Interpreting Test Results

After running a conditional independence (CI) test, the primary output you will work with is the p-value. Understanding what this value represents is key to correctly interpreting the outcome of your analysis.

The P-Value

A CI test starts with a null hypothesis (\(H_0\)), which states that the two variables of interest, X and Y, are conditionally independent given the conditioning set Z.

\(H_0: X \perp Y \mid Z\)

The p-value is the probability of observing a test statistic at least as extreme as the one computed from your data, assuming the null hypothesis is true.

How to Interpret It

High P-Value (e.g., > 0.05): A high p-value suggests that your data is consistent with the null hypothesis. You fail to reject the null hypothesis and conclude that there is not enough evidence to say that X and Y are conditionally dependent. For the purposes of causal discovery algorithms like PC, this is treated as evidence for conditional independence.
Low P-Value (e.g., <= 0.05): A low p-value suggests that your data is unlikely to have occurred if the null hypothesis were true. You reject the null hypothesis and conclude that X and Y are conditionally dependent given Z.

The threshold used to make this decision (commonly 0.05 or 0.01) is known as the significance level (alpha). This is a parameter you set when running algorithms like PC.

Accessing the Results

All tests in citk that are compatible with causal-learn are designed to be called directly by the search algorithm (like pc). The algorithm handles the interpretation internally.

When you call a test yourself, it returns the p-value directly.

import numpy as np
from citk.tests import FisherZ

# Generate some dependent data
data = np.random.randn(200, 2)
data[:, 1] = 2 * data[:, 0] + 0.1 * np.random.randn(200)

# Initialize and run the test
test = FisherZ(data)
p_value = test(0, 1)

print(f"P-value: {p_value:.4f}")

if p_value <= 0.05:
    print("Conclusion: The variables are dependent.")
else:
    print("Conclusion: We cannot reject that the variables are independent.")