階層線性模型 (Hierarchical Linear Modeling) 階層線性模型 HLM
Released已發布Apply Hierarchical Linear Modeling (HLM) to analyze nested data structures with random intercepts and slopes, accounting for intra-class correlation and cross-level interactions. Use this skill when the user has students nested in schools, employees in firms, or repeated measures in individuals, needs to partition variance across levels, or when they ask 'how do I handle nested data', 'what is ICC', or 'do group-level factors moderate individual-level relationships'.
學術研究技能：階層線性模型 (Hierarchical Linear Modeling) 分析與應用。
Overview概述
Hierarchical Linear Modeling (HLM), also called multilevel modeling, accounts for the nested structure of data where lower-level units (e.g., students, employees) are clustered within higher-level units (e.g., schools, firms). By partitioning variance into within-group and between-group components and allowing intercepts and slopes to vary randomly, HLM produces unbiased estimates and correct standard errors.
When to Use使用時機
- Data has a hierarchical or nested structure (individuals within groups)
- Intra-class correlation (ICC) is non-trivial (rule of thumb: ICC > 0.05)
- Research questions involve cross-level interactions (group-level moderators of individual-level effects)
- Repeated measures or longitudinal data nested within subjects (growth models)
When NOT to Use不適用時機
- Data are not nested or clustering is negligible (ICC near zero)
- Number of groups is very small (fewer than 20 Level-2 units)
- Interest is purely in fixed effects with no group-level predictors
- The nesting structure is crossed, not hierarchical (use crossed random effects instead)
Assumptions前提假設
IRON LAW: Ignoring nested structure when ICC is non-trivial produces
UNDERESTIMATED standard errors — leading to inflated Type I error rates.
OLS treats clustered observations as independent, overstating precision.
Key assumptions:
- Level-1 residuals are normally distributed with constant variance within groups
- Random effects (intercepts, slopes) are normally distributed across groups
- Random effects are independent of Level-1 and Level-2 predictors (unless modeled)
- Sufficient number of Level-2 units for stable variance component estimation
Framework 框架
Step 1 — Estimate the Null Model (Unconditional)
Run an intercept-only model to compute ICC = τ₀₀ / (τ₀₀ + σ²). This tells you what proportion of total variance lies between groups. If ICC is near zero, HLM may be unnecessary.
Step 2 — Add Level-1 Predictors (Random Intercept Model)
Include individual-level predictors with a random intercept. Group-mean center Level-1 predictors if the research question distinguishes within-group from between-group effects. See references/ for centering decisions and equations.
Step 3 — Add Level-2 Predictors and Cross-Level Interactions
Include group-level predictors to explain between-group variance in intercepts. Add cross-level interactions to test whether group characteristics moderate individual-level slopes. Allow slopes to vary randomly if theoretically justified.
Step 4 — Evaluate Model and Report
Compare models using deviance (-2LL), AIC, BIC. Report fixed effects with robust standard errors, variance components, and proportion of variance explained at each level.
Output Format輸出格式
Gotchas注意事項
- Grand-mean centering and group-mean centering answer fundamentally different research questions
- Too few Level-2 units (< 20) yields biased variance component estimates
- Adding random slopes without theoretical justification can cause non-convergence
- Pseudo-R² at Level 2 can be negative if adding Level-1 predictors redistributes variance
- Ignoring Level-3 nesting (students in classrooms in schools) when it exists biases Level-2 estimates
- Multicollinearity between Level-1 and Level-2 predictors inflates standard errors of cross-level interactions
References參考資料
- Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical Linear Models (2nd ed.). Sage.
- Hox, J. J., Moerbeek, M., & van de Schoot, R. (2018). Multilevel Analysis (3rd ed.). Routledge.
- Snijders, T. A. B., & Bosker, R. J. (2012). Multilevel Analysis (2nd ed.). Sage.