Log-contrast regression with compositional predictor variables {Compositional} R Documentation

## Log-contrast regression with compositional predictor variables

### Description

Log-contrast regression with compositional predictor variables.

### Usage

```lc.reg(y, x, z = NULL, xnew = NULL, znew = NULL)
```

### Arguments

 `y` A numerical vector containing the response variable values. This must be a continuous variable. `x` A matrix with the predictor variables, the compositional data. No zero values are allowed. `z` A matrix, data.frame, factor or a vector with some other covariate(s). `xnew` A matrix containing the new compositional data whose response is to be predicted. If you have no new data, leave this NULL as is by default. `znew` A matrix, data.frame, factor or a vector with the values of some other covariate(s). If you have no new data, leave this NULL as is by default.

### Details

The function performs the log-contrast regression model as described in Aitchison (2003), pg. 84-85. The logarithm of the compositional predictor variables is used (hence no zero values are allowed). The response variable is linked to the log-transformed data with the constraint that the sum of the regression coefficients equals 0. Hence, we apply constrained least squares, which has a closed form solution. The constrained least squares is described in Chapter 8.2 of Hansen (2019). The idea is to minimise the sum of squares of the residuals under the constraint R^T β = c, where c=0 in our case. If you want the regression without the zum-to-zero contraints see `ulc.reg`. Extra predictors variables are allowed as well, for instance categorical or continuous.

### Value

A list including:

 `be` The constrained regression coefficients. Their sum equals 0. `covbe` If covariance matrix of the constrained regression coefficients. `va` The estimated regression variance. `residuals` The vector of residuals. `est` If the arguments "xnew" and znew were given these are the predicted or estimated values, otherwise it is NULL.

### Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

### References

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

Hansen, B. E. (2019). Econometrics. https://www.ssc.wisc.edu/~bhansen/econometrics/Econometrics.pdf

```ulc.reg, lcreg.aov, lc.reg2, alfa.pcr, alfa.knn.reg ```
```y <- iris[, 1]