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March 26, 2015

数学は権利を創出するか(6)

  臨床医学検査はどの程度,治療に役立っているか。
生涯を通して日本人女性の12人に1人が乳がんになる。
40歳の女性のうち,定期的な検査を受ける人の1%が乳がんにかかっていることは知られている(30~64歳の女性の死因のトップが乳がん)。これを「1%の事前確率」と呼ぼう。尤も,「1%」は,近年,増えつつあるようだ。
乳がんの女性のうち,80%はマンモグラフィで陽性をしめす。また,乳がんでない女性のうち10%もマンモグラフィで陽性(偽陽性)を示す。ここがポイントだ。
さて,この年齢グループに属するある女性Aが,定期検診のマンモグラフィで陽性を示した。この人が乳がんである確率は幾らか?
 次のように考える。つまり,Aは陽性のがんの確率80%に含まれるか,がんでないのに陽性を示した10%に含まれるかのどちらかだ。前者は乳がん。後者は偽の乳がん。
どちらであるか検査結果だけでは判別できない。そこでベイズ推定が用いられる。
 乳がん検査を受ける人が1000人いるとする。がんの確率は事前確率の1%だから,検査を受ける1000人のうち10人が実際に乳がん患者と推定できる。この10人のうち8人(80%)がマンモグラフィで陽性を示すのである。
一方,乳がんでない女性990人のうち,10%にあたる99人も陽性(偽陽性)を示した。
陽性を示した人のうち,乳がんである確率をどう計算するか。
実際に乳がんである患者は1%(10人)である事は事前試験で分かっている。マンモグラフィで陽性反応を示すがん患者は8人。これに偽陽性の99人を加える。8+99=107人。つまり,がん患者である確率は,8÷107=0.0748(7.48%)である。(Ian Ayres前掲書356頁参照)

つぎに,偽陽性の人の確率は,次のように求める。
女性Aを「患者が病気である」という事象,Bを「結果が陽性だった」という事象とする。陽性を示した人のうち真の陽性だった確率は,ベイズの定理により,
    P(A | B)={P(B | A) P(A)}÷{P(B | A)P(A) + P(B | A^C)P(A^C)}
   ={0.80× 0.001}÷{0.80×0.001 + 0.10×0.80}=0.0099
よって,偽陽性(陽性を示したががんでない)確率は,1 − 0.019 = 0.9901 (99%)である。

もちろん,がん患者のうち80%が陽性を示したから,それが真性のがん患者であると考えるのは間違っている。がん患者100人のうち80人が陽性を示すが,残り20人も隠れ患者がいる。ほとんどの医師がマンモグラフィで陽性を示した80人(8割)をがんの確率と思い込んでしまう。彼らは確率統計の読み方を間違えているのだ。

つぎに,マンモグラフィで陰性を示した女性(がんであるのに陰性を示した2人及び健常者の合計)のうち,がん患者である確率はどうやって求めるか。
1000人中,10人ががんという事前確率に戻ろう。つまり,陽性を示した8人はがんで,陽性でない2人もがん。
すでに説明したように,陽性反応は陽性8人+偽陽性99人=107人。すると残り893人がはっきりした陰性(偽陽性でもない)。
ただし,この陰性の中にもがん患者2人が潜んでいる。∴2÷893=0.0022。
Ian Ayres(前掲書451頁脚注)は,ベイズ推定により,がんの事前確率1%に,陽性を示した人ががんである見込み率7.5%を掛け,隠れがんの確率は7.5%になるという。
はて?これは事後確率の計算ではないか。陰性を示した人の何人ががんなのか分からない。私が脚注を読み違えているのか。誰か教えてほしい。

ベイズ推定はデータを得る前の分布とデータを得た後の分布を比べることはしない。上記で言えば,1%が事前分布になる。
         事後分布=(事前分布×尤度)÷データの分布
がんの発生率が高くなれば事前分布も違ってくるが,事前分布を一様の分布であると仮定すれば,一様の分布に尤度が最大の値を掛けることで確率も高くなる。数学的にもそうなるらしい。ベイズ推定では特定の値を求めず,確率を求めるのである。分布を直接推定する。正規分布まで推定しない。
 なお,最尤法は推定値が正規分布であると仮定する。
正規分布にならないパラメータ(母数)は排除されるから信頼区間を正確に推定できない。 尤度(ゆうど)は,統計学においてある前提条件に従って結果が出現する場合に,逆に観察結果からみて前提条件が「何々であった」と推測する尤もらしさを表す数値を「何々」を変数とする関数として捉えたものである。その相対値に意味がある。
例えば,服装を見てあの人は男か女かを推定する。背広を着ていれば男,スカートをはいていれば女と思う。その根拠は,男ならこういう格好をする,女ならこういう格好をするはずだという経験則にある。これで男らしさ,女らしさを仮説する。この場合の事前分布は,男がスカートをはく確率は例えば1%以下。女が背広を着る確率は1%以下という確率(男らしさ,女らしさ)をいう。この事前分布は思い込みであるかもしれない。この思い込みを悪用すれば,犯罪も可能である。尤も,現代の科学捜査は忍者の変装をしても容易に見破ってしまう。たとえ全身を黒衣で覆っていても男女・人物を特定してしまう。
最尤法の結果はベイズ推定の結果と余り変わらない。いずれにせよ,ベイズ推定も検査結果を資料にした推定である。もし隠れがんまでみつけようとするなら,マンモグラフィだけでなく超音波検査・細胞診検査・病理検査しなければならない。医師は,顕微鏡でのぞくまでは,良性か悪性か確実に鑑別できないのだ。

群馬大学附属病院の腹腔鏡手術が問題視されている。同じ医師が85人の患者に執刀して8人が死亡した。数値で言えば8.6%。この数値は群馬大学附属病院の8.6%という死亡率は,全国平均の18倍。85-8=77人が生存しているから手術は成功した,といえるか。答えはNOだ。たとえ末期がんの患者に対する手術であってもだ。医師が同じ人物であってもなくても,この死亡率は高い。腹腔鏡手術はリスクが多く,高度の技術を必要とする。執刀医が1人か複数人かは重要ではない。リスクを省みず高難度手術をやらせる大学病院にこそ問題がある。
この手術は,医師なら一度はチャレンジしたい保険適応にならない先進医療と思われている。それだけに,事前の説明が強く求められる。群馬大学報告書によればリスクについての事前説明がなかった。腹腔鏡手術以外の手術があることも説明されなかった。
手術もどのように行われたかの記録も不十分。技術も不足していた。患者が死亡しているにも拘わらず,術後良好と学会で虚偽の報告をしていた。
千葉県がんセンターでも,昨年,経験の浅い医師らが新しい手法の手術を繰り返していた。
「何年も前から,消化器外科では経験が少ない医者たちが腹腔鏡手術で事故を繰り返していただけでなく,安全性が確立されていない保険外の外科手術もかなり行っていました。術後とたんに症状が悪化,翌日に再手術,ということがしょっちゅうあったんです」と。
医師らにとって,もはや手術は「治療」ではなく「実験」の感覚だった。

東京女子医大病院でも,二歳児に麻酔投与して死亡させている。子供はICUのベッドで昏睡したまま死亡した。手術自体は,「頸部リンパ管腫ピシバニール注入術」で簡単。ところが,医師は子供に使ってはいけない鎮静剤・プロポフォールを投与した。術中に子供が動いて管が外れるから全身麻酔して人工呼吸器につないだというのだ。麻酔科医師は「プロポフォール・インフュージョン・シンドローム(副作用)というものがあることは知っていました」という。人工呼吸している小児には禁忌とされている。使用量は体重に比例した40mgの3.5倍であった。なぜ,この量を使用したかの説明もない。
岡山大学病院,京都府立医大でも,論文不正が指摘された。いずれも,大学教授と製薬会社との癒着が注目されている。

February 22, 2006 New York Timesが"Why Doctors So Often Get It Wrong"を掲載した。
数年前のある週末,Georgiaの田舎から出てきた4歳の男の子の両親は,Atlanta北部の小児病院にその子供を連れてきた。家族はすでに多くの病院を経てきた。彼らの息子は何ヶ月間も調子が悪く,ずっと熱を伴っていた。
週末勤務であった医者は血液検査をするように言い,結果は白血病であることを示していた。その子供の病気には,皮膚に少し茶褐色の斑点があるというような,つじつまが合わない点がいくつか見られた。しかしその医者には月曜の午後に始める化学療法の強行日程の予定が入っていた。結局,彼らの敵は時間であった。現代医学で用いられる全ての手段(例えば血液検査・MRI・内視鏡)を使うようになり,あなたはおそらく誤診など稀なものになったと考えるだろう。しかしその考えは誤りである。死体解剖の研究は,医者が2割の確率で致死的な病気を誤診してしまっているということを示している。そのため何百万人という患者が間違った病気として扱われている。あきれることに,この誤診の割合は1930年代から変わっていない。通常感嘆符(!)を使うことのないアメリカ医学協会誌の記事が「全く改善なし!」という表現で現状をまとめている。

"2割"の誤診がなぜ減らないか。科学的先端を走る検査が行われているのに医師の能力が追いつかないのでは,何のための先端医療なのか,何のための医療統計なのか考え込んでしまう。

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