9.2 Types of errors and power of a test

Hypothesis testing involves making decisions based on data, but these decisions can be wrong. In this section, we'll explore two types of errors: Type I (rejecting a true null hypothesis) and Type II (failing to reject a false null hypothesis).

We'll also dive into the power of a test, which is the probability of correctly rejecting a false null hypothesis. Understanding these concepts helps us interpret results and design better studies.

Type I vs Type II Errors

Defining Type I and Type II Errors

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• Type I error rejects a true null hypothesis resulting in a false positive
• Type II error fails to reject a false null hypothesis resulting in a false negative
• Significance level (α) represents the probability of committing a Type I error
• Beta (β) denotes the probability of committing a Type II error
• These errors remain mutually exclusive in hypothesis testing
• Consequences of errors vary based on research context (medical diagnoses, quality control)
• Recognizing these errors proves crucial for interpreting statistical test results

Implications and Considerations

• Error types impact decision-making processes differently (product recalls, drug approvals)
• Balancing Type I and Type II errors involves trade-offs in research design
• Researchers must consider the relative costs of each error type
• Sample size affects the likelihood of both error types
• Multiple testing increases the chance of Type I errors (Bonferroni correction)
• Replication studies help validate findings and reduce overall error rates
• Understanding these errors aids in critical evaluation of scientific literature

Probability of Errors

Calculating Error Probabilities

• Type I error probability directly set by chosen significance level (α)
• Type II error probability (β) calculated as 1 minus the test's power
• Inverse relationship exists between α and β, assuming other factors remain constant
• β calculation requires knowledge of true population parameter, effect size, sample size, and α level
• Operating characteristic (OC) curve graphically represents Type II error probability for different population parameter values
• Power analysis software or tables determine β for various combinations of factors
• Understanding these calculations proves essential for designing studies with appropriate statistical power

Factors Influencing Error Probabilities

• Effect size impacts both Type I and Type II error probabilities
• Larger sample sizes generally decrease both error types
• Chosen statistical test affects error probabilities (parametric vs non-parametric)
• Assumptions of statistical tests influence error rates when violated
• Directionality of alternative hypothesis affects error probabilities (one-tailed vs two-tailed tests)
• Data quality and measurement precision impact error probabilities
• Understanding these factors helps researchers optimize study designs for minimal error rates

Power of a Test

Defining and Calculating Power

• Power represents the probability of correctly rejecting a false null hypothesis (1 - β)
• Conventionally, researchers aim for a power of 0.80 (80%) or higher
• Power calculation involves effect size, sample size, significance level, and specific statistical test
• Power function graphically represents test's power for different population parameter values
• A priori power analysis determines required sample size for desired power level
• Post hoc power analysis assesses achieved power after study completion
• Understanding power calculations aids in interpreting both significant and non-significant results

Interpreting and Applying Power

• High-powered studies increase confidence in both significant and non-significant results
• Low power increases risk of Type II errors and reduces reproducibility
• Power analysis guides resource allocation in research design (sample size determination)
• Interpreting power involves understanding trade-offs between Type I and Type II errors
• Power considerations affect meta-analyses and systematic reviews (publication bias)
• Reporting power alongside results enhances transparency in scientific communication
• Researchers use power analysis to justify sample sizes in grant proposals and publications

Sample Size, Significance, and Power

Relationships Between Factors

• Increasing sample size generally enhances test power, assuming other factors remain constant
• Higher significance level (α) typically leads to increased power but elevates Type I error risk
• Effect size interacts with sample size and significance level to determine power
• Trade-off exists between Type I and Type II errors when adjusting these factors
• Power analysis determines optimal sample size for desired power, given effect size and significance level
• Researchers balance these factors to design studies with adequate power and acceptable error rates
• Understanding these relationships aids in interpreting results within the context of potential errors

Practical Applications in Research Design

• A priori power analysis guides sample size determination in study planning
• Researchers adjust significance levels for multiple comparisons (Bonferroni correction)
• Effect size estimation from pilot studies or literature informs power calculations
• Sequential analysis allows for sample size adjustment during ongoing studies
• Adaptive designs incorporate power considerations to optimize resource allocation
• Multi-stage studies use interim analyses to reassess power and sample size requirements
• Understanding these applications helps researchers design more efficient and informative studies