grad-hlm
用階層線性模型處理巢狀資料，拆解組內組間變異與跨層級效果
中文翻譯
--- name: "grad-hlm" description: "用階層線性模型（HLM）分析具有隨機截距與隨機斜率的巢狀資料結構，並處理組內相關與跨層級交互作用。當使用者的資料是學生巢套在學校、員工巢套在公司，或個人身上的重複測量，需要把變異拆到不同層級，或問到「巢狀資料怎麼處理」「ICC 是什麼」「群體層級的因素會調節個人層級的關係嗎」時，就用這個 skill。" metadata: category: "WP-31 量化方法" tags: ["HLM", "multilevel-model", "nested-data", "ICC", "random-intercept", "random-slope", "cross-level"] --- # 階層線性模型 (Hierarchical Linear Modeling) ## Overview 階層線性模型（HLM），也叫多層次模型，處理資料的巢狀結構，也就是低層級單位（例如學生、員工）集中在高層級單位（例如學校、公司）之內。它把變異拆成組內和組間兩個部分，並允許截距和斜率隨機變動，所以能產生無偏的估計和正確的標準誤。 ## When to Use - 資料有階層或巢狀結構（個人在群體之內） - 組內相關（ICC）不可忽略（經驗法則：ICC > 0.05） - 研究問題涉及跨層級交互作用（群體層級的變數調節個人層級的效果） - 巢套在受試者之內的重複測量或縱貫資料（成長模型） ## When NOT to Use - 資料沒有巢狀，或集群效應可以忽略（ICC 接近零） - 群體數很少（Level-2 單位少於 20 個） - 只關心固定效果，沒有群體層級的預測變數 - 巢狀結構是交叉的，不是階層式的（改用交叉隨機效果） ## 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. ``` 關鍵假設： 1. Level-1 殘差呈常態分布，而且組內變異數固定 2. 隨機效果（截距、斜率）在各組之間呈常態分布 3. 隨機效果和 Level-1、Level-2 的預測變數獨立（除非有建模處理） 4. 有足夠數量的 Level-2 單位，才能穩定估計變異成分 ## Methodology ### Step 1 — Estimate the Null Model (Unconditional) 跑只有截距的模型，計算 ICC = τ₀₀ / (τ₀₀ + σ²)。這告訴你總變異中有多大比例落在組間。如果 ICC 接近零，可能就不需要 HLM。 ### Step 2 — Add Level-1 Predictors (Random Intercept Model) 加入個人層級的預測變數，並用隨機截距。如果研究問題要區分組內和組間效果，就對 Level-1 預測變數做組平均置中。置中的決定與公式見 `references/`。 ### Step 3 — Add Level-2 Predictors and Cross-Level Interactions 加入群體層級的預測變數，解釋截距的組間變異。加入跨層級交互作用，檢驗群體特徵是否調節個人層級的斜率。如果理論上說得通，就讓斜率隨機變動。 ### Step 4 — Evaluate Model and Report 用 deviance（-2LL）、AIC、BIC 比較模型。報告含穩健標準誤的固定效果、變異成分，以及各層級解釋的變異比例。 ## Output Format ```markdown ## 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 - 總平均置中和組平均置中回答的是根本不同的研究問題 - Level-2 單位太少（< 20）會讓變異成分的估計有偏 - 沒有理論依據就加入隨機斜率，可能造成不收斂 - 加入 Level-1 預測變數會重新分配變異時，Level-2 的 pseudo-R² 可能是負的 - 如果存在 Level-3 巢狀（學生在班級、班級在學校）卻忽略它，會讓 Level-2 的估計有偏 - Level-1 和 Level-2 預測變數之間的共線性，會膨脹跨層級交互作用的標準誤 ## 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.
英文原文
--- name: "grad-hlm" description: "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'." metadata: category: "WP-31 量化方法" tags: ["HLM", "multilevel-model", "nested-data", "ICC", "random-intercept", "random-slope", "cross-level"] --- # 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 ```markdown ## 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.