Calculating power of hypothesis tests.

The code is from the Data Science from Scratch book.

Libraries and helper functions

from typing import Tuple
import math as m
def calc_normal_cdf(x: float, mu: float = 0, sigma: float = 1) -> float:
    return (1 + m.erf((x - mu) / m.sqrt(2) / sigma)) / 2

normal_probability_below = calc_normal_cdf
def normal_probability_between(lo: float, hi: float, mu: float = 0, sigma: float = 1) -> float:
    return normal_probability_below(hi, mu, sigma) - normal_probability_below(lo, mu, sigma)
def calc_normal_cdf(x: float, mu: float = 0, sigma: float = 1) -> float:
    return (1 + m.erf((x - mu) / m.sqrt(2) / sigma)) / 2
def calc_inverse_normal_cdf(p: float, mu:float = 0, sigma: float = 1, tolerance: float = 1E-5, show_steps=False) -> float:

    if p == 0: return -np.inf
    if p == 1: return np.inf

    # In case it is not a standard normal distribution, calculate the standard normal first and then rescale
    if mu != 0 or sigma != 1:
        return mu + sigma * calc_inverse_normal_cdf(p, tolerance=tolerance)

    low_z = -10
    hi_z = 10

    if show_steps: print(f"{'':<19}".join(['low_z', 'mid_z', 'hi_z']), "\n")

    while hi_z - low_z > tolerance:
        mid_z = (low_z + hi_z) / 2
        mid_p = calc_normal_cdf(mid_z)

        if mid_p < p:
            low_z = mid_z
        else:
            hi_z = mid_z

        if show_steps: print("\t".join(map(to_string, [low_z, mid_z, hi_z])))

    return mid_z
def normal_upper_bound(probabilty: float, mu: float = 0, sigma: float = 1) -> float:
    return calc_inverse_normal_cdf(probabilty, mu, sigma)
def normal_lower_bound(probabilty: float, mu: float = 0, sigma: float = 1) -> float:
    return calc_inverse_normal_cdf(1 - probabilty, mu, sigma)
def normal_two_sided_bounds(probability: float, mu: float = 0, sigma: float = 1) -> float:
    if probability == 0: return 0, 0

    tail_probability = (1 - probability) / 2

    lower_bound = normal_upper_bound(tail_probability, mu, sigma)
    upper_bound = normal_lower_bound(tail_probability, mu, sigma)
    
    return lower_bound, upper_bound
def normal_approximation_to_binomial(n: int, p: float) -> Tuple[float, float]:
    mu = p * n
    sigma = m.sqrt(p * (1 - p) * n)

    return mu, sigma

Type 1 Error and Tolerance

Let's make our null hypothesis ($H_0$) that the probability of head is 0.5

mu_0, sigma_0 = normal_approximation_to_binomial(1000, 0.5) 
mu_0, sigma_0
(500.0, 15.811388300841896)

We define our tolerance at 5%. That is, we accept our model to produce 'type 1' errors (false positive) in 5% of the time. With the coin flipping example, we expect to receive 5% of the results to fall outsied of our defined interval.

lo, hi = normal_two_sided_bounds(0.95, mu_0, sigma_0)
lo, hi
(469.01026640487555, 530.9897335951244)

Type 2 Error and Power

At type 2 error we consider false negatives, that is, those cases where we fail to reject our null hypothesis even though we should.

Let's assume that contra $H_0$ the actual probability is 0.55.

mu_1, sigma_1 = normal_approximation_to_binomial(1000, 0.55)
mu_1, sigma_1
(550.0, 15.732132722552274)

In this case we get our Type 2 probability as the overlapping of the real distribution and the 95% probability region of $H_0$. In this particular case, in 11% of the cases we will wrongly fail to reject our null hypothesis.

type_2_probability = normal_probability_between(lo, hi, mu_1, sigma_1)
type_2_probability
0.1134519987046329

The power of the test is then the probability of rightly rejecting the $H_0$

power = 1 - type_2_probability
power
0.886548001295367

Now, let's redefine our null hypothesis so that we expect the probability of head to be less than or equal to 0.5.

In this case we have a one-sided test.

hi = normal_upper_bound(0.95, mu_0, sigma_0)
hi
526.0073585242053

Because this is a less strict hypothesis than our previus one, it has a smaller T2 probability and a greater power.

type_2_probability = normal_probability_below(hi, mu_1, sigma_1)
type_2_probability
0.06362051966928273
power = 1 - type_2_probability
power
0.9363794803307173