# Multilevel Model with Compositional Outcomes

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)

# 1 Multilevel model with compositional outcomes.

## 1.1 Computing compositions and isometric log ratio coordinates.

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

    cilr <- complr(data = mcompd, sbp = sbp,
                    parts = c("TST", "WAKE", "MVPA", "LPA", "SB"), idvar = "ID", total = 1440)

    head(cilr$TotalILR)
    #> NULL

## 1.2 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(complr = cilr,
                  formula = mvbind(z1_1, z2_1, z3_1, z4_1) ~ 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.

Here is a `summary()` of the model. We can see that stress significantly predicted `z1_1` and `z2_1`.

    summary(mv)
    #>  Family: MV(gaussian, gaussian, gaussian, gaussian)
    #>   Links: mu = identity; sigma = identity
    #>          mu = identity; sigma = identity
    #>          mu = identity; sigma = identity
    #>          mu = identity; sigma = identity
    #> Formula: z1_1 ~ Stress + (1 | ID)
    #>          z2_1 ~ Stress + (1 | ID)
    #>          z3_1 ~ Stress + (1 | ID)
    #>          z4_1 ~ Stress + (1 | ID)
    #>    Data: tmp (Number of observations: 3540)
    #>   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
    #>          total post-warmup draws = 4000
    #>
    #> Multilevel Hyperparameters:
    #> ~ID (Number of levels: 266)
    #>                    Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    #> sd(z1_1_Intercept)     0.33      0.02     0.30     0.37 1.00     1228     2060
    #> sd(z2_1_Intercept)     0.30      0.01     0.28     0.33 1.00     1134     1929
    #> sd(z3_1_Intercept)     0.39      0.02     0.35     0.43 1.00     1731     2763
    #> sd(z4_1_Intercept)     0.30      0.02     0.27     0.33 1.00     1710     2483
    #>
    #> Regression Coefficients:
    #>                Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    #> z1_1_Intercept    -0.44      0.02    -0.48    -0.39 1.00      942     1598
    #> z2_1_Intercept     1.47      0.02     1.42     1.51 1.01      876     1608
    #> z3_1_Intercept    -0.88      0.03    -0.94    -0.82 1.00     1740     2565
    #> z4_1_Intercept     0.65      0.02     0.60     0.69 1.00     1648     2562
    #> z1_1_Stress       -0.01      0.00    -0.02    -0.00 1.00     5967     3302
    #> z2_1_Stress        0.01      0.00     0.00     0.01 1.00     5446     3533
    #> z3_1_Stress        0.00      0.01    -0.01     0.01 1.00     6975     3101
    #> z4_1_Stress        0.01      0.00    -0.00     0.01 1.00     6794     3387
    #>
    #> Further Distributional Parameters:
    #>            Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    #> sigma_z1_1     0.44      0.01     0.43     0.45 1.00     6228     3088
    #> sigma_z2_1     0.38      0.00     0.37     0.39 1.00     6496     3182
    #> sigma_z3_1     0.70      0.01     0.68     0.71 1.00     5403     3385
    #> sigma_z4_1     0.53      0.01     0.51     0.54 1.00     5484     3418
    #>
    #> Residual Correlations:
    #>                   Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    #> rescor(z1_1,z2_1)    -0.54      0.01    -0.57    -0.52 1.00     5503     3659
    #> rescor(z1_1,z3_1)    -0.18      0.02    -0.21    -0.14 1.00     5662     3335
    #> rescor(z2_1,z3_1)    -0.05      0.02    -0.08    -0.02 1.00     4974     3475
    #> rescor(z1_1,z4_1)     0.11      0.02     0.07     0.14 1.00     6096     3480
    #> rescor(z2_1,z4_1)    -0.05      0.02    -0.08    -0.01 1.00     5544     3305
    #> rescor(z3_1,z4_1)     0.56      0.01     0.54     0.58 1.00     5224     3252
    #>
    #> 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).

# 2 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(complr = cilr,
                   formula = mvbind(z1_1, z2_1, z3_1, z4_1) ~ 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.
    # full model
    mv <- brmcoda(complr = cilr,
                  formula = mvbind(z1_1, z2_1, z3_1, z4_1) ~ 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.

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

    bayes_factor(mv$model, mv0$model)
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    #> Estimated Bayes factor in favor of mv$model over mv0$model: 0.00015

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.