Log-Ratio transformations for interval responses

Transform interval responses from the simplex space to the unbounded space using either Isometric Log-Ratio (ILR) or Sum Log-Ratio (SLR) transformations, as described by Smithson & Broomell (2024). These transformations preserve the dimensional conceptualization of the interval responses in terms of a location and a width. See also inv_ilr(), inv_slr() for the inverse transformations.

ILR

The ILR transformation equations are: $$x_{loc} = \sqrt{\frac{1}{2}} \log\left(\frac{x_1}{x_3}\right)$$ $$x_{wid} = \sqrt{\frac{2}{3}} \log\left(\frac{x_2}{\sqrt{x_1 x_3}}\right)$$

SLR

The SLR transformation equations are: $$x_{loc} = \log\left(\frac{x_1}{x_3}\right)$$ $$x_{wid} = \log\left(\frac{x_2}{x_1 + x_3}\right)$$

where \((x_1, x_2, x_3)\) is the interval response in the simplex format and \((x_{loc}, x_{wid})\) are the transformed values representing the unbounded location and width.

Usage

ilr(simplex)

slr(simplex)

Arguments

simplex

A numeric vector that is a 2-simplex (3 elements that sum to 1) or a dataframe where each of the rows is a 2-simplex.

Value

A numeric vector with 2 elements, the unbounded interval location and width, or a dataframe where each of the rows is a numeric vector with these 2 elements.

References

Smithson, M., & Broomell, S. B. (2024). Compositional data analysis tutorial. Psychological Methods, 29(2), 362–378.

See also

inv_ilr(), inv_slr()

Examples

# Generate some simplex data
simplex <- data.frame(rbind(c(.1, .2, .7), c(.4, .5, .1)))

# ILR transformation
ilr(simplex)
#>        x_loc      x_wid
#> 1 -1.3759663 -0.2284622
#> 2  0.9802581  0.7481482

# SLR transformation
slr(simplex)
#>       x_loc     x_wid
#> 1 -1.945910 -1.386294
#> 2  1.386294  0.000000