Hierarchical Bayesian Regression
v1.0.2
Sport Sciences context — multilevel regression with a random intercept that varies across hakem. It estimates the puanlama_tutarlilik ~ deneyim_yili relationship separately for each referee and reports the posterior distribution + 95% CrI. PyMC NOT REQUIRED — statsmodels MixedLM + Empirical Bayes BLUP.
🎯 What is it for?
If your data are hierarchical (student-school, patient-hospital, measurement-device, etc.), OLS regression ignores within-group correlation → type-I error increases. Hierarchical Bayesian learns each group’s own intercept and reports it together with the 95% CrI. With BF₁₀ (LMM vs OLS), whether group-level variance is needed is tested.
📌 When is it used?
- Referee-level clustered data in Sport Sciences (≥10-20 individuals per hakem)
- ICC > 0.05 — group-level variance should be taken into account
- For the question “Which hakem shows high/low performance?”, shrinkage-corrected ranking
⚙ Assumptions
- Random effect ~ Normal(0, σ²hakem).
- Within-group residual ~ Normal(0, σ²).
- ≥10 observations per group are recommended; if fewer, an informative prior is critical.
📊 How to Run in MerQur
Panel assignments (form fields in the program):
- Columns:
{'dv': 'Y_yanit', 'fixed': ['X_kovariat'], 'group': 'grup_id'} - Parameters:
{}
📊 Sample Dataset — Sport Sciences
ℹ Note: The scenario, MerQur output and interpretation below were produced by actually running the real example dataset in MerQur. Numeric results on your own data will differ; the goal is to show how the analysis is set up and interpreted end-to-end.
🎬 Example File
This analysis is demonstrated on the following example dataset for Sport Sciences:
Spor_Bilimleri/104_hierarchical_bayesian_LMM.xlsx
🎬 Scenario
We analyze group-nested data with a Bayesian hierarchical model. For stable
estimates in small groups, this is appropriate.
⚙️ Variable Selection
- Dependent variable: Y_response
- Cluster: group_id
- Predictor(s): X_covariate
Data Preview (First 5 Rows)
| group_id | X_covariate | Y_response |
|---|---|---|
| G1 | 48.3 | 50.08 |
| G1 | 54.3 | 52.72 |
| G1 | 56.66 | 63.58 |
| G1 | 53.67 | 55.23 |
| G1 | 54.54 | 55.56 |
n = 200 · Columns: group_id, X_covariate, Y_response
📈 MerQur Output
─────────────────────────────────────────────
sigma^2_u (group) = 2.90 (8.6%) sigma^2_eps (residual) = 30.79 (91.4%) outcome: Y_response, group: group_id
💬 Interpretation
We analyzed group-nested data with a Bayesian hierarchical (multilevel) model: 8.6% of variability is
between-group, 91.4% within-group (ICC ~ 0.09). The Bayesian hierarchical model is the Bayesian version of LMM — it
estimates group effects with posterior distributions and balances small groups via “partial pooling”. In sports
science it is powerful for producing stable estimates even in small groups within multilevel (club/team/athlete)
data.
⚠ Common Mistakes
- If there are <5 observations per group, the group-level estimate is unreliable — a more informative prior is required.
- A random slope (the slope of random deneyim_yili) is not the default; if there is a cross-level interaction, it must be added explicitly.
- If ICC < 0.05, OLS may be sufficient — check BF₁₀.
📚 MerQur’a Atıf
Örücü, Ö. K. (2026). MerQur: Integrated Academic Data Analysis & Reporting Platform [Computer software] (Version 1.0.0). https://doi.org/10.53463/merqur.2026001
📝 Üretim Notu — Bu sayfadaki örnek veri sentetik olarak üretilmiştir (sabit SEED=42, generator: samples/Ileri_Duzey_v102/_generate_v102_datasets.py). Sayfa içeriği Anthropic Claude desteği ile hazırlanmış, akademik doğruluk yazar tarafından kontrol edilmiştir.