階層線性模型 (Hierarchical Linear Modeling)

Overview

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

• 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

• 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

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:

1. Level-1 residuals are normally distributed with constant variance within groups
2. Random effects (intercepts, slopes) are normally distributed across groups
3. Random effects are independent of Level-1 and Level-2 predictors (unless modeled)
4. Sufficient number of Level-2 units for stable variance component estimation

Methodology

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

## HLM Analysis: [Study Title]

### Data Structure
| Level | Unit | N |
|-------|------|---|
| Level 1 | [individual] | xxx |
| Level 2 | [group] | xxx |

### ICC (Null Model)
- ICC = x.xx (x% of variance is between groups)

### Fixed Effects
| Predictor | Level | γ | S.E. | t | p-value |
|-----------|-------|---|------|---|---------|
| Intercept | — | x.xx | x.xx | x.xx | x.xx |
| [L1 var] | 1 | x.xx | x.xx | x.xx | x.xx |
| [L2 var] | 2 | x.xx | x.xx | x.xx | x.xx |
| [Cross-level] | 1×2 | x.xx | x.xx | x.xx | x.xx |

### Random Effects
| Component | Variance | SD | p-value |
|-----------|----------|-----|---------|
| Intercept (τ₀₀) | x.xx | x.xx | x.xx |
| Slope (τ₁₁) | x.xx | x.xx | x.xx |
| Residual (σ²) | x.xx | x.xx | — |

### Model Comparison
| Model | -2LL | AIC | Parameters | Δ deviance (p) |
|-------|------|-----|------------|---------------|
| Null | x.xx | x.xx | x | — |
| Final | x.xx | x.xx | x | x.xx (x.xx) |

### Limitations
- [Note any assumption violations]

Gotchas

• 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

• 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.