Please note that this package (`codaredistlm`, Compositional Data Analysis [CoDA] redistribution linear model) is the new actively maintained package previously known as deltacomp.

# The `codaredistlm` package

Functions to analyse compositional data and produce predictions (with confidence intervals) for relative increases and decreases in the compositional parts.

## 1. Background

For an outcome variable `Y`, D compositional parts (`x_1, ..., x_D`) and C covariates (`z_1, ..., z_C`); this package fits the compositional data analysis model (notation inexact):

`Y = b_0 + b_1 ilr_1 + ... + b_{D-1} ilr_{D-1} + a_1 z_1 + ... + a_C z_C + e`

where `ilr_i` are the D-1 isometric log ratio variables derived from the D compositional parts (`x_1, ..., x_D`), `b_0, ..., b_{D-1}, a_1, ..., a_C` are D+C parameters to be estimated and `e ~ N(0, sigma)` is the error. The package then makes predictions in alterations of the time-use variables (the linearly dependent set of compositional parts) based on this model.

For a starting point to learn about compositional data analysis please see Aitchison (1982) or van den Boogaart and Tolosana-Delgado (2013). However the articles Dumuid et al. (2017a) and Dumuid et al. (2017b) may be more approachable introductions.

## 2. Reallocation of time-use component options

Please note that the use of ‘mean composition’ means the geometric mean on the compositional simplex and not the arithmetic mean. If these words have little meaning to you, that is no problems as these differently calculated means likely do not differ much in your dataset. `codaredistlm` only uses the simplex geometric mean in its calculations from version 0.2.0 onwards.

### 2.1. Option `comparisons = "prop-realloc"`

Information on outcome prediction with time-use exchange between one part and the remaining compositional parts proportionally (`comparisons = "prop-realloc"` option of the `predict_delta_comps()` function), please see Dumuid et al. (2017a).

### 2.1.1. Example

Suppose you have three (predictor) parts in a day summing to 1 (e.g., a day) to predict an outcome variable. The three parts are `sedentary`, `sleep` and `activity`. Let’s assume the mean sampled composition is:

• `sedentary = 0.5` (i.e., half a day)
• `sleep = 0.3` (i.e., 30% a day)
• `activity = 0.2` (i.e., 20% a day)

If you wanted to predict the change in the outcome variable from the above mean composition with `delta = +0.05` (5% of the day) is added to `sedentary`, the option `comparisons = "prop-realloc"` reduces the remaining parts by the 5% proportionately based on their mean values, illustrated below:

• `sedentary* = 0.5 + delta = 0.5 + 0.05 = 0.55`
• `sleep* = 0.3 - delta * sleep / (sleep + activity) = 0.3 - 0.05 * 0.3 / (0.3 + 0.2) = 0.3 - 0.03 = 0.27`
• `activity* = 0.2 - delta * activity / (sleep + activity) = 0.2 - 0.05 * 0.2 / (0.3 + 0.2) = 0.2 - 0.02 = 0.18`

Noting that the new compsition: `sedentary* + sleep* + activity* = 0.55 + 0.27 + 0.18 = 1`.

Note for the example above, the option `comparisons = "prop-realloc"` in `predict_delta_comps()` will actually automatically produce separate predictions for a `delta = +0.05` on each of the parts against the remaining parts. i.e., not only the `sedentary* = 0.5 + delta` scenario as illustrated above but also `sleep* = 0.3 + delta` and `activity* = 0.2 + delta` cases.

### 2.2. Option `comparisons = "one-v-one"`

For information on outcome prediction with time-use exchange between two compositional parts (i.e., the `comparisons = "one-v-one"` option of the `predict_delta_comps()` function), please see Dumuid et al. (2017b).

### 2.2.1. Example

Similarly to the previous example, suppose you have three (predictor) parts in a day summing to 1 (i.e. a day) to predict an outcome variable. The three parts are `sedentary`, `sleep` and `activity`. Let’s assume the mean sampled composition is:

• `sedentary = 0.5` (i.e., half a day)
• `sleep = 0.3` (i.e., 30% a day)
• `activity = 0.2` (i.e., 20% a day)

If you wanted to predict the change in the outcome variable from the above mean composition with `delta = +0.05` (5% of the day), the option `comparisons = "one-v-one"` looks at all pairwise exchanges between the parts `(sedentary*, sleep*, activity*)`:

• `(0.5 + 0.05, 0.3 - 0.05, 0.2 )`
• `(0.5 + 0.05, 0.3 , 0.2 - 0.05)`
• `(0.5 , 0.3 + 0.05, 0.2 - 0.05)`
• `(0.5 - 0.05, 0.3 + 0.05, 0.2 )`
• `(0.5 - 0.05, 0.3 , 0.2 + 0.05)`
• `(0.5 , 0.3 - 0.05, 0.2 + 0.05)`

## 3. Datasets in package

Two datasets are supplied with the package:

• `fairclough` and
• `fat_data`.

The `fairclough` dataset was kindly provided by the authors of Fairclough et al. (2017). `fat_data` is a randomly generated test dataset that might roughly mimic a real dataset.

## 4. Example usage

``````library(devtools) # see https://www.r-project.org/nosvn/pandoc/devtools.html
devtools::install_github('tystan/codaredistlm')
library(codaredistlm)
### see help file to run example
?predict_delta_comps

predict_delta_comps(
dataf = fat_data,
y = "fat",
comps = c("sl", "sb", "lpa", "mvpa"),
covars = c("sibs", "parents", "ed"),
deltas = seq(-60, 60, by = 5) / (24 * 60),
comparisons = "prop-realloc",
alpha = 0.05
)

# OR

predict_delta_comps(
dataf = fat_data,
y = "fat",
comps = c("sl", "sb", "lpa", "mvpa"),
covars = c("sibs", "parents", "ed"),
deltas = seq(-60, 60, by = 5) / (24 * 60),
comparisons = "one-v-one",
alpha = 0.05
)``````

## 5. Output and plotting results

Output is a `data.frame` that can be turned into the plot below using the following code.

``````
pred_df <-
predict_delta_comps(
dataf = fairclough,
y = "z_bmi",
comps = c("sleep", "sed", "lpa", "mvpa"),
covars = c("decimal_age", "sex"),
# careful deltas greater than 25 min in magnitude induce negative compositions
deltas =  seq(-20, 20, by = 5) / (24 * 60),
comparisons = "prop-realloc", # or try "one-v-one"
alpha = 0.05
)

plot_delta_comp(
pred_df, # provide the returned object from predict_delta_comps()
# x-axis can be converted from propotion of composition to meaningful units
comp_total = 24 * 60, # minutes available in the composition
units_lab = "min" # just a label for plotting
)
``````

### 5.1. Prediction for the mean composition

The function `predict_delta_comps()` now outputs the predicted outcome value (with `100 * (1 - alpha)`% confidence interval). This data is printed to the console but also can be extracted from the output of `predict_delta_comps()` as per the below code:

``````
# produces a 1 line data.frame that contains
#    the (simplex/geometric) mean composition,
#    the "average" covariates (the median of the factor variables in order of the levels are taken as default),
#    the ilr coords of the (simplex/geometric) mean composition, and
#    the predicted outcome value with 100*(1-alpha)% confidence interval
attr(pred_df, "mean_pred")
``````

## 6. Release notes

See /change-notes.md.