Multiple Regression Standard Error Formula
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is used to predict a single dependent variable (Y). The predicted value of Y is a linear transformation of the X variables such that the sum of squared deviations of the observed and predicted Y is a minimum. The computations are more complex, however, because the interrelationships among all the variables must be taken into account in the weights assigned to the variables. The interpretation of the regression with two independent variables in excel results of a multiple regression analysis is also more complex for the same reason. With two independent variables the prediction
Multiple Regression Example Problems
of Y is expressed by the following equation: Y'i = b0 + b1X1i + b2X2i Note that this transformation is similar to the linear transformation of two variables discussed multiple regression equation example in the previous chapter except that the w's have been replaced with b's and the X'i has been replaced with a Y'i. The "b" values are called regression weights and are computed in a way that minimizes the sum of squared deviations in the same manner as in standard error multiple regression simple linear regression. The difference is that in simple linear regression only two weights, the intercept (b0) and slope (b1), were estimated, while in this case, three weights (b0, b1, and b2) are estimated. EXAMPLE DATA The data used to illustrate the inner workings of multiple regression will be generated from the "Example Student." The data are presented below: Homework Assignment 21 Example Student PSY645 Dr. Stockburger Due Date
Y1 Y2 X1 X2 X3 X4 125 113 13 18 25 11 158 115 39Multiple Regression Equation With 3 Variables
18 59 30 207 126 52 50 62 53 182 119 29 43 50 29 196 107 50 37 65 56 175 135 64 19 79 49 145 111 11 27 17 14 144 130 22 23 31 17 160 122 30 18 34 22 175 114 51 11 58 40 151 121 27 15 29 31 161 105 41 22 53 39 200 131 51 52 75 36 173 123 37 36 44 27 175 121 23 48 27 20 162 120 43 15 65 36 155 109 38 19 62 37 230 130 62 56 75 50 162 134 28 30 36 20 153 124 30 25 41 33 The example data can be obtained as a text file and as an SPSS/WIN file from this web page. If a student desires a more concrete description of this data file, meaning could be given the variables as follows: Y1 - A measure of success in graduate school. X1 - A measure of intellectual ability. X2 - A measure of "work ethic." X3 - A second measure of intellectu
i = 1, 2, . . . , n where ui are values of an unobserved error term, u, and. the unknown parameters are constants. Assumptions The multiple correlation coefficient formula error terms ui are mutually independent and identically distributed, with mean = 0 and constant variances E how to calculate multiple regression by hand [ui] = 0 V [ui] = This is so, because the observations y1, y2, . . . ,yn are a random sample, they are mutually
Multiple Correlation Coefficient In R
independent and hence the error terms are also mutually independent The distribution of the error term is independent of the joint distribution of x i, x 2, . . . , x p The unknown parameters b 0, b 1, b http://www.psychstat.missouristate.edu/multibook/mlt06m.html 2, . . . ,b p are constants. Equations relating the n observations can be written as: The parameters b 0, b 1, . . . b p can be estimated using the least squares procedure, which minimizes the sum of squares of errors. Minimizing the sum of squares leads to the following equations, from which the values of b can be computed: Geometrical Representation The problem of multiple regression can be geometrically represented as follows. We can visualize that n http://www.unesco.org/webworld/idams/advguide/Chapt5_2.htm observations (xi1, xi2, …..xip, yi) i = 1, 2, ….n are represented as points in a (p+1) - dimensional space. The regression problem is to determine the possible hyper-planes in the p - dimensional space, which will be the best- fit. We use the least squares criterion and locate the hyper-plane that minimizes the sum of squares of the errors, i.e., the distances from the points around the plane (observations) and the point on the plane. (i.e. the estimate ŷ). ŷ = a+b1x1+b2x2+…+bpxp Standard error of the estimate Se = where yi = the sample value of the dependent variable ŷi = corresponding value estimated from the regression equation n = number observations p = number of predictors or independent variable The denominator of the equation indicates that in multiple regression with p independent variables, the standard error has n-p-1 degrees of freedom. This happens because the degrees of freedom are reduced from n by p+1 numerical constants a, b1, b2, …..bp, that have been estimated from the sample. Fit of the regression model The fit of the multiple regression model can be assessed by the Coefficient of Multiple determination, which is a fraction that represents the proportion of total variation of y that is explained by the regression plane. Sum of squares due to error SSE = Sum of squares due to regression SSR = Total sum of squares SST = Obviously, SST = SSR + SSE The ratio SSR/SST represents the proportion of the
Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might http://stats.stackexchange.com/questions/27916/standard-errors-for-multiple-regression-coefficients have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company Business Learn more about hiring developers or posting ads with us Cross Validated Questions Tags Users Badges Unanswered Ask Question _ Cross Validated is a question and answer site for people interested in statistics, machine learning, data multiple regression analysis, data mining, and data visualization. Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Standard errors for multiple regression coefficients? up vote 7 down vote favorite 3 I realize that this is multiple regression equation a very basic question, but I can't find an answer anywhere. I'm computing regression coefficients using either the normal equations or QR decomposition. How can I compute standard errors for each coefficient? I usually think of standard errors as being computed as: $SE_\bar{x}\ = \frac{\sigma_{\bar x}}{\sqrt{n}}$ What is $\sigma_{\bar x}$ for each coefficient? What is the most efficient way to compute this in the context of OLS? standard-error regression-coefficients share|improve this question asked May 7 '12 at 1:21 Belmont 4083613 add a comment| 1 Answer 1 active oldest votes up vote 12 down vote When doing least squares estimation (assuming a normal random component) the regression parameter estimates are normally distributed with mean equal to the true regression parameter and covariance matrix $\Sigma = s^2\cdot(X^TX)^{-1}$ where $s^2$ is the residual variance and $X^TX$ is the design matrix. $X^T$ is the transpose of $X$ and $X$ is defined by the model equation $Y=X\beta+\epsilon$ with $\beta$ the regression parameters and $\epsilon$ is the error term. The estimated standard
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