Introduction

We propose a new dependence measure \(\nu(Y, \mathbf{X})\) (A New Measure Of Dependence: Integrated R2) to assess how much a random vector \(\mathbf{X}\) explains a univariate response \(Y\). Let \(Y\) be a random variable and \(\mathbf{X} = (X_1, \cdots, X_p)\) a random vector defined on the same probability space. Let \(\mu\) be the probability law of \(Y\), and \(S\) be the support of \(\mu\). Define:

$$ \tilde{S} = \begin{cases} S \setminus \{s_{\max}\} & \text{if } S \text{ has a maximum } s_{\max} \\ S & \text{otherwise} \end{cases} $$

We define the measure \(\tilde{\mu}\) on \(S\) as:

$$ \tilde{\mu}(A) = \frac{\mu(A \cap \tilde{S})}{\mu(\tilde{S})}, \quad \text{for measurable } A \subseteq S $$

Then the irdc dependence coefficient is defined as:

$$ \nu(Y, \mathbf{X}) := \int \frac{\mathrm{Var}(\mathbb{E}[\mathbf{1}{Y > t} \mid \mathbf{X}])}{\mathrm{Var}(\mathbf{1}{Y > t})} d\tilde{\mu}(t) $$

In contrast, A Simple Measure Of Conditional Dependence consider:

$$ T(Y, \mathbf{X}) = \frac{\int \mathrm{Var}(\mathbb{E}[\mathbf{1}{Y \ge t} \mid \mathbf{X}]) d\mu(t)}{\int \mathrm{Var}(\mathbf{1}{Y \ge t}) d\mu(t)} $$

Continuous Case

n <- 1000
x <- matrix(runif(n * 3), nrow = n)
y <- (x[, 1] + x[, 2]) %% 1

irdc(y, x[, 1])
#> [1] 0.001002072
irdc(y, x[, 2])
#> [1] 0.04123161
irdc(y, x[, 3])
#> [1] 0.003291506

Discrete Case

Example 1

n <- 10000
s <- 0.1
x1 <- c(rep(0, n * s), runif(n * (1 - s)))
x2 <- runif(n)
y <- x1

irdc(y, x1, dist.type.X = "discrete")
#> [1] 0.9441587
irdc(y, x2)
#> [1] -0.01085533

Example 2

n <- 10000
x1 <- runif(n)
y1 <- rbinom(n, 1, 0.5)
y2 <- as.numeric(x1 >= 0.5)

irdc(y1, x1, dist.type.X = "discrete")
#> [1] -0.4999146
irdc(y2, x1, dist.type.X = "discrete")
#> [1] 0.003289474

FOCI::codec(y1, x1)
#> [1] -0.006410306
FOCI::codec(y2, x1)
#> [1] 1

Example 3: Hurdle vs Gamma Mixture

r_hurdle_poisson <- function(n, p_zero = 0.3, lambda = 2) {
  is_zero <- rbinom(n, 1, p_zero)
  rztpois <- function(m, lambda) {
    samples <- numeric(m)
    for (i in 1:m) {
      repeat {
        x <- rpois(1, lambda)
        if (x > 0) {
          samples[i] <- x
          break
        }
      }
    }
    samples
  }
  result <- numeric(n)
  result[is_zero == 0] <- rztpois(sum(is_zero == 0), lambda)
  result
}

set.seed(123)
n <- 1000
p_zero <- 0.4
lambda <- 10

hurdle <- r_hurdle_poisson(n, p_zero, lambda)
gamma_mix <- c(rep(0, round(p_zero * n)), rgamma(round((1 - p_zero) * n), shape = lambda, rate = 1))

df <- data.frame(
  value = c(hurdle, gamma_mix),
  source = rep(c("Hurdle Poisson", "Gamma Mixture"), each = n)
)

ggplot(df, aes(x = value, fill = source)) +
  geom_histogram(alpha = 0.5, position = "identity", bins = 40) +
  labs(title = "Comparison: Hurdle Poisson vs Gamma Mixture",
       x = "Value", y = "Count", fill = "Distribution") +
  theme_bw()

Example 3 Continued

x1 <- sort(gamma_mix)
y1 <- rbinom(n, 1, 0.5)
y2 <- sort(hurdle)

irdc(y1, x1, dist.type.X = "discrete")
#> [1] -0.5095727
irdc(y2, x1, dist.type.X = "discrete")
#> [1] 0.5443523

FOCI::codec(y1, x1)
#> [1] 0.04361745
FOCI::codec(y2, x1)
#> [1] 0.9969469

Example 4

x1 <- sort(hurdle)
y1 <- rbinom(n, 1, 0.5)
y2 <- sort(gamma_mix)

irdc(y1, x1, dist.type.X = "discrete")
#> [1] -0.5030198
irdc(y2, x1, dist.type.X = "discrete")
#> [1] 0.6265961

FOCI::codec(y1, x1)
#> [1] -0.02403687
FOCI::codec(y2, x1)
#> [1] 0.9450425

Conclusion

irdc provides a flexible and theoretically grounded dependence measure that works for both continuous and discrete predictors.

For further theoretical details, see our paper:
Azadkia and Roudaki (2025), A New Measure Of Dependence: Integrated R2