18.2 Type I and Type II errors

Hypothesis testing is a crucial tool in statistical analysis, helping us make decisions based on data. It involves two types of errors: Type I (false positive) and Type II (false negative), each with its own implications and probabilities.

Understanding these errors is essential for interpreting research results and making informed decisions. The significance level, sample size, and effect size all play roles in determining the likelihood of these errors, impacting the reliability of our conclusions.

Hypothesis Testing and Error Types

Type I vs Type II errors

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• Type I error (false positive) occurs when rejecting the null hypothesis even though it is actually true
  • Denoted by $\alpha$ (alpha)
  • Example: Convicting an innocent person in a criminal trial
• Type II error (false negative) happens when failing to reject the null hypothesis despite it being false
  • Denoted by $\beta$ (beta)
  • Example: Acquitting a guilty person in a criminal trial
• Null hypothesis ($H_0$) represents the default assumption of no significant effect or difference
  • Example: A new drug has no effect on a disease
• Alternative hypothesis ($H_a$ or $H_1$) contradicts the null hypothesis, suggesting a significant effect or difference
  • Example: The new drug effectively treats the disease

Probability of error types

• Probability of a Type I error equals the significance level ($\alpha$)
  • $P(\text{Type I error}) = P(\text{reject } H_0 | H_0 \text{ is true}) = \alpha$
  • Controlled by the researcher when setting the significance level (commonly 0.05 or 0.01)
• Probability of a Type II error ($\beta$) depends on various factors
  • $P(\text{Type II error}) = P(\text{fail to reject } H_0 | H_0 \text{ is false}) = \beta$
  • Influenced by sample size, effect size, and significance level
• Power of a test represents the probability of correctly rejecting a false null hypothesis
  • $\text{Power} = 1 - \beta$
  • Higher power indicates a lower chance of a Type II error

Significance level and Type I error

• Significance level ($\alpha$) sets the probability threshold for rejecting the null hypothesis
• Increasing the significance level
  • Raises the probability of a Type I error
  • Expands the critical region for rejecting the null hypothesis
  • Example: Setting $\alpha = 0.10$ instead of 0.05 makes it easier to reject $H_0$
• Decreasing the significance level
  • Lowers the probability of a Type I error
  • Shrinks the critical region for rejecting the null hypothesis
  • Example: Setting $\alpha = 0.01$ instead of 0.05 makes it harder to reject $H_0$

Real-world consequences of errors

• Type I errors lead to false alarms or false positives
  • Convicting an innocent person (criminal trial)
  • Approving an ineffective drug (medical study)
  • Issuing a product recall for a non-defective item (quality control)
• Type II errors result in missed opportunities or false negatives
  • Acquitting a guilty person (criminal trial)
  • Rejecting an effective drug (medical study)
  • Failing to identify a defective product (quality control)
• Balancing the risks involves considering the relative consequences of each error type
  • In medical testing, minimizing Type I errors (false positives) may be prioritized
  • In criminal trials, minimizing Type II errors (false acquittals) may be more important