Fisher’s Z Test

Fisher’s Z is the canonical partial-correlation–based conditional independence test for continuous data. Under multivariate normality, conditional independence between \(X\) and \(Y\) given \(Z\) is equivalent to zero partial correlation \(\rho(X, Y \mid Z)\) (Anderson, 2003).

Intuition. A variance-stabilising transform converts the (partial) Pearson correlation into a statistic that is asymptotically standard normal under the null, yielding analytic p-values without resampling (Hotelling, 1953; Anderson, 2003).

Mathematical Formulation

The statistical basis of the Z-transform of the (partial) Pearson correlation coefficient was established by Hotelling (1953); its application to partial correlations and CI testing in Gaussian models is developed in classical multivariate analysis (Anderson, 2003). For sample partial correlation \(r = \rho(X, Y \mid Z)\), the transform is

\[Z(r) = \frac{1}{2} \ln\!\left(\frac{1+r}{1-r}\right)\]

equivalently \(\operatorname{artanh}(r)\) (Hotelling, 1953). Under the null \(X \perp Y \mid Z\) in a Gaussian model, the standardised statistic

\[T = \sqrt{n - |Z| - 3} \cdot |Z(r)|\]

is asymptotically \(N(0, 1)\) (Anderson, 2003), where \(n\) is the sample size and \(|Z|\) is the cardinality of the conditioning set.

Assumptions

  • Multivariate normality. Zero partial correlation is equivalent to conditional independence only under restrictive distributional assumptions; Baba et al. (2004) formalised the precise conditions, showing that violations of Gaussianity or linearity can invalidate correlation-based CI decisions even asymptotically.

  • Linearity. The test targets linear (partial) dependence. Outside the Gaussian / elliptical family the equivalence between zero partial correlation and conditional independence breaks down (Baba et al., 2004).

  • Sample size and conditioning-set size. The effective degrees of freedom are \(n - |Z| - 3\); as \(|Z|\) grows, variance increases and power drops (Anderson, 2003).

  • 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 FisherZ

# Generate data where X and Y are independent given Z
# X -> Z -> Y
n = 500
X = np.random.randn(n)
Z = 2 * X + np.random.randn(n)
Y = 3 * Z + np.random.randn(n)
data = np.vstack([X, Y, Z]).T

# Initialize the test
fisher_z_test = FisherZ(data)

# Test for conditional independence of X and Y given Z
# Expected: p-value is large (cannot reject H0 of independence)
p_value_conditional = fisher_z_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 = fisher_z_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.partial_correlation_tests.FisherZ.

References

Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis (3rd ed.). Wiley-Interscience.

Baba, K., Shibata, R., & Sibuya, M. (2004). Partial correlation and conditional correlation as measures of conditional independence. Australian & New Zealand Journal of Statistics, 46(4), 657-664.

Hotelling, H. (1953). New light on the correlation coefficient and its transforms. Journal of the Royal Statistical Society: Series B (Methodological), 15(2), 193-225.