Multilevel Model with Compositional Outcomes • multilevelcoda

In this vignette, we discuss how to specify multilevel models with compositional outcomes using multilevelcoda. In addition to multilevelcoda, we will use brms package (to fit models) and bayestestR package (to compute useful indices and compare models). We will also attach built in datasets mcompd (simulated compositional sleep and wake variables) and sbp (sequential binary partition).

library(multilevelcoda)
library(brms)
library(bayestestR)

data("mcompd") 
data("sbp") 

options(digits = 3)

Multilevel model with compositional outcomes.

Computing compositions and isometric log ratio coordinates.

The ILR coordinates outcomes can be calculated using the compilr() functions.

cilr <- compilr(data = mcompd, sbp = sbp,
                parts = c("TST", "WAKE", "MVPA", "LPA", "SB"), idvar = "ID")

head(cilr$TotalILR)
#>        ilr1 ilr2    ilr3  ilr4
#> [1,]  0.287 1.20  0.6270 1.702
#> [2,] -0.472 1.57 -0.8336 0.984
#> [3,] -0.486 1.33  1.3344 2.659
#> [4,] -0.316 1.37 -0.0332 0.551
#> [5,]  0.205 1.43 -0.6893 0.733
#> [6,] -0.446 1.16 -0.0950 0.670

Fitting model

A model with multilevel compositional outcomes is multivariate, as it has multiple ILR coordinate outcomes,each of which is predicted by a set of predictors. Our brms model can be then fitted using the brmcoda() function.

mv <- brmcoda(compilr = cilr,
              formula = mvbind(ilr1, ilr2, ilr3, ilr4) ~ STRESS + (1 | ID),
              cores = 8, seed = 123, backend = "cmdstanr")
#> Warning: In the future, 'rescor' will be set to FALSE by default for all models. It is thus
#> recommended to explicitely set 'rescor' via 'set_rescor' instead of using the default.
#> Warning: CmdStan's precompiled header (PCH) files may need to be rebuilt.
#> If your model failed to compile please run rebuild_cmdstan().
#> If the issue persists please open a bug report.
#> Error: An error occured during compilation! See the message above for more information.

Here is a summary() of the model. We can see that stress significantly predicted ilr1 and ilr2.

summary(mv$Model)
#>  Family: MV(gaussian, gaussian, gaussian, gaussian) 
#>   Links: mu = identity; sigma = identity
#>          mu = identity; sigma = identity
#>          mu = identity; sigma = identity
#>          mu = identity; sigma = identity 
#> Formula: ilr1 ~ STRESS + (1 | ID) 
#>          ilr2 ~ STRESS + (1 | ID) 
#>          ilr3 ~ STRESS + (1 | ID) 
#>          ilr4 ~ STRESS + (1 | ID) 
#>    Data: tmp (Number of observations: 3540) 
#>   Draws: 8 chains, each with iter = 6000; warmup = 1000; thin = 1;
#>          total post-warmup draws = 40000
#> 
#> Group-Level Effects: 
#> ~ID (Number of levels: 266) 
#>                    Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sd(ilr1_Intercept)     0.33      0.02     0.30     0.37 1.00    13574    21989
#> sd(ilr2_Intercept)     0.30      0.02     0.28     0.34 1.00    11200    19827
#> sd(ilr3_Intercept)     0.39      0.02     0.35     0.43 1.00    17013    24936
#> sd(ilr4_Intercept)     0.30      0.02     0.27     0.33 1.00    16596    25482
#> 
#> Population-Level Effects: 
#>                Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> ilr1_Intercept    -0.43      0.02    -0.48    -0.39 1.00    11069    19841
#> ilr2_Intercept     1.47      0.02     1.42     1.51 1.00     9583    18405
#> ilr3_Intercept    -0.87      0.03    -0.93    -0.81 1.00    17975    27392
#> ilr4_Intercept     0.65      0.02     0.60     0.69 1.00    18563    26992
#> ilr1_STRESS       -0.01      0.00    -0.02    -0.00 1.00    64970    35109
#> ilr2_STRESS        0.01      0.00     0.00     0.01 1.00    56689    36287
#> ilr3_STRESS        0.00      0.01    -0.01     0.01 1.00    67676    33111
#> ilr4_STRESS        0.01      0.00    -0.00     0.01 1.00    62003    35093
#> 
#> Family Specific Parameters: 
#>            Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sigma_ilr1     0.44      0.01     0.43     0.45 1.00    57828    33519
#> sigma_ilr2     0.38      0.00     0.37     0.39 1.00    59436    34195
#> sigma_ilr3     0.70      0.01     0.68     0.71 1.00    52722    34899
#> sigma_ilr4     0.53      0.01     0.51     0.54 1.00    52690    33648
#> 
#> Residual Correlations: 
#>                   Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> rescor(ilr1,ilr2)    -0.54      0.01    -0.57    -0.52 1.00    57259    33739
#> rescor(ilr1,ilr3)    -0.18      0.02    -0.21    -0.14 1.00    56781    34166
#> rescor(ilr2,ilr3)    -0.05      0.02    -0.09    -0.02 1.00    56093    32472
#> rescor(ilr1,ilr4)     0.11      0.02     0.07     0.14 1.00    56993    31275
#> rescor(ilr2,ilr4)    -0.05      0.02    -0.08    -0.01 1.00    54469    33148
#> rescor(ilr3,ilr4)     0.56      0.01     0.54     0.58 1.00    53173    34399
#> 
#> Draws were sampled using sample(hmc). 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).

Bayes Factor for compositional multilevel modelling

We are often interested in whether a predictor significantly predict the overall composition, in addition to the individual ILR coordinates. In Bayesian, this can be done by comparing the marginal likelihoods of two models. Bayes Factors (BFs) are indices of relative evidence of one model over another. In the context of compositional multilevel modelling, Bayes Factors provide two main useful functions:

Testing single parameters within a model
Comparing models

We can utilize Bayes factors to answer the following question: “Which model (i.e., set of composition predictors, expressed as ILRs) is more likely to have produced the observed data?”

Let’s examine whether stress predicts the overall sleep-wake composition.

Note: To use Bayes factors, brmsfit models must be fitted with an additional non-default argument save_pars = save_pars(all = TRUE).

# intercept only
mv0 <- brmcoda(compilr = cilr,
               formula = mvbind(ilr1, ilr2, ilr3, ilr4) ~ 1 + (1 | ID),
               iter = 6000, chains = 8, cores = 8, seed = 123, warmup = 1000,
               backend = "cmdstanr", save_pars = save_pars(all = TRUE))
#> Warning: In the future, 'rescor' will be set to FALSE by default for all models. It is thus
#> recommended to explicitely set 'rescor' via 'set_rescor' instead of using the default.
#> Warning: CmdStan's precompiled header (PCH) files may need to be rebuilt.
#> If your model failed to compile please run rebuild_cmdstan().
#> If the issue persists please open a bug report.
#> Error: An error occured during compilation! See the message above for more information.
# full model
mv <- brmcoda(compilr = cilr,
              formula = mvbind(ilr1, ilr2, ilr3, ilr4) ~ STRESS + (1 | ID),
              iter = 6000, chains = 8, cores = 8, seed = 123, warmup = 1000,
              backend = "cmdstanr", save_pars = save_pars(all = TRUE))
#> Warning: In the future, 'rescor' will be set to FALSE by default for all models. It is thus recommended to explicitely set 'rescor' via 'set_rescor' instead of using the default.

#> Warning: CmdStan's precompiled header (PCH) files may need to be rebuilt.
#> If your model failed to compile please run rebuild_cmdstan().
#> If the issue persists please open a bug report.
#> Error: An error occured during compilation! See the message above for more information.

We can now compare these models with the bayesfactor_models() function

comparison <- bayesfactor_models(mv$Model, denominator = mv0$Model)

comparison
#> Bayes Factors for Model Comparison
#> 
#>     Model       BF
#> [1]       2.90e-05
#> 
#> * Against Denominator: [2]
#> *   Bayes Factor Type: marginal likelihoods (bridgesampling)

With a \(BF\) < 1, our data favours the intercept only model, showing that there is insufficient evidence for stress predicting the overall sleep-wake composition.

Bayes factors provide a intuitive measure of the strength of evidence of one model over the other or among different models. Check out the bayestestR packages for several other useful functions related to BFs.