{"id":315,"date":"2021-03-23T09:05:50","date_gmt":"2021-03-23T09:05:50","guid":{"rendered":"https:\/\/bcisnotes.com\/fourthsemester\/?p=315"},"modified":"2021-03-31T05:58:47","modified_gmt":"2021-03-31T05:58:47","slug":"multiple-regression-analysis","status":"publish","type":"post","link":"https:\/\/bcisnotes.com\/fourthsemester\/multiple-regression-analysis\/","title":{"rendered":"Multiple Regression Analysis || Data Analysis and Modeling || BCIS NOTES"},"content":{"rendered":"

Multiple Regression Analysis<\/strong><\/h1>\n

Concept of Multiple Regression Analysis:<\/strong><\/h2>\n

In simple regression analysis, the regression model is fit between one dependent and one independent variable to estimate the dependent variable or to measure the effectiveness of an independent variable on the dependent variable. When the regression model is fit between one dependent and more the one independent variable then this regression model is called the multiple regression model. In multiple regression analysis, the effect of the different independent variables on the dependent variable can be measured simultaneously.<\/p>\n

Let \u2018y\u2019 be the dependent variable and x1<\/sub>, x2, <\/sub>x3<\/sub> \u2026\u2026\u2026\u2026\u2026 xk <\/sub>be the \u2018k\u2019 independent variables. Then the multiple regression model is defined as<\/p>\n

\"Multiple<\/p>\n

Where<\/p>\n

y = dependent variable.
\nx1<\/sub>, x2, <\/sub>x3<\/sub> \u2026\u2026\u2026\u2026\u2026 xk<\/sub>\u00a0 are independent variables.
\n\u03b20<\/sub> = y-intercept.
\n\u03b21<\/sub> = Slope of y with variable x1<\/sub> holding the remaining variables x2, <\/sub>x3<\/sub> \u2026,xk<\/sub> constant or Regression coefficient of y on x1<\/sub> holding the remaining variables x2, <\/sub>x3<\/sub> \u2026\u2026\u2026\u2026\u2026 xk <\/sub>constant.
\n\u03b22<\/sub> = Slope of y with variable x2<\/sub> holding the remaining variables x1, <\/sub>x3<\/sub> \u2026,xk<\/sub> constant or Regression coefficient of y on x2<\/sub> holding the remaining variables x1, <\/sub>x3<\/sub> \u2026\u2026\u2026\u2026\u2026 xk <\/sub>constant.
\n\u03b23<\/sub> = Slope of y with variable x3<\/sub> holding the remaining variables x1, <\/sub>x2, <\/sub>x4, <\/sub>\u00a0\u2026,xk<\/sub> constant or Regression coefficient of y on x3<\/sub> holding the remaining variables x1, <\/sub>x2, <\/sub>x4, <\/sub>\u00a0\u2026,xk<\/sub> constant.
\nAnd so on. Similarly,
\n\u03b2k<\/sub> = Slope of y with variable xk<\/sub> holding the remaining variables\u00a0 x1, <\/sub>x2, <\/sub>x3<\/sub> \u2026,xk-1<\/sub> constant or Regression coefficient of y on xk<\/sub> holding the remaining variables x1, <\/sub>x2, <\/sub>x3<\/sub> \u2026,xk-1<\/sub> constant.
\ne = error term or residual.<\/p>\n

Multiple Regression Model with Two Independent Variables:<\/strong><\/h2>\n

Let \u2018y\u2019 be the dependent variable and x1<\/sub> and x2<\/sub> are two independent variables. The multiple regression model with two independent variables is defined as,<\/p>\n

\"Multiple<\/p>\n

Where<\/p>\n

y = dependent variable.
\nx1<\/sub>and x2<\/sub> are independent variables.
\n\u03b20<\/sub> = y-intercept.
\n\u03b21<\/sub> = Slope of y with variable x1<\/sub> holding the remaining variable x2 <\/sub>constant or Regression coefficient of y on x1<\/sub> holding the remaining variable x2 <\/sub>constant.
\n\u03b22<\/sub> = Slope of y with variable x2<\/sub> holding the remaining variable x1 <\/sub>constant or Regression coefficient of y on x2<\/sub> holding the remaining variable x1<\/sub> constant.
\ne = error term or residual.<\/p>\n

To fit the regression model (2), we have to estimate the value of \u03b2<\/sub>0<\/sub>, \u03b2<\/sub>1,<\/sub> and \u03b2<\/sub>2<\/sub>. To estimate the value of these parameters we use the principle of lest square. The following three normal equations of equation (2) can be obtained using the method of least square.<\/p>\n

Now normal equations of equation (2) are<\/p>\n

\"Multiple<\/p>\n

On solving these above three normal equations we can estimate the value of \u03b2<\/sub>0<\/sub>, \u03b2<\/sub>1,<\/sub> and \u03b2<\/sub>2<\/sub>. Hence the fitted multiple regression model is<\/p>\n

\"Normal<\/p>\n

Where,
\nThe estimated value of the dependent variable for a given value of the independent variables.
\n\u03b2<\/sub>0 = y-intercept (or Estimated value of \u03b2<\/sub>0.)
\nb1 = Regression coefficients of y on x1 holding the effect of x2 constant (or Estimated
\nvalue of \u03b2<\/sub>1.)
\nb2 = Regression coefficients of y on x2 holding the effect of x1 constant (or Estimated
\nvalue of \u03b2<\/sub>2.)<\/p>\n

Interpreting the multiple regression coefficients:<\/h2>\n

Suppose we have the following multiple regression model<\/p>\n

\"\"<\/p>\n

    \n
  1. The y-intercept \u03b2<\/sub>0<\/sub> represents the average of the dependent variable when the value of independents variables are zero i.e. x1<\/sub> = x2<\/sub> =0. for example<\/li>\n<\/ol>\n

    Here \u03b2<\/sub>0<\/sub> = 20, this means, the average value of dependent variable is 20 when x1<\/sub> = x2<\/sub> =0<\/p>\n

    2. The multiple regression coefficients b1<\/sub> measure the average rate of increased or decreased in the value of a dependent variable (y) while\u00a0 \u00a0 \u00a0 \u00a0 \u00a0increasing the value of independent variable \u2018x1<\/sub>\u2019 by unit, by keeping the effect of another independent variable \u2018x2<\/sub>\u2019 constant. For example,<\/p>\n

    Here, \u03b2<\/sub>1<\/sub> = 2.5, this means, the value of a dependent variable (y) is increased by 2.5 when the value of an independent variable (x1<\/sub>) is\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0increased by 1, by keeping the effect of x2<\/sub> constant.<\/p>\n

    3. The multiple regression coefficients \u03b2<\/sub>2<\/sub> measure the average rate of increased or decreased in the value of a dependent variable (y)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0while increasing the value of independent variable \u2018x2<\/sub>\u2019 by unit, by keeping the effect of other independent variables \u2018x1<\/sub>\u2019 constant. For\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0example,<\/p>\n

    Here, \u03b2<\/sub>2<\/sub> = -3.8, this means, the value of the dependent variable (y) is decreased by 3.8 when the value of an independent variable (x2<\/sub>) is increased by 1, by keeping the effect of x1<\/sub> constant.<\/p>\n

    Error term or Residual:<\/h2>\n

    The difference between the observed and estimated value of the dependent variable (y) is called error or residual and it is denoted by \u2018e\u2019<\/p>\n

    Where<\/p>\n

    \"Error<\/p>\n

    e = Error term<\/p>\n

    y= Observed value of the dependent variable.<\/p>\n

    \"Error= Estimated value of the dependent variable for a given value of a set of independent variables.<\/p>\n

     <\/p>\n

    You may also like:\u00a0Confidence Interval Estimate<\/a><\/p>\n

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

    Multiple Regression Analysis Concept of Multiple Regression Analysis: In simple regression analysis, the regression model is fit between one dependent and one independent variable to […]<\/a><\/p>\n<\/div>","protected":false},"author":4,"featured_media":335,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[137,139,138,136],"yoast_head":"\nMultiple Regression Analysis || Data Analysis and Modeling || BCIS NOTES<\/title>\n<meta name=\"description\" content=\"In simple regression analysis, the regression model is fit between one dependent and one independent variable to estimate the dependent variable or to measure the effectiveness of an independent variable on the dependent variable.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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