Atlas
statminds
Effect Size (Variance-Partitioning Model)The underlying model family class (e.g. GLM, linear model, categorical matrix, log-linear).Parametric ReferenceStatistical methods that assume a specific probability distribution family (typically normal).12-stage workflow
Eta-Squared (η²)
The engine for Variance Discovery. Eta-Squared (η²) audits the total percentage of outcome variability 'owned' by a categorical factor, reveal the structural dominance of grouping in ANOVA designs.
G1
Variance Ownership Audit
Determine exactly what percentage of the 'Outcome Pie' is captured by your treatment grouping.
G2
Factorial Synergy Discovery
Quantify the relative importance of different predictors in explaining global data behavior.
G3
Global Impact Mapping
Identify the dominant drivers of success in complex, multi-factor ANOVA grids.
Visual Overview Dashboard
1
What is it?
Eta-Squared (η²) is designed to mathematically isolate and quantify the magnitude of an observed outcome or model factor, independently of sample size.
The engine for Variance Discovery. Eta-Squared (η²) audits the total percentage of outcome variability 'owned' by a categorical factor, reveal the structural dominance of grouping in ANOVA designs.
2
Goals & Indications
- Variance Ownership Audit: Determine exactly what percentage of the 'Outcome Pie' is captured by your treatment grouping.
- Factorial Synergy Discovery: Quantify the relative importance of different predictors in explaining global data behavior.
- Global Impact Mapping: Identify the dominant drivers of success in complex, multi-factor ANOVA grids.
3
Core Idea Diagram
4
Claims tested
H₀: H₀: Effect size = 0 (the factor explains no variance in the outcome)
Hₐ: Hₐ: Effect size > 0 (the factor explains some proportion of variance)
5
How it works
- Execute analysis of variance (ANOVA) to obtain sum of squares.
- Identify Group Sum of Squares (SS_effect) and Total Sum of Squares (SS_total).
- Calculate proportion of explained variance: eta² = SS_effect / SS_total.
- Reference eta² values to assess effect size magnitude.
6
Assumptions
Valid ANOVA conducted: Effect size requires valid ANOVA
DV is continuous: Outcome measured on continuous scale
Independence of observations: No clustering, nesting, or repeated measures
7
Important Note
Eta-squared is a descriptive effect size metric, not a hypothesis test. It quantifies the proportion of total variance in the DV attributable to the IV. Always report with ANOVA results.
8
Worked Example
| SS effect | SS error | η² |
|---|---|---|
| 10.0 | 90.0 | 10.0% (Small) |
| 40.0 | 60.0 | 40.0% (Large) |
Interactive Sandbox
ANOVA Variance Segmentation Laboratory
Slide effect and error sum of squares (SS). Watch how the explained variance partition changes in real-time.
Variance Explained
Eta-squared (η²): 40.0%
SS Total: 100
Magnitude Grade: Large
Variance Partition Bar
The 12-Stage Precision Workflow
01Variance Parity
Hypotheses
We test the null of 'Zero Variance Explained' against the discovery of a systematic categorical capture of the outcome.
02Omnibus Logic
Assumptions
Ensuring the underlying ANOVA is significant—Eta-Squared is most meaningful when it describes a mathematically verified signal.
03Additive Bias
Diagnostics
Identifying if your model is 'Over-claiming' importance—η² tends to be upwardly biased in small samples (unlike Omega-squared).
04focus
Determining that FlowMotion participation explains 45% of the total variability in functional mobility scores.
05Partial η² Pivot
Alternatives
Knowing when to switch to Partial η² if you specifically need the 'Unique' contribution while ignoring other predictors.
06Non-zero η²
Significance
Understanding that η² is always non-negative; discovery hinges on its distance from the zero-variance baseline.
07The Benchmarks
Effect Size
Interpreting η² values: .01 (Small), .06 (Medium), .14 (Large)—the authoritative standards for categorical dominance.
08Stability Buffer
Sample Size
Accounting for the N required to ensure that the variance partitioning doesn't shift wildly with a few new participants.
09The Explained Narrative
Reporting
Reporting η² alongside F and p: 'The intervention significantly influenced mobility, F(df)=X.XX, p=.XXX, η²=.15.'
10lsr / etaSquared Logic
Software
Executing 'etaSquared()' commands, ensuring the algorithm uses the 'Global' rather than 'Partial' calculation when required.
11focus
Avoiding the error of summing η² values in factorial designs—pure η² values always add up to less than or equal to the Total R².
12focus
Tracing the model back to Karl Pearson's Correlation Ratio and the foundational shift from linear slopes to categorical variance.
01Hypothesis test logic
Hypotheses
Pragmatic null and alternative hypotheses defined in mathematical notation.
A hypothesis is a question sharpened to a point. Ambiguity is the enemy of inference.
Logic CoreNull · H₀
H₀: Effect size = 0 (the factor explains no variance in the outcome)
Alternative · Hₐ
Hₐ: Effect size > 0 (the factor explains some proportion of variance)
Why it matters descriptive
Eta-squared is a descriptive effect size metric, not a hypothesis test. It quantifies the proportion of total variance in the DV attributable to the IV. Always report with ANOVA results.
02Model diagnostics
Assumptions
The core mathematical criteria needed to ensure that statistical testing remains unbiased and valid.
Build your analysis on rock, not sand. Verify the mathematical foundation before building the model.
Integrity Shield6
Assumptions
4
Critical / High Severity
How to check
Quick
Verify ANOVA was run and assumptions checked; η² is computed from SS values in ANOVA table
Rigorous
Review ANOVA diagnostics: Levene's test, Q-Q plots, boxplots. Ensure ANOVA F-test is valid before interpreting η²
If violated
If ANOVA assumptions violated: (1) For unequal variances → use ε² (epsilon-squared) with Welch's ANOVA; (2) For non-normality → report median-based effect sizes or rank-biserial correlation from Kruskal-Wallis; (3) For outliers → use robust effect sizes with trimmed means
cohens dglass delta
How to check
Quick
Confirm variable type; variance-based effect sizes require continuous DV with meaningful variance
Rigorous
Verify measurement scale is truly interval/ratio; check that DV has adequate range (not artificially restricted)
If violated
If ordinal DV → use rank-biserial correlation or rank epsilon-squared from Kruskal-Wallis. If binary DV → use odds ratio, risk ratio, or Cramér's V. If categorical → use Cramér's V or Cohen's w
cramers v
How to check
Quick
Design review; check if any subject IDs appear multiple times; verify no hierarchical structure
Rigorous
ICC analysis to detect clustering; verify each observation is from unique independent unit
If violated
If repeated measures → use partial η² from RM-ANOVA (partials out subject variance). If clustered/nested data → use R² from multilevel models or report ICC alongside effect size. Standard η² assumes independence and will be biased if violated
partial eta squaredintraclass correlation
How to check
Quick
Q-Q plots per group; visual inspection for severe skewness
Rigorous
Shapiro-Wilk test per group; check skewness and kurtosis values
If violated
η² is reasonably robust to moderate non-normality if n is adequate (n ≥ 30 per group). If severe violation: (1) Transform DV and recompute; (2) Use bootstrap CI for η²; (3) Report rank-based effect size from Kruskal-Wallis (rank epsilon-squared); (4) Use robust ω² with trimmed means
How to check
Quick
Levene's test (p > .05); boxplots show similar IQR across groups
Rigorous
Brown-Forsythe test; residual plots; variance ratio checks (largest/smallest < 3)
If violated
If heteroscedastic: (1) Compute ε² (epsilon-squared) from Welch's ANOVA instead of η²; (2) Use generalized η² that adjusts for unequal variances; (3) Transform DV if variance ∝ mean. Standard η² and ω² assume homoscedasticity
How to check
Quick
Boxplots by group; standardized residuals |z| > 3 flagged
Rigorous
Cook's distance, leverage, DFFITS; influence diagnostics from ANOVA model
If violated
Outliers inflate SS_total and can bias η². Solutions: (1) Report η² with and without outliers (sensitivity analysis); (2) Use robust ω² with 20% trimmed means; (3) Winsorize extreme values; (4) Transform DV if outliers due to skewness. Never remove outliers without transparency
03Residual Forensics
Diagnostics
Checking residual plots and indices to examine model deviations and ensure standard error integrity.
Trust, but verify. The outliers often hold more truth than the averages.
System HealthEssential checks
- ANOVA assumption checks (Levene's, Q-Q plots, boxplots)
- Sample size adequacy (n ≥ 20 per group for stable estimates)
- SS decomposition verification (SS_total = SS_effect + SS_error)
Recommended checks
- Bootstrap 95% CI for η² (addresses sampling variability)
- Compare η², ηp², and ω² to understand bias
- Sensitivity analysis: effect size with/without outliers
- Visual inspection of variance components
- Cross-validation of effect size in holdout sample
04Live Instances
Applied Minds
Review concrete study examples, data layout guidelines, and copy executable syntax scripts.
Theory is the map. Practice is the terrain. Simulation bridges the gap.
Applied WisdomExample 01
Teaching Methods and Test Scores (One-way ANOVA)
Research question: Do different teaching methods affect student test performance? Design: 90 students randomized to 3 methods (Lecture n=30, Active Learning n=30, Flipped Classroom n=30). Outcome: Final exam score (0-100, continuous). Compute η², ηp², and ω² from ANOVA.
DesignBetween-subjects RCT
GroupsLecture · Active Learning · Flipped Classroom
Total n90
Outcome ScaleFinal exam score (0-100)
# Eta-squared effect sizes from one-way ANOVA
library(effectsize)
library(tidyverse)
# Simulate data
set.seed(2025)
data <- data.frame(
method = rep(c("Lecture", "Active", "Flipped"), each=30),
score = c(
rnorm(30, 72, 12), # Lecture: M=72, SD=12
rnorm(30, 78, 11), # Active: M=78, SD=11
rnorm(30, 82, 10) # Flipped: M=82, SD=10
)
)
# ANOVA
model <- aov(score ~ method, data=data)
summary(model)
# Extract SS values
anova_summary <- summary(model)[[1]]
SS_effect <- anova_summary["method", "Sum Sq"]
SS_error <- anova_summary["Residuals", "Sum Sq"]
SS_total <- SS_effect + SS_error
df_effect <- anova_summary["method", "Df"]
MS_error <- anova_summary["Residuals", "Mean Sq"]
# Manual computation
eta_sq <- SS_effect / SS_total
partial_eta_sq <- SS_effect / (SS_effect + SS_error) # Same as η² in one-way
omega_sq <- (SS_effect - df_effect * MS_error) / (SS_total + MS_error)
cat("=== Manual Calculation ===", "\n")
cat(sprintf("η² = %.3f (%.1f%% variance explained)\n", eta_sq, eta_sq*100))
cat(sprintf("ηp² = %.3f\n", partial_eta_sq))
cat(sprintf("ω² = %.3f (less biased estimate)\n", omega_sq))
# Using effectsize package (recommended)
cat("\n=== effectsize Package ===", "\n")
eta_squared(model, partial=FALSE) # η²
eta_squared(model, partial=TRUE) # ηp²
omega_squared(model) # ω²
# Interpretation
cat("\n=== Cohen(1988) Benchmarks ===", "\n")
cat("Small: .01, Medium: .06, Large: .14\n")
cat(sprintf("Result: ω² = %.3f → %s effect\n",
omega_sq,
ifelse(omega_sq >= .14, "LARGE",
ifelse(omega_sq >= .06, "Medium", "Small"))))
# Bootstrap CI for η²
library(boot)
boot_eta_sq <- function(data, indices) {
d <- data[indices, ]
model <- aov(score ~ method, data=d)
SS <- summary(model)[[1]][, "Sum Sq"]
SS[1] / sum(SS)
}
boot_results <- boot(data, boot_eta_sq, R=1000)
boot_ci <- boot.ci(boot_results, type="perc")
cat("\n95% CI for η²:", boot_ci$percent[4:5], "\n")
# APA report
cat("\n=== APA Report ===", "\n")
cat(sprintf("F(2, 87) = %.2f, p < .001, ω² = %.2f (large effect, 95%% CI [%.2f, %.2f])\n",
anova_summary["method", "F value"],
omega_sq,
boot_ci$percent[4], boot_ci$percent[5]))Interpretation Blueprint
F(2, 87) = 8.45, p < .001, ω² = .13 (large effect, approaching Cohen's .14 threshold). Teaching method explains ~13% of variance in exam scores. Flipped classroom (M=82) outperformed lecture (M=72), d=0.91. Effect size indicates educationally meaningful differences.
05Tactical Pivots
Alternatives
Structured fallback pathways for choosing alternative tests when normality or slopes requirements fail.
When the path is blocked, pivot. Rigor is not rigidity; it is the intelligent adaptation to reality.
Adaptive StrategyMeasurement Precision Ladder Ideal · Continuous / Interval Matrix
Ratio
Maintain Eta-Squared. Optimal for quantifying total variance captured in high-fidelity designs.
Peak SignalInterval
Ideal for Primary Scales. Ensure the 'Omnibus F' is significant before interpreting variance-explained.
Standard PrecisionOrdinal / Nominal
Abandon η². Use Epsilon-Squared or non-parametric rank-variance metrics.
Model CollapseTemporal Trajectory Audit Static Variance Partitioning
Static Audit
Cross-sectional pool.
Stay with Eta-Squared. Measure the 'Ownership' of the outcome pie.
Longitudinal
Trajectory variance.
Pivot to Partial Eta-Squared to isolate the unique contribution of time while neutralizing subject noise.
Adaptive Technical Safeguards · adaptive safeguards
sample size bias
- Omega-Squared (ω²) — The required pivot for small samples to subtract random sampling error.
- Epsilon-Squared (ε²) — A more conservative alternative for unbiased variance-explained.
factorial designs
- Partial Eta-Squared — Isolate the unique variance of one factor while ignoring others.
- Generalized Eta-Squared — The standard for comparing effects across different experimental designs.
06Adjusted Comparisons
Post-hoc
Group mean comparisons and correction controls (e.g. Tukey HSD, Bonferroni) to protect against Family-Wise Error Rates.
The omnibus test opens the door; post-hoc analysis explores the room.
Forensic DetailAdjusted Comparisons
- Compare with omega-squared (less biased estimator)
- Compare with partial eta-squared for factorial designs
- Bootstrap confidence intervals
- Convert to Cohen's f for power analysis
- Examine epsilon-squared as alternative for unequal groups
Interpretation Guidelines
Eta-squared is an effect size measure for ANOVA, not a hypothesis test. Post-hoc tests are not applicable.
07Standardized scale impact
Effect Size
Understanding effect sizes (e.g., Cohen's d, Partial Eta-Squared) and clinical impact benchmarks.
Significance is noise. Magnitude is the signal. Measure the impact, not just the probability.
Impact MagnitudeProportion of total variance explained. Small: .01, Medium: .06, Large: .14 (Cohen, 1988). Biased upward in small samples.
Proportion of variance explained controlling for other factors. Same benchmarks (.01/.06/.14). Standard in factorial designs.
Less biased population estimate. Small: .01, Medium: .06, Large: .14. RECOMMENDED for reporting.
For Welch's ANOVA with unequal variances. Similar interpretation to ω².
Comparable across different research designs. Adjusts for manipulated vs measured factors.
Recommended Metric: omega_squared (ω²) - less biased estimate of population effect, especially for small samples. Report with 95% bootstrap CI.
Small
0.2
0.2
Medium
0.5
0.5
Large
0.8
0.8
0.50
omega_squared (ω²) - less biased estimate of population effect, especially for small samples. Report with 95% bootstrap CI.
Recommended Measure
5
Available Metrics
ReportUse omega_squared (ω²) - less biased estimate of population effect, especially for small samples. Report with 95% bootstrap CI. to represent clinical impact magnitude.
08Statistical Power
Sample Size
Guidelines for minimum sample requirements and power analysis parameters.
An underpowered study is an ethical failure. Respect the data by collecting enough of it.
Power ProtocolFloor Requirements
The 'Variance Stability' Minimum: A minimum of 25 participants per factor level is essential. Eta-squared systematically overestimates variance-explained in small samples; robust estimates require cell saturation.
| Effect Size | Parameters | Required n |
|---|---|---|
| Small Effect | η² = .01 (Small) | n ≈ 1000 total |
| Medium Effect | η² = .06 (Medium) | n ≈ 160 total |
| Large Effect | η² = .14 (Large) | n ≈ 60 total |
Key considerations
The 'Sample Bias' Penalty: Pure η² counts random sampling error as 'Explained Variance'. In samples with N < 50, η² can claim 10% explained even if the true population effect is zero. Use Omega-squared for small samples to maintain rigor.
G*Power StrategyBenchmark: F-tests → ANOVA (Omnibus). Parameters: Effect size f (converted from η²), α = .05, Power = .80. Note: Power depends on the total number of groups being compared.
09APA narrative blueprint
Reporting
How to compile statistical results into publication prose matching APA and journal style guides.
Data does not speak for itself. It requires a translator. Be clear, be precise, be honest.
Narrative ArcWorked APA paragraph example
A one-way ANOVA revealed a significant effect of teaching method on exam scores, F(2, 87) = 8.45, p < .001, ω² = .13, 95% CI [.04, .24], indicating a large effect. Approximately 13% of the variance in exam scores was attributable to teaching method. Flipped classroom instruction (M = 82.1, SD = 10.2) produced higher scores than traditional lecture (M = 72.3, SD = 11.8), with active learning intermediate (M = 78.4, SD = 10.9).
Reusable template
A one-way/factorial ANOVA revealed a significant effect of IV on DV, F(df_between, df_within) = X.XX, p < .XXX, ω² = .XX, 95% CI .XX, .XX, indicating a small/medium/large effect. Alternatively: η² = .XX (sample estimate) and ω² = .XX (population estimate). Approximately X% of the variance in DV was attributable to IV.
Essential statistics to report
- F-statistic with degrees of freedom
- p-value
- Effect size (ω² preferred, η² acceptable)
- 95% confidence interval for effect size
- Effect size interpretation (small/medium/large)
- Descriptive statistics per group (M, SD, n)
10Exhibit Builder
Manuscript Lab
Copy standard summary tables and forensic reporting grids to outline analysis details.
Table 1: Eta-Squared (η²) Variance Partitioning
| Source | Sum of Squares | Eta-Squared (η²) | Interpretation |
|---|---|---|---|
| Factor A | 45.2 | 0.12 | Small-Medium |
| Factor B | 112.8 | 0.30 | Large |
| Error | 214.1 | 0.58 | Residual Variance |
Note. Reporting total variance explained (not partial). N = 150.
Factor B (0.30)Powerful Dominance. Factor B is the primary lever in this system, explaining 30% of all variation in the data.
Header glossary
The 'Ownership' Ratio. Tells you what percentage of the TOTAL variation in the outcome is owned by this specific factor. Sum of all η² + Error η² = 1.0.
The 'Unknown'. Represents the proportion of variance left unexplained by the model.
11Algorithmic Logic
Command Center
Syntax libraries and function parameters for executing calculations in stats packages.
Code is the modern laboratory. Clean execution ensures reproducible discovery.
Execution Engine# 1. Extract η² from ANOVA object
effectsize::eta_squared(anova_model, partial = FALSE)
# 2. Comparison with Partial η²
lsr::etaSquared(anova_model)Library stack
R
effectsizelsr
Python
pingouin
Elite Forensic Strike
Eta-squared is biased upward in small samples. If your sample is small, report 'Omega-Squared' (ω²) for a more accurate, conservative estimate of variance explained.
# Execute Unbiased Variance Audit
effectsize::omega_squared(anova_model)12The Over-adjustment Trap
Common Mistakes
Analytical caveats and corrections to maintain modeling integrity.
Wisdom is learning from the failures of others. Anticipate the error before it occurs.
Defensive LogicWhy it's wrong
η² is positively biased, especially in small samples. It describes sample variance but overestimates population effect. With small n, η²=.20 might correspond to ω²=.10 in population.
The correction
ALWAYS report ω² (omega-squared) for population inference. Report η² only as descriptive statistic for your specific sample. Include: 'ω² = .12 (95% CI [.05, .21])' with bootstrap CI.
Why it's wrong
ηp² depends on what covariates/factors are in the model. A study with many covariates will show larger ηp² (variance is partitioned out). ηp² from 1-way ANOVA ≠ ηp² from 3-way ANOVA for same effect.
The correction
Only compare ηp² within same design. For cross-study comparison, convert to Cohen's f or d, or report generalized η² (adjusts for design differences).
Why it's wrong
η² quantifies association (variance explained), not causation. In observational data, confounds inflate η². 'Yoga explains 15% of happiness variance' doesn't mean yoga CAUSES 15% of happiness—self-selection and lifestyle confounds not controlled.
The correction
State: 'η² = .06 indicates teaching method is associated with 6% of test score variance' (not 'causes'). For causal claims, use RCT + control for confounds + report adjusted effect sizes.
Why it's wrong
η² has severe positive bias in small samples. Can be 2-3x larger than true population effect. E.g., true ω²=.05, but observed η²=.15 with n=10 per group (sampling error inflates η²).
The correction
With small n: (1) Report ω² instead (less biased); (2) Compute bootstrap 95% CI to show uncertainty; (3) Pre-register and replicate; (4) Use Kelley's (2007) bias-corrected η². Minimum n ≈ 25 per group for stable η² estimates.
Why it's wrong
Cohen's benchmarks (.01/.06/.14) are heuristics, not absolutes. In physics/chemistry, η²=.30 is expected (tight experimental control). In social sciences, η²=.05 can be meaningful (many confounds, noisy measures). Field norms vary widely.
The correction
Contextualize: 'η²=.08 is medium by Cohen but large for educational interventions (typical η²≈.03; Kraft, 2020).' Compare to prior meta-analyses in your domain. Report practical significance, not just benchmarks.
Why it's wrong
Point estimate η²=.12 could have 95% CI [.02, .28]—huge uncertainty! Without CI, readers assume precision. Effect size estimates have sampling error just like means do.
The correction
ALWAYS report CI: 'ω²=.12, 95% CI [.05, .22]'. Use bootstrap (recommended) or analytic formulas. Wide CI → need larger sample. Narrow CI → precise estimate.
Why it's wrong
Post-hoc t-tests use MS_error from subset of data, not full model. This inflates/deflates effect sizes unpredictably. η² should reflect omnibus effect (all groups), not pairwise comparison.
The correction
Compute η²/ω² from omnibus ANOVA F-test. For pairwise effect sizes, use Cohen's d from post-hoc tests. Report: 'Omnibus ω²=.14; post-hoc: Group A vs B, d=0.68.'
Why it's wrong
If model fits poorly (F non-significant), η² can be positive due to sampling error while true effect is zero. η²>.00 doesn't guarantee meaningful variance explained—it can be noise.
The correction
Check ANOVA p-value first. If p > .05, report 'η²=.02, p=.38, indicating no evidence of meaningful effect.' Don't interpret η² if omnibus test is non-significant.
Why it's wrong
η² assumes homogeneous variance. With heteroscedasticity, groups with larger variance contribute disproportionately to SS_total, biasing η² unpredictably. Levene's p<.05 invalidates standard η².
The correction
If heteroscedastic: (1) Use ε² (epsilon-squared) from Welch's ANOVA; (2) Use generalized η² that weights by inverse variance; (3) Transform DV and recompute. Report: 'Levene's p=.03; used ε²=.11 from Welch's ANOVA.'
Why it's wrong
While numerically identical in one-way designs (ηp² = η² when only one factor), they have different conceptual meanings. ηp² is 'variance controlling for other effects' but there are no other effects in one-way. This creates confusion when extending to factorial designs.
The correction
In one-way ANOVA, report η² (not ηp²) for clarity: 'η²=.12 (12% of total variance).' Reserve ηp² for factorial/ANCOVA where multiple effects exist. Be explicit: 'We report η² as no covariates were included.'
13Academic Lineage
References
Scholarly lineage and citation keys grounding the statistical framework.
We stand on the shoulders of giants. Honor the source of the method.
Academic Lineage[1]
Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum Associates.
Seminal source for η² benchmarks (.01/.06/.14) and conversion to Cohen's f for power analysis.
[2]
Richardson, J. T. E. (2011). Eta squared and partial eta squared as measures of effect size in educational research. Educational Research Review, 6(2), 135-147.
Comprehensive tutorial on η² vs ηp² in ANOVA, including calculation, interpretation, and common errors.
doi: 10.1016/j.edurev.2010.12.001
[3]
Lakens, D. (2013). Calculating and reporting effect sizes to facilitate cumulative science: A practical primer for t-tests and ANOVAs. Frontiers in Psychology, 4, 863.
Practical guide to effect size computation in R and SPSS, with worked examples and code.
doi: 10.3389/fpsyg.2013.00863
[4]
Olejnik, S., & Algina, J. (2003). Generalized eta and omega squared statistics: Measures of effect size for some common research designs. Psychological Methods, 8(4), 434-447.
Advanced paper on bias-corrected effect sizes and generalized η² for complex designs (repeated measures, mixed models).
doi: 10.1037/1082-989X.8.4.434
[5]
Kelley, K. (2007). Confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20(8), 1-24.
Methods for computing confidence intervals for η² and other effect sizes, with R package (MBESS).
doi: 10.18637/jss.v020.i08
If you don't know who owns the variance, you don't know who owns the story. Use Eta-Squared to find the dominant signal in the noise.
The Interpretive Rigor Directive
statminds · Eta-SquaredMind reference · v2.2 · updated 2026-01-1715 of 15 sections