參考資料 · REFERENCES

權威書籍、原始論文與套件文件

為 Statistical Inference Interactive Tutorial 17 個主題完整佐證的學術出處，附 DOI 連結。

Fully cited sources for all 17 modules of the Statistical Inference Interactive Tutorial, with DOI links.

概覽

如何使用這份資料？

本頁針對教學中提到的每個概念、檢定、套件與演算法整理學術出處。引用標籤含義：

For every concept, test, package, and algorithm mentioned in the tutorial, this page lists the canonical source. Tag legend:

📚

Book

教科書 / 線上免費書

Textbooks / free online books

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Paper

原始論文（含 DOI / PubMed）

Original papers (with DOI / PubMed)

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Doc

官方文件、vignette、tutorial

Official docs, vignettes, tutorials

⭐

Best Practice

綜述與 community 推薦

Reviews and community recommendations

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Benchmark

獨立 benchmarking 評比

Independent benchmarking studies

本頁目錄

★核心教科書 Step 1Probability / Distributions Step 2Point Estimation Step 3CI / Bootstrap Step 4Hypothesis Testing Step 5Common Tests Step 6Linear Regression Step 7ANOVA / Mixed Step 8Logistic Step 9GLM Counts Step 10Bayes Basics Step 11MCMC / HMC Step 12Posterior / PPC Step 13Empirical Bayes Step 14FDR Step 15DE Statistics Step 16Permutation / GSEA Step 17Survival
核心教科書

⭐ 貫穿全教程的權威教科書

這幾本書幾乎覆蓋了本教程所有章節的理論基礎，是進階學習的最佳起點。

These books cover the theoretical foundation of nearly every topic in this tutorial — the best starting point for going deeper.

Step 1

🎲 機率與分布

Step 2

🎯 點估計與抽樣分布

Step 3

📏 信賴區間與 Bootstrap

Step 4

⚖️ 假設檢定與錯誤

Step 5

🧪 常見檢定方法

Step 6

📈 線性迴歸與診斷

Step 7

🔀 ANOVA / 混合效應

Step 8

🎚️ Logistic 迴歸與 OR

Step 9

🔢 GLM 計數模型

Step 10

🧠 貝氏定理 / 先驗 / 後驗

Step 11

⛓️ MCMC / HMC / 診斷

Step 12

📐 後驗摘要 / BF / PPC

Step 13

🪢 階層 / 經驗貝氏

Step 14

🚦 多重檢定與 FDR

Step 15

🧬 DE 統計核心

Step 16

🔄 排列 / Bootstrap / GSEA

Step 17

⏳ 存活分析

教學註記

📌 教學註記與細節

以下整理本頁引用脈絡中容易被忽略、但會直接影響貝氏/頻率推論判讀的補充說明。內容皆為延伸閱讀，所引文獻已收錄於上方各 Step；目的在於把零散注記集中於一處，方便日後查閱。

Below is a consolidated set of notes that often slip past readers but materially affect how Bayesian/frequentist inference is read. These are extended discussion items — the underlying references already appear in the Step sections above, and gathering them here is meant to make later look-up easier.

E1 · Welch's t as the New Default (Delacre, Lakens, Leys 2017, IRSP)

傳統教學常先教 Student's t-test (assumes equal variances)，再以 Levene's test 決定是否改用 Welch's t-test。Delacre, Lakens & Leys 2017 (Int Rev Soc Psychol; 10.5334/irsp.82) 透過 Monte Carlo simulation 證明這個流程實際上 inferior — 因為 (1) Levene's test 本身有 type II error，會在實際存在 variance heterogeneity 時失敗；(2) Welch's t-test 在 variances 確實相等時，僅損失少量 power (df 略小)；(3) Welch's t-test 對 unequal n + unequal variance 的組合穩健。結論：應將 Welch's t-test (R: `t.test()` 預設、Python: `scipy.stats.ttest_ind(equal_var=False)`) 作為兩組比較的 *default*，不需先做 Levene/F-test。Ruxton 2006 (Behav Ecol) 為更早的同方向論證。注意：Welch-Satterthwaite df 為非整數，這是正確行為。對 paired design，仍應使用 paired t-test (差值的 one-sample t)。在嚴重 non-normality + small n 下，Wilcoxon signed-rank / Mann-Whitney 或 permutation test 為較佳選擇。

Traditional teaching first introduces Student's t-test (equal variances) and uses Levene's test to decide whether to switch to Welch's t-test. Delacre, Lakens & Leys 2017 (Int Rev Soc Psychol; 10.5334/irsp.82) showed via Monte Carlo simulation that this workflow is actually inferior because (1) Levene's test itself has type II error and can fail when variance heterogeneity is real; (2) Welch's t-test loses only a small amount of power (slightly smaller df) when variances are equal; (3) Welch's t-test is robust to combinations of unequal n + unequal variance. Conclusion: Welch's t-test (R: `t.test()` default; Python: `scipy.stats.ttest_ind(equal_var=False)`) should be the *default* for two-group comparisons — no preliminary Levene/F-test required. Ruxton 2006 (Behav Ecol) made an earlier version of the same argument. Note: the Welch-Satterthwaite df is non-integer — that is correct behavior. For paired designs, use the paired t-test (one-sample t on differences). Under severe non-normality + small n, Wilcoxon signed-rank / Mann-Whitney or a permutation test is preferable.

E2 · MCMC Convergence ≠ Truth (Vehtari et al. 2021, Bayesian Analysis)

MCMC 診斷 (Gelman-Rubin R̂、effective sample size ESS、trace plot) 的角色是判斷「chain 是否從 stationary distribution 抽樣」— 即 *computational convergence*。但 convergence 不代表後驗分布反映真實 — 模型若 misspecified、prior 若主導、likelihood 若計算錯誤，MCMC 仍可能 converge 到一個錯誤的後驗。Vehtari et al. 2021 (Bayesian Analysis 16(2):667-718) 提出 rank-normalized split-R̂ + folded R̂ 為改良診斷：(1) split chain 為兩半比較 within/between variance；(2) rank-transform 處理重尾或非 Gaussian posterior；(3) folded 版本檢測 scale 而非 location 差異。建議 R̂ < 1.01 (舊閾值 1.1 過於寬鬆) 且 ESS > 400 per parameter (對 quantile 估計，bulk-ESS 與 tail-ESS 應分開檢視)。配套：用 prior predictive check (Gelman 2017) 確認 prior 合理；posterior predictive check (PPC; Gabry 2019, JRSS A) 確認 model 可生成類似觀察資料；leave-one-out cross-validation (LOO; Vehtari 2017, Stats Comput) 比較 model 預測力。實作：ArviZ `az.summary()`、`az.plot_trace()`、`az.loo()`；bayesplot `mcmc_trace()`、`ppc_dens_overlay()`。

MCMC diagnostics (Gelman-Rubin R̂, effective sample size ESS, trace plot) tell you whether the chain is sampling from the stationary distribution — i.e., *computational convergence*. Convergence does not imply the posterior reflects reality — if the model is misspecified, the prior dominates, or the likelihood is coded wrong, MCMC can still converge to an incorrect posterior. Vehtari et al. 2021 (Bayesian Analysis 16(2):667-718) propose rank-normalized split-R̂ + folded R̂ as improved diagnostics: (1) split each chain in half and compare within/between variance; (2) rank-transform to handle heavy-tailed or non-Gaussian posteriors; (3) the folded version detects scale rather than location differences. Recommend R̂ < 1.01 (the old 1.1 threshold is too lax) and ESS > 400 per parameter (for quantile estimation, examine bulk-ESS and tail-ESS separately). Complement with: prior predictive checks (Gelman 2017) to validate the prior, posterior predictive checks (PPC; Gabry 2019, JRSS A) to confirm the model can generate data similar to observed, and leave-one-out cross-validation (LOO; Vehtari 2017, Stats Comput) to compare predictive power. Implementation: ArviZ `az.summary()`, `az.plot_trace()`, `az.loo()`; bayesplot `mcmc_trace()`, `ppc_dens_overlay()`.

E3 · BIC vs AIC — Asymptotic Limits

AIC = −2·log L + 2k；BIC = −2·log L + k·log(n)。雖然兩者形式相似，背後 asymptotic justification 完全不同：AIC (Akaike 1973) 為估計 *out-of-sample predictive accuracy* (Kullback-Leibler divergence)，假設 "true model" *不在* candidate set 中 — AIC 選最接近真實的 approximation。BIC (Schwarz 1978) 為估計 *posterior model probability*，假設 "true model" *在* candidate set 中 — BIC 在 n → ∞ 時 consistent (選到真實 model 機率 → 1)，但代價是在中等 n 時 *under-fit* (penalty 過重)。實用上：(1) 預測為目的 → AIC 或 LOO-CV (Vehtari 2017)；(2) 變數選擇 + 真實 model 假設 → BIC；(3) hierarchical/random effects 模型 → DIC 已被淘汰，改用 WAIC 或 PSIS-LOO；(4) 雙者不可直接比較數值 — AIC 與 BIC 不在同一尺度。Burnham & Anderson 2002 為頻率主義模型平均的標準參考。對 Bayesian 模型，建議 LOO-CV (Vehtari 2017, Stats Comput 27:1413) + ELPD 為主要工具，因其考慮整個 posterior 而非單一 point estimate。Stan/PyMC + ArviZ 的 `az.compare()` 提供標準輸出。

AIC = −2·log L + 2k; BIC = −2·log L + k·log(n). The forms look similar, but the asymptotic justifications differ completely: AIC (Akaike 1973) estimates *out-of-sample predictive accuracy* (Kullback-Leibler divergence) under the assumption that the "true model" is *not* in the candidate set — AIC selects the best approximation. BIC (Schwarz 1978) estimates *posterior model probability* under the assumption that the "true model" *is* in the candidate set — BIC is consistent as n → ∞ (probability of selecting the true model → 1), but the cost is *under-fitting* at moderate n (heavy penalty). Practical guidance: (1) prediction → AIC or LOO-CV (Vehtari 2017); (2) variable selection assuming the true model is in the set → BIC; (3) hierarchical / random-effects models → DIC is deprecated, use WAIC or PSIS-LOO; (4) AIC and BIC values cannot be directly compared — they are not on the same scale. Burnham & Anderson 2002 is the standard reference for frequentist multi-model inference. For Bayesian models, prefer LOO-CV (Vehtari 2017, Stats Comput 27:1413) + ELPD as the primary tool because it uses the full posterior rather than a point estimate. Stan/PyMC + ArviZ `az.compare()` provides standardized output.

E4 · Stan vs PyMC — NUTS / brms / JAX Backends

兩大主流 probabilistic programming 平台各有定位：(1) Stan (Carpenter et al. 2017, JSS 76(1):1-32) 以 C++ 實作 NUTS (No-U-Turn Sampler, Hoffman & Gelman 2014) 的 HMC，提供 R 介面 (rstan、rstanarm、brms — Bürkner 2017 JSS 80(1):1-28) 與 Python (PyStan、cmdstanpy)。優勢：成熟 (2012-)、社群活躍、warning system 完整 (divergent transitions、E-BFMI)。劣勢：模型需編譯 (cold start ~30s)。(2) PyMC (v5+; Salvatier 2016, PeerJ CS) 以 Python + PyTensor，後端可切換 (default Numba, JAX via `pymc-jax`)。優勢：(a) JAX backend 提供 GPU/TPU 加速；(b) variational inference (ADVI、Pathfinder) 與 SMC sampler 內建；(c) 與 deep learning ecosystem (Flax、Haiku) 整合。NumPyro (Phan 2019) 為 Uber 開發的 JAX-native PPL，速度更快但 API 較底層。BlackJAX 為 JAX-based 採樣器函式庫。選擇建議：(i) hierarchical model with random slopes → brms (Stan)，syntax 最接近 lme4；(ii) deep generative model / amortized inference → NumPyro/Pyro；(iii) 大型 dataset + GPU → PyMC + JAX 或 NumPyro。所有平台都支援 ArviZ 為共用 diagnostics 介面 (Kumar 2019)。

The two mainstream probabilistic programming platforms have different niches: (1) Stan (Carpenter et al. 2017, JSS 76(1):1-32) implements NUTS (No-U-Turn Sampler, Hoffman & Gelman 2014) HMC in C++, with R interfaces (rstan, rstanarm, brms — Bürkner 2017 JSS 80(1):1-28) and Python (PyStan, cmdstanpy). Pros: mature (since 2012), active community, comprehensive warning system (divergent transitions, E-BFMI). Cons: models must be compiled (cold start ~30s). (2) PyMC (v5+; Salvatier 2016, PeerJ CS) uses Python + PyTensor with switchable backends (Numba by default, JAX via `pymc-jax`). Pros: (a) JAX backend gives GPU/TPU acceleration; (b) variational inference (ADVI, Pathfinder) and SMC sampler built in; (c) integrates with deep-learning ecosystems (Flax, Haiku). NumPyro (Phan 2019, Uber) is a JAX-native PPL — faster but lower-level API. BlackJAX is a JAX-based sampler library. Choosing: (i) hierarchical model with random slopes → brms (Stan), syntax closest to lme4; (ii) deep generative model / amortized inference → NumPyro/Pyro; (iii) large dataset + GPU → PyMC + JAX or NumPyro. All platforms support ArviZ as a common diagnostics interface (Kumar 2019).

E5 · p-value vs Posterior — Jeffreys-Lindley Paradox

在大樣本下，frequentist test 與 Bayesian model comparison 可給出截然相反的結論 — 即 Jeffreys-Lindley paradox (Lindley 1957, Biometrika)。經典例子：H₀: μ = 0 vs H₁: μ ≠ 0，n = 10⁶，sample mean = 0.001 (z = 2.0, p = 0.046 → reject H₀)；但若 prior on μ under H₁ 為 N(0, 1)，Bayes factor BF₁₀ ≈ 0.05 → strong evidence *for* H₀。原因：p-value 隨 n 增加而對任何 ε ≠ 0 的偏離都敏感；Bayes factor 則 penalize "預測力分散到大區間" 的 alternative，即 Occam's razor。實作建議：(1) 大樣本 + 點 null 場景，p-value 容易誤導 — 改用 region of practical equivalence (ROPE; Kruschke 2018) 或 minimum effect size of interest (SESOI)。(2) 報告 effect size + 95% CI/CrI 比單一 p-value/BF 更 informative。(3) Bayes factor 對 prior on alternative *敏感* (e.g., Cauchy(0, r=√2/2) 為 BayesFactor 套件 default; Rouder 2009)，需 sensitivity analysis。(4) ASA 2016/2019 與 Greenland 2016 (Eur J Epidemiol) 為 p-value 誤解的標準參考。Berger & Sellke 1987 (JASA) 量化 p ≈ 0.05 對應的 minimum BF₁₀ ≈ 2.5，遠小於 Jeffreys "substantial" 的 3。

At large sample sizes, a frequentist test and a Bayesian model comparison can give opposite conclusions — the Jeffreys-Lindley paradox (Lindley 1957, Biometrika). Classic example: H₀: μ = 0 vs H₁: μ ≠ 0, n = 10⁶, sample mean = 0.001 (z = 2.0, p = 0.046 → reject H₀); but with prior μ ~ N(0, 1) under H₁, the Bayes factor BF₁₀ ≈ 0.05 → strong evidence *for* H₀. Reason: the p-value becomes sensitive to any ε ≠ 0 deviation as n grows, while the Bayes factor penalizes alternatives that "spread predictive mass over a wide region" — Occam's razor. Recommendations: (1) under large-n + point-null settings, p-values are easy to misread — use a region of practical equivalence (ROPE; Kruschke 2018) or a minimum effect size of interest (SESOI). (2) Reporting effect size + 95% CI/CrI is more informative than a single p-value or BF. (3) Bayes factors are *sensitive* to the prior on the alternative (e.g., Cauchy(0, r=√2/2), the BayesFactor-package default, Rouder 2009) — perform sensitivity analysis. (4) ASA 2016/2019 and Greenland 2016 (Eur J Epidemiol) are the canonical references on p-value misinterpretations. Berger & Sellke 1987 (JASA) quantified that p ≈ 0.05 corresponds to a minimum BF₁₀ ≈ 2.5, far below Jeffreys' "substantial" threshold of 3.

E6 · Prior Choice Sensitivity (Gabry et al. 2019, JRSS A)

「Bayesian 結果依賴 prior」是頻率主義常見批評。實情更精確：(1) 在 well-identified、large-sample 場景，posterior 由 likelihood 主導，prior 影響輕微；(2) 在 small n、稀疏 data、複雜階層模型中，prior 可能顯著影響 posterior — 此時必須做 prior sensitivity analysis。Gabry et al. 2019 (JRSS A 182(2):389-402) 與 Schad et al. 2021 (Psychol Methods) 提出 Bayesian workflow：(i) prior predictive check (在看到資料前模擬 prior implied data，確認合理範圍)；(ii) fit model；(iii) MCMC diagnostics (E2)；(iv) posterior predictive check (PPC; 模擬與觀察比較)；(v) leave-one-out (LOO; Vehtari 2017) 評估預測力；(vi) prior sensitivity (重新跑 weakly informative vs informative prior 比較)。預設 prior：(a) 對 fixed effects 用 weakly informative (Normal(0, 2.5·SD(x)) for slope; Gelman 2008 Ann Appl Stat)；(b) 對 variance components 避免 inv-gamma(ε, ε)，改用 half-normal/half-cauchy/half-t (Gelman 2006 Bayesian Anal)；(c) brms 預設 prior 為 student-t(3, ...)，比 N(0, 100) 的 "vague" 更穩定。Stan Wiki "Prior Choice Recommendations" 為實用 cheat sheet。

"Bayesian results depend on the prior" is a frequent frequentist critique. More precisely: (1) in well-identified large-sample settings, the posterior is dominated by the likelihood and the prior matters little; (2) at small n, sparse data, or complex hierarchical models, the prior can substantially affect the posterior — prior sensitivity analysis is then required. Gabry et al. 2019 (JRSS A 182(2):389-402) and Schad et al. 2021 (Psychol Methods) propose a Bayesian workflow: (i) prior predictive check (simulate prior-implied data before seeing the actual data, verify the range is plausible); (ii) fit the model; (iii) MCMC diagnostics (E2); (iv) posterior predictive check (PPC; compare simulated vs observed); (v) leave-one-out (LOO; Vehtari 2017) to assess predictive performance; (vi) prior sensitivity (re-run with weakly-informative vs informative priors and compare). Default priors: (a) for fixed effects, weakly informative (Normal(0, 2.5·SD(x)) for slopes; Gelman 2008 Ann Appl Stat); (b) for variance components, avoid inv-gamma(ε, ε) — prefer half-normal / half-Cauchy / half-t (Gelman 2006 Bayesian Anal); (c) brms defaults to student-t(3, ...), which is more stable than the "vague" N(0, 100). The Stan Wiki "Prior Choice Recommendations" is a practical cheat sheet.

E7 · FDR vs FWER — Conceptual Distinction

兩者控制的錯誤類型不同：(1) FWER (family-wise error rate) = P(至少一個 false positive) ≤ α — 適用於「任何一個錯誤都嚴重」的情境 (clinical trial 多重 endpoints、藥物批准)。方法：Bonferroni、Holm 1979、Hochberg 1988、單步/逐步 Šidák。(2) FDR (false discovery rate) = E[V/R | R>0]·P(R>0) ≤ q — 「在被宣告為 significant 的結果中，預期 false positive 比例」。適用於 large-scale screening (transcriptomics、GWAS、proteomics)。方法：Benjamini-Hochberg 1995、Storey 2002 q-value、Ignatiadis 2016 IHW (data-driven hypothesis weighting)。FDR 比 FWER *寬鬆* — m = 10000 時，BH 在 q=0.05 下可保留 ~500 真實 discovery 中的 25 false positive；Bonferroni 在 α=0.05/10000 下幾乎無 power。注意：(a) FDR 不對單一 hypothesis 給 error guarantee — 只對 *整批* 結果；(b) BH 假設 p-value 在 H₀ 下 uniform 且 PRDS (positive regression dependent on subset; Benjamini-Yekutieli 2001) — 對 negatively correlated tests 應用 BY；(c) 對 discrete tests (Fisher exact)，BH 過於保守，可改用 group-FDR 或 adjusted-BH。Local FDR (Efron 2010 Large-Scale Inference) 為單一 test 的後驗 FDR，與 Bayesian framework 對接。在 RNA-seq 中，IHW 與 apeglm (Zhu 2018) 同時使用可顯著提高 power。

FDR and FWER control different errors: (1) FWER (family-wise error rate) = P(at least one false positive) ≤ α — appropriate when "any single error is serious" (multi-endpoint clinical trials, drug approval). Methods: Bonferroni, Holm 1979, Hochberg 1988, single-step / step-down Šidák. (2) FDR (false discovery rate) = E[V/R | R>0]·P(R>0) ≤ q — "expected proportion of false positives among declared significants". Appropriate for large-scale screening (transcriptomics, GWAS, proteomics). Methods: Benjamini-Hochberg 1995, Storey 2002 q-value, Ignatiadis 2016 IHW (data-driven hypothesis weighting). FDR is *more permissive* than FWER — at m = 10000 and q = 0.05, BH can retain ~25 false positives among ~500 true discoveries; Bonferroni at α = 0.05/10000 has essentially no power. Notes: (a) FDR makes no guarantee about a single hypothesis — only about the *batch* of results; (b) BH assumes p-values are uniform under H₀ and PRDS (positive regression dependence on subset; Benjamini-Yekutieli 2001) — use BY for negatively correlated tests; (c) for discrete tests (Fisher exact), BH is over-conservative — use group-FDR or adjusted-BH. Local FDR (Efron 2010 Large-Scale Inference) is the posterior FDR for a single test and connects to the Bayesian framework. In RNA-seq, using IHW together with apeglm (Zhu 2018) significantly increases power.

E8 · James-Stein / Shrinkage — limma / ashr / DESeq2 Connection

James-Stein 1961 (Berkeley Symp) 是 20 世紀統計學最反直覺的結果之一：當同時估計 *三個或更多 ≥ 3* 獨立常態 means 時，shrinkage estimator (向總平均收縮一個比例) 的 *total MSE* 嚴格優於 maximum likelihood (independent sample means)，無論真實 means 為何。其精神為「借力」(borrow strength) — 即使 means 之間 *沒有* 直接關係，pooled information 也能改善整體估計。現代基因體統計大量應用此精神：(1) limma (Smyth 2004; Ritchie 2015) 用 empirical Bayes 收縮每個 gene 的 variance 估計，向 global variance prior 收縮 — 解決 microarray/RNA-seq 中 per-gene n 過小的問題；moderated t 比 ordinary t 顯著更穩健。(2) DESeq2 (Love 2014, Genome Biol) 收縮 dispersion 估計向 mean-dispersion trend；apeglm (Zhu 2018, Bioinformatics) 用 heavy-tailed (Cauchy-like) prior 收縮 log2 fold change，避免 low-count gene 的 noise 主導 ranking。(3) edgeR (Robinson 2010) 透過 weighted likelihood 結合 common dispersion 與 tagwise dispersion。(4) ashr (Stephens 2017, Biostatistics 18(2):275) 將 FDR 框架本身改寫為 adaptive shrinkage — local FDR + effect size 一併估計，使用 unimodal prior 假設提供 LFSR (local false sign rate) 取代 q-value。(5) Bayesian hierarchical model 為更通用的 shrinkage 框架 — group-level random effects 即自動實現 borrow-strength。Efron 2010 (Large-Scale Inference) 為此議題的完整 textbook。實作要點：在小樣本、多 hypothesis、高 variance 場景，always prefer shrinkage estimators over per-test MLE。

James-Stein 1961 (Berkeley Symp) is one of the most counter-intuitive results in 20th-century statistics: when estimating *three or more (≥ 3)* independent normal means simultaneously, a shrinkage estimator (shrinking each sample mean toward the overall mean by a fraction) strictly dominates the MLE (independent sample means) in *total MSE*, regardless of the true means. The intuition is "borrowing strength" — even when the means have *no* direct relationship, pooled information still improves overall estimation. Modern genomic statistics applies this idea extensively: (1) limma (Smyth 2004; Ritchie 2015) uses empirical Bayes to shrink each gene's variance estimate toward a global variance prior, solving the per-gene-small-n problem in microarray/RNA-seq; moderated t is significantly more robust than ordinary t. (2) DESeq2 (Love 2014, Genome Biol) shrinks dispersion estimates toward a mean-dispersion trend; apeglm (Zhu 2018, Bioinformatics) uses a heavy-tailed (Cauchy-like) prior to shrink log2 fold changes, preventing low-count genes from dominating rankings by noise. (3) edgeR (Robinson 2010) combines common and tagwise dispersion through a weighted likelihood. (4) ashr (Stephens 2017, Biostatistics 18(2):275) reformulates the FDR framework itself as adaptive shrinkage — jointly estimating local FDR and effect size under a unimodal-prior assumption and giving LFSR (local false sign rate) in place of the q-value. (5) Bayesian hierarchical models are a more general shrinkage framework — group-level random effects implement borrow-strength automatically. Efron 2010 (Large-Scale Inference) is the complete textbook on this topic. Practice: in small-n, many-hypothesis, high-variance settings, always prefer shrinkage estimators over per-test MLE.

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