Effect Size

Effect Size

Purpose and Rationale

Why Effect Size Matters

Effect size is crucial because:

Practical Significance
- Measures magnitude of relationships
- Indicates real-world importance
- Complements statistical significance
Research Interpretation
- Provides context for findings
- Helps compare across studies
- Enables meta-analyses
Study Planning
- Guides sample size decisions
- Helps set realistic expectations
- Informs power analysis

The Rationale Behind Effect Size

Why We Need It
- p-values don't measure importance
- Sample size affects significance
- Need standardized measures
How It Works
- Quantifies relationship strength
- Provides standardized metrics
- Enables meaningful comparisons

Types of Effect Sizes

For Mean Differences

Measure	Formula	When to Use
Cohen's d	$\frac{M_{1} - M_{2}}{σ}$	Comparing two means
Hedges' g	Adjusted d	Small samples
Glass's Δ	$\frac{M_{1} - M_{2}}{σ_{c o n t r o l}}$	Control group reference

For Correlations

Measure	Description	When to Use
Pearson's $r$	Linear correlation	Continuous variables
Point-biserial	Binary-continuous	Binary vs continuous
Phi coefficient	Binary-binary	Two binary variables

For Proportions

Measure	Description	When to Use
Risk ratio	$\frac{p_{1}}{p_{2}}$	Comparing risks
Odds ratio	$\frac{p_{1} / (1 - p_{1})}{p_{2} / (1 - p_{2})}$	Case-control studies
Risk difference	$p_{1} - p_{2}$	Direct comparison

Interpretation Guidelines

Standard Benchmarks

Effect Size	Small	Medium	Large
Cohen's $d$	0.2	0.5	0.8
Pearson's $r$	0.1	0.3	0.5
Odds ratio	1.5	2.5	4.0

Contextual Considerations

Field-Specific Standards
- Different fields have different norms
- Consider typical effects in area
- Review literature benchmarks
Practical Importance
- Clinical significance
- Economic impact
- Real-world relevance

Calculation Methods

For Different Study Designs

Design	Common Measures	Considerations
Between-subjects	Cohen's d, η²	Group independence
Within-subjects	Cohen's dz, r	Repeated measures
Mixed designs	Partial η²	Complex designs

Software Implementation

Software	Function	Example
R	cohens_d()	cohens_d(group1, group2)
Python	scipy.stats	effect_size()
SPSS	Analyze → Compare Means	Effect size options

Reporting Guidelines

Essential Elements

What to Report
- Effect size measure used
- Value and confidence interval
- Interpretation context
- Comparison benchmarks
How to Present
- Clear labeling
- Consistent formatting
- Appropriate precision
- Visual aids when helpful

Common Mistakes

Mistake	Problem	Solution
Not reporting	Missing context	Always include
Wrong measure	Inappropriate comparison	Choose carefully
Poor interpretation	Misleading conclusions	Use benchmarks

Applications

Research Planning

Sample Size
- Power analysis
- Resource allocation
- Study design
Meta-Analysis
- Study comparison
- Effect aggregation
- Publication bias assessment

Clinical Practice

Treatment Effects
- Intervention impact
- Clinical significance
- Patient outcomes
Risk Assessment
- Risk quantification
- Decision making
- Policy implications

Best Practices

Selection Guidelines

Choose Appropriate Measure
- Consider research question
- Match study design
- Account for variables
Consider Limitations
- Sample size effects
- Distribution assumptions
- Measurement issues

Interpretation Framework

Statistical Context
- Compare to benchmarks
- Consider confidence intervals
- Account for variability
Practical Context
- Field-specific standards
- Real-world implications
- Stakeholder needs

Hypothesis Testing Basics - Foundation for effect size
Power Analysis - Using effect size in planning
Statistical Significance - Complement to effect size
Sample Size - Relationship to effect size
Confidence Interval - Precision of effect estimates
Meta-Analysis - Combining effect sizes
P-value - Different from effect size
ANOVA - Effect sizes in ANOVA
Correlation - Effect size for relationships
Regression - Effect size in regression