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href="/atom.xml" title="Magicland" type="application/atom+xml"><link rel="alternate" href="/rss2.xml" title="Magicland" type="application/rss+xml"><title>辛普森悖论|你还相信数据吗 - Magicland</title><meta name="keywords" content="随笔, 探索,MHuiG, @MHuiG, Blog, 博客, Magicland, 魔法世界"><meta desc name="description" content="辛普森悖论揭示数据整体趋势与分组趋势可能矛盾，如肾结石治疗案例中，A疗法整体成功率高于B疗法，但分组看B对大小结石成功率均高于A。其根源是忽略混杂变量，警示需通过科学分组与因果分析避免数据误读。 - MHuiG - Magicland"><meta property="og:type" content="article"><meta property="og:title" content="辛普森悖论|你还相信数据吗"><meta property="og:url" content="https://blog.mhuig.top/p/2f345bb/"><meta property="og:site_name" content="Magicland"><meta property="og:description" content="辛普森悖论揭示数据整体趋势与分组趋势可能矛盾，如肾结石治疗案例中，A疗法整体成功率高于B疗法，但分组看B对大小结石成功率均高于A。其根源是忽略混杂变量，警示需通过科学分组与因果分析避免数据误读。"><meta property="og:locale"><meta property="og:image" content="https://blog.mhuig.top/lib/favicon/android-chrome-192x192.png"><meta property="article:published_time" content="2022-09-22T08:14:00.000Z"><meta 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content="辛普森悖论|你还相信数据吗"><meta itemprop="description" content="辛普森悖论揭示数据整体趋势与分组趋势可能矛盾，如肾结石治疗案例中，A疗法整体成功率高于B疗法，但分组看B对大小结石成功率均高于A。其根源是忽略混杂变量，警示需通过科学分组与因果分析避免数据误读。"></span><div class="article-meta" id="top"><h1 class="title allow-select" itemprop="name headline"> 辛普森悖论|你还相信数据吗</h1><div class="new-meta-box"><div class="new-meta-item author" itemprop="author" itemscope itemtype="http://schema.org/Person"> <a itemprop="url" class="author" target="_blank" href="https://mhuig.top/" rel="nofollow noopener" alt="author"><img itemprop="image" src="/lib/avatar/avatar-16.webp" class="lazyload" data-srcset="/lib/avatar/avatar-16.webp" srcset="data:image/gif;base64,R0lGODlhAQABAIAAAP///////yH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" alt="author"><p itemprop="name">MHuiG</p></a></div><div class="new-meta-item wordcount"><a class="notlink"><i class="fa-duotone fa-keyboard fa-fw" aria-hidden="true"></i><p>字数：1.3k 字</p></a></div><div class="new-meta-item readtime"><a class="notlink"><i class="fa-duotone fa-hourglass-half fa-fw" aria-hidden="true"></i><p>时长：4 分钟</p></a></div></div></div><div id="layoutHelper-page-plugins"></div><div id="post-body" itemprop="articleBody"><blockquote><p>数据是一个有力的武器，它既能被用来澄清现实，也能被用来混淆是非</p></blockquote><p>你知不知道，数据也会说谎？</p><div class="story post-story"><h2 id="一个栗子"><a href="#一个栗子" class="headerlink" title="一个栗子"></a>一个栗子</h2><p>假设您患有肾结石并去看医生。医生告诉你有两种治疗方法，治疗 A（开放手术 open surgery）和治疗 B（体外冲击波碎石术 ESWL）。</p><p>你问哪种治疗效果更好，医生说：“一项研究发现治疗 A 的成功概率高于治疗 B。”</p><p>你说：“我会接受治疗 A，谢谢！”</p><p>这时医生打断你，“但同样的研究还研究了哪种治疗效果更好，这取决于患者是大肾结石还是小肾结石。”</p><p>你说：“好吧，我有大肾结石还是小肾结石？”</p><p>你说话的时候，医生又打断了你，说：“其实没关系。你看，他们发现治疗 B 比治疗 A 成功的概率更高，不管你的肾结石是大还是小。”</p><p>你可能想知道你是否没看错。听起来不可能。但这是真的：在<a target="_blank" rel="external nofollow noopener noreferrer" href="/go.html?u=aHR0cDovL3d3dy5uY2JpLm5sbS5uaWguZ292L3BtYy9hcnRpY2xlcy9QTUMxMzM5OTgxLw">一项实际研究</a>中，发现治疗 B 比治疗 A 对大肾结石和小肾结石起作用的概率更高，尽管事实上治疗 A 的总体 概率高于治疗 B。这是研究数据：</p><table><thead><tr><th align="center"></th><th align="center">治疗 A 有帮助</th><th align="center">治疗 B 有帮助</th></tr></thead><tbody><tr><td align="center">大肾结石</td><td align="center">69% (55 / 80)</td><td align="center">73% (192 / 263)</td></tr><tr><td align="center">小肾结石</td><td align="center">87% (234 / 270)</td><td align="center">93% (81 / 87)</td></tr><tr><td align="center">所有患者</td><td align="center">83% (289 / 350)</td><td align="center">78% (273 / 350)</td></tr></tbody></table><p>表中的第一项显示，80 名大肾结石患者接受了 A 治疗，治疗帮助了其中 55 人，成功率为 69%。这不如治疗 B 好，它帮助了 263 名大肾结石患者中的 192 人，成功率为 73%。以类似的方式，第二行显示治疗 B 比治疗 A 对患有小肾结石的人更有效。</p><p>但是当你把每一列的数字加起来时，你会发现治疗 A 确实比治疗 B 整体效果更好。</p><p>值得花时间检查所有数字加起来检验一下，并说服自己我没有欺骗你.</p><p>刚刚展示的这种现象被称为辛普森悖论。如果你和包括我在内的大多数人一样，那么辛普森悖论在你第一次见到它时就会令人震惊。因为它违反了我们对世界推理的本能方式。而且，正如我们看到的那样，辛普森悖论不仅是一种怪异的现象，而且它经常在具有重要决策后果的地方出现。</p></div><div class="story post-story"><h2 id="表达式"><a href="#表达式" class="headerlink" title="表达式"></a>表达式</h2><p>我们抽离出符号表达：</p><table><thead><tr><th align="center"></th><th align="center">统计对象1</th><th align="center">统计对象2</th></tr></thead><tbody><tr><td align="center">分项指标1</td><td align="center"><mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -1.091ex;" xmlns="http://www.w3.org/2000/svg" width="2.499ex" height="2.878ex" role="img" focusable="false" viewbox="0 -789.7 1104.4 1271.9"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{q_1}{p_1}"><g data-mml-node="mfrac" data-latex="\frac{q_1}{p_1}"><g data-mml-node="msub" transform="translate(240.2,477.2) scale(0.707)" data-latex="q_1"><g data-mml-node="mi" data-latex="q"><path data-c="1D45E" d="M372 377C352 420 321 442 280 442C215 442 158 409 109 342C63 280 40 216 40 149C40 62 90-13 173-13C209-13 245 4 282 38L241-126C238-141 229-150 215-153C210-154 197-155 174-155C168-156 164-156 162-156C153-157 148-165 148-179L148-183C151-190 157-194 165-194C184-194 244-191 263-191C282-191 345-194 364-194C378-194 385-186 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419 186 442 137 442C96 442 66 409 45 344C35 311 30 292 30 286C30 277 35 272 46 272C51 272 54 273 57 275C62 284 65 291 66 298C84 374 107 412 134 412C152 412 161 398 161 371C161 356 159 339 154 321L44-118C35-151 34-155-5-155C-23-155-32-163-32-178C-32-189-26-194-15-194C0-194 52-191 67-191C86-191 146-194 165-194C180-194 187-186 187-170C187-160 178-155 159-155C141-155 114-156 114-144C114-131 155 26 159 43C179 6 209-13 249-13C314-13 371 20 421 87C467 148 490 213 490 280C490 367 438 442 355 442M352 412C391 412 411 382 411 323C411 296 405 260 394 215C371 128 342 71 307 42C286 25 267 16 248 16C219 16 199 29 188 56C179 77 174 92 174 100L225 309C230 331 248 354 276 377C304 400 329 412 352 412Z"/></g><g data-mml-node="mn" transform="translate(536,-150) scale(0.707)" data-latex="2"><path data-c="32" d="M237 666C186 666 143 648 106 612C69 576 50 534 50 483C50 449 75 424 106 424C136 424 161 450 161 480C161 513 137 536 105 536C102 536 100 536 98 535C117 584 161 627 224 627C306 627 352 556 352 470C352 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391 372 377M340 373C349 352 354 338 354 329C354 328 353 322 351 313L327 216C308 144 299 107 298 105C279 70 224 16 176 16C137 16 117 46 117 105C117 151 148 263 162 300C181 347 225 412 280 412C307 412 327 399 340 373Z"/></g><g data-mml-node="mn" transform="translate(479,-150) scale(0.707)" data-latex="4"><path data-c="34" d="M353 677C344 677 336 672 330 663L28 199L28 163L289 163L289 81C289 63 285 51 278 46C271 41 252 39 219 39L194 39L194 0C223 2 269 3 331 3C393 3 439 2 468 0L468 39L443 39C410 39 391 41 384 46C377 51 373 63 373 81L373 163L471 163L471 202L373 202L373 660C373 670 366 677 353 677M295 553L295 202L67 202Z"/></g></g><g data-mml-node="msub" transform="translate(220,-345) scale(0.707)" data-latex="p_4"><g data-mml-node="mi" data-latex="p"><path data-c="1D45D" d="M355 442C311 442 269 419 228 373C217 419 186 442 137 442C96 442 66 409 45 344C35 311 30 292 30 286C30 277 35 272 46 272C51 272 54 273 57 275C62 284 65 291 66 298C84 374 107 412 134 412C152 412 161 398 161 371C161 356 159 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427C452 436 447 441 438 441C420 441 382 391 372 377M340 373C349 352 354 338 354 329C354 328 353 322 351 313L327 216C308 144 299 107 298 105C279 70 224 16 176 16C137 16 117 46 117 105C117 151 148 263 162 300C181 347 225 412 280 412C307 412 327 399 340 373Z"/></g><g data-mml-node="mn" transform="translate(479,-150) scale(0.707)" data-latex="2"><path data-c="32" d="M237 666C186 666 143 648 106 612C69 576 50 534 50 483C50 449 75 424 106 424C136 424 161 450 161 480C161 513 137 536 105 536C102 536 100 536 98 535C117 584 161 627 224 627C306 627 352 556 352 470C352 403 318 331 250 255L62 43C49 28 50 29 50 0L421 0L450 180L417 180C409 129 402 100 396 91C391 86 361 84 306 84L139 84L236 179C304 243 390 312 419 365C439 400 449 435 449 470C449 588 357 666 237 666Z"/></g></g><g data-mml-node="mo" data-latex="+" transform="translate(882.6,0)"><path data-c="2B" d="M698 274L413 274L413 559C413 575 405 583 389 583C373 583 365 575 365 559L365 274L80 274C64 274 56 266 56 250C56 234 64 226 80 226L365 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249-13C314-13 371 20 421 87C467 148 490 213 490 280C490 367 438 442 355 442M352 412C391 412 411 382 411 323C411 296 405 260 394 215C371 128 342 71 307 42C286 25 267 16 248 16C219 16 199 29 188 56C179 77 174 92 174 100L225 309C230 331 248 354 276 377C304 400 329 412 352 412Z"/></g><g data-mml-node="mn" transform="translate(536,-150) scale(0.707)" data-latex="2"><path data-c="32" d="M237 666C186 666 143 648 106 612C69 576 50 534 50 483C50 449 75 424 106 424C136 424 161 450 161 480C161 513 137 536 105 536C102 536 100 536 98 535C117 584 161 627 224 627C306 627 352 556 352 470C352 403 318 331 250 255L62 43C49 28 50 29 50 0L421 0L450 180L417 180C409 129 402 100 396 91C391 86 361 84 306 84L139 84L236 179C304 243 390 312 419 365C439 400 449 435 449 470C449 588 357 666 237 666Z"/></g></g><g data-mml-node="mo" data-latex="+" transform="translate(939.6,0)"><path data-c="2B" d="M698 274L413 274L413 559C413 575 405 583 389 583C373 583 365 575 365 559L365 274L80 274C64 274 56 266 56 250C56 234 64 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data-latex="
\frac{q_1}{p_1}>\frac{q_2}{p_2},\frac{q_3}{p_3}>\frac{q_4}{p_4}
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92 174 100L225 309C230 331 248 354 276 377C304 400 329 412 352 412Z"/></g><g data-mml-node="mn" transform="translate(536,-150) scale(0.707)" data-latex="4"><path data-c="34" d="M353 677C344 677 336 672 330 663L28 199L28 163L289 163L289 81C289 63 285 51 278 46C271 41 252 39 219 39L194 39L194 0C223 2 269 3 331 3C393 3 439 2 468 0L468 39L443 39C410 39 391 41 384 46C377 51 373 63 373 81L373 163L471 163L471 202L373 202L373 660C373 670 366 677 353 677M295 553L295 202L67 202Z"/></g></g><rect width="1139.6" height="60" x="120" y="220"/></g></g></g></g></svg><svg data-labels="true" preserveaspectratio="xMaxYMid" viewbox="1278 -1249 1 1998"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:1" transform="translate(0,879)"><text data-id-align="true"/><g data-idbox="true" transform="translate(0,-748)"><g data-mml-node="mtext" data-latex="\text{(}"><path data-c="28" d="M318-248C327-248 332-243 332-234C332-231 330-227 327-223C275-183 233-117 202-26C175 53 161 131 161 208L161 292C161 369 175 447 202 526C233 617 275 683 327 723C330 726 332 730 332 734C332 743 327 748 318 748C317 748 314 747 311 745C251 699 201 631 160 540C121 453 101 371 101 292L101 208C101 129 121 47 160-40C201-131 251-199 311-245C314-247 317-248 318-248Z"/></g><g data-mml-node="mtext" data-latex="\text{1}" transform="translate(389,0)"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47C195 42 170 39 130 39L95 39L95 0C120 2 174 3 257 3C340 3 394 2 419 0L419 39L384 39C343 39 318 42 310 47C302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"/></g><g data-mml-node="mtext" data-latex="\text{)}" transform="translate(889,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 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\frac{q_1+q_3}{p_1+p_3}>\frac{q_2+q_4}{p_2+p_4}
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小肾结石恢复率 = h(疗法)
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stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -750)"><g data-mml-node="math" data-latex="
大肾结石恢复率 = i(疗法)
"><g data-mml-node="mtable" data-latex="大肾结石恢复率 = i(疗法)" transform="translate(2078,0) translate(-2078,0)"><g transform="translate(0 750) matrix(1 0 0 -1 0 0) scale(55.25)"><svg data-table="true" preserveaspectratio="xMidYMid" viewbox="5728.3 -750 1 1000"><g transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mlabeledtr"><g data-mml-node="mtd"><g data-mml-node="mi" data-latex="大"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">大</text></g><g data-mml-node="mi" data-latex="肾" transform="translate(1000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">肾</text></g><g data-mml-node="mi" data-latex="结" transform="translate(2000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">结</text></g><g data-mml-node="mi" data-latex="石" transform="translate(3000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">石</text></g><g 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565C257 565 284 593 284 621M259 138C237 59 205 19 164 19C151 19 144 28 144 47C144 64 173 150 232 306C240 329 244 347 244 360C244 409 210 445 161 445C118 445 84 420 59 369C39 328 29 301 29 288C29 279 34 275 45 275C58 275 60 281 64 295C87 375 118 415 158 415C171 415 178 406 178 387C178 373 175 357 168 338C145 275 121 208 101 155C86 115 78 88 78 74C78 25 114-11 162-11C205-11 239 14 264 64C283 103 293 130 293 145C293 154 288 159 277 159C274 159 259 150 259 138Z"/></g><g data-mml-node="mo" data-latex="(" transform="translate(8678.6,0)"><path data-c="28" d="M318-248C327-248 332-243 332-234C332-231 330-227 327-223C275-183 233-117 202-26C175 53 161 131 161 208L161 292C161 369 175 447 202 526C233 617 275 683 327 723C330 726 332 730 332 734C332 743 327 748 318 748C317 748 314 747 311 745C251 699 201 631 160 540C121 453 101 371 101 292L101 208C101 129 121 47 160-40C201-131 251-199 311-245C314-247 317-248 318-248Z"/></g><g data-mml-node="mi" data-latex="疗" transform="translate(9067.6,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">疗</text></g><g data-mml-node="mi" data-latex="法" transform="translate(10067.6,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">法</text></g><g data-mml-node="mo" data-latex=")" transform="translate(11067.6,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g></g></g></g></svg><svg data-labels="true" preserveaspectratio="xMaxYMid" viewbox="1278 -750 1 1000"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:6"><g data-mml-node="mtext" data-latex="\text{(}"><path data-c="28" d="M318-248C327-248 332-243 332-234C332-231 330-227 327-223C275-183 233-117 202-26C175 53 161 131 161 208L161 292C161 369 175 447 202 526C233 617 275 683 327 723C330 726 332 730 332 734C332 743 327 748 318 748C317 748 314 747 311 745C251 699 201 631 160 540C121 453 101 371 101 292L101 208C101 129 121 47 160-40C201-131 251-199 311-245C314-247 317-248 318-248Z"/></g><g data-mml-node="mtext" data-latex="\text{6}" transform="translate(389,0)"><path data-c="36" d="M383 504C416 504 432 521 432 555C432 627 378 666 304 666C221 666 155 627 106 548C63 480 42 403 42 316C42 189 65 100 112 47C152 1 198-22 251-22C312-22 362 1 401 47C438 91 457 144 457 205C457 266 439 318 403 361C365 407 316 431 257 431C205 431 165 402 138 346L138 352C138 465 166 561 226 605C252 624 279 633 306 633C342 633 368 623 385 602C351 602 334 583 334 553C334 525 355 504 383 504M344 340C355 317 361 272 361 206C361 141 356 98 345 76C325 35 294 14 251 14C222 14 200 24 184 44C171 60 162 74 158 85C146 116 140 163 140 227C140 255 144 282 151 308C164 355 201 399 256 399C295 399 325 379 344 340Z"/></g><g data-mml-node="mtext" data-latex="\text{)}" transform="translate(889,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p>这样看来激进的治疗 B（体外冲击波碎石术 ESWL）效果更好.</p><p>如果有潜在变量（特别是混合变量）存在，牢记：整体数据未必可靠，要通过科学合理的分组来查看具体细致的数据。</p></div><div class="story post-story"><h2 id="启示"><a href="#启示" class="headerlink" title="启示"></a>启示</h2><ul><li>数据从来都不是完全客观的. 我们必须对这些数字持怀疑态度.</li><li>辛普森悖论的出现是因为人们忽略了研究的因果关系，一旦我们理解了数据生成的机制，我们就可以寻找影响结果的其他因素，而图表不会告诉你这些.</li><li>充分考察事件的潜在影响因素和维度，系数消除分组数据基数差异造成的影响.</li><li>要求我们具备科学辩证思维，客观看待关联现象。很多时候，我们选择相信直觉，因为我们的直觉往往很准。但是，在信息不全或者信息非对称的情况下，直觉常常是是值得怀疑的。</li></ul></div></div><div class="footer"><div class="copyright license"><div class="license-title">辛普森悖论|你还相信数据吗</div><div class="license-link"><a href="https://blog.mhuig.top/p/2f345bb/">https://blog.mhuig.top/p/2f345bb/</a></div><div class="license-meta"><div class="license-meta-item"><div class="license-meta-title">本文作者</div><div class="license-meta-text">MHuiG</div></div><div class="license-meta-item"><div class="license-meta-title">发布于</div><div class="license-meta-text">Sep 22, 2022</div></div><div class="license-meta-item"><div class="license-meta-title">更新于</div><div class="license-meta-text">Nov 20, 2025</div></div><div class="license-meta-item"><div class="license-meta-title">许可协议</div><div class="license-meta-text"><a 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