Author: Paul Bürkner
Date: 2021-11-22

# Estimating Multivariate Models with brms

#### Paul Bürkner

#### 2021-11-22

## 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")`). 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

    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     0.58 1.00      789
    sd(back_Intercept)                       0.24      0.08     0.09     0.39 1.01      286
    cor(tarsus_Intercept,back_Intercept)    -0.52      0.23    -0.95    -0.08 1.01      377
                                         Tail_ESS
    sd(tarsus_Intercept)                     1353
    sd(back_Intercept)                        606
    cor(tarsus_Intercept,back_Intercept)      695

    ~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      664
    sd(back_Intercept)                       0.35      0.06     0.23     0.47 1.00      453
    cor(tarsus_Intercept,back_Intercept)     0.68      0.21     0.20     0.98 1.00      316
                                         Tail_ESS
    sd(tarsus_Intercept)                     1163
    sd(back_Intercept)                        952
    cor(tarsus_Intercept,back_Intercept)      557

    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    -0.27 1.00     1356     1387
    back_Intercept      -0.01      0.06    -0.14     0.11 1.00     2247     1577
    tarsus_sexMale       0.77      0.06     0.66     0.87 1.00     3631     1251
    tarsus_sexUNK        0.23      0.13    -0.03     0.48 1.00     3904     1252
    tarsus_hatchdate    -0.04      0.06    -0.16     0.07 1.00     1221     1286
    back_sexMale         0.01      0.07    -0.12     0.13 1.00     3669     1885
    back_sexUNK          0.15      0.15    -0.13     0.44 1.00     3706     1602
    back_hatchdate      -0.09      0.05    -0.19     0.02 1.00     2152     1551

    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     1928     1652
    sigma_back       0.90      0.02     0.85     0.95 1.00     2607     1610

    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     2597     1370

    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 0.4341246 0.02329163 0.3857624 0.4776837
    R2back   0.1977502 0.02836021 0.1408088 0.2523545

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)
      Draws: 2 chains, each with iter = 2000; warmup = 1000; thin = 1;
             total post-warmup draws = 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 1.00      800
    sd(back_Intercept)                       0.25      0.07     0.11     0.39 1.00      392
    cor(tarsus_Intercept,back_Intercept)    -0.49      0.22    -0.91    -0.06 1.00      576
                                         Tail_ESS
    sd(tarsus_Intercept)                     1270
    sd(back_Intercept)                        589
    cor(tarsus_Intercept,back_Intercept)      649

    ~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 1.00      642
    sd(back_Intercept)                       0.35      0.06     0.23     0.45 1.00      511
    cor(tarsus_Intercept,back_Intercept)     0.70      0.20     0.24     0.98 1.00      229
                                         Tail_ESS
    sd(tarsus_Intercept)                      937
    sd(back_Intercept)                       1098
    cor(tarsus_Intercept,back_Intercept)      311

    Population-Level Effects:
                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    tarsus_Intercept    -0.42      0.07    -0.55    -0.28 1.00     1248     1372
    back_Intercept       0.00      0.05    -0.10     0.10 1.00     1552     1414
    tarsus_sexMale       0.77      0.06     0.66     0.88 1.00     3231     1537
    tarsus_sexUNK        0.23      0.13    -0.03     0.49 1.00     2914     1644
    back_hatchdate      -0.08      0.05    -0.19     0.02 1.00     2024     1543

    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     2089     1584
    sigma_back       0.90      0.02     0.85     0.95 1.00     2066     1532

    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     2480     1767

    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  -2127.2 33.5
    p_loo       176.2  7.4
    looic      4254.3 67.1
    ------
    Monte Carlo SE of elpd_loo is 0.4.

    Pareto k diagnostic values:
                             Count Pct.    Min. n_eff
    (-Inf, 0.5]   (good)     814   98.3%   196
     (0.5, 0.7]   (ok)        14    1.7%   76
       (0.7, 1]   (bad)        0    0.0%   <NA>
       (1, Inf)   (very bad)   0    0.0%   <NA>

    All Pareto k estimates are ok (k < 0.7).
    See help('pareto-k-diagnostic') for details.

    Output of model 'fit2':

    Computed from 2000 by 828 log-likelihood matrix

             Estimate   SE
    elpd_loo  -2124.6 33.6
    p_loo       173.5  7.3
    looic      4249.2 67.3
    ------
    Monte Carlo SE of elpd_loo is NA.

    Pareto k diagnostic values:
                             Count Pct.    Min. n_eff
    (-Inf, 0.5]   (good)     813   98.2%   208
     (0.5, 0.7]   (ok)        14    1.7%   107
       (0.7, 1]   (bad)        1    0.1%   32
       (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 -2.5       1.4

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

    Smooth Terms:
                           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    sds(back_shatchdate_1)     2.10      1.11     0.34     4.66 1.00      563      515

    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      476
    sd(back_Intercept)                       0.23      0.07     0.08     0.36 1.00      249
    cor(tarsus_Intercept,back_Intercept)    -0.54      0.24    -0.95    -0.05 1.00      458
                                         Tail_ESS
    sd(tarsus_Intercept)                     1100
    sd(back_Intercept)                        302
    cor(tarsus_Intercept,back_Intercept)      706

    ~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 1.01      362
    sd(back_Intercept)                       0.32      0.06     0.21     0.43 1.00      478
    cor(tarsus_Intercept,back_Intercept)     0.66      0.23     0.16     0.99 1.01      148
                                         Tail_ESS
    sd(tarsus_Intercept)                      518
    sd(back_Intercept)                        696
    cor(tarsus_Intercept,back_Intercept)      306

    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      797     1161
    back_Intercept           0.00      0.05    -0.10     0.11 1.00     1219     1284
    tarsus_sexMale           0.77      0.06     0.66     0.88 1.00     2604     1442
    tarsus_sexUNK            0.21      0.12    -0.02     0.44 1.00     2548     1583
    sigma_tarsus_sexFem     -0.30      0.04    -0.39    -0.21 1.00     2220     1266
    sigma_tarsus_sexMale    -0.24      0.04    -0.32    -0.16 1.00     1723     1329
    sigma_tarsus_sexUNK     -0.39      0.13    -0.63    -0.14 1.00     1636     1461
    back_shatchdate_1       -0.08      3.24    -5.80     7.09 1.00      880     1003

    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     2079     1550
    alpha_tarsus    -1.22      0.43    -1.85     0.10 1.01      922      445

    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.