{"id":9752,"date":"2024-05-14T10:18:49","date_gmt":"2024-05-14T08:18:49","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=9752"},"modified":"2024-05-14T14:00:31","modified_gmt":"2024-05-14T12:00:31","slug":"les-tests-faux-positifs","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/14\/les-tests-faux-positifs\/","title":{"rendered":"Les tests faux positifs"},"content":{"rendered":"\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Les tests faux positifs sont bien connus en probabilit\u00e9s.   C&rsquo;est une sorte de paradoxe contre-intuitif que beaucoup de personnes ont du mal \u00e0 saisir. M\u00eame les m\u00e9decins s&rsquo;y trompent selon une \u00e9tude de l&rsquo;institut Max-Planck d&rsquo;il y a pas mal d&rsquo;ann\u00e9es (je ne retrouve plus les sources malheureusement).<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_86 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/14\/les-tests-faux-positifs\/#Tests_faux_positifs_introduction\" >Tests faux positifs: introduction<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/14\/les-tests-faux-positifs\/#Tests_faux_positifs_un_exemple_medical\" >Tests faux positifs: un exemple m\u00e9dical<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/14\/les-tests-faux-positifs\/#Lenonce\" >L&rsquo;\u00e9nonc\u00e9<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/14\/les-tests-faux-positifs\/#Resolution_du_probleme\" >R\u00e9solution du probl\u00e8me<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/14\/les-tests-faux-positifs\/#Combien_de_tests_au_minimum_pour_etre_surs_detre_malade\" >Combien de tests au minimum pour \u00eatre s\u00fbrs d&rsquo;\u00eatre malade ?<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/14\/les-tests-faux-positifs\/#Tests_faux_positifs_cas_general\" >Tests faux positifs: cas g\u00e9n\u00e9ral<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/14\/les-tests-faux-positifs\/#Une_formule_generale\" >Une formule g\u00e9n\u00e9rale<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/05\/14\/les-tests-faux-positifs\/#Autres_domaines_ou_lon_rencontre_des_faux_positifs\" >Autres domaines o\u00f9 l&rsquo;on rencontre des faux positifs<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Tests_faux_positifs_introduction\"><\/span>Tests faux positifs: introduction<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">J&rsquo;ai parl\u00e9 des m\u00e9decins, mais le mod\u00e8le du \u00ab\u00a0faux-positif\u00a0\u00bb peut s&rsquo;appliquer dans d&rsquo;autres domaines. Quand on effectue un test (biologique ou sociologique par exemple), il y a toujours une marge d&rsquo;erreur. Un test de grossesse peut \u00eatre positif alors que la femme n&rsquo;est pas enceinte. C&rsquo;est ce que l&rsquo;on appelle le \u00ab\u00a0faux-positif\u00a0\u00bb.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Nous allons donc nous pencher sur ce probl\u00e8me. Quelle est la probabilit\u00e9 d&rsquo;\u00eatre r\u00e9ellement malade (par exemple) sachant que le test est positif ? Combien de tests doit-on faire pour \u00eatre assur\u00e9(e)s d&rsquo;\u00eatre r\u00e9ellement malade au seuil de 95% (par exemple) ?<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Avant toute chose, je pr\u00e9cise que l&rsquo;article de <a href=\"https:\/\/fr.wikipedia.org\/wiki\/Faux_positif\" target=\"_blank\" rel=\"noreferrer noopener\">wikipedia<\/a> concernant ces faux positifs peut vous apporter un autre angle.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Tests_faux_positifs_un_exemple_medical\"><\/span>Tests faux positifs: un exemple m\u00e9dical<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Lenonce\"><\/span>L&rsquo;\u00e9nonc\u00e9<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On a un test de s\u00e9rologie pour identifier une maladie qui atteint 0,5% de la population. <\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Sur 99% des malades, le test r\u00e9agit. Autrement dit, 99% des malades sont d\u00e9tect\u00e9s par ce test. Mais sur 2% des personnes saines, le test montre une fausse r\u00e9action positive.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Sur un patient, un test est positif. Quelle est la probabilit\u00e9 d&rsquo;\u00eatre malade ?<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Resolution_du_probleme\"><\/span>R\u00e9solution du probl\u00e8me<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Nous allons noter:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u00ab\u00a0M\u00a0\u00bb l&rsquo;\u00e9v\u00e9nement : \u00ab\u00a0le patient est malade\u00a0\u00bb<\/li>\n\n\n\n<li>\u00ab\u00a0R\u00a0\u00bb l&rsquo;\u00e9v\u00e9nement : \u00ab\u00a0le test r\u00e9agit\u00a0\u00bb<\/li>\n<\/ul>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">L&rsquo;arbre des probabilit\u00e9s correspondant \u00e0 notre situation est le suivant:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-7.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"237\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-7-300x237.png\" alt=\"test faux positif\" class=\"wp-image-9753\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-7-300x237.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-7.png 446w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure>\n<\/div>\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">$$\\begin{align*}P(R) &amp; = P(M \\cap R) + P(\\overline{M}\\cap R)\\\\ &amp; = 0,005\\times0,99+0,995\\times0,02\\\\ &amp; =0,02485.\\end{align*}$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On en d\u00e9duit alors, d&rsquo;apr\u00e8s la formule de Bayes:$$\\begin{align*}P_R(M) &amp;= \\frac{P_M(R)\\times P(M)}{P(R)}\\\\&amp; = \\frac{P(M \\cap R)}{P(R}\\\\&amp; = \\frac{0,005\\times0,99}{0,02485}\\\\&amp;\\approx0,199.\\end{align*}<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On peut alors conclure que le patient est r\u00e9ellement malade avec une probabilit\u00e9 d&rsquo;environ 20%&#8230; ce qui est peu!<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Combien_de_tests_au_minimum_pour_etre_surs_detre_malade\"><\/span>Combien de tests au minimum pour \u00eatre s\u00fbrs d&rsquo;\u00eatre malade ?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On ne peut jamais \u00eatre s\u00fbrs \u00e0 100%, mais on peut se rapprocher de cette probabilit\u00e9 en multipliant les tests. Mais combien de tests au minimum afin que la probabilit\u00e9 atteigne au moins 95% ? Nous allons le calculer. <\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Pour commencer, faisons un autre test. L&rsquo;arbre suivant d\u00e9crit la situation:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-8.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"185\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-8-300x185.png\" alt=\"test faux positifs\" class=\"wp-image-9755\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-8-300x185.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-8-600x369.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-8.png 674w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure>\n<\/div>\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On convenant de noter:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\\(R_1\\) l&rsquo;\u00e9v\u00e9nement : \u00ab\u00a0le premier test r\u00e9agit\u00a0\u00bb<\/li>\n\n\n\n<li>\\(R_2\\) l&rsquo;\u00e9v\u00e9nement : \u00ab\u00a0le second test r\u00e9agit\u00a0\u00bb<\/li>\n<\/ul>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Nous cherchons \u00e0 calculer : $$\\begin{align*}P_{(R_1\\cap R_2)}(M) &amp; = \\frac{P(M \\cap R_1 \\cap R_2)}{P(R_1 \\cap R_2)}\\\\ &amp; = \\frac{0,005 \\times 0,99^2}{0,005\\times0,99^2+0,995\\times0,02^2} \\\\&amp;\\approx0,9249.\\end{align*}$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Nous voyons alors que deux tests suffisent pour assurer une probabilit\u00e9 d&rsquo;au moins 90%. On constate alors la n\u00e9cessit\u00e9 d&rsquo;effectuer un deuxi\u00e8me.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Si on note <em>n<\/em> le nombre de tests, et <em>A<\/em> l&rsquo;\u00e9v\u00e9nement : \u00ab\u00a0tous les tests r\u00e9agissent\u00a0\u00bb,  nous avons:$$\\begin{align*}P_A(M) &amp; = \\frac{P(M \\cap A)}{P(A)}\\\\ &amp; = \\frac{0,005\\times0,99^n}{0,005\\times0,99^n+0,995\\times0,02^n} \\end{align*}$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On calcule alors que pour <em>n<\/em> = 3n cette probabilit\u00e9 atteint 99,8%.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Tests_faux_positifs_cas_general\"><\/span>Tests faux positifs: cas g\u00e9n\u00e9ral<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Une_formule_generale\"><\/span>Une formule g\u00e9n\u00e9rale<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Notons:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><em>n<\/em> le nombre de tests effectu\u00e9s (sur le sch\u00e9ma ci-dessous, je me contente de prendre <em>n<\/em> = 2)<\/li>\n\n\n\n<li><em>p<\/em> la probabilit\u00e9 d&rsquo;\u00eatre malade (ou de pr\u00e9senter une quelconque particularit\u00e9)<\/li>\n\n\n\n<li><em>a<\/em> la probabilit\u00e9 que le test soit positif sachant que la personne pr\u00e9sente la particularit\u00e9 cherch\u00e9e<\/li>\n\n\n\n<li><em>b<\/em> la probabilit\u00e9 que le test soit un faux positif<\/li>\n\n\n\n<li><em>X<\/em> la variable al\u00e9atoire repr\u00e9sentant le nombre de tests positifs<\/li>\n<\/ul>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-9.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"184\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-9-300x184.png\" alt=\"\" class=\"wp-image-9756\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-9-300x184.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-9-600x368.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/05\/image-9.png 682w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure>\n<\/div>\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On peut g\u00e9n\u00e9raliser ce qui a \u00e9t\u00e9 fait en amont:$$P_{(X=k)}(M) = \\frac{p \\times \\binom{n}{k}\\times a^k (1-a)^{n-k}}{p \\times \\binom{n}{k}\\times a^k (1-a)^{n-k} + (1-p)\\times \\binom{n}{k}\\times b^k (1-b)^{n-k}}.$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Autres_domaines_ou_lon_rencontre_des_faux_positifs\"><\/span>Autres domaines o\u00f9 l&rsquo;on rencontre des faux positifs<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Bien entendu, il n&rsquo;y a pas que dans le m\u00e9dical que l&rsquo;on rencontre des faux positifs. Par exemple, si l&rsquo;on analyse le comportement suspect des individus \u00e0 l&rsquo;aide d&rsquo;une vid\u00e9o surveillance et d&rsquo;un algorithme, ce dernier mettra en relief des individus au comportement jug\u00e9 suspect alors que les individus ne pr\u00e9sentent aucun danger.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">En informatique, et plus pr\u00e9cis\u00e9ment lors de l\u2019utilisation d&rsquo;anti-virus, un programme peut \u00eatre d\u00e9tect\u00e9 comme nuisible alors qu&rsquo;il est totalement inoffensif.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Les tests faux positifs sont bien connus en probabilit\u00e9s. C&rsquo;est une sorte de paradoxe contre-intuitif que beaucoup de personnes ont du mal \u00e0 saisir. M\u00eame les m\u00e9decins s&rsquo;y trompent selon une \u00e9tude de l&rsquo;institut Max-Planck d&rsquo;il y a pas mal d&rsquo;ann\u00e9es (je ne retrouve plus les sources malheureusement).<\/p>\n","protected":false},"author":1,"featured_media":9758,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[420,169],"class_list":["post-9752","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques","tag-conditionnement","tag-probabilites"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.2 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Les tests faux positifs - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"Les tests faux positifs sont bien connus en probabilit\u00e9s. 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