# Multilevel Models with Compositional Predictors

In this vignette, we discuss how to use `multilevelcoda` to specify multilevel models where compositional data are used as predictors.

The following table outlines the packages used and a brief description of their purpose.

Package | Purpose
--- | ---
`multilevelcoda` | calculate between and within composition variables, calculate substitutions and plots
`brms` | fit Bayesian multilevel models using Stan as a backend
`bayestestR` | compute Bayes factors used to compare models
`doFuture` | parallel processing to speed up run times

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

    options(digits = 3) # reduce number of digits shown

For the examples, we make use of three built in datasets:

Dataset | Purpose
--- | ---
`mcompd` | compositional sleep and wake variables and additional predictors/outcomes (simulated)
`sbp` | a pre-specified sequential binary partition, used in calculating compositional predictors
`psub` | all possible pairwise substitutions between compositional variables, used for substitution analyses

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

The following table shows a few rows of data from `mcompd`.

TST | WAKE | MVPA | LPA | SB | ID | Age | Female | STRESS
--- | --- | --- | --- | --- | --- | --- | --- | ---
542 | 99.0 | 297.4 | 460 | 41.4 | 185 | 29.7 | 0 | 3.67
458 | 49.4 | 117.3 | 653 | 162.3 | 185 | 29.7 | 0 | 7.21
271 | 41.1 | 488.7 | 625 | 14.5 | 185 | 29.7 | 0 | 2.84
286 | 52.7 | 106.9 | 906 | 89.2 | 184 | 22.3 | 1 | 2.36
281 | 18.8 | 403.0 | 611 | 126.3 | 184 | 22.3 | 1 | 1.18
397 | 26.5 | 39.9 | 587 | 389.8 | 184 | 22.3 | 1 | 0.00

The following table shows the sequential binary partition being used in `sbp`. Columns correspond to the composition variables (TST, WAKE, MVPA, LPA, SB). Rows correspond to distinct ILR coordinates.

1 | 1 | -1 | -1 | -1
1 | -1 | 0 | 0 | 0
0 | 0 | 1 | -1 | -1
0 | 0 | 0 | 1 | -1

The following table shows how all the possible binary substitions contrasts are setup. Time substitions work by taking time from the -1 variable and adding time to the +1 variable.

TST | WAKE | MVPA | LPA | SB
--- | --- | --- | --- | ---
1 | -1 | 0 | 0 | 0
1 | 0 | -1 | 0 | 0
1 | 0 | 0 | -1 | 0
1 | 0 | 0 | 0 | -1
-1 | 1 | 0 | 0 | 0
0 | 1 | -1 | 0 | 0
0 | 1 | 0 | -1 | 0
0 | 1 | 0 | 0 | -1
-1 | 0 | 1 | 0 | 0
0 | -1 | 1 | 0 | 0
0 | 0 | 1 | -1 | 0
0 | 0 | 1 | 0 | -1
-1 | 0 | 0 | 1 | 0
0 | -1 | 0 | 1 | 0
0 | 0 | -1 | 1 | 0
0 | 0 | 0 | 1 | -1
-1 | 0 | 0 | 0 | 1
0 | -1 | 0 | 0 | 1
0 | 0 | -1 | 0 | 1
0 | 0 | 0 | -1 | 1

# Multilevel model with compositional predictors

## Compositions and isometric log ratio (ILR) coordinates.

Compositional data are often expressed as a set of isometric log ratio (ILR) coordinates in regression models. We can use the `compilr()` function to calculate both between- and within-level ILR coordinates for use in subsequent models as predictors. Note: `compilr()` also calculates total ILR coordinates that could be used as outcomes (or predictors) in models, if the decomposition into a between- and within-level ILR coordinates was not desired.

The `compilr()` function for multilevel data requires four arguments:

Argument | Input
--- | ---
`data` | a dataset containing all variables: at a minimum composition variables and an ID variable
`sbp` | a matrix with contrast codes used to calculate the sequential binary partitions for the ILR coordinates
`parts` | a character vector giving the names of the variables in `data` that are compositional
`idvar` | a character string giving the name of the variable that contains the IDs

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

## Fitting model

We now will use output from the `compilr()` to fit our `brms` model, using the `brmcoda()`. Here is a model predicting `STRESS` from between- and within-person sleep-wake behaviours (expressed as ILR coordinates).

Note: make sure you pass the correct names of the ILR coordinates to `brms` model.

    m <- brmcoda(compilr = cilr,
                 formula = STRESS ~ bilr1 + bilr2 + bilr3 + bilr4 +
                   wilr1 + wilr2 + wilr3 + wilr4 + (1 | ID),
                 cores = 8, seed = 123, backend = "cmdstanr")
    #> Compiling Stan program...
    #> Start sampling

Here is a `summary()` of the model results.

    summary(m$Model)
    #>  Family: gaussian
    #>   Links: mu = identity; sigma = identity
    #> Formula: STRESS ~ bilr1 + bilr2 + bilr3 + bilr4 + wilr1 + wilr2 + wilr3 + wilr4 + (1 | ID)
    #>    Data: tmp (Number of observations: 3540)
    #>   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
    #>          total post-warmup draws = 4000
    #>
    #> Group-Level Effects:
    #> ~ID (Number of levels: 266)
    #>               Estimate Est.Error l-95% CI u-95% CI
    #> sd(Intercept)     1.00      0.06     0.88     1.13
    #>               Rhat Bulk_ESS Tail_ESS
    #> sd(Intercept) 1.00     1370     2369
    #>
    #> Population-Level Effects:
    #>           Estimate Est.Error l-95% CI u-95% CI Rhat
    #> Intercept     2.56      0.49     1.64     3.50 1.00
    #> bilr1         0.17      0.33    -0.49     0.82 1.00
    #> bilr2         0.41      0.35    -0.28     1.08 1.00
    #> bilr3         0.12      0.21    -0.29     0.54 1.00
    #> bilr4        -0.04      0.29    -0.59     0.52 1.00
    #> wilr1        -0.34      0.12    -0.58    -0.10 1.00
    #> wilr2         0.04      0.13    -0.21     0.29 1.00
    #> wilr3        -0.11      0.08    -0.25     0.04 1.00
    #> wilr4         0.24      0.10     0.05     0.43 1.00
    #>           Bulk_ESS Tail_ESS
    #> Intercept     1011     1819
    #> bilr1          987     1601
    #> bilr2          964     2099
    #> bilr3         1091     1861
    #> bilr4         1001     1775
    #> wilr1         3280     3276
    #> wilr2         3576     3033
    #> wilr3         3196     2789
    #> wilr4         3769     2893
    #>
    #> Family Specific Parameters:
    #>       Estimate Est.Error l-95% CI u-95% CI Rhat
    #> sigma     2.37      0.03     2.31     2.42 1.00
    #>       Bulk_ESS Tail_ESS
    #> sigma     5037     3013
    #>
    #> 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).

Results show that the first and forth within-person ILR coordinate was associated with stress. The interpretation of these outputs depends on how you construct your sequential binary partition. For the built-in sequential binary partition `sbp` (shown previously), the resulting interpretation would be as follows:

ILR coordinates | What it means
--- | ---
`bilr1` | Sleep (TST & WAKE) vs wake behaviours at between-person level
`bilr2` | Sleep vs awake in bed at between-person level
`bilr3` | MVPA vs (LPA and SB) at between-person level
`bilr4` | LPA vs SB at between-person level
`wilr1` | Sleep (TST & WAKE) vs wake behaviours at between-person level
`wilr2` | Sleep vs awake in bed at within-person level
`wilr3` | MVPA vs (LPA and SB) at within-person level
`wilr4` | LPA vs SB at within-person level

Due to the nature of within-person ILR coordinates, it is often challenging to interpret these results in great details. For example, the significant coefficient for `wilr1` shows that the within-person change in sleep behaviours (sleep duration and time awake in bed combined), relative to wake behaviours (moderate to vigorous physical activity, light physical activity, and sedentary behaviour) on a given day, was associated with stress. However, as there are several behaviours involved in this coordinate, we don’t know the within-person change in which of them drives the association. It could be the change in sleep, such that people sleep more than their own average on a given day, but it could also be the change in time awake. Further, we don’t know about the specific changes in time spent across behaviours. That is, if people slept more, what behaviour did they spend less time in?

One approach to gain further insights into these relationships, and the changes in outcomes associated with changes in specific time across compositionl components is the substitution model. We will discuss the substitution model later in this vignette.

## Bayes Factor for significance testing

In the frequentist approach, we usually compare the fits of models using `anova()`. 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 is more likely to have produced the observed data?”

Let’s fit a series of model with `brmcoda()` to predict `STRESS` from sleep-wake composition. For precise Bayes factors, we will use 40,000 posterior draws for each model.

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

    # intercept only model
    m0 <- brmcoda(compilr = cilr,
                 formula = STRESS ~ 1 + (1 | ID),
                 iter = 6000, chains = 8, cores = 8, seed = 123, warmup = 1000,
                 backend = "cmdstanr", save_pars = save_pars(all = TRUE))
    #> Compiling Stan program...
    #> Start sampling

    # between-person composition only model
    m1 <- brmcoda(compilr = cilr,
                 formula = STRESS ~ bilr1 + bilr2 + bilr3 + bilr4 + (1 | ID),
                 iter = 6000, chains = 8, cores = 8, seed = 123, warmup = 1000,
                 backend = "cmdstanr", save_pars = save_pars(all = TRUE))
    #> Compiling Stan program...
    #> Start sampling

    # within-person composition only model
    m2 <- brmcoda(compilr = cilr,
                 formula = STRESS ~ wilr1 + wilr2 + wilr3 + wilr4 + (1 | ID),
                 iter = 6000, chains = 8, cores = 8, seed = 123, warmup = 1000,
                 backend = "cmdstanr", save_pars = save_pars(all = TRUE))
    #> Compiling Stan program...
    #> Start sampling

    # full model
    m <- brmcoda(compilr = cilr,
                 formula = STRESS ~ bilr1 + bilr2 + bilr3 + bilr4 +
                   wilr1 + wilr2 + wilr3 + wilr4 + (1 | ID),
                 iter = 6000, chains = 8, cores = 8, seed = 123, warmup = 1000,
                 backend = "cmdstanr", save_pars = save_pars(all = TRUE))
    #> Compiling Stan program...
    #> Start sampling

We can now compare these models with the `bayesfactor_models()` function, using the intercept-only model as reference.

    comparison <- bayesfactor_models(m$Model, m1$Model, m2$Model, denominator = m0$Model)

    comparison
    #> Bayes Factors for Model Comparison
    #>
    #>     Model                                                                       BF
    #> [1] bilr1 + bilr2 + bilr3 + bilr4 + wilr1 + wilr2 + wilr3 + wilr4 + (1 | ID)  4.01
    #> [2] bilr1 + bilr2 + bilr3 + bilr4 + (1 | ID)                                 0.341
    #> [3] wilr1 + wilr2 + wilr3 + wilr4 + (1 | ID)                                 11.62
    #>
    #> * Against Denominator: [4] 1 + (1 | ID)
    #> *   Bayes Factor Type: marginal likelihoods (bridgesampling)

We can see that model with only within-person composition is the best model - with \(BF\) = 11.86 compared to the null (intercept only).

Let’s compare these models against the full model.

    update(comparison, reference = 1)
    #> Bayes Factors for Model Comparison
    #>
    #>     Model                                       BF
    #> [2] bilr1 + bilr2 + bilr3 + bilr4 + (1 | ID) 0.085
    #> [3] wilr1 + wilr2 + wilr3 + wilr4 + (1 | ID)  2.89
    #> [4] 1 + (1 | ID)                             0.249
    #>
    #> * Against Denominator: [1] bilr1 + bilr2 + bilr3 + bilr4 + wilr1 + wilr2 + wilr3 + wilr4 + (1 | ID)
    #> *   Bayes Factor Type: marginal likelihoods (bridgesampling)

Again, our data favours the within-person composition only model over the full model, giving 2.93 times more support.

# Substitution model

When examining the relationships between compositional data and an outcome, we often are also interested in the changes in an outcomes when a fixed duration of time is reallocated from one compositional component to another, while the other components remain constant. These changes can be examined using the compositional isotemporal substitution model. In `multilevelcoda`, we extend this model to multilevel approach to test both between-person and within-person changes. All substitution models can be computed using the `substitution()` function.

## Between-person substitution model

The below example examines the changes in stress for different pairwise substitution of sleep-wake behaviours for 5 minutes, at between-person level.

    bsubm <- substitution(object = m, delta = 5,
                          level = "between", type = "conditional")

The output contains multiple data sets of results for all compositional components. Here are the results for changes in stress when sleep (TST) is substituted for 5 minutes, averaged across levels of covariates.

    knitr::kable(bsubm$TST)

|| || || ||

None of the results are significant, given that the credible intervals did not cross 0, showing that increasing sleep (TST) at the expense of any other behaviours was not associated in changes in stress. Notice there is no column indicating the levels of convariates, indicating that these results have been averaged.

## Within-person substitution model

Let’s now take a look at how stress changes when different pairwise of sleep-wake behaviours are substituted for 5 minutes, at within-person level.

    # Within-person substitution
    wsubm <- substitution(object = m, delta = 5,
                          level = "within", type = "conditional")

Results for 5 minute substitution.

    knitr::kable(wsubm$TST)

|| || || ||

At within-person level, there were significant results for substitution of sleep (TST) and time awake in bed (WAKE) for 5 minutes, but not other behaviours. Increasing sleep at the expense of time spent awake in bed predicted 0.02 higher stress [95% CI 0.00, 0.03], on a given day. Conversely, less sleep and more time awake in bed predicted less stress (b = -0.016 [95% CI -0.03, -0.00]).

## More interesting substitution models

You can learn more about different types of substitution models at
Compositional Multilevel Substitution Models.