Type I & II Errors - False Alarms & Missed Clues
- Hypothesis Testing Goal: Determine if there's enough evidence to reject the null hypothesis ($H_0$), which presumes no effect or difference exists.
| Error Type | Description | Analogy | Probability |
|---|---|---|---|
| Type I | Rejecting a true $H_0$ | False Alarm | $α$ (alpha) |
| Type II | Failing to reject a false $H_0$ | Missed Clue | $β$ (beta) |
- Probability of a Type I error. You conclude there IS an effect, when there ISN'T.
- Conventionally set at **0.05**.
- β (Beta):
- Probability of a Type II error. You conclude there is NO effect, when there IS.
- Power:
- The probability of correctly detecting a true effect (correctly rejecting a false $H_0$).
- Power = $1 - β$. Desired power is typically >80%.
📌 Mnemonic: The boy who cried wolf was making a Type I error (false positive). The villagers made a Type II error when they ignored the real wolf (false negative).
⭐ Increasing sample size is the most common way to increase the power of a study. A larger sample provides a more precise estimate of the effect, reducing the probability of a Type II error (↓β).
Power & Sample Size - Finding a Real Difference
- Power is the probability of finding a true difference when one exists. It is the ability to avoid a Type II error.
- Power = $1 - β$
- Standard is 80% power.
| Error Type | Reality: No Difference (H₀ True) | Reality: Difference Exists (H₀ False) |
|---|---|---|
| Test Says: Difference | Type I Error (α) False Positive | Correct (Power) True Positive |
| Test Says: No Difference | Correct True Negative | Type II Error (β) False Negative |
- β (beta): Probability of a Type II error.
📌 Mnemonic: You can remember the two types of errors as:
- Type A (I) error is α (alpha) or accusing an innocent person.
- Type B (II) error is β (beta) or blinding yourself to a guilty person.
⭐ Decreasing the risk of a Type I error (↓ α) directly increases the risk of a Type II error (↑ β), and vice-versa.
Error & Power Interplay - The Statistical Balancing Act
Hypothesis testing involves a trade-off between two key errors. Adjusting one directly impacts the other, as well as the study's power to detect a true effect.
- Inverse Relationship: For a fixed sample size (N), decreasing α (risk of Type I error) inevitably increases β (risk of Type II error).
- Power (1-β): The probability of correctly rejecting a false null hypothesis (H₀). The accepted standard for study power is ≥80%.
- Increasing Power: The most effective methods are:
- ↑ Sample size (N): The most common approach.
- ↑ Effect size: Larger differences are easier to detect.
- ↑ α level: Increases power but also the risk of a false positive.
⭐ Decreasing α (e.g., from 0.05 to 0.01) to be more confident in a positive result (reduce Type I error) makes the study more stringent. This increases the chance of missing a true effect (↑β) and thus reduces power.
High‑Yield Points - ⚡ Biggest Takeaways
- Type I error (α) is a false positive-incorrectly rejecting a true null hypothesis (H₀). The p-value represents the probability of committing this error.
- Type II error (β) is a false negative-failing to reject a false null hypothesis when an effect is present.
- Power (1 - β) is the probability of detecting a true effect; it is the ability to correctly reject a false null hypothesis.
- α and β are inversely related; decreasing the risk of one error type increases the risk of the other.
- Power is increased by ↑ sample size, ↑ effect size, or a higher α level.
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