Hierarchical Bayesian Regression
v1.0.2
Education Sciences context — multilevel regression with a random intercept that varies across okuls. It estimates the sinav_basarisi ~ onceki_donem_not_ort relationship separately for each school 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?
- School-level clustered data in Education Sciences (≥10-20 individuals per okul)
- ICC > 0.05 — group-level variance should be taken into account
- Shrinkage-corrected ranking for the question “Which okul performs high/low?”
⚙ Assumptions
- Random effect ~ Normal(0, σ²okul).
- Within-group residual ~ Normal(0, σ²).
- ≥10 observations per group are recommended; with 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 — Education 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 Education Sciences:
Egitim_Bilimleri/104_hierarchical_bayesian_LMM.xlsx
🎬 Scenario
Consider data collected from learners nested within five groups such as
classrooms or schools, labeled G1 through G5, with 200 observations in
total. We have a continuous predictor X_covariate and a continuous response
Y_response, and we suspect that the relationship varies from one group to
another. A Bayesian hierarchical (multilevel) model is ideal because it lets
each group have its own intercept and slope while partially pooling
information across groups, and it gives full posterior distributions for
both the overall effect and the between-group variability. This respects the
clustered structure of educational data and avoids treating students as if
they were independent.
⚙️ Variable Selection
- Grouping / random-effect variable: group_id (G1…G5)
- Predictor (covariate): X_covariate
- Response (dependent variable): Y_response
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
─────────────────────────────────────────────
Group random variance = 2.90 (8.6%) residual = 30.79 (91.4%) ICC = 0.086
Y ~ X covariate + (1 | group) (Bayesian)
💬 Interpretation
We analysed nested data (observations within groups) with a Bayesian hierarchical model — the Bayesian
counterpart of LMM. The group-level variance is only 8.6% of the total (ICC = 0.086); most of the variability
(91%) is within-group/individual. So in this data between-group differences are small, observations behave
largely independently. The Bayesian hierarchical model’s strength: it expresses both within- and between-group
uncertainty with full probability distributions and balances groups with few observations via “partial pooling”.
It is a modern method for multi-centre or nested educational data.
⚠ Common Mistakes
- If there are <5 observations in each group, the group-level estimate is unreliable — a more informative prior is required.
- Random slope (random slope of onceki_donem_not_ort) 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.