Inputs
Enter your model components (fixed effect estimate, uncertainty, and variance terms).
Example Data Table
Illustrative inputs and outputs for a random-intercept model with one fixed predictor.
| β | SE | df | σ | τ₀₀ | Computed d (total SD) | ICC |
|---|---|---|---|---|---|---|
| 0.45 | 0.12 | 48 | 1.10 | 0.35 | 0.37 | 0.22 |
| 0.80 | 0.20 | 60 | 1.40 | 0.50 | 0.52 | 0.20 |
| -0.30 | 0.10 | 90 | 0.90 | 0.15 | -0.32 | 0.16 |
Formula Used
- Total variance (with optional random slope): V_total = σ² + τ₀₀ + τ₁₁ · SDₓ²
- Standardized fixed-effect size (d-like): d = β / SD_target, where SD_target is chosen as total SD, residual SD, or between SD.
- t statistic: t = β / SE
- Semi-partial R² (t-based approximation): R²_partial = t² / (t² + df)
- Cohen’s f² from R²: f² = R²_partial / (1 − R²_partial)
- ICC (random intercept only): ICC = τ₀₀ / (τ₀₀ + σ²)
- Confidence bounds (β-based): β_L,U = β ± z · SE, then standardized with the same target SD to get d_L,U.
How to Use This Calculator
- Enter the fixed effect estimate (β) and its standard error (SE).
- Add df if you want semi-partial R² and f² outputs.
- Provide residual SD (σ) and random intercept variance (τ₀₀).
- Optionally include random slope variance (τ₁₁) and predictor SD (SDₓ).
- Pick a standardization target, then click Submit.
- Use CSV/PDF buttons to export the results table.
Fixed effects as standardized change
Mixed models estimate a fixed coefficient β that represents the expected change in the outcome for a one-unit predictor change, averaged across clusters. This calculator converts β into a standardized effect (d-like) by dividing by a chosen standard deviation, making units comparable across studies and scales. For example, β=0.45 with a total SD 1.22 yields d≈0.37, a small-to-moderate shift.
Variance components drive interpretability
The residual standard deviation σ captures within-cluster noise, while the random intercept variance τ₀₀ captures between-cluster differences. When τ₀₀ is large relative to σ², clustering is strong and standardized effects based on total SD shrink compared with residual-only standardization. A model with σ=1.10 and τ₀₀=0.35 has total SD ≈1.22, about 11% larger than σ alone.
Intra-class correlation as a clustering signal
ICC = τ₀₀ /(τ₀₀ + σ²) summarizes how much outcome variation is attributable to clusters. ICC values around 0.05 suggest mild clustering, 0.10–0.25 moderate clustering, and above 0.25 strong clustering, which often motivates random intercepts and careful power planning. In the example above, ICC≈0.22, meaning roughly one-fifth of variance is between clusters.
Random slopes add predictor-dependent spread
If a random slope variance τ₁₁ is included, the total variance used for standardization becomes σ² + τ₀₀ + τ₁₁·SDₓ². Larger predictor variability (SDₓ) amplifies the contribution of random slopes, reducing the standardized magnitude of β when using total SD. With τ₁₁=0.10 and SDₓ=2, the slope term adds 0.40 variance, which can widen the implied outcome spread.
t-based semi-partial R² for reporting
With SE and df, the calculator estimates a t statistic and a semi-partial R² via t²/(t²+df). This metric helps communicate practical importance for a specific fixed effect within the fitted model, and f² translates the same signal into the familiar effect-size scale used in planning. For β=0.45 and SE=0.12, t≈3.75; with df=48, R²≈0.23 and f²≈0.30.
Confidence bounds and transparent exports
Confidence limits are computed as β ± z·SE at your selected confidence level, then standardized using the same target SD. Exporting CSV and PDF supports reproducible workflows, enabling reviewers to trace assumptions, rounding choices, and the variance terms underlying each reported effect size. Reporting d and its bounds is useful when precision differs across studies or when outcomes are heteroskedastic across clusters.
FAQs
1) What effect size does the calculator report?
Primarily a standardized fixed-effect size d-like value computed as β divided by your chosen SD target (total, residual, or between). It also reports t, optional semi-partial R², f², ICC, and confidence bounds.
2) Which standardization target should I choose?
Use total SD for a conservative, model-wide standardization. Use residual SD for within-cluster interpretation. Use between SD if you want scaling relative to cluster-to-cluster differences. Report the choice explicitly for comparability.
3) Do I need degrees of freedom?
df is optional. Provide it when you want the t-based semi-partial R² and f² outputs. If df is unknown, you can still compute standardized d, ICC, and confidence bounds from β, SE, and variance terms.
4) How are confidence intervals calculated?
The calculator uses a normal-approximation z value for the selected confidence level and computes β ± z·SE. It then divides the bounds by the same SD target used for d, keeping the standardization consistent across results.
5) How do random slopes change the results?
Random slope variance adds τ₁₁·SDₓ² to total variance. As that term grows, total SD increases and standardized d typically decreases. If your model has no random slopes, keep τ₁₁ at 0 and SDₓ is ignored.
6) Is this a replacement for a full mixed-model analysis?
No. It summarizes effect sizes from parameters you already estimated elsewhere. Fit the mixed model in your statistical software, then paste β, SE, df, and variance components here to generate standardized metrics and exports.