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What Are Effect Sizes?
Effect sizes quantify the magnitude of research findings in standardized units, independent of sample size. While p-values tell whether effects are statistically significant, effect sizes reveal how large or meaningful effects are. A statistically significant finding may have trivial practical importance, while large effects in small samples may not reach significance.
Why Effect Sizes Matter
- Practical significance - Distinguish statistical from practical importance
- Standardized comparison - Compare effects across different studies and measures
- Meta-analysis - Essential for combining results across studies
- Power analysis - Required for sample size calculations
- Journal requirements - Most journals mandate effect size reporting
- Effect interpretation - Contextualizes findings beyond p-values
Cohen's d (Standardized Mean Difference)
What It Measures
Cohen's d expresses mean differences in standard deviation units. Used for comparing two group means (t-tests) or pre-post changes within groups.
Formula: d = (M₁ - M₂) / SD_pooled
Interpretation Guidelines
Cohen's Benchmarks:
- d = 0.20 - Small effect
- d = 0.50 - Medium effect
- d = 0.80 - Large effect
More Nuanced Interpretation:
- d < 0.20 - Trivial
- d = 0.20-0.49 - Small
- d = 0.50-0.79 - Medium
- d ≥ 0.80 - Large
When to Use
Calculate Cohen's d for:
- Independent samples t-tests
- Paired samples t-tests
- Pre-test/post-test designs
- Experimental vs. control comparisons
Example
Treatment group mean anxiety = 45, control group = 55, pooled SD = 12 d = (45 - 55) / 12 = -0.83 (large effect, treatment reduces anxiety)
Hedges' g (Bias-Corrected d)
What It Measures
Hedges' g corrects Cohen's d for small sample bias. When n < 50, d overestimates population effect size. Hedges' g applies correction factor producing more accurate estimates.
Formula: g = d × (1 - 3/(4(n₁ + n₂) - 9))
When to Use
Use Hedges' g instead of Cohen's d when:
- Total sample size < 50
- Reporting for meta-analysis (preferred metric)
- Need unbiased effect size estimate
Interpretation identical to Cohen's d benchmarks.
Glass's Δ (Delta)
What It Measures
Glass's Δ uses control group standard deviation only as standardizer, rather than pooled SD. Appropriate when treatment changes variability or when control group represents baseline.
Formula: Δ = (M_treatment - M_control) / SD_control
When to Use
Use Glass's Δ when:
- Treatment affects variability
- Control group provides natural baseline
- Groups have heterogeneous variances
Interpretation similar to Cohen's d, though values may differ when variances unequal.
Eta Squared (η²) for ANOVA
What It Measures
η² represents proportion of total variance explained by factor(s). Used with ANOVA designs having one or more factors.
Formula: η² = SS_effect / SS_total
Interpretation Guidelines
- η² = 0.01 - Small effect
- η² = 0.06 - Medium effect
- η² = 0.14 - Large effect
Limitation
η² is biased in complex designs. Increases artificially as more factors added. Consider partial η² or ω² for multi-factor ANOVA.
Omega Squared (ω²)
What It Measures
ω² is less biased estimate of variance explained than η². Accounts for error variance, providing more conservative and accurate effect size for population.
Formula: ω² = (SS_effect - (df_effect × MS_error)) / (SS_total + MS_error)
When to Use
Prefer ω² over η² for:
- Multi-factor ANOVA
- Unequal sample sizes
- Effect size estimation for populations
- More accurate variance explained estimates
Interpretation uses same benchmarks as η² (0.01, 0.06, 0.14 for small, medium, large).
Cohen's f for ANOVA
What It Measures
Cohen's f expresses ANOVA effect sizes in standardized form useful for power analysis. Unlike η² or ω² which are bounded (0-1), f can exceed 1.
Formula: f = √(η² / (1 - η²))
Interpretation Guidelines
- f = 0.10 - Small effect
- f = 0.25 - Medium effect
- f = 0.40 - Large effect
When to Use
Use Cohen's f for:
- ANOVA power analysis (required by G*Power)
- Comparing effects across different ANOVA designs
- Planning sample sizes for factorial designs
Pearson r (Correlation Effect Size)
What It Measures
Correlation coefficient r serves as effect size for relationships between continuous variables. Already standardized (-1 to +1).
Interpretation Guidelines
Cohen's Benchmarks:
- r = 0.10 - Small effect
- r = 0.30 - Medium effect
- r = 0.50 - Large effect
Field-Specific Context: In some fields (meteorology, economics), r = 0.30 represents strong relationships. Consider disciplinary norms alongside Cohen's benchmarks.
R² Interpretation
R² (coefficient of determination) shows variance explained:
- r = 0.30 means R² = 0.09 (9% variance explained)
- r = 0.50 means R² = 0.25 (25% variance explained)
Even "medium" correlations explain limited variance, highlighting complexity of human behavior.
Odds Ratio (OR)
What It Measures
Odds ratio quantifies association strength in 2×2 contingency tables (chi-square tests). Compares odds of outcome in one group vs. another.
Formula: OR = (a × d) / (b × c)
Where a, b, c, d are cell counts in 2×2 table.
Interpretation Guidelines
- OR = 1.00 - No association
- OR = 1.50 - Small effect
- OR = 2.50 - Medium effect
- OR = 4.00 - Large effect
- OR < 1.00 - Negative association
Example
Disease present: Treatment = 20, Control = 40 Disease absent: Treatment = 80, Control = 60 OR = (20 × 60) / (40 × 80) = 0.375 (treatment reduces disease odds by 62.5%)
Cohen's w for Chi-Square
What It Measures
Cohen's w quantifies effect size for chi-square tests of independence. Indicates degree of association between categorical variables.
Formula: w = √(χ² / N)
Interpretation Guidelines
- w = 0.10 - Small effect
- w = 0.30 - Medium effect
- w = 0.50 - Large effect
Use for chi-square goodness-of-fit and independence tests.
Converting Between Effect Sizes
Common Conversions
Many effect sizes can be converted approximately:
- d to r: r ≈ d / √(d² + 4)
- r to d: d ≈ 2r / √(1 - r²)
- η² to f: f = √(η² / (1 - η²))
- OR to d: d ≈ ln(OR) × √3 / π
Conversions enable comparing effects across studies using different metrics.
Confidence Intervals for Effect Sizes
Why CI Matters
Confidence intervals show effect size precision. Wide intervals indicate uncertainty; narrow intervals suggest precise estimates.
Example: d = 0.45, 95% CI [0.15, 0.75] Effect is medium, but CI includes small effects (0.15) and approaches large (0.75), indicating uncertainty.
Reporting
Report effect sizes with confidence intervals: "Treatment significantly reduced anxiety, t(98) = 3.24, p = .002, d = 0.65, 95% CI [0.25, 1.05]."
Context-Specific Interpretation
Don't Over-Rely on Benchmarks
Cohen's benchmarks are rough guidelines. Consider:
- Field norms: What effect sizes are typical in your discipline?
- Practical significance: Does this effect matter for real-world applications?
- Cost-benefit: Even small effects may be valuable if interventions are inexpensive
- Baseline rates: Small effects on rare outcomes may be important
Example Contexts
Medical interventions: Small effects (d = 0.20) reducing mortality are extremely valuable despite being "small" by Cohen's standards.
Educational interventions: Medium effects (d = 0.50) represent meaningful learning gains equivalent to several months' progress.
Psychological interventions: Large effects (d = 0.80) for depression treatment represent substantial symptom reduction.
Reporting Effect Sizes
APA Style Requirements
Report effect sizes alongside significance tests:
- State which effect size measure used
- Provide effect size values
- Include interpretation when helpful
- Consider confidence intervals
Example: "The intervention significantly improved test scores, t(120) = 4.56, p < .001, Cohen's d = 0.83, 95% CI [0.46, 1.20], representing a large effect."
Meta-Analysis Preparation
When conducting research for potential meta-analysis:
- Report multiple effect size types
- Provide means, SDs, and sample sizes
- Calculate Hedges' g (preferred for meta-analysis)
- Include sufficient detail for future meta-analysts
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