One-Sided vs Two-Sided Tests¶

The choice between a one-sided and two-sided test determines how the rejection region is allocated and directly affects the p-value. A two-sided test checks whether the parameter differs from the null value in either direction, while a one-sided test concentrates all rejection probability in one tail. This choice must be made before looking at the data, based on the research question.

Two-Sided Tests¶

A two-sided (two-tailed) test uses the hypotheses

\[ H_0: \mu = \mu_0 \quad \text{vs} \quad H_1: \mu \neq \mu_0 \]

The rejection region splits \(\alpha\) equally between both tails. For a \(t\)-test, reject \(H_0\) when

\[ |t| > t_{\alpha/2,\, n-1} \]

The two-sided p-value is

\[ p = 2 \cdot P(T \ge |t_{\text{obs}}|) \]

where \(T\) follows the \(t\)-distribution with \(n - 1\) degrees of freedom under \(H_0\).

import numpy as np
from scipy import stats

np.random.seed(42)
data = np.random.normal(loc=52, scale=10, size=30)

# Two-sided test: H0: mu = 50 vs H1: mu != 50
t_stat, p_two = stats.ttest_1samp(data, 50)
print(f"t = {t_stat:.3f}")
print(f"Two-sided p-value: {p_two:.4f}")

One-Sided Tests¶

Right-Tailed (Greater Than)¶

A right-tailed test checks whether the parameter exceeds the null value:

\[ H_0: \mu \le \mu_0 \quad \text{vs} \quad H_1: \mu > \mu_0 \]

The rejection region places all \(\alpha\) in the right tail. Reject \(H_0\) when

\[ t > t_{\alpha,\, n-1} \]

The one-sided p-value is

\[ p = P(T \ge t_{\text{obs}}) \]

# Right-tailed: H0: mu <= 50 vs H1: mu > 50
t_stat, p_two = stats.ttest_1samp(data, 50)

# Convert two-sided to one-sided (right tail)
if t_stat > 0:
    p_right = p_two / 2
else:
    p_right = 1 - p_two / 2

print(f"t = {t_stat:.3f}")
print(f"Right-tailed p-value: {p_right:.4f}")

Left-Tailed (Less Than)¶

A left-tailed test checks whether the parameter is below the null value:

\[ H_0: \mu \ge \mu_0 \quad \text{vs} \quad H_1: \mu < \mu_0 \]

Reject \(H_0\) when \(t < -t_{\alpha,\, n-1}\). The one-sided p-value is

\[ p = P(T \le t_{\text{obs}}) \]

# Left-tailed: H0: mu >= 50 vs H1: mu < 50
if t_stat < 0:
    p_left = p_two / 2
else:
    p_left = 1 - p_two / 2

print(f"Left-tailed p-value: {p_left:.4f}")

Using the alternative Parameter

Several scipy.stats test functions accept an alternative parameter directly, avoiding manual p-value conversion:

# scipy >= 1.7 supports alternative for many tests
result = stats.ttest_1samp(data, 50, alternative='greater')
print(f"Right-tailed p-value: {result.pvalue:.4f}")

result = stats.ttest_1samp(data, 50, alternative='less')
print(f"Left-tailed p-value: {result.pvalue:.4f}")

Comparing Rejection Regions¶

The key difference lies in how the significance level \(\alpha\) is distributed:

Test Type	Rejection Region	Critical Value
Two-sided	Both tails, \(\alpha/2\) each	\(\pm t_{\alpha/2,\, n-1}\)
Right-tailed	Right tail only, \(\alpha\)	\(t_{\alpha,\, n-1}\)
Left-tailed	Left tail only, \(\alpha\)	\(-t_{\alpha,\, n-1}\)

Because a one-sided test places all \(\alpha\) in one tail, its critical value is less extreme than the two-sided critical value. This makes one-sided tests more powerful for detecting effects in the specified direction.

alpha = 0.05
df = len(data) - 1

t_crit_two = stats.t.ppf(1 - alpha / 2, df)
t_crit_one = stats.t.ppf(1 - alpha, df)

print(f"Two-sided critical value: ±{t_crit_two:.3f}")
print(f"One-sided critical value: {t_crit_one:.3f}")
print(f"One-sided threshold is lower by: {t_crit_two - t_crit_one:.3f}")

Power Implications¶

A one-sided test has greater power than a two-sided test for detecting effects in the specified direction, because the one-sided critical value is less extreme. However, it has zero power for detecting effects in the opposite direction.

from statsmodels.stats.power import tt_solve_power

# Power comparison for same effect size
effect_size = 0.5
n = 30

# Two-sided power
power_two = tt_solve_power(effect_size=effect_size, nobs=n,
                           alpha=0.05, alternative='two-sided')
# One-sided power (correct direction)
power_one = tt_solve_power(effect_size=effect_size, nobs=n,
                           alpha=0.05, alternative='larger')

print(f"Two-sided power: {power_two:.3f}")
print(f"One-sided power: {power_one:.3f}")
print(f"Power gain: {power_one - power_two:.3f}")

When to Use Each¶

Pre-Registration Requirement

The choice between one-sided and two-sided must be made before examining the data. Choosing a one-sided test after seeing the data direction is a form of p-hacking that inflates the false positive rate.

Use a two-sided test when:

You do not have a strong directional prediction before data collection
Effects in either direction would be scientifically interesting
You are conducting exploratory research

Use a one-sided test when:

Prior theory or evidence strongly predicts the direction of the effect
Only one direction is practically meaningful (e.g., a drug cannot make patients worse)
You need maximum power to detect an effect in a specific direction

# Example: testing whether a new drug increases blood pressure
# Strong prior: the drug class is known to raise BP
np.random.seed(42)
bp_change = np.random.normal(loc=3, scale=8, size=25)  # Mean increase of 3

# One-sided is appropriate here (prior expectation of increase)
result = stats.ttest_1samp(bp_change, 0, alternative='greater')
print(f"One-sided test (increase): p = {result.pvalue:.4f}")

# Two-sided for comparison
result_two = stats.ttest_1samp(bp_change, 0)
print(f"Two-sided test: p = {result_two.pvalue:.4f}")

Summary¶

Two-sided tests distribute the rejection probability equally in both tails and are appropriate when effects in either direction are of interest. One-sided tests concentrate all rejection probability in one tail, providing greater power for detecting effects in the pre-specified direction at the cost of being unable to detect effects in the opposite direction. The one-sided p-value is exactly half the two-sided p-value when the observed effect is in the hypothesized direction. The choice must always be justified by the research question and made before analyzing the data.