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)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$TotalILR)
#> ilr1 ilr2 ilr3 ilr4
#> [1,] 0.287 1.20 0.6270 1.702
#> [2,] -0.472 1.57 -0.8336 0.984
#> [3,] -0.486 1.33 1.3344 2.659
#> [4,] -0.316 1.37 -0.0332 0.551
#> [5,] 0.205 1.43 -0.6893 0.733
#> [6,] -0.446 1.16 -0.0950 0.670A model with multilevel compositional outcomes is multivariate, as it
has multiple ILR coordinate outcomes,each of which is predicted by a set
of predictors. Our brms model can be then fitted using the
brmcoda() function.
mv <- brmcoda(compilr = cilr,
formula = mvbind(ilr1, ilr2, ilr3, ilr4) ~ STRESS + (1 | ID),
cores = 8, seed = 123, backend = "cmdstanr")
#> Warning: In the future, 'rescor' will be set to FALSE by default for all models. It is thus
#> recommended to explicitely set 'rescor' via 'set_rescor' instead of using the default.
#> Warning: CmdStan's precompiled header (PCH) files may need to be rebuilt.
#> If your model failed to compile please run rebuild_cmdstan().
#> If the issue persists please open a bug report.
#> Error: An error occured during compilation! See the message above for more information.Here is a summary() of the model. We can see that stress
significantly predicted ilr1 and ilr2.
summary(mv$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).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:
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
#> 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.