hstats (version 1.2.2)

h2_threeway: Three-way Interaction Strength

Description

Friedman and Popescu's statistic of three-way interaction strength, see Details. Use plot() to get a barplot. In hstats(), set threeway_m to a value above 2 to calculate this statistic for all feature triples of the threeway_m features with strongest overall interaction.

Usage

h2_threeway(object, ...)
# S3 method for default
h2_threeway(object, ...)
# S3 method for hstats
h2_threeway(
  object,
  normalize = TRUE,
  squared = TRUE,
  sort = TRUE,
  zero = TRUE,
  ...
)

Value

An object of class "hstats_matrix" containing these elements:

• M: Matrix of statistics (one column per prediction dimension), or NULL.

• SE: Matrix with standard errors of M, or NULL. Multiply with sqrt(m_rep) to get standard deviations instead. Currently, supported only for perm_importance().

• m_rep: The number of repetitions behind standard errors SE, or NULL. Currently, supported only for perm_importance().

• statistic: Name of the function that generated the statistic.

• description: Description of the statistic.

Arguments

object

Object of class "hstats".

...

Currently unused.

normalize

Should statistics be normalized? Default is TRUE.

squared

Should squared statistics be returned? Default is TRUE.

sort

Should results be sorted? Default is TRUE. (Multi-output is sorted by row means.)

zero

Should rows with all 0 be shown? Default is TRUE.

Methods (by class)

• h2_threeway(default): Default pairwise interaction strength.

• h2_threeway(hstats): Pairwise interaction strength from "hstats" object.

Details

Friedman and Popescu (2008) describe a test statistic to measure three-way interactions: in case there are no three-way interactions between features \(x_j\), \(x_k\) and \(x_l\), their (centered) three-dimensional partial dependence function \(F_{jkl}\) can be decomposed into lower order terms: $$ F_{jkl}(x_j, x_k, x_l) = B_{jkl} - C_{jkl} $$ with $$ B_{jkl} = F_{jk}(x_j, x_k) + F_{jl}(x_j, x_l) + F_{kl}(x_k, x_l) $$ and $$ C_{jkl} = F_j(x_j) + F_k(x_k) + F_l(x_l). $$

The squared and scaled difference between the two sides of the equation leads to the statistic $$ H_{jkl}^2 = \frac{\frac{1}{n} \sum_{i = 1}^n \big[\hat F_{jkl}(x_{ij}, x_{ik}, x_{il}) - B^{(i)}_{jkl} + C^{(i)}_{jkl}\big]^2}{\frac{1}{n} \sum_{i = 1}^n \hat F_{jkl}(x_{ij}, x_{ik}, x_{il})^2}, $$ where $$ B^{(i)}_{jkl} = \hat F_{jk}(x_{ij}, x_{ik}) + \hat F_{jl}(x_{ij}, x_{il}) + \hat F_{kl}(x_{ik}, x_{il}) $$ and $$ C^{(i)}_{jkl} = \hat F_j(x_{ij}) + \hat F_k(x_{ik}) + \hat F_l(x_{il}). $$ Similar remarks as for h2_pairwise() apply.

References

Friedman, Jerome H., and Bogdan E. Popescu. "Predictive Learning via Rule Ensembles." The Annals of Applied Statistics 2, no. 3 (2008): 916-54.

See Also

hstats(), h2(), h2_overall(), h2_pairwise()

Examples

# MODEL 1: Linear regression
fit <- lm(uptake ~ Type * Treatment * conc, data = CO2)
s <- hstats(fit, X = CO2[, 2:4], threeway_m = 5)
h2_threeway(s)

#' MODEL 2: Multivariate output (taking just twice the same response as example)
fit <- lm(cbind(up = uptake, up2 = 2 * uptake) ~ Type * Treatment * conc, data = CO2)
s <- hstats(fit, X = CO2[, 2:4], threeway_m = 5)
h2_threeway(s)
h2_threeway(s, normalize = FALSE, squared = FALSE)  # Unnormalized H
plot(h2_threeway(s))