Decision Making

Decision Making in Hypothesis Testing

Purpose and Rationale

Core Purpose

The decision-making approach in hypothesis testing serves several essential purposes:

  1. Formal Decision Framework

    • Provides a structured procedure for choosing between competing hypotheses
    • Establishes clear rules for making statistical decisions
    • Enables consistent interpretation of results across studies

  2. Error Control

    • Helps manage the risk of incorrect decisions
    • Provides a framework for controlling Type I errors
    • Enables planning for appropriate sample sizes

  3. Practical Applications

    • Supports decision-making in research and business
    • Guides policy and intervention decisions
    • Helps in quality control and process improvement

The Rationale Behind Decision Making

  1. Why We Need Formal Decisions

    • Research and business often require clear yes/no decisions
    • Need to balance evidence with practical constraints
    • Provides a framework for action when evidence is sufficient

  2. Why Use Significance Levels

    • Controls the rate of false positive errors
    • Provides a standardized approach to decision making
    • Enables comparison across different studies

  3. Why Consider Both Types of Errors

    • Type I errors (false positives) can lead to unnecessary actions
    • Type II errors (false negatives) can miss important effects
    • Need to balance both types of errors based on context

The Decision Making Process

Setting the Significance Level (α)

Aspect Description Considerations
Definition Pre-determined threshold for p-value Usually 0.05 or 0.01
Purpose Controls Type I error rate Should be context-dependent
Common Values 0.05, 0.01, 0.10 Consider consequences of errors

Step-by-Step Process

  1. Pre-Analysis Steps

    • Set significance level (α)
    • Define null and alternative hypotheses
    • Determine required sample size
    • Plan analysis approach

  2. Data Collection and Analysis

    • Gather random sample
    • Check necessary conditions
    • Calculate test statistic
    • Determine p-value

  3. Decision Making

    • Compare p-value to α
    • Make decision (reject/fail to reject H₀)
    • Consider practical significance
    • State conclusion in context

Decision Rules

Condition Decision Interpretation
p-value ≤ α Reject H₀ Sufficient evidence against H₀
p-value > α Fail to reject H₀ Insufficient evidence against H₀

Types of Errors

Error Types and Consequences

Error Type Definition Probability Consequences
Type I (α) Reject true H₀ α False positive, unnecessary action
Type II (β) Fail to reject false H₀ β False negative, missed opportunity

Error Control

Aspect Description How to Control
Type I Error False positive rate Set α level
Type II Error False negative rate Increase sample size
Power 1 - β Plan sample size

Important Considerations

Statistical vs. Practical Significance

Aspect Statistical Practical
Focus P-value and α Effect size and context
Question Is there an effect? Is the effect important?
Dependence Sample size Real-world impact

Common Issues and Solutions

Issue Problem Solution
Arbitrary α Rigid cutoff may not fit context Adjust based on consequences
Multiple Testing Increased Type I error Adjust α or use corrections
P-hacking Data manipulation for significance Pre-specify analyses
Publication Bias Only significant results published Report all studies

Best Practices

  1. Before Analysis

    • Pre-specify hypotheses and analyses
    • Choose appropriate α level
    • Plan sample size for desired power
    • Consider practical significance

  2. During Analysis

    • Check all assumptions
    • Calculate effect size
    • Consider confidence intervals
    • Document all decisions

  3. After Analysis

    • Report exact p-values
    • Include effect sizes
    • Discuss practical significance
    • Consider limitations

Alternative Approaches

Strength of Evidence Approach

Aspect Description Advantages
Focus P-value interpretation More nuanced understanding
Decision Based on evidence strength Less arbitrary
Reporting Exact p-values More informative

Complementary Methods

Method Purpose When to Use
Confidence Intervals Estimate precision Always report
Effect Size Measure importance Always calculate
Power Analysis Plan sample size Before study