Power Analysis

Statistical power is the probability that a test correctly rejects a false null hypothesis. HypoTestX provides functions to calculate required sample sizes before a study and to compute post-hoc power after data collection.

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Required sample size — two-sample t-test

How many participants per group do you need to detect an effect of a given size?

import hypotestx as hx

n = hx.n_ttest_two_sample(effect_size=0.5, alpha=0.05, power=0.8)
print(f"Required n per group: {n}")   # 64

Parameter	Type	Default	Description
effect_size	float	—	Cohen’s d (standardised mean difference)
alpha	float	0.05	Significance level (Type I error rate)
power	float	0.80	Desired power (1 − Type II error rate)
alternative	str	"two-sided"	"two-sided", "greater", or "less"

Common effect size guidelines (Cohen’s d)

d	Magnitude
0.2	Small
0.5	Medium
0.8	Large

# Need to detect a small effect with 90% power
n_small = hx.n_ttest_two_sample(effect_size=0.2, alpha=0.05, power=0.9)
print(f"Small effect, 90% power: n = {n_small} per group")

# One-tailed test (directional hypothesis)
n_one = hx.n_ttest_two_sample(
    effect_size=0.5, alpha=0.05, power=0.8, alternative="greater"
)

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Post-hoc power — two-sample t-test

After collecting data, compute the power actually achieved:

pow_result = hx.power_ttest_two_sample(
    effect_size=0.4,
    n1=30,
    n2=30,
    alpha=0.05,
    alternative="two-sided",
)
print(f"Achieved power: {pow_result.power:.2f}")

Parameter	Type	Description
effect_size	float	Cohen’s d
n1	int	Size of group 1
n2	int	Size of group 2
alpha	float	Significance level used in the test
alternative	str	"two-sided", "greater", or "less"

The returned object has a .power attribute (float 0–1).

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Worked Example: Planning a Study

import hypotestx as hx

# Before the study: power calculation
# Expecting a medium effect (d = 0.5), standard alpha, 80% power
n_required = hx.n_ttest_two_sample(effect_size=0.5, alpha=0.05, power=0.8)
print(f"Collect at least {n_required} participants per group")

# After the study: you collected 45 per group
# Effect observed was d = 0.42 (slightly smaller than expected)
achieved = hx.power_ttest_two_sample(
    effect_size=0.42,
    n1=45,
    n2=45,
    alpha=0.05,
)
print(f"Post-hoc power: {achieved.power:.1%}")

# Run the actual test
result = hx.ttest_2samp(group1, group2, alpha=0.05)
print(result.summary())

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Power Curves

To visualise how power changes with sample size:

import hypotestx as hx

effect = 0.5     # medium effect
alpha  = 0.05

ns     = list(range(10, 200, 5))
powers = [
    hx.power_ttest_two_sample(effect, n1=n, n2=n, alpha=alpha).power
    for n in ns
]

# Find minimum n for 80% power
adequate = [(n, p) for n, p in zip(ns, powers) if p >= 0.80]
if adequate:
    print(f"First n with ≥80% power: {adequate[0][0]}")