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.

Mental Model

A two-sided test guards against surprises in either direction (the parameter could be higher or lower); a one-sided test bets everything on one direction and gains statistical power in exchange. The rule is simple: choose one-sided only when the opposite direction is scientifically meaningless, and always decide before looking at the data.

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 \(H_0\).

```python 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

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

```python

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

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

```python

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:

```python

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.

```python 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.

```python 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

```python

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.

Exercises¶

Exercise 1. Generate 25 samples from \(N(52, 10^2)\) and test \(H_0: \mu = 50\) using both a two-sided test and a right-tailed test via the alternative parameter of ttest_1samp. Compare the p-values and explain the relationship.

Solution to Exercise 1

import numpy as np
from scipy import stats

np.random.seed(42)
data = np.random.normal(52, 10, 25)
result_two = stats.ttest_1samp(data, 50)
result_right = stats.ttest_1samp(data, 50, alternative='greater')
print(f"Two-sided p-value:   {result_two.pvalue:.4f}")
print(f"Right-tailed p-value:{result_right.pvalue:.4f}")
print("Right-tailed p = two-sided p / 2 (when t > 0)")

Exercise 2. Compute the critical values for a two-sided and a one-sided (right-tailed) \(t\)-test at \(\alpha = 0.05\) with 20 degrees of freedom. Show that the one-sided critical value is less extreme.

Solution to Exercise 2

from scipy import stats

df = 20
t_two = stats.t.ppf(1 - 0.05/2, df)
t_one = stats.t.ppf(1 - 0.05, df)
print(f"Two-sided critical value:  +/- {t_two:.4f}")
print(f"One-sided critical value:  {t_one:.4f}")
print(f"Difference: {t_two - t_one:.4f}")

Exercise 3. Using statsmodels.stats.power.tt_solve_power, compute the power of a one-sample t-test with \(n = 20\) and effect size \(d = 0.5\) for both one-sided ('larger') and two-sided alternatives. Confirm that the one-sided test is more powerful.

Solution to Exercise 3

from statsmodels.stats.power import tt_solve_power

power_two = tt_solve_power(effect_size=0.5, nobs=20, alpha=0.05,
                            alternative='two-sided')
power_one = tt_solve_power(effect_size=0.5, nobs=20, alpha=0.05,
                            alternative='larger')
print(f"Two-sided power: {power_two:.4f}")
print(f"One-sided power: {power_one:.4f}")