Horns index ($h$) — horns • unsum

horns() measures the dispersion in ordinal data based on scale limits or min/max values. The result is the actual variance as a proportion of the maximum possible variance. It ranges from 0 to 1:

0 means no variance, i.e., all observations have the same value.
1 means that the observations are evenly split between the extremes, with none in between.

horns_uniform() computes the value that horns() would return for a uniform distribution within given scale limits. This can be useful as a point of reference for horns().

These two functions correspond to the horns column of results and the uniform column of metrics_horns in closure_generate()'s output.

horns_rescaled() is a version of horns() that is normalized by scale length, such that 0.5 always indicates a uniform distribution, independent of the number of scale points. It is meant to enable comparison across scales of different lengths, but it is harder to interpret for an individual scale. This makes it unlikely to be useful in most cases. Even so, the range and the meaning of 0 and 1 are the same as for horns().

Usage

horns(freqs, scale_min, scale_max)

horns_uniform(scale_min, scale_max)

horns_rescaled(freqs, scale_min, scale_max)

Arguments

freqs: Numeric. Vector with the frequencies (relative or absolute) of binned observations; e.g., a vector with 5 elements for a 1-5 scale.
scale_min, scale_max: Numeric (length 1 each). Minimal and maximal possible values. For example, with a 1-7 Likert scale, use scale_min = 1 and scale_max = 7. Prefer the empirical min and max if available: they constrain the possible values further.

Value

Numeric (length 1).

Details

The horns index $h$ is defined as:

$$ h = \frac {\sum_{i=1}^{k} f_i (i - \bar{s})^2} {\frac{1}{4} (k - 1)^2} $$

where $k$ is the number of scale points (i.e., the length of freqs here), $f_i$ is the relative frequency of the $i$th point on an integer scale from $1$ to $k$, and $\bar{s}$ is the weighted mean frequency. The mean is derived as follows:

$$ \bar{s} = \sum_{i=1}^{k} i \ f_i $$

Note that $h$ only depends on the frequency distribution, not the actual values of the scale points. This is why both formulas invariably use a scale from $1$ to $k$. The only reason why the horns() function still takes scale_min and scale_max arguments is safety: if freqs is misstated such that its length is different from the number of points on the scale implied by those two arguments, there will be an error.

Maximum possible variance

The "maximum possible" variance here is based on scale range only (Popoviciu 1935):

$$ \sigma_{\max}^2 = \frac{1}{4} (k - 1)^2 $$

It is deliberately agnostic to the mean and the sample size. In the context of CLOSURE, these statistics are only relevant for generating possible samples, which then enter the equation via the numerator; just like the standard deviation. By contrast, the number of scale points, $k$, is an intrinsic property of those samples. It is sensible to assess the variance of the samples by benchmarking it against the greatest variance that can occur in any sample with the same $k$.

Uniform distribution

Although horns_uniform() is implemented as a wrapper around horns() that constructs a perfect uniform distribution internally, an equivalent closed-form solution can be given as

$$ h_u = \frac{k + 1}{3 (k - 1)} $$

"Horns of no confidence"

The term horns index was inspired by Heathers (2017) which defines the "horns of no confidence" as a reconstructed sample "where an incorrect, impossible or unlikely value set has all its constituents stacked into its highest or lowest bins to try meet a ludicrously high SD". In its purest form, this is a case where $h = 1$, so horns() would return 1. However, note that the implications for the plausibility of any given set of summary statistics depend on the substantive context of the data (Heathers et al. 2018).

Heathers wrote about SPRITE, but the logic applies equally to CLOSURE. Indeed, as CLOSURE is exhaustive but SPRITE is not, it is only CLOSURE that provides certainty about the set of possible samples and their horns indices.

Rust implementation

The metrics_horns tibble that is part of the output of closure_generate() is not based on the R functions presented here. Instead, it relies on efficient Rust implementations of the above formulas. These Rust functions are part of closure-core, which mainly implements CLOSURE but does not currently export horns functions for users.

References

Popoviciu, T. (1935). Sur les équations algébriques ayant toutes leurs racines réelles. Mathematica (Cluj), 9, 129-145.

Examples

# For simplicity, all examples use a 1-5 scale and a total N of 300.

# ---- With all values at the extremes

horns(freqs = c(300, 0, 0, 0, 0), scale_min = 1, scale_max = 5)
#> [1] 0

horns(c(150, 0, 0, 0, 150), 1, 5)
#> [1] 1

horns(c(100, 0, 0, 0, 200), 1, 5)
#> [1] 0.8888889


# ---- With some values in between

horns(c(60, 60, 60, 60, 60), 1, 5)
#> [1] 0.5

horns(c(200, 50, 30, 20, 0), 1, 5)
#> [1] 0.2113889

horns(c(150, 100, 50, 0, 0), 1, 5)
#> [1] 0.1388889

horns(c(100, 40, 20, 40, 100), 1, 5)
#> [1] 0.7333333

Horns index (\(h\))