PearsonrEquivalence

class pgmpy.ci_tests.PearsonrEquivalence(data: DataFrame, delta_threshold: float = 0.1)

Bases: Pearsonr

Pearson equivalence test [1] for conditional independence on continuous data.

This test first computes the partial correlation coefficient \(\hat{\rho}_{XY \mid Z}\) using Pearsonr. Let \(\delta\) denote delta_threshold. The Fisher transform is computed as:

\[z_\rho = \operatorname{arctanh}(\hat{\rho}_{XY \mid Z}), \qquad z_\delta = \operatorname{arctanh}(\delta),\]

and defines

\[c = \sqrt{n - |Z| - 3},\]

where \(n\) is the sample size and \(|Z|\) is the number of conditioning variables.

The test then performs a TOST (two one-sided tests) procedure for the equivalence hypothesis

\[H_0: \rho_{XY \mid Z} \leq -\delta \;\; \text{or} \;\; \rho_{XY \mid Z} \geq \delta \qquad \text{vs.} \qquad H_1: -\delta < \rho_{XY \mid Z} < \delta.\]

The two one-sided test statistics are:

\[T_{\mathrm{lower}} = c (z_\rho + z_\delta), \qquad T_{\mathrm{upper}} = c (z_\rho - z_\delta),\]

with corresponding p-values:

\[p_{\mathrm{lower}} = 1 - \Phi(T_{\mathrm{lower}}), \qquad p_{\mathrm{upper}} = \Phi(T_{\mathrm{upper}}),\]

where \(\Phi\) is the standard normal CDF. The reported p-value is:

\[p = \max(p_{\mathrm{lower}}, p_{\mathrm{upper}}).\]

Parameters:

datapandas.DataFrame

The dataset in which to test the independence condition.

delta_thresholdfloat

The equivalence bound (threshold for practical independence).

Attributes:

statistic_float

Fisher z-transformed correlation coefficient \(z_\rho\). Set after calling the test.

p_value_float

The p-value from the TOST procedure. Independence is concluded when p_value_ < significance_level (opposite of standard CI tests). Set after calling the test.

References

[1]

Malinsky, Daniel. “A cautious approach to constraint-based causal model selection.” arXiv preprint arXiv:2404.18232 (2024).

is_independent(X: str, Y: str, Z: list | tuple = (), significance_level: float = 0.05) → bool

Perform the equivalence CI test.

Note: Independence is concluded when p_value_ < significance_level (rejecting the null of dependence), which is the OPPOSITE of standard CI tests.

Returns:

bool

True if X ⊥⊥ Y | Z (p_value_ < significance_level), else False.

run_test(X: str, Y: str, Z: list)

Compute Pearson equivalence statistic and p-value.

Sets self.statistic_ (Fisher z-transformed partial correlation) and self.p_value_.