{"id":1400,"date":"2015-02-16T15:33:13","date_gmt":"2015-02-16T21:33:13","guid":{"rendered":"http:\/\/www.ssc.wisc.edu\/~jfrees\/?page_id=1400"},"modified":"2015-08-21T13:41:16","modified_gmt":"2015-08-21T18:41:16","slug":"3-1-method-of-least-squares","status":"publish","type":"page","link":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/regression\/chapter-3-multiple-linear-regression\/3-1-method-of-least-squares\/","title":{"rendered":"3.1 Method of Least Squares"},"content":{"rendered":"<p><div class=\"scbb-content-box scbb-content-box-gray\">In this section, you learn how to: \n<ul>\n<li>Interpret correlation coefficients by visualizing a scatterplot matrix<\/li>\n<li>Fit a plane to data using the method of least squares and calculus<\/li>\n<li>Predict an observation using a least squares fitted plane<\/li>\n<\/ul>\n<h2 style=\"text-align: center\"><a href=\"http:\/\/flash.bus.wisc.edu\/data\/act_sci\/Frees\/Regression2015\/Chapter3\/Part1\/MethodLeastSquares.html\"><span class=\"nolink\">Video Overview of the Section <\/span><\/a><a href=\"http:\/\/flash.bus.wisc.edu\/data\/act_sci\/Frees\/Regression2015\/Chapter3\/Part1\/MethodLeastSquares.mp4\"><span class=\"nolink\">(<em>Alternative .mp4 Version &#8211; 10:59 min<\/em>)<\/span><\/a><br \/>\n<\/h2>\n<p><\/p><\/div><br \/>\nConsider data sets where there are <em>k<\/em> explanatory variables and one dependent variable in a sample of size <em>n<\/em>. That is, the data consist of:<br \/>\n$$<br \/>\n\\left\\{<br \/>\n\\begin{eqnarray*}<br \/>\nx_{11},x_{12},\\ldots,x_{1k},y_1 \\\\<br \/>\nx_{21},x_{22},\\ldots,x_{2k},y_2 \\\\<br \/>\n\\vdots~~~~~~~~~~~~~~~~ \\\\<br \/>\nx_{n1},x_{n2},\\ldots,x_{nk},y_n \\\\<br \/>\n\\end{eqnarray*}<br \/>\n\\right\\} .<br \/>\n$$ The <em>i<\/em>th observation corresponds to the <em>i<\/em>th row, consisting of \\( (x_{i1},x_{i2},\\ldots,x_{ik},y_i) \\). For this general case, we take <em>k<\/em>+1 measurements on each entity.<\/p>\n<p>To provide some context, the next page introduces an example that will be referenced throughout this chapter.<br \/>\n<div class=\"alignleft\"><a href=\"https:\/\/users.ssc.wisc.edu\/~ewfrees\/regression\/chapter-3-multiple-linear-regression\/\" title=\"Chapter 3. Multiple Linear Regression\">&#9668 Previous page<\/a><\/div><div class=\"alignright\"><a href=\"https:\/\/users.ssc.wisc.edu\/~ewfrees\/regression\/chapter-3-multiple-linear-regression\/3-1-method-of-least-squares\/example-term-life-insurance\/\" title=\"Protected: Example: Term Life Insurance\">Next page &#9658<\/a><\/div><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Consider data sets where there are k explanatory variables and one dependent variable in a sample of size n. That is, the data consist of: $$ \\left\\{ \\begin{eqnarray*} x_{11},x_{12},\\ldots,x_{1k},y_1 \\\\ x_{21},x_{22},\\ldots,x_{2k},y_2 \\\\ \\vdots~~~~~~~~~~~~~~~~ \\\\ x_{n1},x_{n2},\\ldots,x_{nk},y_n &hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":1395,"menu_order":1,"comment_status":"closed","ping_status":"closed","template":"","meta":{"jetpack_post_was_ever_published":false},"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/P8cLPd-mA","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/1400"}],"collection":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/comments?post=1400"}],"version-history":[{"count":19,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/1400\/revisions"}],"predecessor-version":[{"id":5030,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/1400\/revisions\/5030"}],"up":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/1395"}],"wp:attachment":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/media?parent=1400"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}