Definition and interpretation of p-values

Definition of P-Values - The Probability Probe

• P-value is the probability of obtaining the observed test statistic, or one more extreme, assuming the Null Hypothesis ($H₀$) is true.
• It quantifies how surprising your result is, if there's truly no effect (i.e., only random chance is at play).
  • A small p-value suggests the observed effect is unlikely to be due to chance alone.
  • This provides evidence to potentially reject $H₀$ in favor of the Alternative Hypothesis ($H₁$).

⭐ The entire calculation of a p-value is predicated on the assumption that the Null Hypothesis ($H₀$) is TRUE.

Interpretation of P-Values - The Alpha Decision

• Alpha (α), the significance level, is a pre-determined threshold for deciding if a result is statistically significant. It is the probability of making a Type I error (rejecting a true null hypothesis).
• The most common threshold is α = 0.05.

• Decision Rule:
  • If $p \le \alpha$, the result is statistically significant.
  • If $p > \alpha$, the result is not statistically significant.
• 📌 Mnemonic: If the P is low, the null must go.

⭐ Statistical significance does not imply clinical significance or a large effect size.

P-Value Pitfalls - Common Misconceptions

A p-value is a frequently misunderstood concept. To correctly interpret research findings, it is essential to recognize what a p-value does not indicate.

• Probability of the Null Hypothesis ($H₀$): A p-value is NOT the probability that $H₀$ is true or that the results occurred by chance. For example, a p-value of 0.03 does not mean there is a 3% chance that the null hypothesis is correct.
• Effect Size or Clinical Importance: A small p-value does not imply a large or clinically meaningful effect. A large study might yield a tiny p-value for a trivial effect that has no practical importance.
• Arbitrary Cutoff: The 0.05 threshold is a convention, not a sacred truth. A p-value of 0.06 is not drastically different from 0.04.

⭐ A non-significant p-value (e.g., p > 0.05) does not prove the null hypothesis is true. It only signifies that the study lacks sufficient evidence to reject it. This is a classic exam point.

High‑Yield Points - ⚡ Biggest Takeaways

• The p-value is the probability of an observed result (or more extreme) assuming the null hypothesis (H₀) is true.
• p < 0.05 is deemed statistically significant, leading to rejection of the null hypothesis.
• p ≥ 0.05 is not significant, thus we fail to reject the null hypothesis.
• The p-value is NOT the probability that the null hypothesis is true.
• Statistical significance does NOT equal clinical importance or a large effect size.
• P-values are highly dependent on sample size; larger samples can detect smaller effects.