One-sided vs two-sided tests

Hypothesis Testing - The Core Logic

• Foundation: A formal procedure to accept or reject a statistical hypothesis.
• Null Hypothesis ($H_0$): Assumes no effect, no association, or no difference between groups (e.g., new drug = placebo).
• Alternative Hypothesis ($H_A$): Contradicts the null; assumes an effect or difference exists. The goal of research is often to find evidence for $H_A$.

⭐ The Decision Rule: If the p-value is less than or equal to the pre-specified alpha ($oldsymbol{\alpha}$), typically 0.05, we reject the null hypothesis ($H_0$). This means the observed result is "statistically significant."

One-Sided vs. Two-Sided - The Main Event

The choice between a one-sided and two-sided test hinges on the research hypothesis and must be decided before data collection to maintain integrity.

• A two-sided test is the rigorous standard, as it accounts for the possibility of an effect in the opposite direction of what is expected (e.g., a new drug causing harm instead of benefit).
• ⚠️ Using a one-sided test can inflate the significance of findings; it's easier to get a smaller p-value.

⭐ If a two-sided test yields a p-value of 0.06, a one-sided test on the same data (assuming the effect is in the hypothesized direction) would yield a p-value of 0.03, crossing the threshold for significance.

P-Values & CIs - Reading the Results

• Two-Sided Test (Standard Approach)

  • Tests for a difference in either direction (e.g., drug is better OR worse).
  • Alternative hypothesis: $H_A: \mu_1 \neq \mu_2$.
  • The $\alpha$ level (e.g., 0.05) is split between two tails of the distribution curve.

• One-Sided Test

  • Tests for a difference in one pre-specified direction (e.g., drug is better).
  • Alternative hypothesis: $H_A: \mu_1 > \mu_2$.
  • Concentrates the entire $\alpha$ level in one tail, increasing statistical power for that direction.

⭐ A one-sided p-value is roughly half the two-sided p-value. A result that is non-significant in a two-sided test may become significant in a one-sided test. This requires strong justification before the study begins.

High‑Yield Points - ⚡ Biggest Takeaways

• Two-sided tests evaluate for a difference in either direction (e.g., A ≠ B) and are the most common type in clinical trials.
• One-sided tests assess for a difference in one pre-specified direction only (e.g., A > B or A < B).
• For the same dataset, the p-value of a one-sided test is half that of a two-sided test.
• This gives one-sided tests greater statistical power to detect an effect, but only in the hypothesized direction.
• Use a one-sided test only with a strong, pre-existing hypothesis about the direction of the effect.