# Estimating Multivariate Models with brms
Author: Paul Bürkner
Date: 2024-03-19

#### 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.