為什麼相關係數會騙人？
Pearson r 衡量「線性」關聯，對非線性關係毫無感應（即使是完美 sin 曲線 r 可能 ~0）。Spearman 用排名，對單調非線性穩健。劑量反應曲線本質是 sigmoidal，硬套線性回歸是論文 reviewer 最常質疑的問題之一。
Ritz et al. 2015 (PLOS ONE) 的 R 套件 drc 是劑量反應分析的標準工具，2025 年 dr4pl、drda 在 robust 估計上更進一步。Sebaugh 2011 提醒：可靠的 IC50/EC50 估計必須包含上下平台區。
Pearson r measures linear association — completely blind to non-linear ones (a perfect sine wave gives r≈0). Spearman uses ranks, robust to monotonic non-linearity. Dose-response curves are inherently sigmoidal; forcing a linear fit is among reviewers' top complaints.
The R package drc (Ritz et al. 2015, PLOS ONE) is the dose-response standard; in 2025 dr4pl and drda advance robust estimation. Sebaugh 2011 reminds us: trustworthy IC50/EC50 estimates require the standard to cover both plateaus.
一、相關 vs 回歸
Pearson r
線性、有方向 (−1~+1)、對 outlier 敏感。要求雙變量近似常態。
Linear, directional (−1 to +1), outlier-sensitive. Requires both variables ~normal.
Spearman ρ
排名相關。對單調非線性、outlier 穩健。報告時須註明是 ρ 不是 r。
Rank correlation. Robust to monotonic non-linearity and outliers. Label as ρ, not r.
Linear regression
給出方程式與斜率（含 CI 與 p）。可預測，但有方向假設。
Provides equation, slope (with CI & p), and prediction. Has directional assumption.
Nonlinear regression
4-PL、Hill、Michaelis-Menten、雙指數衰減等。需給合理初始值，否則收斂困難。
4-PL, Hill, Michaelis-Menten, biexponential decay, etc. Needs sensible initial values to converge.
劑量反應曲線參數
調整 Hill slope 與 IC50，觀察曲線形狀變化。實務上 Hill ≈ 1 表示 1:1 binding；Hill > 1 表示協同 (cooperativity)。
Tune Hill slope and IC50 to see curve shape. Hill ≈ 1 indicates 1:1 binding; Hill > 1 indicates cooperativity.
X：log10(μM) | Y：% inhibition
二、各種回歸的 R/Python 程式碼
# 1. 線性回歸 + 假設診斷 fit <- lm(y ~ x, data=df) summary(fit) library(performance); check_model(fit) # Q-Q、殘差、共線性、影響點 # 2. 相關 cor.test(df$x, df$y, method="pearson") cor.test(df$x, df$y, method="spearman") # 3. 劑量反應 (4-PL) — drc library(drc) fit <- drm(response ~ dose, data=df, fct=LL.4()) summary(fit) ED(fit, c(50), interval="delta") # IC50 with 95% CI # 4. Michaelis-Menten — nls mm <- nls(v ~ Vmax*S/(Km+S), data=enz, start=list(Vmax=2, Km=0.5)) summary(mm)
import numpy as np, pandas as pd from scipy import stats from scipy.optimize import curve_fit import statsmodels.api as sm # 1. 線性回歸 + 假設診斷 X = sm.add_constant(df.x) fit = sm.OLS(df.y, X).fit() print(fit.summary()) # Q-Q via fit.resid + statsmodels.graphics.gofplots.qqplot # 2. 相關 stats.pearsonr(df.x, df.y) stats.spearmanr(df.x, df.y) # 3. 4-PL 劑量反應 def hill(x, IC50, n, top, bot): return bot + (top - bot) / (1 + (x/IC50)**n) popt, pcov = curve_fit(hill, df.dose, df.response, p0=[1,1,100,0]) print("IC50 =", popt[0], "Hill n =", popt[1]) # 4. Michaelis-Menten def mm(S, Vmax, Km): return Vmax*S/(Km+S) popt, _ = curve_fit(mm, enz.S, enz.v, p0=[2, 0.5])
三、reviewer 必查清單
❌ r² 當作唯一指標
非線性 fit 用 r² 沒意義（也不在 0-1 之間）。看 AIC / BIC + residual plot。
r² is meaningless for non-linear fits (not bounded 0-1). Use AIC/BIC + residual plot.
❌ IC50 外推
曲線未涵蓋 top/bottom 平台 → IC50 不可信。需設計濃度橫跨 IC50 上下各 1-2 個 log。
Without top & bottom plateaus the IC50 is unreliable. Design doses spanning ~2 logs above and below the expected IC50.
❌ 忽略殘差診斷
檢查 residual vs fitted（線性？同質？）、Q-Q（常態？）、leverage（影響點？）。違反任一需處理。
Check residual-vs-fitted (linearity/homoscedasticity), Q-Q (normality), leverage (influential points).
❌ 相關當因果
相關只反映共變，因果需介入實驗或工具變量。報告時避免「導致」「造成」這類因果語言。
Correlation is co-variation, not causation. Avoid causal language ("causes," "leads to") without an intervention or instrumental variable.
🎯 章末小測驗
1. 非線性單調 + outlier 用？
2. 估 Vmax/Km 正確做法？
3. Hill slope = 2 代表？