Modello logit - Wikipedia

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<span>Definizione</span> </div> </a> <button aria-controls="toc-Definizione-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Attiva/disattiva la sottosezione Definizione</span> </button> <ul id="toc-Definizione-sublist" class="vector-toc-list"> <li id="toc-Varianza" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Varianza"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Varianza</span> </div> </a> <ul id="toc-Varianza-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Effetto_marginale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Effetto_marginale"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Effetto marginale</span> </div> </a> <ul id="toc-Effetto_marginale-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Illustrazione_del_metodo" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Illustrazione_del_metodo"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Illustrazione del metodo</span> </div> </a> <ul id="toc-Illustrazione_del_metodo-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stima_del_modello" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Stima_del_modello"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Stima del modello</span> </div> </a> <button aria-controls="toc-Stima_del_modello-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Attiva/disattiva la sottosezione Stima del modello</span> </button> <ul id="toc-Stima_del_modello-sublist" class="vector-toc-list"> <li id="toc-Funzione_di_verosimiglianza" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Funzione_di_verosimiglianza"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Funzione di verosimiglianza</span> </div> </a> <ul id="toc-Funzione_di_verosimiglianza-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stima_dei_parametri" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stima_dei_parametri"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Stima dei parametri</span> </div> </a> <ul id="toc-Stima_dei_parametri-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stima_della_probabilità" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stima_della_probabilità"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Stima della probabilità</span> </div> </a> <ul id="toc-Stima_della_probabilità-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Note</span> </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voci_correlate" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voci_correlate"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Voci correlate</span> </div> </a> <ul id="toc-Voci_correlate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Altri_progetti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Altri_progetti"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Altri progetti</span> </div> </a> <ul id="toc-Altri_progetti-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Indice" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Mostra/Nascondi l'indice" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Mostra/Nascondi l'indice</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Modello logit</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Vai a una voce in un'altra lingua. Disponibile in 25 lingue" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-25" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">25 lingue</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D9%86%D8%AD%D8%AF%D8%A7%D8%B1_%D9%84%D9%88%D8%AC%D8%B3%D8%AA%D9%8A" title="انحدار لوجستي - arabo" lang="ar" hreflang="ar" data-title="انحدار لوجستي" data-language-autonym="العربية" data-language-local-name="arabo" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Regressi%C3%B3_log%C3%ADstica" title="Regressió logística - catalano" lang="ca" hreflang="ca" data-title="Regressió logística" data-language-autonym="Català" data-language-local-name="catalano" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Logistick%C3%A1_regrese" title="Logistická regrese - ceco" lang="cs" hreflang="cs" data-title="Logistická regrese" data-language-autonym="Čeština" data-language-local-name="ceco" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Logistische_Regression" title="Logistische Regression - tedesco" lang="de" hreflang="de" data-title="Logistische Regression" data-language-autonym="Deutsch" data-language-local-name="tedesco" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Logistic_regression" title="Logistic regression - inglese" lang="en" hreflang="en" data-title="Logistic regression" data-language-autonym="English" data-language-local-name="inglese" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Regresi%C3%B3n_log%C3%ADstica" title="Regresión logística - spagnolo" lang="es" hreflang="es" data-title="Regresión logística" data-language-autonym="Español" data-language-local-name="spagnolo" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Logistiline_regressioon" title="Logistiline regressioon - estone" lang="et" hreflang="et" data-title="Logistiline regressioon" data-language-autonym="Eesti" data-language-local-name="estone" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Erregresio_logistiko" title="Erregresio logistiko - basco" lang="eu" hreflang="eu" data-title="Erregresio logistiko" data-language-autonym="Euskara" data-language-local-name="basco" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B1%DA%AF%D8%B1%D8%B3%DB%8C%D9%88%D9%86_%D9%84%D8%AC%D8%B3%D8%AA%DB%8C%DA%A9" title="رگرسیون لجستیک - persiano" lang="fa" hreflang="fa" data-title="رگرسیون لجستیک" data-language-autonym="فارسی" data-language-local-name="persiano" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Logistinen_regressio" title="Logistinen regressio - finlandese" lang="fi" hreflang="fi" data-title="Logistinen regressio" data-language-autonym="Suomi" data-language-local-name="finlandese" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/R%C3%A9gression_logistique" title="Régression logistique - francese" lang="fr" hreflang="fr" data-title="Régression logistique" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A8%D7%92%D7%A8%D7%A1%D7%99%D7%94_%D7%9C%D7%95%D7%92%D7%99%D7%A1%D7%98%D7%99%D7%AA" title="רגרסיה לוגיסטית - ebraico" lang="he" hreflang="he" data-title="רגרסיה לוגיסטית" data-language-autonym="עברית" data-language-local-name="ebraico" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Regresi_logistik" title="Regresi logistik - indonesiano" lang="id" hreflang="id" data-title="Regresi logistik" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiano" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%AD%E3%82%B8%E3%82%B9%E3%83%86%E3%82%A3%E3%83%83%E3%82%AF%E5%9B%9E%E5%B8%B0" title="ロジスティック回帰 - giapponese" lang="ja" hreflang="ja" data-title="ロジスティック回帰" data-language-autonym="日本語" data-language-local-name="giapponese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A1%9C%EC%A7%80%EC%8A%A4%ED%8B%B1_%ED%9A%8C%EA%B7%80" title="로지스틱 회귀 - coreano" lang="ko" hreflang="ko" data-title="로지스틱 회귀" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Logistische_regressie" title="Logistische regressie - olandese" lang="nl" hreflang="nl" data-title="Logistische regressie" data-language-autonym="Nederlands" data-language-local-name="olandese" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Regresja_logistyczna" title="Regresja logistyczna - polacco" lang="pl" hreflang="pl" data-title="Regresja logistyczna" data-language-autonym="Polski" data-language-local-name="polacco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Regress%C3%A3o_log%C3%ADstica" title="Regressão logística - portoghese" lang="pt" hreflang="pt" data-title="Regressão logística" data-language-autonym="Português" data-language-local-name="portoghese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D1%81%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%80%D0%B5%D0%B3%D1%80%D0%B5%D1%81%D1%81%D0%B8%D1%8F" title="Логистическая регрессия - russo" lang="ru" hreflang="ru" data-title="Логистическая регрессия" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Logistic_regression" title="Logistic regression - Simple English" lang="en-simple" hreflang="en-simple" data-title="Logistic regression" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Enkel_logistisk_regression" title="Enkel logistisk regression - svedese" lang="sv" hreflang="sv" data-title="Enkel logistisk regression" data-language-autonym="Svenska" data-language-local-name="svedese" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%96%E0%B8%94%E0%B8%96%E0%B8%AD%E0%B8%A2%E0%B9%82%E0%B8%A5%E0%B8%88%E0%B8%B4%E0%B8%AA%E0%B8%95%E0%B8%B4%E0%B8%81" title="การถดถอยโลจิสติก - thailandese" lang="th" hreflang="th" data-title="การถดถอยโลจิสติก" data-language-autonym="ไทย" data-language-local-name="thailandese" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D1%96%D1%81%D1%82%D0%B8%D1%87%D0%BD%D0%B0_%D1%80%D0%B5%D0%B3%D1%80%D0%B5%D1%81%D1%96%D1%8F" title="Логістична регресія - ucraino" lang="uk" hreflang="uk" data-title="Логістична регресія" data-language-autonym="Українська" data-language-local-name="ucraino" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%82%8F%E8%BC%AF%E6%96%AF%E8%AB%A6%E8%BF%B4%E6%AD%B8" title="邏輯斯諦迴歸 - cinese" lang="zh" hreflang="zh" data-title="邏輯斯諦迴歸" data-language-autonym="中文" data-language-local-name="cinese" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%82%8F%E8%BC%AF%E8%BF%B4%E6%AD%B8" title="邏輯迴歸 - cantonese" lang="yue" hreflang="yue" data-title="邏輯迴歸" data-language-autonym="粵語" data-language-local-name="cantonese" 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class="vector-pinnable-header-label">Aspetto</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">nascondi</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Da Wikipedia, l'enciclopedia libera.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="it" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Logistic-sigmoid-vs-scaled-probit.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Logistic-sigmoid-vs-scaled-probit.svg/220px-Logistic-sigmoid-vs-scaled-probit.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Logistic-sigmoid-vs-scaled-probit.svg/330px-Logistic-sigmoid-vs-scaled-probit.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Logistic-sigmoid-vs-scaled-probit.svg/440px-Logistic-sigmoid-vs-scaled-probit.svg.png 2x" data-file-width="630" data-file-height="630" /></a><figcaption>Il modello logit è rappresentato in blu.</figcaption></figure> <p>In <a href="/wiki/Statistica" title="Statistica">statistica</a>, il <b>modello logit</b>, noto anche come <b>modello logistico</b> o <b>regressione logistica</b>, è un modello di <a href="/wiki/Regressione_nonlineare" title="Regressione nonlineare">regressione nonlineare</a> utilizzato quando la <a href="/wiki/Variabile_dipendente" class="mw-redirect" title="Variabile dipendente">variabile dipendente</a> è di tipo <a href="/wiki/Variabile_dicotomica" title="Variabile dicotomica">dicotomico</a>. L'obiettivo del modello è di stabilire la probabilità con cui un'osservazione può generare uno o l'altro valore della variabile dipendente; può inoltre essere utilizzato per classificare le osservazioni, in base alla caratteristiche di queste, in due categorie.<sup id="cite_ref-Definizione_1-0" class="reference"><a href="#cite_note-Definizione-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Il modello logit fa parte della classe dei <a href="/wiki/Modello_lineare_generalizzato" title="Modello lineare generalizzato">modelli lineari generalizzati</a>, così come il <a href="/wiki/Modello_probit" title="Modello probit">modello probit</a> ed il <a href="/w/index.php?title=Modello_loglineare&action=edit&redlink=1" class="new" title="Modello loglineare (la pagina non esiste)">modello loglineare</a>, dai quali differisce essenzialmente per la scelta della funzione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }"></span>.<sup id="cite_ref-Definizione_1-1" class="reference"><a href="#cite_note-Definizione-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Scelta_della_funzione">Scelta della funzione</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_logit&veaction=edit&section=1" title="Modifica la sezione Scelta della funzione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_logit&action=edit&section=1" title="Edit section's source code: Scelta della funzione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Mplwp_logit.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Mplwp_logit.svg/220px-Mplwp_logit.svg.png" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Mplwp_logit.svg/330px-Mplwp_logit.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Mplwp_logit.svg/440px-Mplwp_logit.svg.png 2x" data-file-width="600" data-file-height="400" /></a><figcaption>La funzione logit. L'inversa di questa funzione è utilizzata nella regressione logistica.</figcaption></figure> <p>Un modello di regressione dove la variabile dipendente è dicotomica, ossia una variabile che può avere come unici valori 0 e 1 o riconducibili ad essi, calcola la probabilità che questa variabile acquisisca valore 1. Poiché le probabilità per definizione sono limitate ad un intervallo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=\left[0,1\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=\left[0,1\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dc3019b82e440a7031193b9060a4f1e9d45780e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.517ex; height:2.843ex;" alt="{\displaystyle C=\left[0,1\right]}"></span>, l'utilizzo di un modello di <a href="/wiki/Regressione_lineare" title="Regressione lineare">regressione lineare</a> non sarebbe appropriato, infatti esso restituirebbe dei valori appartenenti all'intero insieme <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>.<sup id="cite_ref-Scelta_funzione_2-0" class="reference"><a href="#cite_note-Scelta_funzione-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Si supponga infatti il seguente modello lineare: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(Y=1\mid X=x)=\beta _{0}+\beta _{1}X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>=</mo> <mn>1</mn> <mo>∣</mo> <mi>X</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(Y=1\mid X=x)=\beta _{0}+\beta _{1}X.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f998c4dc2f8d13844b24ea10d691b0303a422515" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.989ex; height:2.843ex;" alt="{\displaystyle \Pr(Y=1\mid X=x)=\beta _{0}+\beta _{1}X.}"></span></dd></dl> <p>La <a href="/wiki/Derivata" title="Derivata">derivata</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial }{\partial X}}\Pr(Y=1\mid X=x)=\beta _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>X</mi> </mrow> </mfrac> </mrow> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>=</mo> <mn>1</mn> <mo>∣</mo> <mi>X</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial X}}\Pr(Y=1\mid X=x)=\beta _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9233daf30bbaf5e8e2b2cd2388eb03710e0fd757" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.673ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial }{\partial X}}\Pr(Y=1\mid X=x)=\beta _{1}}"></span></dd></dl> <p>essendo costante e uguale al parametro <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeeccd8b585b819e38f9c1fe5e9816a3ea01804c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\displaystyle \beta _{1}}"></span>, non permette alla funzione di cambiare pendenza in base al valore di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> e quindi di poter avere come codominio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>. Questa caratteristica è invece posseduta, ad esempio, dalle <a href="/wiki/Funzione_di_ripartizione" title="Funzione di ripartizione">funzioni di ripartizione</a>.<sup id="cite_ref-Scelta_funzione_2-1" class="reference"><a href="#cite_note-Scelta_funzione-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> L'utilizzo infatti di una funzione non lineare permette di avere una derivata prima dipendente da <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> e quindi in grado di cambiare al variare di questa variabile. Se si considera infatti il seguente modello: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(Y=1\mid X=x)=F(\alpha _{0}+\alpha _{1}X),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>=</mo> <mn>1</mn> <mo>∣</mo> <mi>X</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>X</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(Y=1\mid X=x)=F(\alpha _{0}+\alpha _{1}X),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9efd30d77689399666bacee5249459e7b964cf6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.883ex; height:2.843ex;" alt="{\displaystyle \Pr(Y=1\mid X=x)=F(\alpha _{0}+\alpha _{1}X),}"></span></dd></dl> <p>dove la derivata è la seguente </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial }{\partial X}}\Pr(Y=1\mid X=x)=f(\alpha _{0}+\alpha _{1}X)\alpha _{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>X</mi> </mrow> </mfrac> </mrow> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>=</mo> <mn>1</mn> <mo>∣</mo> <mi>X</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>X</mi> <mo stretchy="false">)</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial X}}\Pr(Y=1\mid X=x)=f(\alpha _{0}+\alpha _{1}X)\alpha _{1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b0a038694913ad23a331dc020f6f53e97a2f718" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:42.484ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial }{\partial X}}\Pr(Y=1\mid X=x)=f(\alpha _{0}+\alpha _{1}X)\alpha _{1}.}"></span></dd></dl> <p>Si nota come la pendenza della curva ora possa variare al variare di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>, potendo quindi possedere un codominio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>. Per il modello logit si utilizza come funzione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> la funzione di ripartizione della <a href="/wiki/Distribuzione_logistica" title="Distribuzione logistica">distribuzione logistica</a> standard.<sup id="cite_ref-Definizione_1-2" class="reference"><a href="#cite_note-Definizione-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Definizione">Definizione</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_logit&veaction=edit&section=2" title="Modifica la sezione Definizione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_logit&action=edit&section=2" title="Edit section's source code: Definizione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Il modello di regressione logit per la <a href="/wiki/Popolazione_statistica" title="Popolazione statistica">popolazione</a> è:<sup id="cite_ref-Definizione_1-3" class="reference"><a href="#cite_note-Definizione-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} [Y\mid \mathbf {X} ]=\Pr(Y=1\mid X_{1},\ldots ,X_{k})=\Lambda (\mathbf {X} ^{T}{\boldsymbol {\beta }})={\frac {e^{\beta _{0}+\beta _{1}X_{1}+\ldots +\beta _{k}X_{k}}}{1+e^{\beta _{0}+\beta _{1}X_{1}+\ldots +\beta _{k}X_{k}}}}=p,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <mi>Y</mi> <mo>∣</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>=</mo> <mn>1</mn> <mo>∣</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>…</mo> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>…</mo> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>p</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} [Y\mid \mathbf {X} ]=\Pr(Y=1\mid X_{1},\ldots ,X_{k})=\Lambda (\mathbf {X} ^{T}{\boldsymbol {\beta }})={\frac {e^{\beta _{0}+\beta _{1}X_{1}+\ldots +\beta _{k}X_{k}}}{1+e^{\beta _{0}+\beta _{1}X_{1}+\ldots +\beta _{k}X_{k}}}}=p,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09b6e9f4a4b3279ee9c3b5b715d5956de4a17aad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:76.236ex; height:6.176ex;" alt="{\displaystyle \mathbb {E} [Y\mid \mathbf {X} ]=\Pr(Y=1\mid X_{1},\ldots ,X_{k})=\Lambda (\mathbf {X} ^{T}{\boldsymbol {\beta }})={\frac {e^{\beta _{0}+\beta _{1}X_{1}+\ldots +\beta _{k}X_{k}}}{1+e^{\beta _{0}+\beta _{1}X_{1}+\ldots +\beta _{k}X_{k}}}}=p,}"></span></dd></dl> <p>dove: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50be466e52dac236652eedbbfb5660e570f2a5e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle \Pr }"></span> indica la probabilità;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> è la <a href="/wiki/Variabile_dipendente" class="mw-redirect" title="Variabile dipendente">variabile dipendente</a> <a href="/wiki/Variabile_dicotomica" title="Variabile dicotomica">dicotomica</a> con una <a href="/wiki/Distribuzione_bernoulliana" class="mw-redirect" title="Distribuzione bernoulliana">distribuzione bernoulliana</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\sim {\mathcal {B}}(p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\sim {\mathcal {B}}(p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f22d7710d86ae70d6f23039084a06c1bb83cd46d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.394ex; height:2.843ex;" alt="{\displaystyle Y\sim {\mathcal {B}}(p)}"></span>;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }"></span> è il <a href="/wiki/Vettore_(matematica)" title="Vettore (matematica)">vettore</a> di <a href="/wiki/Variabili_indipendenti" class="mw-redirect" title="Variabili indipendenti">variabili indipendenti</a> o regressori <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},\ldots ,X_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},\ldots ,X_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b6ecebf8cf8de4436ef9a6310a4937a5cef6d6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.17ex; height:2.509ex;" alt="{\displaystyle X_{1},\ldots ,X_{k}}"></span>;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/702cafc420cc00c54896f6d125112820956aaf6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.534ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {\beta }}}"></span> è il vettore di parametri <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{0},\ldots ,\beta _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{0},\ldots ,\beta _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd5c4bf26f233d51db3c4f43e1894d0d51a40caf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.953ex; height:2.509ex;" alt="{\displaystyle \beta _{0},\ldots ,\beta _{k}}"></span>;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }"></span> è la funzione di ripartizione della distribuzione logistica standard;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span> è il <a href="/wiki/E_(costante_matematica)" title="E (costante matematica)">numero di Eulero</a>, circa uguale a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2,71828}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>,</mo> <mn>71828</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2,71828}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02ab25afb1f656b7a1bb031a65c51e9e6e93309e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.009ex; height:2.509ex;" alt="{\displaystyle 2,71828}"></span>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Varianza">Varianza</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_logit&veaction=edit&section=3" title="Modifica la sezione Varianza" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_logit&action=edit&section=3" title="Edit section's source code: Varianza"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La <a href="/wiki/Varianza" title="Varianza">varianza</a> della variabile dipendente risulta dipendere dal vettore dei regressori <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }"></span>. Infatti </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Var} (Y\mid \mathbf {X} )=\mathbb {E} \left[Y^{2}\mid \mathbf {X} \right]-\mathbb {E} \left[Y\mid \mathbf {X} \right]^{2}=\Lambda (\mathbf {X} ^{T}{\boldsymbol {\beta }})\cdot (1-\Lambda (\mathbf {X} ^{T}{\boldsymbol {\beta }})).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>∣</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mrow> <mo>[</mo> <mrow> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>∣</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> <mo>]</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <msup> <mrow> <mo>[</mo> <mrow> <mi>Y</mi> <mo>∣</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Var} (Y\mid \mathbf {X} )=\mathbb {E} \left[Y^{2}\mid \mathbf {X} \right]-\mathbb {E} \left[Y\mid \mathbf {X} \right]^{2}=\Lambda (\mathbf {X} ^{T}{\boldsymbol {\beta }})\cdot (1-\Lambda (\mathbf {X} ^{T}{\boldsymbol {\beta }})).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c77edfdd74194d021166e2017345200ebf355c62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:65.678ex; height:3.509ex;" alt="{\displaystyle \mathrm {Var} (Y\mid \mathbf {X} )=\mathbb {E} \left[Y^{2}\mid \mathbf {X} \right]-\mathbb {E} \left[Y\mid \mathbf {X} \right]^{2}=\Lambda (\mathbf {X} ^{T}{\boldsymbol {\beta }})\cdot (1-\Lambda (\mathbf {X} ^{T}{\boldsymbol {\beta }})).}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Effetto_marginale">Effetto marginale</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_logit&veaction=edit&section=4" title="Modifica la sezione Effetto marginale" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_logit&action=edit&section=4" title="Edit section's source code: Effetto marginale"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>L'effetto sulla variabile dipendente <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> dato da un cambiamento in un regressore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3cb1ef7c9f25e85e1957e4eb58a72fa16a0066" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.834ex; height:2.843ex;" alt="{\displaystyle X_{j}}"></span>, chiamato effetto marginale, è calcolato come la derivata del <a href="/wiki/Valore_atteso" title="Valore atteso">valore atteso</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> rispetto a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3cb1ef7c9f25e85e1957e4eb58a72fa16a0066" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.834ex; height:2.843ex;" alt="{\displaystyle X_{j}}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial }{\partial X_{j}}}\mathbb {E} [Y\mid \mathbf {X} ]={\frac {\partial }{\partial X_{j}}}\Lambda (\mathbf {X} ^{T}{\boldsymbol {\beta }})={\frac {\partial }{\partial X_{j}}}{\frac {e^{\mathbf {X} ^{T}{\boldsymbol {\beta }}}}{1+e^{\mathbf {X} ^{T}{\boldsymbol {\beta }}}}}={\frac {e^{\mathbf {X} ^{T}{\boldsymbol {\beta }}}}{1+e^{\mathbf {X} ^{T}{\boldsymbol {\beta }}}}}\cdot {\frac {1}{1+e^{\mathbf {X} ^{T}{\boldsymbol {\beta }}}}}\cdot \beta _{j},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <mi>Y</mi> <mo>∣</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> </mrow> </msup> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> </mrow> </msup> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>⋅</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial X_{j}}}\mathbb {E} [Y\mid \mathbf {X} ]={\frac {\partial }{\partial X_{j}}}\Lambda (\mathbf {X} ^{T}{\boldsymbol {\beta }})={\frac {\partial }{\partial X_{j}}}{\frac {e^{\mathbf {X} ^{T}{\boldsymbol {\beta }}}}{1+e^{\mathbf {X} ^{T}{\boldsymbol {\beta }}}}}={\frac {e^{\mathbf {X} ^{T}{\boldsymbol {\beta }}}}{1+e^{\mathbf {X} ^{T}{\boldsymbol {\beta }}}}}\cdot {\frac {1}{1+e^{\mathbf {X} ^{T}{\boldsymbol {\beta }}}}}\cdot \beta _{j},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ed025f146d2606488494fc5c335f25251da2ea3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:76.743ex; height:6.843ex;" alt="{\displaystyle {\frac {\partial }{\partial X_{j}}}\mathbb {E} [Y\mid \mathbf {X} ]={\frac {\partial }{\partial X_{j}}}\Lambda (\mathbf {X} ^{T}{\boldsymbol {\beta }})={\frac {\partial }{\partial X_{j}}}{\frac {e^{\mathbf {X} ^{T}{\boldsymbol {\beta }}}}{1+e^{\mathbf {X} ^{T}{\boldsymbol {\beta }}}}}={\frac {e^{\mathbf {X} ^{T}{\boldsymbol {\beta }}}}{1+e^{\mathbf {X} ^{T}{\boldsymbol {\beta }}}}}\cdot {\frac {1}{1+e^{\mathbf {X} ^{T}{\boldsymbol {\beta }}}}}\cdot \beta _{j},}"></span></dd></dl> <p>dove <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83edf0558c67ad56ca5c05096b550bd733d62c4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.225ex; height:2.843ex;" alt="{\displaystyle \beta _{j}}"></span> è il parametro associato al regressore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3cb1ef7c9f25e85e1957e4eb58a72fa16a0066" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.834ex; height:2.843ex;" alt="{\displaystyle X_{j}}"></span>.<sup id="cite_ref-Definizione_1-4" class="reference"><a href="#cite_note-Definizione-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Per il calcolo della derivata il regressore deve essere continuo. </p> <div class="mw-heading mw-heading2"><h2 id="Illustrazione_del_metodo">Illustrazione del metodo</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_logit&veaction=edit&section=5" title="Modifica la sezione Illustrazione del metodo" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_logit&action=edit&section=5" title="Edit section's source code: Illustrazione del metodo"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Per ogni osservazione <a href="/wiki/Campione_statistico" class="mw-redirect" title="Campione statistico">campionaria</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,\ldots ,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,\ldots ,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5726d00b79af1b4666a6319c45381579dc85a9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.636ex; height:2.509ex;" alt="{\displaystyle i=1,\ldots ,n}"></span> si dispone di una determinazione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> e di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> determinazioni <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},\ldots ,X_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},\ldots ,X_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b6ecebf8cf8de4436ef9a6310a4937a5cef6d6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.17ex; height:2.509ex;" alt="{\displaystyle X_{1},\ldots ,X_{k}}"></span>. Il modello cerca una relazione non lineare, utilizzando la funzione di ripartizione della distribuzione logistica standard, tra la variabile dipendente e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> variabili indipendenti, stimando il valore dei coefficienti <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{0},\ldots ,\beta _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{0},\ldots ,\beta _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd5c4bf26f233d51db3c4f43e1894d0d51a40caf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.953ex; height:2.509ex;" alt="{\displaystyle \beta _{0},\ldots ,\beta _{k}}"></span> tramite il metodo della massima verosimiglianza.<sup id="cite_ref-Definizione_1-5" class="reference"><a href="#cite_note-Definizione-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Stima_del_modello">Stima del modello</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_logit&veaction=edit&section=6" title="Modifica la sezione Stima del modello" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_logit&action=edit&section=6" title="Edit section's source code: Stima del modello"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Il vettore di parametri <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/702cafc420cc00c54896f6d125112820956aaf6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.534ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {\beta }}}"></span> è di norma stimato con il <a href="/wiki/Metodo_della_massima_verosimiglianza" title="Metodo della massima verosimiglianza">metodo della massima verosimiglianza</a>, con il quale si ottengono stimatori <a href="/wiki/Efficienza_(statistica)" title="Efficienza (statistica)">efficienti</a>, <a href="/wiki/Consistenza_(statistica)" title="Consistenza (statistica)">consistenti</a> e distribuiti normalmente nel caso in cui il <a href="/wiki/Campione_statistico" class="mw-redirect" title="Campione statistico">campione statistico</a> sia abbastanza grande.<sup id="cite_ref-Stimatori_4-0" class="reference"><a href="#cite_note-Stimatori-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Queste proprietà permettono di calcolare il <a href="/wiki/Test_t" title="Test t">test t</a> su un parametro, il <a href="/wiki/Test_F" title="Test F">test F</a> nel caso di restrizioni multiple e gli <a href="/wiki/Intervallo_di_confidenza" title="Intervallo di confidenza">intervalli di confidenza</a>.<sup id="cite_ref-Stimatori_4-1" class="reference"><a href="#cite_note-Stimatori-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Alla stima dei parametri segue la stima della probabilità <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Funzione_di_verosimiglianza">Funzione di verosimiglianza</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_logit&veaction=edit&section=7" title="Modifica la sezione Funzione di verosimiglianza" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_logit&action=edit&section=7" title="Edit section's source code: Funzione di verosimiglianza"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Nel modello logit la variabile dipendente <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> è dicotomica e con distribuzione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\sim {\mathcal {B}}(p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\sim {\mathcal {B}}(p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f22d7710d86ae70d6f23039084a06c1bb83cd46d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.394ex; height:2.843ex;" alt="{\displaystyle Y\sim {\mathcal {B}}(p)}"></span>. Si consideri un campione di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> osservazioni dove ciascuna di esse è identificata con <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,\ldots ,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,\ldots ,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5726d00b79af1b4666a6319c45381579dc85a9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.636ex; height:2.509ex;" alt="{\displaystyle i=1,\ldots ,n}"></span>. Per la definizione del modello, la probabilità che questa variabile sia 1 per una data osservazione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> è </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(Y_{i}=1\mid X_{1i},\ldots ,X_{ki})=\Lambda (\beta _{0}+\beta _{1}X_{1i}+\ldots +\beta _{k}X_{ki})=p_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>∣</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mo>…</mo> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(Y_{i}=1\mid X_{1i},\ldots ,X_{ki})=\Lambda (\beta _{0}+\beta _{1}X_{1i}+\ldots +\beta _{k}X_{ki})=p_{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee1e66937cb6836014f4a26d0cbe361c090dcc7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:62.707ex; height:2.843ex;" alt="{\displaystyle \Pr(Y_{i}=1\mid X_{1i},\ldots ,X_{ki})=\Lambda (\beta _{0}+\beta _{1}X_{1i}+\ldots +\beta _{k}X_{ki})=p_{i},}"></span></dd></dl> <p>mentre la probabilità che sia 0 è </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(Y_{i}=0\mid X_{1i},\ldots ,X_{ki})=1-\Lambda (\beta _{0}+\beta _{1}X_{1i}+\ldots +\beta _{k}X_{ki})=1-p_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>∣</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mo>…</mo> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(Y_{i}=0\mid X_{1i},\ldots ,X_{ki})=1-\Lambda (\beta _{0}+\beta _{1}X_{1i}+\ldots +\beta _{k}X_{ki})=1-p_{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e96e9d2f52185d79bafe5e1d09c2134209a149a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:70.713ex; height:2.843ex;" alt="{\displaystyle \Pr(Y_{i}=0\mid X_{1i},\ldots ,X_{ki})=1-\Lambda (\beta _{0}+\beta _{1}X_{1i}+\ldots +\beta _{k}X_{ki})=1-p_{i}.}"></span></dd></dl> <p>La distribuzione di <a href="/wiki/Probabilit%C3%A0_condizionata" title="Probabilità condizionata">probabilità condizionata</a> per ogni elemento <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> può essere scritta come </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(Y_{i}=y_{i}\mid X_{1i},\ldots ,X_{ki})=p_{i}^{y_{i}}(1-p_{i})^{1-y_{i}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∣</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(Y_{i}=y_{i}\mid X_{1i},\ldots ,X_{ki})=p_{i}^{y_{i}}(1-p_{i})^{1-y_{i}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdac905080fbae0d850f2903b2ebc9d0367ed732" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.855ex; height:3.343ex;" alt="{\displaystyle \Pr(Y_{i}=y_{i}\mid X_{1i},\ldots ,X_{ki})=p_{i}^{y_{i}}(1-p_{i})^{1-y_{i}}.}"></span></dd></dl> <p>Si considera ora l'intero campione e sia assume che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1i},X_{2i},\ldots ,X_{ki},Y_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1i},X_{2i},\ldots ,X_{ki},Y_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/275c5b7a05791d4a2a4575d77d8673f77dd237cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.069ex; height:2.509ex;" alt="{\displaystyle X_{1i},X_{2i},\ldots ,X_{ki},Y_{i}}"></span> siano <a href="/wiki/Variabili_indipendenti_e_identicamente_distribuite" title="Variabili indipendenti e identicamente distribuite">indipendenti e identicamente distribuite</a> per ogni osservazione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>. Risulta quindi che la distribuzione di <a href="/wiki/Probabilit%C3%A0_congiunta" class="mw-redirect" title="Probabilità congiunta">probabilità congiunta</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Y_{1},\ldots ,Y_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Y_{1},\ldots ,Y_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbb8bb995dc83de7a95b59a50eeb1f03b32ea097" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.961ex; height:2.843ex;" alt="{\displaystyle (Y_{1},\ldots ,Y_{n})}"></span> è il prodotto delle probabilità condizionate di ogni osservazione: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Pr(Y_{1}=y_{1},\ldots ,Y_{n}=y_{n}\mid X_{1i},\ldots ,X_{ki})&=\Pr(Y_{1}=y_{1}\mid X_{11},\ldots ,X_{k1})\cdot \ldots \cdot \Pr(Y_{n}=y_{n}\mid X_{1n},\ldots ,X_{kn})=\\&=p_{1}^{y_{1}}(1-p_{1})^{1-y_{1}}\cdot \ldots \cdot p_{n}^{y_{n}}(1-p_{n})^{1-y_{n}}=\prod _{i=1}^{n}p_{i}^{y_{i}}(1-p_{i})^{1-y_{i}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∣</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∣</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo>…</mo> <mo>⋅</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∣</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>⋅</mo> <mo>…</mo> <mo>⋅</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Pr(Y_{1}=y_{1},\ldots ,Y_{n}=y_{n}\mid X_{1i},\ldots ,X_{ki})&=\Pr(Y_{1}=y_{1}\mid X_{11},\ldots ,X_{k1})\cdot \ldots \cdot \Pr(Y_{n}=y_{n}\mid X_{1n},\ldots ,X_{kn})=\\&=p_{1}^{y_{1}}(1-p_{1})^{1-y_{1}}\cdot \ldots \cdot p_{n}^{y_{n}}(1-p_{n})^{1-y_{n}}=\prod _{i=1}^{n}p_{i}^{y_{i}}(1-p_{i})^{1-y_{i}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ce333780f4f0724ab60d1c4390847f74f403dd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:105.99ex; height:10.176ex;" alt="{\displaystyle {\begin{aligned}\Pr(Y_{1}=y_{1},\ldots ,Y_{n}=y_{n}\mid X_{1i},\ldots ,X_{ki})&=\Pr(Y_{1}=y_{1}\mid X_{11},\ldots ,X_{k1})\cdot \ldots \cdot \Pr(Y_{n}=y_{n}\mid X_{1n},\ldots ,X_{kn})=\\&=p_{1}^{y_{1}}(1-p_{1})^{1-y_{1}}\cdot \ldots \cdot p_{n}^{y_{n}}(1-p_{n})^{1-y_{n}}=\prod _{i=1}^{n}p_{i}^{y_{i}}(1-p_{i})^{1-y_{i}}.\end{aligned}}}"></span></dd></dl> <p>Si riprende ora la definizione del modello logit e la si sostituisce al posto di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}"></span>, ottenendo quindi la <a href="/wiki/Funzione_di_verosimiglianza" title="Funzione di verosimiglianza">funzione di verosimiglianza</a><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{l}{\mathcal {L}}_{\textrm {logit}}(\beta _{0},\ldots ,\beta _{k};Y_{1},\ldots ,Y_{n}\mid X_{1i},\ldots ,X_{ki})=\\\qquad \displaystyle \prod _{i=1}^{n}\left[\Lambda (\beta _{0}+\beta _{1}X_{1i}+\ldots +\beta _{k}X_{ki})\right]^{Y_{i}}\left[1-\Lambda (\beta _{0}+\beta _{1}X_{1i}+\ldots +\beta _{k}X_{ki})\right]^{1-Y_{i}}.\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>logit</mtext> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>;</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∣</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> </mtd> </mtr> <mtr> <mtd> <mspace width="2em" /> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mrow> <mo>[</mo> <mrow> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mo>…</mo> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> <msup> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mo>…</mo> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> <mo>.</mo> </mstyle> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{l}{\mathcal {L}}_{\textrm {logit}}(\beta _{0},\ldots ,\beta _{k};Y_{1},\ldots ,Y_{n}\mid X_{1i},\ldots ,X_{ki})=\\\qquad \displaystyle \prod _{i=1}^{n}\left[\Lambda (\beta _{0}+\beta _{1}X_{1i}+\ldots +\beta _{k}X_{ki})\right]^{Y_{i}}\left[1-\Lambda (\beta _{0}+\beta _{1}X_{1i}+\ldots +\beta _{k}X_{ki})\right]^{1-Y_{i}}.\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/805e17d32b7e57225f018bb01b56e56267a2b801" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.365ex; margin-bottom: -0.306ex; width:79.59ex; height:10.509ex;" alt="{\displaystyle {\begin{array}{l}{\mathcal {L}}_{\textrm {logit}}(\beta _{0},\ldots ,\beta _{k};Y_{1},\ldots ,Y_{n}\mid X_{1i},\ldots ,X_{ki})=\\\qquad \displaystyle \prod _{i=1}^{n}\left[\Lambda (\beta _{0}+\beta _{1}X_{1i}+\ldots +\beta _{k}X_{ki})\right]^{Y_{i}}\left[1-\Lambda (\beta _{0}+\beta _{1}X_{1i}+\ldots +\beta _{k}X_{ki})\right]^{1-Y_{i}}.\end{array}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Stima_dei_parametri">Stima dei parametri</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_logit&veaction=edit&section=8" title="Modifica la sezione Stima dei parametri" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_logit&action=edit&section=8" title="Edit section's source code: Stima dei parametri"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Per calcolare gli stimatori <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\beta }}_{0},{\hat {\beta }}_{1},\ldots ,{\hat {\beta }}_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\beta }}_{0},{\hat {\beta }}_{1},\ldots ,{\hat {\beta }}_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40839d77576b5bc39a2db06c0ffae8dff9a377c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.763ex; height:3.343ex;" alt="{\displaystyle {\hat {\beta }}_{0},{\hat {\beta }}_{1},\ldots ,{\hat {\beta }}_{k}}"></span> dei parametri <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{0},\beta _{1},\ldots ,\beta _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{0},\beta _{1},\ldots ,\beta _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4239fa5c2053c4a5cab496ae91b7205615c1534" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.357ex; height:2.509ex;" alt="{\displaystyle \beta _{0},\beta _{1},\ldots ,\beta _{k}}"></span> risulta conveniente calcolare la funzione di log-verosimiglianza poiché in questo modo si riesce a eliminare la <a href="/wiki/Produttoria" title="Produttoria">produttoria</a>. Si applica quindi il logaritmo alla funzione di verosimiglianza: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{l}{\mathcal {l}}_{\textrm {logit}}(\beta _{0},\ldots ,\beta _{k};Y_{1},\ldots ,Y_{n}\mid X_{1i},\ldots ,X_{ki})=\ln {\mathcal {L}}_{\textrm {logit}}(\beta _{0},\ldots ,\beta _{k};Y_{1},\ldots ,Y_{n}\mid X_{1i},\ldots ,X_{ki})\\\qquad \displaystyle =\sum _{i=1}^{n}Y_{i}\ln \left[\Lambda (\beta _{0}+\beta _{1}X_{1i}+\ldots +\beta _{k}X_{ki})\right]+\sum _{i=1}^{n}(1-Y_{i})\ln \left[1-\Lambda (\beta _{0}+\beta _{1}X_{1i}+\ldots +\beta _{k}X_{ki})\right]\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>logit</mtext> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>;</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∣</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>logit</mtext> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>;</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∣</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mspace width="2em" /> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mo>…</mo> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mo>…</mo> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{l}{\mathcal {l}}_{\textrm {logit}}(\beta _{0},\ldots ,\beta _{k};Y_{1},\ldots ,Y_{n}\mid X_{1i},\ldots ,X_{ki})=\ln {\mathcal {L}}_{\textrm {logit}}(\beta _{0},\ldots ,\beta _{k};Y_{1},\ldots ,Y_{n}\mid X_{1i},\ldots ,X_{ki})\\\qquad \displaystyle =\sum _{i=1}^{n}Y_{i}\ln \left[\Lambda (\beta _{0}+\beta _{1}X_{1i}+\ldots +\beta _{k}X_{ki})\right]+\sum _{i=1}^{n}(1-Y_{i})\ln \left[1-\Lambda (\beta _{0}+\beta _{1}X_{1i}+\ldots +\beta _{k}X_{ki})\right]\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/514761f7af3892d976abacc2547224cd90a15128" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.365ex; margin-bottom: -0.306ex; width:97.018ex; height:10.509ex;" alt="{\displaystyle {\begin{array}{l}{\mathcal {l}}_{\textrm {logit}}(\beta _{0},\ldots ,\beta _{k};Y_{1},\ldots ,Y_{n}\mid X_{1i},\ldots ,X_{ki})=\ln {\mathcal {L}}_{\textrm {logit}}(\beta _{0},\ldots ,\beta _{k};Y_{1},\ldots ,Y_{n}\mid X_{1i},\ldots ,X_{ki})\\\qquad \displaystyle =\sum _{i=1}^{n}Y_{i}\ln \left[\Lambda (\beta _{0}+\beta _{1}X_{1i}+\ldots +\beta _{k}X_{ki})\right]+\sum _{i=1}^{n}(1-Y_{i})\ln \left[1-\Lambda (\beta _{0}+\beta _{1}X_{1i}+\ldots +\beta _{k}X_{ki})\right]\end{array}}}"></span></dd></dl> <p>Gli stimatori calcolati con il metodo della massima verosimiglianza massimizzano la funzione precedente risolvendo il seguente problema: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{{\hat {\beta }}_{0},{\hat {\beta }}_{1},\ldots ,{\hat {\beta }}_{k}\right\}_{MV}=\arg \max _{\beta _{0},\ldots ,\beta _{k}}{\mathcal {l}}_{\textrm {logit}}(\beta _{0},\ldots ,\beta _{k};Y_{1},\ldots ,Y_{n}\mid X_{1i},\ldots ,X_{ki}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>{</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mi>V</mi> </mrow> </msub> <mo>=</mo> <mi>arg</mi> <mo>⁡</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>logit</mtext> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>;</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∣</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\hat {\beta }}_{0},{\hat {\beta }}_{1},\ldots ,{\hat {\beta }}_{k}\right\}_{MV}=\arg \max _{\beta _{0},\ldots ,\beta _{k}}{\mathcal {l}}_{\textrm {logit}}(\beta _{0},\ldots ,\beta _{k};Y_{1},\ldots ,Y_{n}\mid X_{1i},\ldots ,X_{ki}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2405e716d8d7405e379babfb6d06921090afe232" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:75.549ex; height:5.509ex;" alt="{\displaystyle \left\{{\hat {\beta }}_{0},{\hat {\beta }}_{1},\ldots ,{\hat {\beta }}_{k}\right\}_{MV}=\arg \max _{\beta _{0},\ldots ,\beta _{k}}{\mathcal {l}}_{\textrm {logit}}(\beta _{0},\ldots ,\beta _{k};Y_{1},\ldots ,Y_{n}\mid X_{1i},\ldots ,X_{ki}).}"></span><sup id="cite_ref-metodo_MV_6-0" class="reference"><a href="#cite_note-metodo_MV-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></dd></dl> <p>Per semplificare la scrittura consideriamo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/702cafc420cc00c54896f6d125112820956aaf6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.534ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {\beta }}}"></span> un vettore dei parametri <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{0},\beta _{1},\ldots ,\beta _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{0},\beta _{1},\ldots ,\beta _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4239fa5c2053c4a5cab496ae91b7205615c1534" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.357ex; height:2.509ex;" alt="{\displaystyle \beta _{0},\beta _{1},\ldots ,\beta _{k}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> la derivata di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }"></span>, ossia la <a href="/wiki/Funzione_di_densit%C3%A0_di_probabilit%C3%A0" title="Funzione di densità di probabilità">funzione di densità di probabilità</a> della distribuzione logistica, e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> il numero di osservazioni nel campione. Le condizioni per la massimizzazione sono due: quella di primo ordine dove la <a href="/wiki/Derivata" title="Derivata">derivata</a> prima rispetto ai parametri deve essere posta uguale a zero per trovare i punti estremanti, la seconda invece pone la derivata seconda, sempre rispetto ai parametri, minore di zero per determinare le <a href="/wiki/Concavit%C3%A0" class="mw-redirect" title="Concavità">concavità</a> della funzione e quindi garantire che quelli trovati siano solo <a href="/wiki/Punto_di_massimo" class="mw-redirect" title="Punto di massimo">punti di massimo</a>: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial }{\partial {\boldsymbol {\beta }}}}{\mathcal {l}}_{\textrm {logit}}({\boldsymbol {\beta }};\mathbf {y} )=0\Longleftrightarrow \sum _{i=1}^{n}\left\{{\frac {y_{i}-\Lambda (\mathbf {x} _{i}'{\boldsymbol {\beta }})}{\Lambda (\mathbf {x} _{i}'{\boldsymbol {\beta }})\left[1-\Lambda (\mathbf {x} _{i}'{\boldsymbol {\beta }})\right]}}\cdot \lambda (\mathbf {x} _{i}'{\boldsymbol {\beta }})\right\}=0;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>logit</mtext> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⟺</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> <mo stretchy="false">)</mo> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mi>λ</mi> <mo stretchy="false">(</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial {\boldsymbol {\beta }}}}{\mathcal {l}}_{\textrm {logit}}({\boldsymbol {\beta }};\mathbf {y} )=0\Longleftrightarrow \sum _{i=1}^{n}\left\{{\frac {y_{i}-\Lambda (\mathbf {x} _{i}'{\boldsymbol {\beta }})}{\Lambda (\mathbf {x} _{i}'{\boldsymbol {\beta }})\left[1-\Lambda (\mathbf {x} _{i}'{\boldsymbol {\beta }})\right]}}\cdot \lambda (\mathbf {x} _{i}'{\boldsymbol {\beta }})\right\}=0;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/313711c15f9ce20e379ce9391894d04442429325" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:65.952ex; height:7.509ex;" alt="{\displaystyle {\frac {\partial }{\partial {\boldsymbol {\beta }}}}{\mathcal {l}}_{\textrm {logit}}({\boldsymbol {\beta }};\mathbf {y} )=0\Longleftrightarrow \sum _{i=1}^{n}\left\{{\frac {y_{i}-\Lambda (\mathbf {x} _{i}'{\boldsymbol {\beta }})}{\Lambda (\mathbf {x} _{i}'{\boldsymbol {\beta }})\left[1-\Lambda (\mathbf {x} _{i}'{\boldsymbol {\beta }})\right]}}\cdot \lambda (\mathbf {x} _{i}'{\boldsymbol {\beta }})\right\}=0;}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}}{\partial {\boldsymbol {\beta }}\partial {\boldsymbol {\beta '}}}}{\mathcal {l}}_{\textrm {logit}}({\boldsymbol {\beta }};\mathbf {y} )<0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold-italic">β</mi> <mo mathvariant="bold">′</mo> </msup> </mrow> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>logit</mtext> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">)</mo> <mo><</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}}{\partial {\boldsymbol {\beta }}\partial {\boldsymbol {\beta '}}}}{\mathcal {l}}_{\textrm {logit}}({\boldsymbol {\beta }};\mathbf {y} )<0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38234323b8447a2ad6d40b17e9a50d1edb283571" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.045ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial ^{2}}{\partial {\boldsymbol {\beta }}\partial {\boldsymbol {\beta '}}}}{\mathcal {l}}_{\textrm {logit}}({\boldsymbol {\beta }};\mathbf {y} )<0.}"></span></li></ul> <p>Solitamente le soluzioni di queste condizioni non sono semplici da determinare oppure non possono essere trovate affatto, ma per ovviare a questo problema si possono utilizzare dei programmi statistici per computer che, attraverso alcuni <a href="/wiki/Algoritmi" class="mw-redirect" title="Algoritmi">algoritmi</a>, trovano delle loro approssimazioni.<sup id="cite_ref-metodo_MV_6-1" class="reference"><a href="#cite_note-metodo_MV-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Stima_della_probabilità"><span id="Stima_della_probabilit.C3.A0"></span>Stima della probabilità</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_logit&veaction=edit&section=9" title="Modifica la sezione Stima della probabilità" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_logit&action=edit&section=9" title="Edit section's source code: Stima della probabilità"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Quando è stato calcolato il vettore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\hat {\beta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">β</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\hat {\beta }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db74a112c44daacba85b2a69f8daabdf4208e155" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.658ex; height:3.176ex;" alt="{\displaystyle {\boldsymbol {\hat {\beta }}}}"></span>, ossia la stima del vettore dei parametri <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/702cafc420cc00c54896f6d125112820956aaf6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.534ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {\beta }}}"></span>, è possibile procedere alla stima della probabilità <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>. Per definizione del modello, questa probabilità è anche il valore atteso di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {p}}={\hat {\mathbb {E} }}\left[Y\mid \mathbf {X} \right]=\Lambda (\mathbf {X} ^{T}{\hat {\boldsymbol {\beta }}})={\frac {e^{\mathbf {X} ^{T}{\hat {\boldsymbol {\beta }}}}}{1+e^{\mathbf {X} ^{T}{\boldsymbol {\hat {\beta }}}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow> <mo>[</mo> <mrow> <mi>Y</mi> <mo>∣</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">β</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">β</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </msup> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">β</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {p}}={\hat {\mathbb {E} }}\left[Y\mid \mathbf {X} \right]=\Lambda (\mathbf {X} ^{T}{\hat {\boldsymbol {\beta }}})={\frac {e^{\mathbf {X} ^{T}{\hat {\boldsymbol {\beta }}}}}{1+e^{\mathbf {X} ^{T}{\boldsymbol {\hat {\beta }}}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29857322e44d99ab4658e33c937c17de21c1a094" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; margin-left: -0.089ex; width:38.678ex; height:7.176ex;" alt="{\displaystyle {\hat {p}}={\hat {\mathbb {E} }}\left[Y\mid \mathbf {X} \right]=\Lambda (\mathbf {X} ^{T}{\hat {\boldsymbol {\beta }}})={\frac {e^{\mathbf {X} ^{T}{\hat {\boldsymbol {\beta }}}}}{1+e^{\mathbf {X} ^{T}{\boldsymbol {\hat {\beta }}}}}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_logit&veaction=edit&section=10" title="Modifica la sezione Note" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_logit&action=edit&section=10" title="Edit section's source code: Note"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Definizione-1"><span class="mw-cite-backlink"><b>^</b> <sup><i><a href="#cite_ref-Definizione_1-0">a</a></i></sup> <sup><i><a href="#cite_ref-Definizione_1-1">b</a></i></sup> <sup><i><a href="#cite_ref-Definizione_1-2">c</a></i></sup> <sup><i><a href="#cite_ref-Definizione_1-3">d</a></i></sup> <sup><i><a href="#cite_ref-Definizione_1-4">e</a></i></sup> <sup><i><a href="#cite_ref-Definizione_1-5">f</a></i></sup></span> <span class="reference-text"><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) James H. Stock e Mark W. Watson, <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoec0000stoc_z0a9"><span style="font-style:italic;">Regression with a Binary Dependent Variable</span></a>, in <span style="font-style:italic;">Introduction to Econometrics</span>, 3ª ed., Pearson, 2015, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoec0000stoc_z0a9/page/442">442</a>-443, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-1-292-07131-2" title="Speciale:RicercaISBN/978-1-292-07131-2">978-1-292-07131-2</a>.</cite></span> </li> <li id="cite_note-Scelta_funzione-2"><span class="mw-cite-backlink"><b>^</b> <sup><i><a href="#cite_ref-Scelta_funzione_2-0">a</a></i></sup> <sup><i><a href="#cite_ref-Scelta_funzione_2-1">b</a></i></sup></span> <span class="reference-text"><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) James H. Stock e Mark W. Watson, <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoec0000stoc_z0a9"><span style="font-style:italic;">Regression with a Binary Dependent Variable</span></a>, in <span style="font-style:italic;">Introduction to Econometrics</span>, 3ª ed., Pearson, 2015, p. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoec0000stoc_z0a9/page/437">437</a>, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-1-292-07131-2" title="Speciale:RicercaISBN/978-1-292-07131-2">978-1-292-07131-2</a>.</cite></span> </li> <li id="cite_note-3"><a href="#cite_ref-3"><b>^</b></a> <span class="reference-text">Il valore attes</span> </li> <li id="cite_note-Stimatori-4"><span class="mw-cite-backlink"><b>^</b> <sup><i><a href="#cite_ref-Stimatori_4-0">a</a></i></sup> <sup><i><a href="#cite_ref-Stimatori_4-1">b</a></i></sup></span> <span class="reference-text"><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) James H. Stock e Mark W. Watson, <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoec0000stoc_z0a9"><span style="font-style:italic;">Regression with a Binary Dependent Variable</span></a>, in <span style="font-style:italic;">Introduction to Econometrics</span>, 3ª ed., Pearson, 2015, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoec0000stoc_z0a9/page/441">441</a>-442, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-1-292-07131-2" title="Speciale:RicercaISBN/978-1-292-07131-2">978-1-292-07131-2</a>.</cite></span> </li> <li id="cite_note-5"><a href="#cite_ref-5"><b>^</b></a> <span class="reference-text">L'intera derivazione della funzione di verosimiglianza è consultabile alle pagine qui riportate. <cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) James H. Stock e Mark W. Watson, <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoec0000stoc_z0a9"><span style="font-style:italic;">Regression with a Binary Dependent Variable</span></a>, in <span style="font-style:italic;">Introduction to Econometrics</span>, 3ª ed., Pearson, 2015, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoec0000stoc_z0a9/page/465">465</a>-466, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-1-292-07131-2" title="Speciale:RicercaISBN/978-1-292-07131-2">978-1-292-07131-2</a>.</cite></span> </li> <li id="cite_note-metodo_MV-6"><span class="mw-cite-backlink"><b>^</b> <sup><i><a href="#cite_ref-metodo_MV_6-0">a</a></i></sup> <sup><i><a href="#cite_ref-metodo_MV_6-1">b</a></i></sup></span> <span class="reference-text"><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) James H. Stock e Mark W. Watson, <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoec0000stoc_z0a9"><span style="font-style:italic;">Regression with a Binary Dependent Variable</span></a>, in <span style="font-style:italic;">Introduction to Econometrics</span>, 3ª ed., Pearson, 2015, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoec0000stoc_z0a9/page/465">465</a>-466, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-1-292-07131-2" title="Speciale:RicercaISBN/978-1-292-07131-2">978-1-292-07131-2</a>.</cite></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_logit&veaction=edit&section=11" title="Modifica la sezione Bibliografia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_logit&action=edit&section=11" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Alan Agresti, <span style="font-style:italic;">Categorical Data Analysis</span>, Wiley, 2003, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-0-471-36093-3" title="Speciale:RicercaISBN/978-0-471-36093-3">978-0-471-36093-3</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) William H. Greene, <span style="font-style:italic;">Econometric Analysis</span>, 4ª ed., Prentice Hall, 1999 <abbr title="Data di edizione originale">[1993]</abbr>, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-0-130-13297-0" title="Speciale:RicercaISBN/978-0-130-13297-0">978-0-130-13297-0</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) James H. Stock e Mark W. Watson, <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoec0000stoc_z0a9"><span style="font-style:italic;">Regression with a Binary Dependent Variable</span></a>, in <span style="font-style:italic;">Introduction to Econometrics</span>, 3ª ed., Pearson, 2015, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-1-292-07131-2" title="Speciale:RicercaISBN/978-1-292-07131-2">978-1-292-07131-2</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) P. McCullagh e John A. Nelder, <span style="font-style:italic;">Generalized Linear Models</span>, 2ª ed., Chapman and Hall/CRC, 1989, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-0-412-31760-6" title="Speciale:RicercaISBN/978-0-412-31760-6">978-0-412-31760-6</a>.</cite></li></ul> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_logit&veaction=edit&section=12" title="Modifica la sezione Voci correlate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_logit&action=edit&section=12" title="Edit section's source code: Voci correlate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Logit" title="Logit">Logit</a></li> <li><a href="/wiki/Modello_probit" title="Modello probit">Modello probit</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Altri_progetti">Altri progetti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_logit&veaction=edit&section=13" title="Modifica la sezione Altri progetti" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_logit&action=edit&section=13" title="Edit section's source code: Altri progetti"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div id="interProject" class="toccolours" style="display: none; 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