Regression with compositional data using the alpha-transformation {Compositional} R Documentation

## Regression with compositional data using the α-transformation

### Description

Regression with compositional data using the α-transformation.

### Usage

```alfa.reg(y, x, a, xnew = NULL, yb = NULL, seb = FALSE)
```

### Arguments

 `y` A matrix with the compositional data. `x` A matrix with the continuous predictor variables or a data frame including categorical predictor variables. `a` The value of the power transformation, it has to be between -1 and 1. If zero values are present it has to be greater than 0. If α=0 the isometric log-ratio transformation is applied and the solution exists in a closed form, since it the classical mutivariate regression. `xnew` If you have new data use it, otherwise leave it NULL. `yb` If you have already transformed the data using the α-transformation with the same α as given in the argument "a", put it here. Othewrise leave it NULL. This is intended to be used in the function `alfareg.tune` in order to speed up the process. The time difference in that function is small for small samples. But, if you have a few thousands and or a few more components, there will be bigger differences. `seb` Do you want the standard error of the coefficients to be returned? In the `alfareg.tune` function this extra computation that is avoided can save some time.

### Details

The α-transformation is applied to the compositional data first and then multivariate regression is applied. This involves numerical optimisation.

### Value

A list including:

 `runtime` The time required by the regression. `be` The beta coefficients. `seb` The standard error of the beta coefficients. `est` The fitted values for xnew if xnew is not NULL.

### Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Giorgos Athineou <gioathineou@gmail.com>.

### References

Tsagris M. (2015). Regression analysis with compositional data containing zero values. Chilean Journal of Statistics, 6(2): 47-57. https://arxiv.org/pdf/1508.01913v1.pdf

Tsagris M.T., Preston S. and Wood A.T.A. (2011). A data-based power transformation for compositional data. In Proceedings of the 4th Compositional Data Analysis Workshop, Girona, Spain. https://arxiv.org/pdf/1106.1451.pdf

Mardia K.V., Kent J.T., and Bibby J.M. (1979). Multivariate analysis. Academic press.

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

```alfareg.tune, diri.reg, js.compreg, kl.compreg, ols.compreg, comp.reg ```
```library(MASS)