Grad Hlm

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.

Example: 員工工作自主性對工作滿意度的影響——跨層次主管支持調節效果

Scenario

台灣一家人力資源顧問公司 TalentBridge 受委託分析一份問卷資料，資料來自 35 家中型製造業企業、共 892 名正職員工。研究問題：

個人層次的「工作自主性」是否預測「工作滿意度」？公司層次的「主管支持氣候」是否調節這個關係？

研究者懷疑員工間的相似性部分來自公司文化，因此擔心 OLS 迴歸會低估標準誤。

資料結構：

• Level 1（個人）：工作自主性（autonomy，1-7 分）、年資（tenure，年）、工作滿意度（satisf，1-7 分，依變數）
• Level 2（公司）：主管支持氣候（mgr_climate，公司平均分、grand-mean centered）、公司規模（firm_size，log 員工數）

---

Analysis

Step 1 — Null Model (Unconditional)

僅含截距，估計組間變異：

Level 1: satisf_ij = β₀j + r_ij
Level 2: β₀j = γ₀₀ + u₀j

估計結果：

ICC = 0.31 / (0.31 + 0.87) = 0.263

26.3% 的工作滿意度變異在公司層次。ICC 遠超 0.05 門檻——若用 OLS，標準誤將被嚴重低估，必須使用 HLM。

---

Step 2 — Random Intercept Model（加入 Level-1 預測變數）

autonomy 採組內平均中心化（group-mean centering），分離「個人相對於公司同事的自主性」效果，避免與公司層次效果混淆。tenure 同樣組內中心化。

Level 1: satisf_ij = β₀j + β₁j(autonomy_ij − ā_j) + β₂j(tenure_ij) + r_ij
Level 2: β₀j = γ₀₀ + u₀j
         β₁j = γ₁₀              ← 固定斜率（先不允許隨機）
         β₂j = γ₂₀

固定效果結果：

Level-1 殘差變異 σ² 從 0.87 降至 0.64，Level-1 解釋比例 ≈ 26%。

---

Step 3 — 加入 Level-2 預測變數與跨層次交互作用

理論上，高主管支持氣候的公司，員工的自主性-滿意度斜率可能更陡（因為自主性在支持文化中更能發揮）。因此允許 autonomy 斜率隨機變動，並以 mgr_climate 加以解釋：

Level 1: satisf_ij = β₀j + β₁j(autonomy_cwc) + β₂j(tenure_cwc) + r_ij

Level 2: β₀j = γ₀₀ + γ₀₁(mgr_climate_j) + γ₀₂(firm_size_j) + u₀j
         β₁j = γ₁₀ + γ₁₁(mgr_climate_j) + u₁j   ← 跨層次交互
         β₂j = γ₂₀

固定效果：

隨機效果：

---

Step 4 — 模型比較

Level-2 截距變異從 0.31（null）降至 0.09，Level-2 解釋比例 ≈ 71%。

---

Result

## HLM Analysis: 員工工作自主性與工作滿意度

### Data Structure
| Level | Unit | N |
|-------|------|---|
| Level 1 | 員工 | 892 |
| Level 2 | 製造業公司 | 35 |

### ICC (Null Model)
- ICC = 0.26（26% 的滿意度變異在公司層次）
- 結論：ICC 顯著，OLS 會低估標準誤，HLM 為必要選擇

### Fixed Effects (Final Model)
| 預測變數 | Level | γ | S.E. | t | p |
|----------|-------|---|------|---|---|
| 截距 | — | 4.82 | 0.09 | 53.6 | < .001 |
| 主管支持氣候 | 2 | 0.38 | 0.12 | 3.17 | .003 |
| 公司規模 | 2 | −0.07 | 0.06 | −1.17 | .251 |
| 工作自主性 (cwc) | 1 | 0.43 | 0.05 | 8.6 | < .001 |
| 自主性 × 主管支持氣候 | 1×2 | 0.19 | 0.07 | 2.71 | .008 |
| 年資 (cwc) | 1 | 0.03 | 0.01 | 3.1 | .002 |

### Random Effects
| 成分 | 變異 | SD | p |
|------|------|-----|---|
| 截距 (τ₀₀) | 0.09 | 0.30 | < .001 |
| autonomy 斜率 (τ₁₁) | 0.04 | 0.20 | .012 |
| 殘差 (σ²) | 0.63 | 0.79 | — |

### Model Comparison
| 模型 | −2LL | AIC | 參數 | Δ deviance (p) |
|------|------|-----|------|----------------|
| Null | 2341.6 | 2347.6 | 3 | — |
| Final | 2152.7 | 2174.7 | 11 | 188.9 (< .001) |

### 實質解釋
- 工作自主性正向預測滿意度（γ = 0.43），在其他條件不變下，
  自主性每提升 1 分，滿意度平均提升 0.43 分
- 跨層次交互顯著（γ = 0.19）：主管支持氣候越高的公司，
  自主性對滿意度的效果越強——在 +1 SD 氣候公司中效果為 0.62，
  在 −1 SD 氣候公司中效果為 0.24
- 公司規模對滿意度無顯著效果，可從最終模型精簡時考慮移除

### Limitations
- 35 個 Level-2 單位位於可接受下限，τ₁₁ 估計有較大不確定性
- 橫斷面資料，無法排除員工自我選擇進入高自主文化公司的反向因果
- 未控制個人層次的人格特質（如自我效能），可能造成自主性效果高估