power.py

from scipy.stats import norm


def detectable_difference(alpha, power, proportion, n_positives):
    """
    Calculate the detectable difference in proportions for a given sample size, power, significance level, and proportion.

    Parameters:
    alpha: Significance level
    power: Desired power of the test
    proportion: Proportion in the control group (e.g., existing provider detection rate)
    n_positives: Number of positive examples

    Returns:
    detectable_difference: Detectable difference in proportions
    """
    z_alpha = norm.ppf(1 - alpha / 2)
    z_power = norm.ppf(power)

    p1 = proportion
    p2 = p1  # Assuming null hypothesis that the proportions are equal
    p_bar = (p1 + p2) / 2
    q_bar = 1 - p_bar

    n1 = n_positives
    n2 = n_positives  # Assuming equal sample sizes

    delta = z_alpha * ((p_bar * q_bar * (1 / n1 + 1 / n2)) ** 0.5) + z_power * (
        (p1 * (1 - p1) / n1 + p2 * (1 - p2) / n2) ** 0.5
    )
    detectable_difference = delta / (1 + (p1 * (1 - p1) / p2 / (1 - p2)) ** 0.5)

    return detectable_difference


# Given parameters
alpha = 0.05
power = 0.80
provider_detection_rate = 0.8
total_images = 2000
occult_femoral_fracture_incidence = 0.03
positive_cases = int(total_images * occult_femoral_fracture_incidence)

# Calculate detectable difference
difference = detectable_difference(
    alpha, power, provider_detection_rate, positive_cases
)
print(
    f"The detectable difference in detection rate is approximately {difference * 100:.2f}%"
)