari.hpp:5,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math/rev/core.hpp:12,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math/rev/mat.hpp:4,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math.hpp:4,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/src/stan/model/model_header.hpp:4,
                     from file37b44bb3235c.cpp:8:
    C:/Users/paulb/Documents/R/win-library/3.5/BH/include/boost/config/compiler/gcc.hpp:186:0: warning: "BOOST_NO_CXX11_RVALUE_REFERENCES" redefined
     #  define BOOST_NO_CXX11_RVALUE_REFERENCES
     ^
    <command-line>:0:0: note: this is the location of the previous definition
    In file included from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math/rev/core.hpp:44:0,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math/rev/mat.hpp:4,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math.hpp:4,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/src/stan/model/model_header.hpp:4,
                     from file37b44bb3235c.cpp:8:
    C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math/rev/core/set_zero_all_adjoints.hpp:14:17: warning: 'void stan::math::set_zero_all_adjoints()' defined but not used [-Wunused-function]
         static void set_zero_all_adjoints() {
                     ^

As can be seen in the model code, we have used `cbind` 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

    add_ic(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 Eff.Sample Rhat
    sd(tarsus_Intercept)                     0.48      0.05     0.39     0.58        798 1.00
    sd(back_Intercept)                       0.24      0.07     0.09     0.39        359 1.00
    cor(tarsus_Intercept,back_Intercept)    -0.53      0.23    -0.93    -0.09        609 1.00

    ~fosternest (Number of levels: 104)
                                         Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
    sd(tarsus_Intercept)                     0.27      0.05     0.16     0.38        487 1.00
    sd(back_Intercept)                       0.35      0.06     0.24     0.47        537 1.00
    cor(tarsus_Intercept,back_Intercept)     0.71      0.19     0.29     0.98        261 1.01

    Population-Level Effects:
                     Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
    tarsus_Intercept    -0.41      0.07    -0.54    -0.27       1336 1.00
    back_Intercept      -0.01      0.06    -0.14     0.11       2000 1.00
    tarsus_sexMale       0.77      0.06     0.66     0.89       2000 1.00
    tarsus_sexUNK        0.23      0.13    -0.02     0.48       2000 1.00
    tarsus_hatchdate    -0.04      0.06    -0.15     0.08       1410 1.00
    back_sexMale         0.01      0.06    -0.12     0.14       2000 1.00
    back_sexUNK          0.15      0.15    -0.15     0.45       2000 1.00
    back_hatchdate      -0.09      0.05    -0.19     0.01       2000 1.00

    Family Specific Parameters:
                 Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
    sigma_tarsus     0.76      0.02     0.72     0.80       2000 1.00
    sigma_back       0.90      0.02     0.86     0.95       2000 1.00

    Residual Correlations:
                        Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
    rescor(tarsus,back)    -0.05      0.04    -0.13     0.02       2000 1.00

    Samples were drawn using sampling(NUTS). For each parameter, Eff.Sample
    is a crude measure of effective sample size, 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
    R2_tarsus 0.4339561 0.02386071 0.3850141 0.4781046
    R2_back   0.1983159 0.02795730 0.1424196 0.2531194

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 `cbind` 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)

    In file included from C:/Users/paulb/Documents/R/win-library/3.5/BH/include/boost/config.hpp:39:0,
                     from C:/Users/paulb/Documents/R/win-library/3.5/BH/include/boost/math/tools/config.hpp:13,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math/rev/core/var.hpp:7,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math/rev/core/gevv_vvv_vari.hpp:5,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math/rev/core.hpp:12,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math/rev/mat.hpp:4,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math.hpp:4,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/src/stan/model/model_header.hpp:4,
                     from file37b47cc44075.cpp:8:
    C:/Users/paulb/Documents/R/win-library/3.5/BH/include/boost/config/compiler/gcc.hpp:186:0: warning: "BOOST_NO_CXX11_RVALUE_REFERENCES" redefined
     #  define BOOST_NO_CXX11_RVALUE_REFERENCES
     ^
    <command-line>:0:0: note: this is the location of the previous definition
    In file included from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math/rev/core.hpp:44:0,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math/rev/mat.hpp:4,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math.hpp:4,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/src/stan/model/model_header.hpp:4,
                     from file37b47cc44075.cpp:8:
    C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math/rev/core/set_zero_all_adjoints.hpp:14:17: warning: 'void stan::math::set_zero_all_adjoints()' defined but not used [-Wunused-function]
         static void set_zero_all_adjoints() {
                     ^

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.

    add_ic(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 Eff.Sample Rhat
    sd(tarsus_Intercept)                     0.48      0.05     0.39     0.59        781 1.00
    sd(back_Intercept)                       0.25      0.07     0.11     0.39        363 1.00
    cor(tarsus_Intercept,back_Intercept)    -0.49      0.22    -0.92    -0.06        492 1.00

    ~fosternest (Number of levels: 104)
                                         Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
    sd(tarsus_Intercept)                     0.26      0.05     0.16     0.36        656 1.00
    sd(back_Intercept)                       0.34      0.06     0.22     0.46        464 1.00
    cor(tarsus_Intercept,back_Intercept)     0.68      0.21     0.20     0.98        372 1.00

    Population-Level Effects:
                     Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
    tarsus_Intercept    -0.41      0.07    -0.54    -0.27       1746 1.00
    back_Intercept      -0.00      0.05    -0.11     0.10       2000 1.00
    tarsus_sexMale       0.77      0.06     0.66     0.88       2000 1.00
    tarsus_sexUNK        0.22      0.12    -0.01     0.46       2000 1.00
    back_hatchdate      -0.08      0.05    -0.18     0.02       2000 1.00

    Family Specific Parameters:
                 Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
    sigma_tarsus     0.76      0.02     0.72     0.80       2000 1.00
    sigma_back       0.90      0.02     0.85     0.95       2000 1.00

    Residual Correlations:
                        Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
    rescor(tarsus,back)    -0.05      0.04    -0.12     0.02       2000 1.00

    Samples were drawn using sampling(NUTS). For each parameter, Eff.Sample
    is a crude measure of effective sample size, 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)

                  LOOIC    SE
    fit1        4254.18 67.50
    fit2        4251.49 67.37
    fit1 - fit2    2.70  2.79

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)
    )

    In file included from C:/Users/paulb/Documents/R/win-library/3.5/BH/include/boost/config.hpp:39:0,
                     from C:/Users/paulb/Documents/R/win-library/3.5/BH/include/boost/math/tools/config.hpp:13,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math/rev/core/var.hpp:7,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math/rev/core/gevv_vvv_vari.hpp:5,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math/rev/core.hpp:12,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math/rev/mat.hpp:4,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math.hpp:4,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/src/stan/model/model_header.hpp:4,
                     from file37b458db5ab1.cpp:8:
    C:/Users/paulb/Documents/R/win-library/3.5/BH/include/boost/config/compiler/gcc.hpp:186:0: warning: "BOOST_NO_CXX11_RVALUE_REFERENCES" redefined
     #  define BOOST_NO_CXX11_RVALUE_REFERENCES
     ^
    <command-line>:0:0: note: this is the location of the previous definition
    In file included from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math/rev/core.hpp:44:0,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math/rev/mat.hpp:4,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math.hpp:4,
                     from C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/src/stan/model/model_header.hpp:4,
                     from file37b458db5ab1.cpp:8:
    C:/Users/paulb/Documents/R/win-library/3.5/StanHeaders/include/stan/math/rev/core/set_zero_all_adjoints.hpp:14:17: warning: 'void stan::math::set_zero_all_adjoints()' defined but not used [-Wunused-function]
         static void set_zero_all_adjoints() {
                     ^

Again, we summarize the model and look at some posterior-predictive checks.

    add_ic(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 Eff.Sample Rhat
    sds(back_shatchdate_1)     2.59      1.44     0.46     6.17        570 1.00

    Group-Level Effects:
    ~dam (Number of levels: 106)
                                         Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
    sd(tarsus_Intercept)                     0.47      0.05     0.38     0.58        820 1.00
    sd(back_Intercept)                       0.23      0.07     0.10     0.37        369 1.00
    cor(tarsus_Intercept,back_Intercept)    -0.53      0.23    -0.95    -0.08        611 1.00

    ~fosternest (Number of levels: 104)
                                         Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
    sd(tarsus_Intercept)                     0.26      0.05     0.16     0.36        506 1.01
    sd(back_Intercept)                       0.31      0.06     0.19     0.42        467 1.00
    cor(tarsus_Intercept,back_Intercept)     0.64      0.22     0.16     0.98        259 1.00

    Population-Level Effects:
                         Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
    tarsus_Intercept        -0.41      0.07    -0.54    -0.27        917 1.00
    back_Intercept           0.00      0.05    -0.10     0.10       1430 1.00
    tarsus_sexMale           0.77      0.06     0.66     0.88       2000 1.00
    tarsus_sexUNK            0.21      0.12    -0.02     0.44       2000 1.00
    sigma_tarsus_sexFem     -0.30      0.04    -0.38    -0.22       2000 1.00
    sigma_tarsus_sexMale    -0.24      0.04    -0.32    -0.16       2000 1.00
    sigma_tarsus_sexUNK     -0.40      0.13    -0.65    -0.13       2000 1.00
    back_shatchdate_1        0.04      0.54    -0.86     1.27       1137 1.00

    Family Specific Parameters:
                 Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
    sigma_back       0.90      0.02     0.86     0.95       2000 1.00
    alpha_tarsus    -1.24      0.41    -1.90    -0.18        802 1.00

    Samples were drawn using sampling(NUTS). For each parameter, Eff.Sample
    is a crude measure of effective sample size, 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

    marginal_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.