Odds Ratios

The odds ratio (OR) measures the association between a binary predictor and a binary outcome. It is the ratio of two odds.

import numpy as np

import pandas as pd

from scipy import stats



# 2×2 contingency table

# Smoking vs Heart Disease

data = np.array([[80, 120],   # Smokers: disease, no disease

                 [30, 270]])  # Non-smokers: disease, no disease



smoker_odds = data[0,0] / data[0,1]

nonsmoker_odds = data[1,0] / data[1,1]

OR = smoker_odds / nonsmoker_odds



print("Smoking and Heart Disease:")

print(f"  Smokers: {data[0,0]} disease, {data[0,1]} no disease -> odds = {smoker_odds:.3f}")

print(f"  Non-smokers: {data[1,0]} disease, {data[1,1]} no disease -> odds = {nonsmoker_odds:.3f}")

print(f"  Odds Ratio = {OR:.3f}")

print(f"  Smokers have {OR:.1f}× the odds of heart disease vs non-smokers")



# 95% CI for OR (log-method)

log_OR = np.log(OR)

SE_log_OR = np.sqrt(sum(1/x for x in data.flatten()))

CI_lower = np.exp(log_OR - 1.96*SE_log_OR)

CI_upper = np.exp(log_OR + 1.96*SE_log_OR)

print(f"  95% CI: ({CI_lower:.3f}, {CI_upper:.3f})")



# Fisher's exact test

oddsratio_fisher, p_fisher = stats.fisher_exact(data)

print(f"  Fisher's exact p-value: {p_fisher:.6f}")



# OR from logistic regression

import statsmodels.api as sm

np.random.seed(42)

n = 500

smoking = np.random.binomial(1, 0.4, n)

heart_disease = np.random.binomial(1, 0.1 + 0.2*smoking)



X = sm.add_constant(smoking)

logit_model = sm.Logit(heart_disease, X).fit(disp=False)

or_logit = np.exp(logit_model.params['x1'])

ci_logit = np.exp(logit_model.conf_int().loc['x1'])

print(f"\nOR from logistic regression: {or_logit:.3f}")

print(f"95% CI: ({ci_logit[0]:.3f}, {ci_logit[1]:.3f})")



# OR vs Risk Ratio

p1 = data[0,0] / data[0].sum()

p2 = data[1,0] / data[1].sum()

RR = p1 / p2

print(f"\nOR = {OR:.3f}, Risk Ratio (RR) = {RR:.3f}")

print("OR overestimates effect when outcome is common (>10%)")

print("Use RR for cohort studies, OR for case-control studies")

Key Takeaways