Discretised Chi-Squared (DiscChiSq) Test

DiscChiSq is an adapter that applies equal-frequency discretisation to continuous variables before running the standard :doc:/tests/chi_sq_test (Agresti, 2013). It provides a practical baseline for applying classical contingency-table CI tests to continuous or mixed data, at the cost of the information loss inherent in binning.

Intuition. Once continuous variables are mapped to discrete bins, CI testing reduces to verifying the factorisation of a finite joint probability table — the setting Pearson’s \(\chi^2\) test was designed for (Agresti, 2013). The trade-off is that the null hypothesis being tested is no longer conditional independence in the continuous space, but rather independence between discretised representations of the variables.

Mathematical Formulation

For each continuous column \(V\) of the input data, the adapter produces a discrete version

\[\widetilde{V}_i = \mathrm{quantile\_bin}(V_i; b)\]

where quantile_bin assigns each observation to one of \(b\) equal-frequency bins (i.e., bins defined by the empirical \(1/b, 2/b, \ldots, (b-1)/b\) quantiles). Columns that are already categorical are left unchanged. The Pearson chi-squared statistic is then computed on the binned data (Agresti, 2013):

\[\chi^2 = \sum_{i} \frac{(O_i - E_i)^2}{E_i}\]

where \(O_i\) and \(E_i\) are the observed and expected cell counts in the binned contingency table under the null \(X \perp Y \mid Z\) (Agresti, 2013). Under the null and standard regularity conditions, the statistic is asymptotically \(\chi^2\)-distributed with degrees of freedom matching the binned cardinalities (Agresti, 2013); see :doc:/tests/chi_sq_test.

Assumptions

  • Discretisation preserves dependence. The number of bins must be coarse enough to keep cells populated, but fine enough that the underlying dependence is still detectable after binning (Agresti, 2013).

  • Independent observations. Same multinomial sampling assumption as Pearson’s \(\chi^2\) (Agresti, 2013).

  • Adequate cell counts. Cochran (1954) recommends that the test may be unreliable if more than 20% of cells have an expected frequency below 5 or any cell has expected frequency below 1; the same rule applies after binning (Agresti, 2013).

  • Approximate CI procedure. Discretisation tests independence between binned representations rather than the underlying continuous CI null; CI verdicts should be treated as approximate (Agresti, 2013).

  • Dtype validation is opt-in. Passing data outside the declared dtype produces undefined results; call Test.validate_data(data) to check. citk does not enforce supported_dtypes at construction.

Code Example

import numpy as np
from citk.tests import DiscChiSq

# Continuous chain: X -> Z -> Y
n = 500
X = np.random.randn(n)
Z = 0.8 * X + 0.5 * np.random.randn(n)
Y = 0.8 * Z + 0.5 * np.random.randn(n)
data = np.vstack([X, Y, Z]).T

# Initialize the test with 4 equal-frequency bins per continuous variable
disc_chisq_test = DiscChiSq(data, n_bins=4)

# Test for conditional independence of X and Y given Z
# Expected: p-value is large (cannot reject H0 of independence)
p_value_conditional = disc_chisq_test(0, 1, [2])
print(f"P-value for X _||_ Y | Z: {p_value_conditional:.4f}")

# Test for unconditional independence of X and Y
# Expected: p-value is small (reject H0 of independence)
p_value_unconditional = disc_chisq_test(0, 1)
print(f"P-value for X _||_ Y: {p_value_unconditional:.4f}")

API Reference

For a full list of parameters, see the API documentation: :class:citk.tests.adapter_tests.DiscChiSq.

References

Agresti, A. (2013). Categorical Data Analysis (3rd ed.). Wiley.

Cochran, W. G. (1954). Some methods for strengthening the common \(\chi^2\) tests. Biometrics, 10(4), 417-451.