Anesthesia & Analgesia

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content="Articles,Open Access Articles" /><meta name="wkhealth_journal_title" content="Anesthesia & Analgesia" /><meta name="wkhealth_journal_title_legacy" content="Anesthesia & Analgesia" /><meta name="wkhealth_issue" content="5" /><meta name="wkhealth_article_publication_date" content="May 2018" /><meta name="wkhealth_volume" content="126" /><meta name="wkhealth_ovid_accession_number" content="00000539-201805000-00050" /><meta name="wkhealth_issn" content="0003-2999" /><meta name="wkhealth_title" content="Correlation Coefficients: Appropriate Use and Interpretation" /><meta name="wkhealth_authors" content="Schober, Patrick;Boer, Christa;Schwarte, Lothar A." /><meta name="wkhealth_doi" content="10.1213/ANE.0000000000002864" /><meta name="wkhealth_abstract_html_url" content="https://journals.lww.com/anesthesia-analgesia/abstract/2018/05000/correlation_coefficients__appropriate_use_and.50.aspx" /><meta name="wkhealth_fulltext_html_url" content="https://journals.lww.com/anesthesia-analgesia/fulltext/2018/05000/correlation_coefficients__appropriate_use_and.50.aspx" /><meta name="wkhealth_toc_section" content="General Articles" /><meta name="wkhealth_toc_sub_section" /><meta name="wkhealth_eJpId" content="anesthesia-analgesia:2018:05000:00050" /><meta name="wkhealth_firstpage" content="1763" /><meta name="wkhealth_date" content="May 2018" /><meta name="wkhealth_pdf_url" content="https://journals.lww.com/anesthesia-analgesia/_layouts/15/oaks.journals/downloadpdf.aspx?trckng_src_pg=Other&an=00000539-201805000-00050" /><meta name="Description" content="Correlation in the broadest sense is a measure of an association between variables. In correlated da" /><meta property="og:site_name" content="LWW" /><meta property="og:type" content="article" /><meta property="og:title" content="Correlation Coefficients: Appropriate Use and Interpretation : Anesthesia & Analgesia" /><meta property="og:description" content="irection. Most often, the term correlation is used in the context of a linear relationship between 2 continuous variables and expressed as Pearson product-moment correlation. The Pearson correlation coefficient is typically used for jointly normally distributed data (data that follow a bivariate normal distribution). For nonnormally distributed continuous data, for ordinal data, or for data with relevant outliers, a Spearman rank correlation can be used as a measure of a monotonic association. Both correlation coefficients are scaled such that they range from –1 to +1, where 0 indicates that there is no linear or monotonic association, and the relationship gets stronger and ultimately approaches a straight line (Pearson correlation) or a constantly increasing or decreasing curve (Spearman correlation) as the coefficient approaches an absolute value of 1. Hypothesis tests and confidence intervals can be used to address the statistical significance of the results and to estimate the strength of the relationship in the population from which the data were sampled. The aim of this tutorial is to guide researchers and clinicians in the appropriate use and interpretation of correlation coefficients...." /><meta property="og:image" content="https://images.journals.lww.com/anesthesia-analgesia/SocialThumb.00000539-201805000-00050.F1.jpeg" /><meta property="og:image:width" content="1200" /><meta property="og:image:height" content="849" /><meta property="og:url" content="https://journals.lww.com/anesthesia-analgesia/fulltext/2018/05000/correlation_coefficients__appropriate_use_and.50.aspx" /><meta property="twitter:card" content="summary" /><meta name="robots" content="index, follow, noarchive" /><meta name="WT.sv" content="ZUSE1PEJPWEB02" /><meta name="Title" content="Anesthesia & Analgesia" /><style id="SPThemeHideForms" type="text/css">body {opacity:0 !important}</style>  <link href="https://fonts.googleapis.com/css?family=Arvo:400,400i,700|Fira+Sans:300,300i,400,400i,500,500i,600,600i,700,700i&display=swap" rel="stylesheet" /> <script> // temporary 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Note that the scatter approaches a straight line as the coefficient approaches –1 or +1, whereas there is no linear relationship when the coefficient is 0 (D). E shows by example that the correlation depends on the range of the assessed values. While the coefficient is +0.6 for the whole range of data shown in E, it is only +0.34 when calculated for the data in the shaded area.","https://images.journals.lww.com/anesthesia-analgesia/Rollover.00000539-201805000-00050.F1.jpeg","https://images.journals.lww.com/anesthesia-analgesia/Gallery.00000539-201805000-00050.F1.jpeg","https://journals.lww.com/anesthesia-analgesia/_layouts/oaks.journals/ImageView.aspx?k=anesthesia-analgesia:2018:05000:00050&i=F1&year=2018&issue=05000&article=00050&type=Fulltext","https://images.journals.lww.com/anesthesia-analgesia/Original.00000539-201805000-00050.F1.jpeg","/anesthesia-analgesia/pages/imagegallery.aspx?year=2018&issue=05000&article=00050", "anesthesia-analgesia_2018_05000_00050_F1");images["T1"]=new image("T1","F2","F1","Table.","Example of a Conventional Approach to Interpreting a Correlation Coefficient","https://images.journals.lww.com/anesthesia-analgesia/Rollover.00000539-201805000-00050.T1.jpeg","https://images.journals.lww.com/anesthesia-analgesia/Gallery.00000539-201805000-00050.T1.jpeg","https://journals.lww.com/anesthesia-analgesia/_layouts/oaks.journals/ImageView.aspx?k=anesthesia-analgesia:2018:05000:00050&i=T1&year=2018&issue=05000&article=00050&type=Fulltext","https://images.journals.lww.com/anesthesia-analgesia/Original.00000539-201805000-00050.T1.jpeg","/anesthesia-analgesia/pages/imagegallery.aspx?year=2018&issue=05000&article=00050", "anesthesia-analgesia_2018_05000_00050_T1");images["F2"]=new image("F2","F3","T1","Figure 2.","A, A strictly monotonic curve with a Pearson correlation coefficient (<em xmlns:mrws=\"http://webservices.ovid.com/mrws/1.0\">r</em>) of +0.84. Also in the left-side flat part, the curve is continuously slightly increasing. After ranking the values of both variables from lowest to highest, the ranks show a perfect linear relationship (B). Spearman rank correlation is Pearson correlation calculated with the data ranks instead of their actual values. Hence, Spearman coefficient (<em xmlns:mrws=\"http://webservices.ovid.com/mrws/1.0\">ρ</em>) of +1.0 in A corresponds to Pearson correlation of +1.0 in B.","https://images.journals.lww.com/anesthesia-analgesia/Rollover.00000539-201805000-00050.F2.jpeg","https://images.journals.lww.com/anesthesia-analgesia/Gallery.00000539-201805000-00050.F2.jpeg","https://journals.lww.com/anesthesia-analgesia/_layouts/oaks.journals/ImageView.aspx?k=anesthesia-analgesia:2018:05000:00050&i=F2&year=2018&issue=05000&article=00050&type=Fulltext","https://images.journals.lww.com/anesthesia-analgesia/Original.00000539-201805000-00050.F2.jpeg","/anesthesia-analgesia/pages/imagegallery.aspx?year=2018&issue=05000&article=00050", "anesthesia-analgesia_2018_05000_00050_F2");images["F3"]=new image("F3","","F2","Figure 3.","Constructed examples to illustrate that the relationship between data should also be assessed by visual inspection of plots, rather than relying only on correlation coefficients. A, A correlation coefficient close to 0 does not necessarily mean that the <em xmlns:mrws=\"http://webservices.ovid.com/mrws/1.0\">x</em> axis and the <em xmlns:mrws=\"http://webservices.ovid.com/mrws/1.0\">y</em> axis variable are not related. In fact, the graph suggests a strong quadratic relationship. B–D, Pearson correlation coefficient (<em xmlns:mrws=\"http://webservices.ovid.com/mrws/1.0\">r</em>) is +0.84, just as in <a href=\"javascript:void(0);\" onclick=\"slideShow_NavigateToUrl(\'/anesthesia-analgesia/fulltext/2018/05000/correlation_coefficients__appropriate_use_and.50.aspx#F2\')\" xmlns:ssr=\"http://ssr-eus-go-csi.cloudapp.net/v1/assets\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" xmlns:xhtml=\"http://www.w3.org/1999/xhtml\">Figure 2A</a>, yet the actual relationship between the data is quite different in each panel. 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href="javascript:;" style="display: none;"></a>  <div id="ctl00_TopAdvertisementContainer" class="advertisement-content_top top-banner"> <div class="top-banner__content--centered top-banner--low-opacity"> </div> </div> <div class="main-wrapper"> <div id="main-container">  <header class="header"> <div class="header__holder"> <div class="header__layout-top society wk-grid-page-container"> <script type="text/javascript" src="https://lww.com/_layouts/15/1033/Scripts/wk/UserToolbarDropdown.js?v=05.24.00.023" defer="true"> </script> <section class="user-action-toolbar"> <ul class="user-menu user-info" role="tablist"> <li id="ctl00_ctl51_Header_UserActionTools_liLogin" class="user-menu__item user-menu__item--popup-container user-menu__item--login-popup-container dib"> <button class="user-menu__link user-menu__link--login js-ejp-login-btn" type="button" data-config="{"eventName":"login","eventDetail":{"source":"Masthead Dropdown"},"isNull":false}"> <i class="fa fa-sign-in"></i> <span>Log in</span> </button> </li> <li id="ctl00_ctl51_Header_UserActionTools_liOr" class="dib or-spacing"><span>or</span></li> <li id="ctl00_ctl51_Header_UserActionTools_liRegister2" class="user-menu__item dib"> <a id="ctl00_ctl51_Header_UserActionTools_RegisterUserPopupLnk" class="user-menu__link js-register-link" data-config="{"eventName":"Register","eventDetail":{"source":"Masthead Dropdown"},"isNull":false}" href="https://journals.lww.com/anesthesia-analgesia/pages/register.aspx?ContextUrl=%2fanesthesia-analgesia%2ffulltext%2f2018%2f05000%2fcorrelation_coefficients__appropriate_use_and.50.aspx">Register</a> </li> <li id="ctl00_ctl51_Header_UserActionTools_liIssueAlerts" class="user-menu__item dib"> <i class="fa fa-envelope"></i> <a rel="nofollow" title href="javascript:javascript: alerts_ShowSubscribeeTOCPopup('Anesthesia & Analgesia','anesthesia-analgesia','eTOC', 'eTOC');;" onclick="javascript: alerts_ShowSubscribeeTOCPopup('Anesthesia & Analgesia','anesthesia-analgesia','eTOC', 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alerts_ShowUnSubscribeeTOCPopup_Citation(title, journalId, ovidId) { var unSubscribeeTocPopup = $find('unSubscribeToAlerts'); var lblUnsubscribeMessage = document.getElementById('ctl00_ctl51_Header_UserActionTools_ucUnsubscribeToAlertsUserControl_lblUnsubscribeMessage'); lblUnsubscribeMessage.innerHTML = title; document.getElementById('ctl00_ctl51_Header_UserActionTools_ucUnsubscribeToAlertsUserControl_hiddenUnsubscribeMessage').value = title; document.getElementById('ctl00_ctl51_Header_UserActionTools_ucUnsubscribeToAlertsUserControl_hiddenUnSubscribeAlertType').value = 'Citation'; document.getElementById('ctl00_ctl51_Header_UserActionTools_ucUnsubscribeToAlertsUserControl_hiddenUnsubscribeJournalId').value = journalId; document.getElementById('ctl00_ctl51_Header_UserActionTools_ucUnsubscribeToAlertsUserControl_hiddenOvidId').value = ovidId; $('#' + 'ctl00_ctl51_Header_UserActionTools_ucUnsubscribeToAlertsUserControl_pUnsubscribeButtons').show(); if (unSubscribeeTocPopup != null) { unSubscribeeTocPopup.show(); } } function alerts_ShowUnSubscribeeTOCPopup_Topical(title, journalId, topicalCollectionId) { var unSubscribeeTocPopup = $find('unSubscribeToAlerts'); var lblUnsubscribeMessage = document.getElementById('ctl00_ctl51_Header_UserActionTools_ucUnsubscribeToAlertsUserControl_lblUnsubscribeMessage'); lblUnsubscribeMessage.innerHTML = title; document.getElementById('ctl00_ctl51_Header_UserActionTools_ucUnsubscribeToAlertsUserControl_hiddenUnsubscribeMessage').value = title; document.getElementById('ctl00_ctl51_Header_UserActionTools_ucUnsubscribeToAlertsUserControl_hiddenUnSubscribeAlertType').value = 'Topical'; document.getElementById('ctl00_ctl51_Header_UserActionTools_ucUnsubscribeToAlertsUserControl_hiddenUnsubscribeJournalId').value = journalId; document.getElementById('ctl00_ctl51_Header_UserActionTools_ucUnsubscribeToAlertsUserControl_hiddenTopicalCollectionId').value = topicalCollectionId; $('#' + 'ctl00_ctl51_Header_UserActionTools_ucUnsubscribeToAlertsUserControl_pUnsubscribeButtons').show(); if (document.getElementById('ctl00_ctl51_Header_UserActionTools_ucUnsubscribeToAlertsUserControl_hiddenJournalId').value == null || document.getElementById('ctl00_ctl51_Header_UserActionTools_ucUnsubscribeToAlertsUserControl_hiddenJournalId').value == '') { document.getElementById('ctl00_ctl51_Header_UserActionTools_ucUnsubscribeToAlertsUserControl_hiddenJournalId').value = journalId; } if (unSubscribeeTocPopup != null) { unSubscribeeTocPopup.show(); } } function alerts_ShowUnSubscribeeTOCSuccessPopup() { var unSubscribeeTocPopup = $find('unSubscribeToAlerts'); if (unSubscribeeTocPopup != null) { unSubscribeeTocPopup.show(); } } $('.js-close').on('click', function() { var unSubscribeeTocPopup = $find('unSubscribeToAlerts'); if (unSubscribeeTocPopup != null) { unSubscribeeTocPopup.hide(); } return false; }); </script> <span id="ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_dummyTargetControl"></span> <input type="hidden" name="ctl00$ctl51$Header$UserActionTools$SubscribeToEtocPopupControl$hiddenSubscribeToEtocPopupState" id="ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_hiddenSubscribeToEtocPopupState" value="showemail" /> <input type="hidden" name="ctl00$ctl51$Header$UserActionTools$SubscribeToEtocPopupControl$hiddenConfirmSubscriptionPageUrl" id="ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_hiddenConfirmSubscriptionPageUrl" /> <input type="hidden" name="ctl00$ctl51$Header$UserActionTools$SubscribeToEtocPopupControl$hiddenJournalId" id="ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_hiddenJournalId" /> <input type="hidden" name="ctl00$ctl51$Header$UserActionTools$SubscribeToEtocPopupControl$hiddenSubscriptionType" id="ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_hiddenSubscriptionType" /> <div id="divPopupOverLay" class="Popup-Overlay" style="display: none; position: fixed; left: 0; top: 0; z-index: 3800; width: 1920px; height: 4517px;"></div> <div id="ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_pnlSubscribeToEtoc" class="etoc-popup ej-box-modal-drop-shadow" style="display: none;"> <div tabindex="-1" role="dialog" class="wk-modal horizontally-centered small wk-modal-with-overlay wk-modal-small ejp-modal"> <div id="ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_pnlMain" onkeypress="javascript:return WebForm_FireDefaultButton(event, 'ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_btnContinue')"> <section class="wk-modal-content"> <header class="wk-modal-header ejp-modal-header"> <div class="wk-modal-title ejp-modal-title"> <span id="ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_lblTitle"></span> </div> <button aria-label="close dialog" class="js-close primary-button wk-button-icon wk-modal-close ejp-modal-close "> <i class="fa fa-times"></i> </button> </header> <div class="wk-modal-body"> <div class="wk-field-item"> <div class="wk-field-body"> <strong> <span id="ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_lblThankYouMessage"></span></strong> </div> <div class="wk-field-body"> <span id="ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_cvEmailError" style="color:Black;display:none;"> <div class="ej-error-message"> <p class="ej-error-message-icon error-red"> </p> </div> </span> <label for="Email input">Enter your Email address:</label> <div class="wk-field-body"> <input name="ctl00$ctl51$Header$UserActionTools$SubscribeToEtocPopupControl$txtEmailAddress" type="text" id="ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_txtEmailAddress" class="ej-box-modal-style-1-input" /> </div> </div> <div class="wk-button-bar"> <div class="wk-button-group-left"> <input type="submit" name="ctl00$ctl51$Header$UserActionTools$SubscribeToEtocPopupControl$btnContinue" value="Continue" onclick="javascript:WebForm_DoPostBackWithOptions(new WebForm_PostBackOptions("ctl00$ctl51$Header$UserActionTools$SubscribeToEtocPopupControl$btnContinue", "", true, "", "", false, false))" id="ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_btnContinue" class="primary-button" /> <input type="button" value="Cancel" id="btnCancel" class="js-close secondary-button" /> </div> </div> </div> <div class="wk-field-item"> <div class="wk-field-body"> <a id="ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_hypPrivacy" href="https://journals.lww.com/anesthesia-analgesia/_layouts/oaks.journals/privacy.aspx" target="_blank">Privacy Policy</a> </div> </div> </div> </section> </div> <div id="ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_pnlAuthenticatedUser" style="display: none;"> <section class="wk-modal-content"> <header class="wk-modal-header ejp-modal-header"> <div class="wk-modal-title ejp-modal-title"></div> <button aria-label="close dialog" class="js-close primary-button wk-button-icon wk-modal-close ejp-modal-close "><i class="fa fa-times"></i></button> </header> <div class="wk-modal-body"> <div class="wk-field-item"> <p> <strong> <span id="ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_lblAuthenticatedUserTitle"></span> </strong> </p> </div> <p id="ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_pAuthenticatedUserMessage"> </p> </div> </section> </div> </div> </div> <div id="ej-clear-float"> </div> <script type="text/javascript"> function alerts_SaveSubscriptionTypeInHiddenField(url, subscriptionType, journalId, yearId, issueId, articleId) { if (journalId && yearId && issueId && articleId) { logCounterDataAlertWhenCitedSynchronously(journalId, yearId, issueId, articleId); } $('#' + 'ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_hiddenSubscriptionType').val(subscriptionType); $('#' + 'ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_hiddenJournalId').val(journalId); if (url) window.location = url; else __doPostBack('hypSubscribeToAlerts'); return false; } function alerts_ShowSubscribeeTOCPopup(journalTitle, journalId, subscriptionType, displaySubscriptionType) { $('#' + 'ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_hiddenSubscriptionType').val(subscriptionType); var subscribeToEtocPopup = $find('subscribeToEtoc'); subscribeToEtocPopup.show(); document.getElementsByTagName( 'html' )[0].classList.add('no-scroll'); var modelPopupState = document.getElementById('ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_hiddenSubscribeToEtocPopupState'); var emailTextBox = document.getElementById('ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_txtEmailAddress'); var thanksPanel = document.getElementById('ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_pnlAuthenticatedUser'); var mainPanel = document.getElementById('ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_pnlMain'); var thankYouMessageUnAuthenticated = document.getElementById('ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_lblThankYouMessage'); var thankYouMessageAuthenticated = document.getElementById('ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_lblAuthenticatedUserTitle'); if (!displaySubscriptionType) { displaySubscriptionType = subscriptionType; } if (modelPopupState.value == 'showthanksmessage') { thanksPanel.style.display = "block"; mainPanel.style.display = "none"; thankYouMessageAuthenticated.innerHTML = "Thank you for subscribing to the " + displaySubscriptionType + " for " + journalTitle + "."; } if (modelPopupState.value == 'showemail') { thanksPanel.style.display = "none"; mainPanel.style.display = "block"; thankYouMessageUnAuthenticated.innerHTML = "Thank you for choosing to subscribe to the " + displaySubscriptionType + " for " + journalTitle + "."; } if ($('.etoc-popup') != null) { document.getElementById('ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_hiddenJournalId').value = journalId; if (modelPopupState.value == 'showemail') emailTextBox.focus(); } } $('.js-close').on('click', function () { document.getElementById('ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_txtEmailAddress').value = ""; document.getElementById('ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_cvEmailError').style.display = "none"; var pageUrl = document.getElementById('ctl00_ctl51_Header_UserActionTools_SubscribeToEtocPopupControl_hiddenConfirmSubscriptionPageUrl'); var subscribeeTocPopup = $find('subscribeToEtoc'); if (subscribeeTocPopup != null) { subscribeeTocPopup.hide(); document.getElementsByTagName( 'html' )[0].classList.remove('no-scroll'); if (pageUrl.value !== '' && pageUrl.value !== window.location.pathname) window.location = pageUrl.value; } return false; }); </script> <div id="ctl00_ctl51_Header_UserActionTools_eTocTopContainer"> <div id="eTocTop" style="display: none;"> <a rel="nofollow" title href="javascript:javascript: alerts_ShowSubscribeeTOCPopup('Anesthesia & Analgesia','anesthesia-analgesia','eTOC', 'eTOC');;" onclick="javascript: alerts_ShowSubscribeeTOCPopup('Anesthesia & Analgesia','anesthesia-analgesia','eTOC', 'eTOC');"><span id="ctl00_ctl51_Header_UserActionTools_alertLinkControlTop_defaultText" class="hideOnFixed">Subscribe to eTOC</span></a> </div> </div> <script> $(function () { //find recent searches link and move container var recentSearchesControl = $("a[id$='RecentSearches']"); if (recentSearchesControl.length) { recentSearchesControl.click(function (e) { e.stopPropagation(); if ($("#ResentSearchContainer").is(":hidden")) { $("#ResentSearchContainer").slideDown("slow"); recentSearchesControl.addClass('open'); } else { $("#ResentSearchContainer").hide(); recentSearchesControl.removeClass('open'); } }); } var eTocAccountLink = $("ul.js-user-action-toolbar__dropdown>li>a[id='SubscribeToEToc']"); if (eTocAccountLink.length) { eTocAccountLink.parent().html($("#eTocTop>a")); } }); </script> <a class="logo-secondary black" href="https://iars.org/" target="_blank"> <h3 class="logo-secondary-text">Secondary Logo</h3> </a> <a href="#" class="sticky-search"><i class="fa fa-search"></i></a> </div> <div class="header__layout-bottom wk-grid-page-container"> <a id="hypJournals" class="logo-main" href="https://journals.lww.com/anesthesia-analgesia/pages/default.aspx"> <h3 class="logo-main-text">Journal Logo </h3> </a> <div class="search-v2"> <div class="search-wrapper"> <select name="ctl00$ctl51$Header$SearchTopBoxControl$ddlContentScope" id="ctl00_ctl51_Header_SearchTopBoxControl_ddlContentScope"> <option value="All">All</option> <option selected="selected" value="Articles">Articles</option> <option value="Images">Images</option> <option value="Videos">Videos</option> <option value="Podcasts">Podcasts</option> <option value="Blogs">Blogs</option> </select> <div class="input-group" id="globalSearch"> <input name="ctl00$ctl51$Header$SearchTopBoxControl$txtKeywords" type="text" value="Search" id="ctl00_ctl51_Header_SearchTopBoxControl_txtKeywords" class="form-control searchText" /> <span class="input-group-btn"> <button id="ctl00_ctl51_Header_SearchTopBoxControl_btnGlobalSearchMagnifier" type="submit" class="btn btn-default"><i class="fa fa-search"></i></button> </span> <input type="hidden" name="ctl00$ctl51$Header$SearchTopBoxControl$IsAutoCompleteEnabled" id="ctl00_ctl51_Header_SearchTopBoxControl_IsAutoCompleteEnabled" value="true" /> <input type="hidden" name="ctl00$ctl51$Header$SearchTopBoxControl$AutocompleteApiUrl" id="ctl00_ctl51_Header_SearchTopBoxControl_AutocompleteApiUrl" value="https://platformservices.wolterskluwerhealth.com/autocomplete/ASAG,CRAA/shingle/" /> </div> </div> <div class="advance-search" style=""> <a id="ctl00_ctl51_Header_SearchTopBoxControl_lnkAdvanceSearch" href="https://journals.lww.com/anesthesia-analgesia/pages/advancedsearch.aspx">Advanced Search</a> <input type="submit" name="ctl00$ctl51$Header$SearchTopBoxControl$btnGlobalSearch" value="" id="ctl00_ctl51_Header_SearchTopBoxControl_btnGlobalSearch" disabled="disabled" class="aspNetDisabled btnGlobalSearch" style="display: none;" /> <input name="ctl00$ctl51$Header$SearchTopBoxControl$hfldKeywords" type="hidden" id="ctl00_ctl51_Header_SearchTopBoxControl_hfldKeywords" class="hfldKeywords" value="disableButton" /> </div> </div> <script> $(function() { try { var objSearchBox = { btnGlobalSearchMagnifierId: 'ctl00_ctl51_Header_SearchTopBoxControl_btnGlobalSearchMagnifier', searchButtonId: 'ctl00_ctl51_Header_SearchTopBoxControl_btnGlobalSearch', txtKeywordsId: 'ctl00_ctl51_Header_SearchTopBoxControl_txtKeywords', hfldKeywordsHiddenId: 'ctl00_ctl51_Header_SearchTopBoxControl_hfldKeywords', defaultSearchText: 'Search' }; new Searchbox(objSearchBox).Init(); } catch (e) { console.error(e); } }); //autocomplete functionality for basic search //this functionality will call the SBA url and start suggesting after typing 3rd character var textKeywords = null; var hiddenAutocompleteApiUrl = null; var buttonGlobalSearch = null; $(function () { try { var objAutocomplete = { hiddenIsAutoCompleteEnabled: $('#ctl00_ctl51_Header_SearchTopBoxControl_IsAutoCompleteEnabled'), textKeywords: $('#ctl00_ctl51_Header_SearchTopBoxControl_txtKeywords'), hiddenAutocompleteApiUrl: $('#ctl00_ctl51_Header_SearchTopBoxControl_AutocompleteApiUrl'), buttonGlobalSearch: $('#ctl00_ctl51_Header_SearchTopBoxControl_btnGlobalSearch'), enableNewAutoComplete: true, newAutoCompleteUrl: 'https://dxg-search-na-prd.wkgposvc.cloud/Query.svc/Query_Expand', newAutoCompleteMaxSuggestions: '10', newAutoCompleteDictionaryIds: 'ovid_sted28_mesh22' }; autoComplete(objAutocomplete); } catch (e) { console.error(e); } }); $(function () { let searchSelect = $('select[id$="ddlContentScope"]');; if (searchSelect.length){ 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'ANE', 'Correlation Coefficients: Appropriate Use and Interpretation', '05012018', 'Patrick Schober, Christa Boer, and Lothar A. Schwarte', '126', '5', 'Copyright \u00A9 2018 The Author(s). Published by Wolters Kluwer Health, Inc. on behalf of the International Anesthesia Research Society.', '00000539-201805000-00050', '', true, ''); } </script> <div id="ArticleContainer"> <article id="ej-article-view" class="js-article-view ej-article-view" xmlns:ssr="http://ssr-eus-go-csi.cloudapp.net/v1/assets" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xhtml="http://www.w3.org/1999/xhtml"><section class="ejp-r-article-subsection"><div class="ejp-r-article-subsection__text">General Articles</div></section><header class="ejp-article-header"><h1 class="ejp-article-title">Correlation Coefficients: Appropriate Use and Interpretation</h1></header><section id="ejp-article-authors"><p id="P7">Schober, Patrick MD, PhD, MMedStat; Boer, Christa PhD, MSc; Schwarte, Lothar A. MD, PhD, MBA</p><section><a href="#" id="ejp-article-authors-link"> Author Information <i class="wk-icon-arrow-down-r"></i></a></section></section><section id="ejp-article-authors-info" class="ejp-article-authors-info ejp-article-authors-info--hidden"><div class="ejp-article-authors-info-holder"> <p id="">From the Department of Anesthesiology, VU University Medical Center, Amsterdam, the Netherlands.</p> <p id="">Published ahead of print February 23, 2018.</p> <p id="">Accepted for publication January 11, 2018.</p> <p id="">Funding: None.</p> <p id="">The authors declare no conflicts of interest.</p> <p id="">Reprints will not be available from the authors.</p> <p id="c1">Address correspondence to Patrick Schober, MD, PhD, MMedStat, Department of Anesthesiology, VU University Medical Center, De Boelelaan 1117, 1081HV Amsterdam, the Netherlands. Address e-mail to <span><a href="/cdn-cgi/l/email-protection#08656961647c673278267b6b60676a6d7a487e7d656b266664" onclick="javascript:window.open('mailto:mailto:p.schober@vumc.nl');return false"><span class="__cf_email__" data-cfemail="87f7a9f4e4efe8e5e2f5c7f1f2eae4a9e9eb">[email protected]</span></a></span>.</p> <p id="">This is an open-access article distributed under the terms of the <span><a href="http://creativecommons.org/licenses/by-nc-nd/4.0/" onclick="javascript:window.open('http://creativecommons.org/licenses/by-nc-nd/4.0/');return false">Creative Commons Attribution-Non Commercial-No Derivatives License 4.0 (CCBY-NC-ND)</a></span>, where it is permissible to download and share the work provided it is properly cited. The work cannot be changed in any way or used commercially without permission from the journal.</p> </div></section><section id="article-journal-info"><div class="ej-journal-info"><span class="ej-journal-name">Anesthesia & Analgesia </span><span id="ej-journal-date-volume-issue-pg"><a href="/anesthesia-analgesia/toc/2018/05000">126(5):p 1763-1768, May 2018.</a></span><span> | </span><span class="ej-journal-doi">DOI: </span>10.1213/ANE.0000000000002864<a id="ej-article-indicators-ccl" onmouseover="javascript:Tooltip_CCL('This is an open-access article distributed under the terms of the Creative Commons Attribution-Non Commercial-No Derivatives License 4.0 (CCBY-NC-ND), where it is permissible to download and share the work provided it is properly cited. The work cannot be changed in any way or used commercially without permission from the journal.')" onmouseout="javascript:Tooltip_MouseOut()" href="http://creativecommons.org/licenses/by-nc-nd/4.0/" target="_blank"><span class="ejp-article-indicators__ccl"></span></a></div></section><section id="ejp-article-indicators" class="ejp-article-indicators"><ul class="ejp-article-indicators__list"> <li id="ej-article-indicators-open" class="ejp-article-indicators__list-item" onmouseover="javascript:Tooltip_Open()" onmouseout="javascript:Tooltip_MouseOut()"><span class="ejp-indicator">Open</span></li> <li id="ej-article-indicators-ce" class="ejp-article-indicators__list-item" onmouseover="javascript:Tooltip_CE()" onmouseout="javascript:Tooltip_MouseOut()"><span class="ejp-indicator">CE Test</span></li> </ul><button id="openArticleMetricsPopupButton" type="button" class="wk-button wk-button-ghost wk-button-icon-left ejp-article-metrics-btn" data-articleMetricsEndpoint="//services.journals.lww.com/IntegrationServices/ArticleMetricsService.svc" data-articleMetricsDetailsUrl="https://wolterskluwer.altmetric.com/details/" data-doi="10.1213/ANE.0000000000002864" data-suiteId="vrs_woltere_journalsglobalvrs" data-articlePublicationDate="2018-05-01" data-articlePublicationDateString="May 2018" data-receivedDate="" data-revisionReceivedDate="" data-acceptedDate="" data-crossrefRefLinkRoot="https://doi.org/" data-medlineRefLinkRoot="http://www.ncbi.nlm.nih.gov/pubmed/" data-pubMedId="29481436"><i class="wk-icon-filled-chart-r"></i><i class="wk-icon-filled-chart-white-r"></i> Metrics </button></section><section class="ejp-article-tabs" id="abstractWrap"><h2 class="ejp-article-tabs-heading">Abstract</h2> <div class="ejp-article-tabs__tabpanel" id="panel1" role="tabpanel" aria-labelledby="tab1"> <div id="article-abstract-content1"> <div class="ejp-article-text-abstract"> <p>Correlation in the broadest sense is a measure of an association between variables. In correlated data, the change in the magnitude of 1 variable is associated with a change in the magnitude of another variable, either in the same (positive correlation) or in the opposite (negative correlation) direction. Most often, the term correlation is used in the context of a linear relationship between 2 continuous variables and expressed as Pearson product-moment correlation. The Pearson correlation coefficient is typically used for jointly normally distributed data (data that follow a bivariate normal distribution). For nonnormally distributed continuous data, for ordinal data, or for data with relevant outliers, a Spearman rank correlation can be used as a measure of a monotonic association. Both correlation coefficients are scaled such that they range from –1 to +1, where 0 indicates that there is no linear or monotonic association, and the relationship gets stronger and ultimately approaches a straight line (Pearson correlation) or a constantly increasing or decreasing curve (Spearman correlation) as the coefficient approaches an absolute value of 1. Hypothesis tests and confidence intervals can be used to address the statistical significance of the results and to estimate the strength of the relationship in the population from which the data were sampled. The aim of this tutorial is to guide researchers and clinicians in the appropriate use and interpretation of correlation coefficients.</p> </div> </div> </div></section><div class="ejp-fulltext-content js-ejp-fulltext-content"><section id="ArticleBody"><p id="JCL-P-6">Researchers often aim to study whether there is some association between 2 observed variables and to estimate the strength of this relationship. For example, Nishimura et al<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R1">1</a></sup></sup> assessed whether the volume of infused crystalloid fluid is related to the amount of interstitial fluid leakage during surgery, and Kim et al<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R2">2</a></sup></sup> studied whether opioid growth factor receptor (OGFR) expression is associated with cell proliferation in cancer cells. These and similar research objectives can be quantitatively addressed by correlation analysis, which provides information about not only the strength but also the direction of a relationship (eg, an increase in OGFR expression is associated with an increase or a decrease in cell proliferation).</p> <p id="JCL-P-7">As part of the ongoing series in <em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">Anesthesia & Analgesia</em>, this basic statistical tutorial discusses the 2 most commonly used correlation coefficients in medical research, the Pearson coefficient and the Spearman coefficient.<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R3">3</a></sup></sup> It is important to note that these correlation coefficients are frequently misunderstood and misused.<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R4">4</a></sup></sup><sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0">,</sup><sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R5">5</a></sup></sup> We thus focus on how they should and should not be used and correctly interpreted.</p> <h2 class="ejp-article-outline-heading" data-level="1" id="JCL0">PEARSON PRODUCT-MOMENT CORRELATION</h2> <p id="JCL-P-8">Correlation is a measure of a monotonic association between 2 variables. A monotonic relationship between 2 variables is a one in which either (1) as the value of 1 variable increases, so does the value of the other variable; or (2) as the value of 1 variable increases, the other variable value decreases.</p> <p id="JCL-P-9">In correlated data, therefore, the change in the magnitude of 1 variable is associated with a change in the magnitude of another variable, either in the same or in the opposite direction. In other words, higher values of 1 variable tend to be associated with either higher (positive correlation) or lower (negative correlation) values of the other variable, and vice versa.</p> <p id="JCL-P-10">A linear relationship between 2 variables is a special case of a monotonic relationship. Most often, the term “correlation” is used in the context of such a linear relationship between 2 continuous, random variables, known as a Pearson product-moment correlation, which is commonly abbreviated as “<em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">r</em>.”<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R6">6</a></sup></sup></p> <p id="JCL-P-11">The degree to which the change in 1 continuous variable is associated with a change in another continuous variable can mathematically be described in terms of the covariance of the variables.<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R7">7</a></sup></sup> Covariance is similar to variance, but whereas variance describes the variability of a single variable, covariance is a measure of how 2 variables vary together.<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R7">7</a></sup></sup> However, covariance depends on the measurement scale of the variables, and its absolute magnitude cannot be easily interpreted or compared across studies. To facilitate interpretation, a Pearson correlation coefficient is commonly used. This coefficient is a dimensionless measure of the covariance, which is scaled such that it ranges from –1 to +1.<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R7">7</a></sup></sup></p> <p id="JCL-P-12"><a href="javascript:void(0)" onclick="javascript:showSlideShowByImageID('F1', '00000539-201805000-00050');">Figure 1</a> shows scatterplots with examples of simulated data sampled from bivariate normal distributions with different Pearson correlation coefficients. As illustrated, <em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">r</em> = 0 indicates that there is no linear relationship between the variables, and the relationship becomes stronger (ie, the scatter decreases) as the absolute value of <em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">r</em> increases and ultimately approaches a straight line as the coefficient approaches –1 or +1.</p><section class="ejp-r-article-images"><figure class="ejp-r-article-images__figure"><a href="javascript:void(0)" class="ejp-r-article-images__image-link" onclick="showSlideShowByImageID('F1', '00000539-201805000-00050')"><img class="ejp-r-article-images__img js-lazy-load lazy-load" src="javascript:void(0);" data-src="https://images.journals.lww.com/anesthesia-analgesia/ArticleViewerPreview.00000539-201805000-00050.F1.jpeg" data-srcset="https://images.journals.lww.com/anesthesia-analgesia/ArticleViewerPreview@2.00000539-201805000-00050.F1.jpeg 2x" srcset="" alt="F1"></a><figcaption class="ejp-r-article-images__figcaption"><a href="javascript:void(0)" class="ejp-r-article-images__figcaption-link" onclick="showSlideShowByImageID('F1', '00000539-201805000-00050')">Figure 1.: </a><div class="ejp-r-article-images__figcaption-text">A–F, Scatter plots with data sampled from simulated bivariate normal distributions with varying Pearson correlation coefficients (<em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">r</em>). Note that the scatter approaches a straight line as the coefficient approaches –1 or +1, whereas there is no linear relationship when the coefficient is 0 (D). E shows by example that the correlation depends on the range of the assessed values. While the coefficient is +0.6 for the whole range of data shown in E, it is only +0.34 when calculated for the data in the shaded area.</div></figcaption></figure></section><p id="JCL-P-13">A perfect correlation of –1 or +1 means that all the data points lie exactly on the straight line, which we would expect, for example, if we correlate the weight of samples of water with their volume, assuming that both quantities can be measured very accurately and precisely. However, such absolute relationships are not typical in medical research due to variability of biological processes and measurement error.</p> <h3 class="ejp-article-outline-heading" data-level="2" id="JCL1">Assumptions of a Pearson Correlation</h3> <p id="JCL-P-14">Assumptions of a Pearson correlation have been intensely debated.<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R8 R9 R10">8–10</a></sup></sup> It is therefore not surprising, but nonetheless confusing, that different statistical resources present different assumptions. In reality, the coefficient can be calculated as a measure of a linear relationship without any assumptions.</p> <p id="JCL-P-15">However, proper inference on the strength of the association in the population from which the data were sampled (what one is usually interested in) does require that some assumptions be met:<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R9 R10 R11">9–11</a></sup></sup></p> <ol class="wk-article__list wk-article__list--arabic-numeral"> <li class="wk-article__list-item wk-article__list-item--arabic-numeral">As is actually true for any statistical inference, the data are derived from a random, or at least representative, sample. If the data are not representative of the population of interest, one cannot draw meaningful conclusions about that population.</li> <li class="wk-article__list-item wk-article__list-item--arabic-numeral">Both variables are continuous, jointly normally distributed, random variables. They follow a bivariate normal distribution in the population from which they were sampled. The bivariate normal distribution is beyond the scope of this tutorial but need not be fully understood to use a Pearson coefficient.</li> </ol> <p id="JCL-P-18">Two typical properties of the bivariate normal distribution can be relatively easily assessed, and researchers should check approximate compliance of their data with these properties:</p> <ul id="JCL1-6" class="wk-article__list wk-article__list--plain"> <li class="wk-article__list-item wk-article__list-item--plain">i. Both variables are normally distributed. Methods to assess this assumption have recently been reviewed in this series of basic statistical tutorials.<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R12">12</a></sup></sup></li> <li class="wk-article__list-item wk-article__list-item--plain">ii. If there is a relationship between jointly normally distributed data, it is always linear.<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R13">13</a></sup></sup> Therefore, if the data points in a scatter plot seem to lie close to some curve, the assumption of a bivariate normal distribution is violated.</li> </ul> <p id="JCL-P-21">There are several possibilities to deal with violations to this assumption. First, variables can often be transformed to approach a normal distribution and to linearize the relationship between the variables.<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R12">12</a></sup></sup> Second, in contrast to a Pearson correlation, a Spearman correlation (see below) does not require normally distributed data and can be used to analyze nonlinear monotonic (ie, continuously increasing or decreasing) relationships.<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R14">14</a></sup></sup></p> <ul id="JCL1-8" class="wk-article__list wk-article__list--plain"> <li class="wk-article__list-item wk-article__list-item--plain">3. There are no relevant outliers. Extreme outliers may have undue influence on the Pearson correlation coefficient. While it is generally not legitimate to simply exclude outliers,<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R15">15</a></sup></sup> running the correlation analysis with and without the outlier(s) and comparing the coefficients is a possibility to assess the actual influence of the outlier on the analysis. For data with relevant outliers, Spearman correlation is preferred as it tends to be relatively robust against outliers.<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R14">14</a></sup></sup></li> <li class="wk-article__list-item wk-article__list-item--plain">4. Each pair of <em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">x</em>–<em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">y</em> values is measured independently from each other pair. Multiple observations from the same subjects would violate this assumption.<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R11">11</a></sup></sup> The way to deal with such data depends on whether we are interested in correlations within subjects or between subjects as reviewed previously.<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R16">16</a></sup></sup><sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0">,</sup><sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R17">17</a></sup></sup></li> </ul> <h3 class="ejp-article-outline-heading" data-level="2" id="JCL2">Interpretation of the Correlation Coefficient</h3> <p id="JCL-P-24">Several approaches have been suggested to translate the correlation coefficient into descriptors like “weak,” “moderate,” or “strong” relationship (see the <a href="javascript:void(0)" onclick="javascript:showSlideShowByImageID('T1', '00000539-201805000-00050');">Table</a> for an example).<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R3">3</a></sup></sup><sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0">,</sup><sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R18">18</a></sup></sup> These cutoff points are arbitrary and inconsistent and should be used judiciously. While most researchers would probably agree that a coefficient of <0.1 indicates a negligible and >0.9 a very strong relationship, values in-between are disputable. For example, a correlation coefficient of 0.65 could either be interpreted as a “good” or “moderate” correlation, depending on the applied rule of thumb. It is also quite capricious to claim that a correlation coefficient of 0.39 represents a “weak” association, whereas 0.40 is a “moderate” association.</p><section class="ejp-r-article-images"><figure class="ejp-r-article-images__figure"><a href="javascript:void(0)" class="ejp-r-article-images__image-link" onclick="showSlideShowByImageID('T1', '00000539-201805000-00050')"><img class="ejp-r-article-images__img js-lazy-load lazy-load" src="javascript:void(0);" data-src="https://images.journals.lww.com/anesthesia-analgesia/ArticleViewerPreview.00000539-201805000-00050.T1.jpeg" data-srcset="https://images.journals.lww.com/anesthesia-analgesia/ArticleViewerPreview@2.00000539-201805000-00050.T1.jpeg 2x" srcset="" alt="T1"></a><figcaption class="ejp-r-article-images__figcaption"><a href="javascript:void(0)" class="ejp-r-article-images__figcaption-link" onclick="showSlideShowByImageID('T1', '00000539-201805000-00050')">Table.: </a><div class="ejp-r-article-images__figcaption-text">Example of a Conventional Approach to Interpreting a Correlation Coefficient</div></figcaption></figure></section><p id="JCL-P-25">Rather than using oversimplified rules, we suggest that a specific coefficient should be interpreted as a measure of the strength of the relationship in the context of the posed scientific question. Note that the range of the assessed values should be considered in the interpretation, as a wider range of values tends to show a higher correlation than a smaller range (<a href="javascript:void(0)" onclick="javascript:showSlideShowByImageID('F1', '00000539-201805000-00050');">Figure 1E</a>).<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R19">19</a></sup></sup></p> <p id="JCL-P-26">The observed correlation may also not necessarily be a good estimate for the population correlation coefficient, because samples are inevitably affected by chance. Therefore, the observed coefficient should always be accompanied by a confidence interval, which provides the range of plausible values of the coefficient in the population from which the data were sampled.<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R20">20</a></sup></sup></p> <p id="JCL-P-27">In the study by Nishimura et al,<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R1">1</a></sup></sup> the authors report a correlation coefficient of 0.42 for the relationship between the infused crystalloid volume and the amount of interstitial fluid leakage, so there appears to be a considerable association between the 2 variables. However, the 95% confidence interval, which ranges from 0.03 to 0.70, suggests that the results are also compatible with a negligible (<em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">r</em> = 0.03) and hence clinically unimportant relationship. On the other hand, the data are also compatible with a quite strong association (<em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">r</em> = 0.70). Data with such a wide confidence interval do not allow a definitive conclusion about the strength of the relationship between the variables.</p> <p id="JCL-P-28">Researchers typically also aim to determine whether their result is “statistically significant.” A <em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">t</em> test is available to test the null hypothesis that the correlation coefficient is zero.<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R13">13</a></sup></sup> Note that the <em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">P</em> value derived from the test provides no information on how strongly the 2 variables are related. With large datasets, very small correlation coefficients can be “statistically significant.” Therefore, a statistically significant correlation must not be confused with a clinically relevant correlation. For further information on how results of hypothesis tests and confidence intervals should be interpreted, we refer the reader to previous tutorials in <em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">Anesthesia & Analgesia</em>.<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R20">20</a></sup></sup><sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0">,</sup><sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R21">21</a></sup></sup></p> <h3 class="ejp-article-outline-heading" data-level="2" id="JCL3">Coefficient of Determination</h3> <p id="JCL-P-29">The correlation coefficient is sometimes criticized as having no obvious intrinsic interpretation,<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R6">6</a></sup></sup> and researchers sometimes report the square of the correlation coefficient. This <em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">R</em><sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0">2</sup> is termed the “coefficient of determination.” It can be interpreted as the proportion of variance in 1 variable that is accounted for by the other.<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R6">6</a></sup></sup></p> <p id="JCL-P-30">The correlation coefficient of 0.42 reported by Nishimura et al<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R1">1</a></sup></sup> corresponds to a coefficient of determination (<em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">R</em><sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0">2</sup>) of 0.18, suggesting that about 18% of the variability of the amount of interstitial fluid leakage can be “explained” by the relationship with the amount of infused crystalloid fluid. As more than 80% of the variability is yet unexplained, there must be 1 or more other relevant factors that are related to interstitial leakage.</p> <p id="JCL-P-31">In interpreting the coefficient of determination, note that the squared correlation coefficient is always a positive number, so information on the direction of a relationship is lost. The landmark publication by Ozer<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R22">22</a></sup></sup> provides a more complete discussion on the coefficient of determination.</p> <h3 class="ejp-article-outline-heading" data-level="2" id="JCL4">Pearson Correlation Versus Linear Regression</h3> <p id="JCL-P-32">Due to similarities between a Pearson correlation and a linear regression, researchers sometimes are uncertain as to which test to use. Both techniques have a close mathematical relationship, but distinct purposes and assumptions.</p> <p id="JCL-P-33">Linear regression will be covered in a subsequent tutorial in this series. Briefly, simple linear regression has only 1 independent variable (<em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">x</em>) and 1 dependent variable (<em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">y</em>). It fits a line through the data points of the scatter plot, which allows estimates of <em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">y</em> values from <em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">x</em> values.<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R23">23</a></sup></sup> However, the regression line itself provides no information about how strongly the variables are related. In contrast, a correlation does not fit such a line and does not allow such estimations, but it describes the strength of the relationship. The choice of a correlation or a linear regression thus depends on the research objective: strength of relationship versus estimation of <em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">y</em> values from <em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">x</em> values.</p> <p id="JCL-P-34">However, additional factors should be considered. In a Pearson correlation analysis, both variables are assumed to be normally distributed. The observed values of these variables are subject to natural random variation. In contrast, in linear regression, the values of the independent variable (<em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">x</em>) are considered known constants.<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R23">23</a></sup></sup> Therefore, a Pearson correlation analysis is conventionally applied when both variables are observed, while a linear regression is generally, but not exclusively, used when fixed values of the independent variable (<em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">x</em>) are chosen by the investigators in an experimental protocol.</p> <p id="JCL-P-35">To illustrate the difference, in the study by Nishimura et al,<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R1">1</a></sup></sup> the infused volume and the amount of leakage are observed variables. However, had the investigators chosen different infusion regimes to which they assigned patients (eg, 500, 1000, 1500, and 2000 mL), the independent variable would no longer be random, and a Pearson correlation analysis would have been inappropriate.</p> <h2 class="ejp-article-outline-heading" data-level="1" id="JCL5">SPEARMAN RANK CORRELATION</h2> <p id="JCL-P-36">In the previously mentioned study by Kim et al,<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R2">2</a></sup></sup> the scatter plot of OGFR expression and cell growth does not seem compatible with a bivariate normal distribution, and the relationship appears to be monotonic but nonlinear. Spearman rank correlation can be used for an analysis of the association between such data.<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R14">14</a></sup></sup></p> <p id="JCL-P-37">Basically, a Spearman coefficient is a Pearson correlation coefficient calculated with the ranks of the values of each of the 2 variables instead of their actual values (<a href="javascript:void(0)" onclick="javascript:showSlideShowByImageID('F2', '00000539-201805000-00050');">Figure 2</a>).<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R13">13</a></sup></sup> A Spearman coefficient is commonly abbreviated as ρ (rho) or “<em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">r</em><sub xmlns:mrws="http://webservices.ovid.com/mrws/1.0">s</sub>.” Because ordinal data can also be ranked, use of a Spearman coefficient is not restricted to continuous variables. By using ranks, the coefficient quantifies strictly monotonic relationships between 2 variables (ranking of the data converts a nonlinear strictly monotonic relationship to a linear relationship, see <a href="javascript:void(0)" onclick="javascript:showSlideShowByImageID('F2', '00000539-201805000-00050');">Figure 2</a>). Moreover, this property makes a Spearman coefficient relatively robust against outliers (<a href="javascript:void(0)" onclick="javascript:showSlideShowByImageID('F3', '00000539-201805000-00050');">Figure 3</a>).</p><section class="ejp-r-article-images"><figure class="ejp-r-article-images__figure"><a href="javascript:void(0)" class="ejp-r-article-images__image-link" onclick="showSlideShowByImageID('F2', '00000539-201805000-00050')"><img class="ejp-r-article-images__img js-lazy-load lazy-load" src="javascript:void(0);" data-src="https://images.journals.lww.com/anesthesia-analgesia/ArticleViewerPreview.00000539-201805000-00050.F2.jpeg" data-srcset="https://images.journals.lww.com/anesthesia-analgesia/ArticleViewerPreview@2.00000539-201805000-00050.F2.jpeg 2x" srcset="" alt="F2"></a><figcaption class="ejp-r-article-images__figcaption"><a href="javascript:void(0)" class="ejp-r-article-images__figcaption-link" onclick="showSlideShowByImageID('F2', '00000539-201805000-00050')">Figure 2.: </a><div class="ejp-r-article-images__figcaption-text">A, A strictly monotonic curve with a Pearson correlation coefficient (<em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">r</em>) of +0.84. Also in the left-side flat part, the curve is continuously slightly increasing. After ranking the values of both variables from lowest to highest, the ranks show a perfect linear relationship (B). Spearman rank correlation is Pearson correlation calculated with the data ranks instead of their actual values. Hence, Spearman coefficient (<em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">ρ</em>) of +1.0 in A corresponds to Pearson correlation of +1.0 in B.</div></figcaption></figure><figure class="ejp-r-article-images__figure"><a href="javascript:void(0)" class="ejp-r-article-images__image-link" onclick="showSlideShowByImageID('F3', '00000539-201805000-00050')"><img class="ejp-r-article-images__img js-lazy-load lazy-load" src="javascript:void(0);" data-src="https://images.journals.lww.com/anesthesia-analgesia/ArticleViewerPreview.00000539-201805000-00050.F3.jpeg" data-srcset="https://images.journals.lww.com/anesthesia-analgesia/ArticleViewerPreview@2.00000539-201805000-00050.F3.jpeg 2x" srcset="" alt="F3"></a><figcaption class="ejp-r-article-images__figcaption"><a href="javascript:void(0)" class="ejp-r-article-images__figcaption-link" onclick="showSlideShowByImageID('F3', '00000539-201805000-00050')">Figure 3.: </a><div class="ejp-r-article-images__figcaption-text">Constructed examples to illustrate that the relationship between data should also be assessed by visual inspection of plots, rather than relying only on correlation coefficients. A, A correlation coefficient close to 0 does not necessarily mean that the <em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">x</em> axis and the <em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">y</em> axis variable are not related. In fact, the graph suggests a strong quadratic relationship. B–D, Pearson correlation coefficient (<em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">r</em>) is +0.84, just as in <a href="javascript:void(0)" onclick="javascript:showSlideShowByImageID('F2', '00000539-201805000-00050');">Figure 2A</a>, yet the actual relationship between the data is quite different in each panel. Note that the outlier in B has a relevant influence on Pearson coefficient as excluding this extreme value yields a perfect linear relationship, whereas it has virtually no influence on Spearman coefficient (with the outlier, <em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">ρ</em> is very close to +1, without the outlier, it is exactly +1). C, The relationship is neither linear nor monotonic, and neither Pearson nor Spearman coefficient captures the sinusoid relationship. D, Sampled from a bivariate normal distribution.</div></figcaption></figure></section><p id="JCL-P-38">Analogous to Pearson coefficient, a Spearman coefficient also ranges from –1 to +1. It can be interpreted as describing anything between no association (<em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">ρ</em> = 0) to a perfect monotonic relationship (<em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">ρ</em> = –1 or +1). Analogous considerations as described above for a Pearson correlation also apply to the interpretation of confidence intervals and <em xmlns:mrws="http://webservices.ovid.com/mrws/1.0">P</em> values for a Spearman coefficient.</p> <h2 class="ejp-article-outline-heading" data-level="1" id="JCL6">PITFALLS AND MISINTERPRETATIONS</h2> <p id="JCL-P-39">Correlations are frequently misunderstood and misused.<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R4">4</a></sup></sup><sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0">,</sup><sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R5">5</a></sup></sup> It is important to note that an observed correlation (ie, association) does not assure that the relationship between 2 variables is causal. Ice cream sales increase as the temperature increases during summer, and so does the sales of fans. Hence, fan sales tend to increase along with ice cream sales, but this positive correlation does not justify the conclusion that eating ice cream causes people to buy fans. While the fallacy is easily detected in this example, it might be tempting to conclude that infusion of large amounts of crystalloid fluid causes fluid leakage into the interstitium. While it is certainly possible that a causal relationship exists, we would not be justified to conclude this based on a correlation analysis. The distinction between association and causation is discussed in detail in a previous tutorial.<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R24">24</a></sup></sup></p> <p id="JCL-P-40">Correlations also do not describe the strength of agreement between 2 variables (eg, the agreement between the readings from 2 measurement devices, diagnostic tests, or observers/raters).<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R25">25</a></sup></sup> Two variables can exhibit a high degree of correlation but can at the same time disagree substantially, for example if 1 technique measures consistently higher than the other.</p> <p id="JCL-P-41">Another misconception is that a correlation coefficient close to zero demonstrates that the variables are not related. Correlation should be used to describe a linear or monotonic association, but this does not exclude that researchers might deliberately or inadvertently misuse the correlation coefficient for relationships that are not adequately characterized by correlation analysis (eg, quadratic relationship as in <a href="javascript:void(0)" onclick="javascript:showSlideShowByImageID('F3', '00000539-201805000-00050');">Figure 3A</a>). Very different relationships can result in similar correlation coefficients (<a href="javascript:void(0)" onclick="javascript:showSlideShowByImageID('F2', '00000539-201805000-00050');">Figures 2A</a> and <a href="javascript:void(0)" onclick="javascript:showSlideShowByImageID('F3', '00000539-201805000-00050');">3B–D</a>). Therefore, researchers are well advised not only to rely on the correlation coefficient but also to plot the data for a visual inspection of the relationship.<sup xmlns:mrws="http://webservices.ovid.com/mrws/1.0"><sup><a class="ejp-citation-link js-ejp-citation-link" data-reference-links="R26">26</a></sup></sup> Graphing data are generally a good first step before performing any numerical analysis.</p> <h2 class="ejp-article-outline-heading" data-level="1" id="JCL7">CONCLUSIONS</h2> <p id="JCL-P-42">Correlation coefficients describe the strength and direction of an association between variables. A Pearson correlation is a measure of a linear association between 2 normally distributed random variables. A Spearman rank correlation describes the monotonic relationship between 2 variables. It is (1) useful for nonnormally distributed continuous data, (2) can be used for ordinal data, and (3) is relatively robust to outliers. Hypothesis tests are used to test the null hypothesis of no correlation, and confidence intervals provide a range of plausible values of the estimate.</p> <p id="JCL-P-43">Researchers should avoid inferring causation from correlation, and correlation is unsuited for analyses of agreement. Visual inspection of scatter plots is always advisable, as correlation fails to adequately describe nonlinear or nonmonotonic relationships, and different relationships between variables can result in similar correlation coefficients.</p> <h2 class="ejp-article-outline-heading" data-level="1" id="JCL8">DISCLOSURES</h2> <p id="JCL-P-44"><strong xmlns:mrws="http://webservices.ovid.com/mrws/1.0">Name:</strong> Patrick Schober, MD, PhD, MMedStat.</p> <p id="JCL-P-45"><strong xmlns:mrws="http://webservices.ovid.com/mrws/1.0">Contribution:</strong> This author helped write and revise the article.</p> <p id="JCL-P-46"><strong xmlns:mrws="http://webservices.ovid.com/mrws/1.0">Name:</strong> Christa Boer, PhD, MSc.</p> <p id="JCL-P-47"><strong xmlns:mrws="http://webservices.ovid.com/mrws/1.0">Contribution:</strong> This author helped write and revise the article.</p> <p id="JCL-P-48"><strong xmlns:mrws="http://webservices.ovid.com/mrws/1.0">Name:</strong> Lothar A. Schwarte, MD, PhD, MBA.</p> <p id="JCL-P-49"><strong xmlns:mrws="http://webservices.ovid.com/mrws/1.0">Contribution:</strong> This author helped write and revise the article.</p> <p id="JCL-P-50"><strong xmlns:mrws="http://webservices.ovid.com/mrws/1.0">This manuscript was handled by:</strong> Thomas R. Vetter, MD, MPH.</p> <p id="JCL-P-51"><strong xmlns:mrws="http://webservices.ovid.com/mrws/1.0">Acting EIC on final acceptance:</strong> Thomas R. Vetter, MD, MPH.</p></section><section id="article-references" class="article-references js-article-references"><h2 class="ejp-article-outline-heading" data-level="1" id="JCL-L-3">REFERENCES</h2> <div id="R1" class="article-references__item js-article-reference">1. Nishimura A, Tabuchi Y, Kikuchi M, Masuda R, Goto K, Iijima T. The amount of fluid given during surgery that leaks into the interstitium correlates with infused fluid volume and varies widely between patients. 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