# 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)
    #>
    #> Attaching package: 'multilevelcoda'
    #> The following objects are masked _by_ '.GlobalEnv':
    #>
    #>     psub, sbp
    library(brms)
    #> Loading required package: Rcpp
    #> Loading 'brms' package (version 2.21.0). Useful instructions
    #> can be found by typing help('brms'). A more detailed introduction
    #> to the package is available through vignette('brms_overview').
    #>
    #> Attaching package: 'brms'
    #> The following object is masked from 'package:stats':
    #>
    #>     ar
    library(bayestestR)
    library(doFuture)
    #> Loading required package: foreach
    #> Loading required package: future

    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`.

ID | Time | Stress | TST | WAKE | MVPA | LPA | SB | Age | Female
--- | --- | --- | --- | --- | --- | --- | --- | --- | ---
185 | 1 | 4 | 542 | 99 | 297 | 460 | 41 | 30 | 0
185 | 2 | 7 | 458 | 49 | 117 | 653 | 162 | 30 | 0
185 | 3 | 3 | 271 | 41 | 489 | 625 | 15 | 30 | 0
184 | 12 | 2 | 286 | 53 | 107 | 906 | 89 | 22 | 1
184 | 13 | 1 | 281 | 19 | 403 | 611 | 126 | 22 | 1
184 | 14 | 0 | 397 | 26 | 40 | 587 | 390 | 22 | 1

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.

TST | WAKE | MVPA | LPA | SB
--- | --- | --- | --- | ---
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 substitutions contrasts are setup. Time substitutions 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

# 1 Multilevel model with compositional predictors

## 1.1 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 `complr()` function to calculate both between- and within-level ILR coordinates for use in subsequent models as predictors.

Notes: `complr()` also calculates total ILR coordinates to be used as outcomes (or predictors) in models, if the decomposition into a between- and within-level ILR coordinates was not desired.

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

Argument | Description
--- | ---
`data` | A long data set containing all variables needed to fit the multilevel models,
| including the repeated measure compositional predictors and outcomes, along with any additional covariates.
`sbp` | A Sequential Binary Partition to calculate \(ilr\) coordinates.
`parts` | The name of the compositional components in `data`.
`idvar` | The grouping factor on `data` to compute the between-person and within-person composition and \(ilr\) coordinates.
`total` | Optional argument to specify the amount to which the compositions should be closed.

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

## 1.2 Fitting model

We now will use output from the `complr()` 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).

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

    m <- brmcoda(complr = cilr,
                 formula = Stress ~ bz1_1 + bz2_1 + bz3_1 + bz4_1 +
                   wz1_1 + wz2_1 + wz3_1 + wz4_1 + (1 | ID),
                 cores = 8, seed = 123, backend = "cmdstanr")

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

    summary(m)
    #>  Family: gaussian
    #>   Links: mu = identity; sigma = identity
    #> Formula: Stress ~ bz1_1 + bz2_1 + bz3_1 + bz4_1 + wz1_1 + wz2_1 + wz3_1 + wz4_1 + (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(Intercept)     1.00      0.06     0.88     1.12 1.00     1169     2516
    #>
    #> Regression Coefficients:
    #>           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    #> Intercept     2.61      0.49     1.66     3.60 1.00     1101     1761
    #> bz1_1         0.19      0.32    -0.46     0.82 1.00      943     1863
    #> bz2_1         0.39      0.35    -0.29     1.06 1.00     1061     1922
    #> bz3_1         0.14      0.22    -0.28     0.56 1.01      895     1673
    #> bz4_1        -0.04      0.28    -0.59     0.51 1.00      976     1923
    #> wz1_1        -0.33      0.12    -0.56    -0.09 1.00     3226     3037
    #> wz2_1         0.06      0.13    -0.20     0.33 1.00     3469     2836
    #> wz3_1        -0.09      0.08    -0.25     0.06 1.00     3088     2932
    #> wz4_1         0.23      0.10     0.04     0.42 1.00     3225     2965
    #>
    #> Further Distributional Parameters:
    #>       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    #> sigma     2.38      0.03     2.33     2.44 1.00     5896     2788
    #>
    #> 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 | Interpretation
--- | ---
`bz1_1` | Between-person sleep (`TST` & `WAKE`) vs wake (`MVPA`, `LPA`, & `SB`) behaviours
`bz2_1` | Between-person `TST` vs `WAKE`
`bz3_1` | Between-person `MVPA` vs (`LPA` and `SB`)
`bz4_1` | Between-person `LPA` vs `SB`
`wz1_1` | Within-person `Sleep` (`TST` & `WAKE`) vs wake (`MVPA`, `LPA`, & `SB`) behaviours
`wz2_1` | Within-person `TST` vs `WAKE`
`wz3_1` | Within-person `MVPA` vs (`LPA` and `SB`)
`wz4_1` | Within-person `LPA` vs `SB`

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 `wz1_1` 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.

## 1.3 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 may utilize Bayes factors to answer the following question: “Which model (i.e., set of ILR predictors) 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(complr = cilr,
                 formula = Stress ~ 1 + (1 | ID),
                 iter = 6000, chains = 8, cores = 8, seed = 123, warmup = 1000,
                 backend = "cmdstanr", save_pars = save_pars(all = TRUE))

    # between-person composition only model
    m1 <- brmcoda(complr = cilr,
                 formula = Stress ~ bz1_1 + bz2_1 + bz3_1 + bz4_1 + (1 | ID),
                 iter = 6000, chains = 8, cores = 8, seed = 123, warmup = 1000,
                 backend = "cmdstanr", save_pars = save_pars(all = TRUE))

    # within-person composition only model
    m2 <- brmcoda(complr = cilr,
                 formula = Stress ~ wz1_1 + wz2_1 + wz3_1 + wz4_1 + (1 | ID),
                 iter = 6000, chains = 8, cores = 8, seed = 123, warmup = 1000,
                 backend = "cmdstanr", save_pars = save_pars(all = TRUE))

    # full model
    m <- brmcoda(complr = cilr,
                 formula = Stress ~ bz1_1 + bz2_1 + bz3_1 + bz4_1 +
                   wz1_1 + wz2_1 + wz3_1 + wz4_1 + (1 | ID),
                 iter = 6000, chains = 8, cores = 8, seed = 123, warmup = 1000,
                 backend = "cmdstanr", save_pars = save_pars(all = TRUE))

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)
    #> Error in `[.data.frame`(x, i, j, drop): undefined columns selected

    comparison
    #> Bayes Factors for Model Comparison
    #>
    #>     Model                                                                       BF
    #> [1] bz1_1 + bz2_1 + bz3_1 + bz4_1 + wz1_1 + wz2_1 + wz3_1 + wz4_1 + (1 | ID)  3.45
    #> [2] bz1_1 + bz2_1 + bz3_1 + bz4_1 + (1 | ID)                                 0.294
    #> [3] wz1_1 + wz2_1 + wz3_1 + wz4_1 + (1 | ID)                                 11.63
    #>
    #> * 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 \(BF\) = 11.00 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] bz1_1 + bz2_1 + bz3_1 + bz4_1 + (1 | ID) 0.085
    #> [3] wz1_1 + wz2_1 + wz3_1 + wz4_1 + (1 | ID)  3.37
    #> [4] 1 + (1 | ID)                             0.290
    #>
    #> * Against Denominator: [1] bz1_1 + bz2_1 + bz3_1 + bz4_1 + wz1_1 + wz2_1 + wz3_1 + wz4_1 + (1 | ID)
    #> *   Bayes Factor Type: marginal likelihoods (bridgesampling)

Again, our data favours the within-person composition only model over the full model, giving 2.79 times more support.

# 2 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, with the following arguments:

Argument | Description
--- | ---
`object` | A fitted `brmcoda` object
`base` | A `data.frame` or `data.table` of possible substitution of variables.
| This data set can be computed using function `possub`
`delta` | A integer, numeric value or vector indicating the amount of change in compositional parts for substitution
`level` | A character value or vector to specify whether the change in composition should be at `between`-person and/or `within`-person levels
`type` | A character value or vector to specify whether the estimated change in outcome should be `conditional` or `marginal`
`regrid` | Optional reference grid consisting of combinations of covariates over which predictions are made. If not provided, the default reference grid is used.
`summary` | A logical value to indicate whether the prediction at each level of the reference grid or an average of them should be returned.
`...` | Additional arguments to be passed to `describe_posterior`

## 2.1 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", ref = "grandmean")

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(summary(bsubm, level = "between", to = "TST"))

Mean | CI_low | CI_high | Delta | From | To | Level | Reference
--- | --- | --- | --- | --- | --- | --- | ---
0.02 | -0.01 | 0.05 | 5 | WAKE | TST | between | grandmean
0.00 | -0.01 | 0.02 | 5 | MVPA | TST | between | grandmean
0.01 | -0.01 | 0.02 | 5 | LPA | TST | between | grandmean
0.01 | -0.01 | 0.02 | 5 | SB | TST | between | grandmean

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.

## 2.2 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", ref = "grandmean")

Results for 5 minute substitution.

    knitr::kable(summary(wsubm, level = "within", to = "TST"))

Mean | CI_low | CI_high | Delta | From | To | Level | Reference
--- | --- | --- | --- | --- | --- | --- | ---
0.02 | 0.00 | 0.03 | 5 | WAKE | TST | within | grandmean
0.00 | -0.01 | 0.00 | 5 | MVPA | TST | within | grandmean
0.00 | -0.01 | 0.00 | 5 | LPA | TST | within | grandmean
0.00 | -0.01 | 0.00 | 5 | SB | TST | within | grandmean

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.

## 2.3 More interesting substitution models

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