{"id":219491,"date":"2025-05-26T08:37:51","date_gmt":"2025-05-26T08:37:51","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=219491"},"modified":"2025-05-26T08:37:53","modified_gmt":"2025-05-26T08:37:53","slug":"consider-the-simple-linear-regression-model","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/26\/consider-the-simple-linear-regression-model\/","title":{"rendered":"Consider the simple linear regression model"},"content":{"rendered":"\n

Consider the simple linear regression model Y =B0 + B1x + e suppose that the analyst wants to use z = x \u2013 x as the regress or variable.
(a) Using the data in Exercise 11-12, construct one scatter plot of the (xi, yi) points and then another of the (zi = xi \u2013 x, y) points. Use the two plots to intuitively explain how the two models, Y = B0 + Bix + E and, are related.
(b) Find the least squares estimates of B0 and B<\/em>I and in the model Y = B0 + B<\/em>Iz + E. How do they relate to the least squares estimates and B0 and B1?<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

To address this question properly, we\u2019ll break it into the two parts: (a)<\/strong> visual and intuitive explanation, and (b)<\/strong> algebraic derivation and relationship of coefficients.<\/p>\n\n\n\n

Let\u2019s first clarify the two models:<\/p>\n\n\n\n

    \n
  1. Original regression model:
    Y=\u03b20+\u03b21x+\u03b5Y = \\beta_0 + \\beta_1 x + \\varepsilon<\/li>\n\n\n\n
  2. Centered variable model (where z=x\u2212x\u02c9z = x – \\bar{x}):
    Y=\u03b20\u2217+\u03b21\u2217z+\u03b5Y = \\beta_0^* + \\beta_1^* z + \\varepsilon<\/li>\n<\/ol>\n\n\n\n
    \n\n\n\n

    (a) Scatter Plots and Intuitive Explanation<\/h3>\n\n\n\n

    The variable z=x\u2212x\u02c9z = x – \\bar{x} is simply a centered version of xx, meaning we shift the x-values so that their mean is zero. This transformation does not<\/strong> change the shape<\/strong> or slope<\/strong> of the scatter plot; it just repositions the data horizontally.<\/p>\n\n\n\n

      \n
    • The scatter plot of (xi,yi)(x_i, y_i) shows the original data and its linear relationship.<\/li>\n\n\n\n
    • The scatter plot of (zi,yi)=(xi\u2212x\u02c9,yi)(z_i, y_i) = (x_i – \\bar{x}, y_i) is identical in shape but shifted so the x-axis is centered around zero.<\/li>\n<\/ul>\n\n\n\n

      Intuition:<\/strong> Since we’re just shifting the x-axis, the slope \u03b21\\beta_1 of the line doesn’t change. However, because the x-values now average to zero, the intercept of the new model \u03b20\u2217\\beta_0^* is the value of YY when x=x\u02c9x = \\bar{x}. This simplifies interpretation and calculations.<\/p>\n\n\n\n

      \n\n\n\n

      (b) Least Squares Estimates and Relationship<\/h3>\n\n\n\n

      From the original model:<\/p>\n\n\n\n

        \n
      • \u03b2^1=\u2211(xi\u2212x\u02c9)(yi\u2212y\u02c9)\u2211(xi\u2212x\u02c9)2\\hat{\\beta}_1 = \\frac{\\sum (x_i – \\bar{x})(y_i – \\bar{y})}{\\sum (x_i – \\bar{x})^2}<\/li>\n\n\n\n
      • \u03b2^0=y\u02c9\u2212\u03b2^1x\u02c9\\hat{\\beta}_0 = \\bar{y} – \\hat{\\beta}_1 \\bar{x}<\/li>\n<\/ul>\n\n\n\n

        In the transformed model with zi=xi\u2212x\u02c9z_i = x_i – \\bar{x}:<\/p>\n\n\n\n

          \n
        • \u03b2^1\u2217=\u03b2^1\\hat{\\beta}_1^* = \\hat{\\beta}_1 (the slope remains the same)<\/li>\n\n\n\n
        • \u03b2^0\u2217=y\u02c9\\hat{\\beta}_0^* = \\bar{y} (since the mean of zz is 0)<\/li>\n<\/ul>\n\n\n\n

          Explanation<\/h3>\n\n\n\n

          In simple linear regression, centering the independent variable xx by subtracting its mean (z=x\u2212x\u02c9z = x – \\bar{x}) simplifies the regression model without altering its predictive power. This transformation merely redefines the horizontal axis to be centered at the average of xx, which enhances interpretability.<\/p>\n\n\n\n

          The slope of the regression line, \u03b2^1\\hat{\\beta}_1, represents the rate of change in YY for a unit change in xx. This rate remains unchanged in the centered model because the relationship between xx and yy is linear and invariant under shifting. Therefore, \u03b2^1\u2217=\u03b2^1\\hat{\\beta}_1^* = \\hat{\\beta}_1.<\/p>\n\n\n\n

          However, the intercept does change. In the original model, \u03b2^0\\hat{\\beta}_0 is the predicted value of YY when x=0x = 0, which may lie outside the data range, making interpretation less meaningful. In the centered model, the intercept \u03b2^0\u2217\\hat{\\beta}_0^* equals the mean of YY, y\u02c9\\bar{y}, because z=0z = 0 when x=x\u02c9x = \\bar{x}. This simplifies the model and interpretation: the intercept now represents the average response at the average xx.<\/p>\n\n\n\n

          In practical terms, centering helps reduce multicollinearity in multiple regression and can improve numerical stability. It doesn\u2019t change predictions or goodness-of-fit; it only repositions the model\u2019s baseline to be more meaningful and interpretable.<\/p>\n\n\n\n

          Thus, the relationship is:<\/p>\n\n\n\n

            \n
          • \u03b2^1\u2217=\u03b2^1\\hat{\\beta}_1^* = \\hat{\\beta}_1<\/li>\n\n\n\n
          • \u03b2^0\u2217=\u03b2^0+\u03b2^1x\u02c9=y\u02c9\\hat{\\beta}_0^* = \\hat{\\beta}_0 + \\hat{\\beta}_1 \\bar{x} = \\bar{y}<\/li>\n<\/ul>\n\n\n\n

            This shows that centering leads to a more intuitive intercept while preserving the slope.<\/p>\n\n\n\n

            \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

            Consider the simple linear regression model Y =B0 + B1x + e suppose that the analyst wants to use z = x \u2013 x as the regress or variable.(a) Using the data in Exercise 11-12, construct one scatter plot of the (xi, yi) points and then another of the (zi = xi \u2013 x, y) […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-219491","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/219491","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=219491"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/219491\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=219491"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=219491"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=219491"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}