Multiple Regression Standard Error Of Estimate
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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 results regression with two independent variables in excel of a multiple regression analysis is also more complex for the same reason. With two independent variables the prediction of multiple regression example problems 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 in the
Multiple Regression Equation Example
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 simple linear
Multiple Regression Equation With 3 Variables
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 39 18 how to calculate multiple regression by hand 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 intellectual ability. X4 - A measuthe estimate from a scatter plot Compute the standard error of the estimate based on errors of prediction Compute the standard error using Pearson's correlation Estimate the standard error
Multiple Correlation Coefficient Formula
of the estimate based on a sample Figure 1 shows two regression multiple correlation coefficient in r examples. You can see that in Graph A, the points are closer to the line than they are multiple correlation example in Graph B. Therefore, the predictions in Graph A are more accurate than in Graph B. Figure 1. Regressions differing in accuracy of prediction. The standard error of the estimate is http://www.psychstat.missouristate.edu/multibook/mlt06m.html a measure of the accuracy of predictions. Recall that the regression line is the line that minimizes the sum of squared deviations of prediction (also called the sum of squares error). The standard error of the estimate is closely related to this quantity and is defined below: where σest is the standard error of the estimate, Y is an actual score, http://onlinestatbook.com/lms/regression/accuracy.html Y' is a predicted score, and N is the number of pairs of scores. The numerator is the sum of squared differences between the actual scores and the predicted scores. Note the similarity of the formula for σest to the formula for σ.  It turns out that σest is the standard deviation of the errors of prediction (each Y - Y' is an error of prediction). Assume the data in Table 1 are the data from a population of five X, Y pairs. Table 1. Example data. X Y Y' Y-Y' (Y-Y')2 1.00 1.00 1.210 -0.210 0.044 2.00 2.00 1.635 0.365 0.133 3.00 1.30 2.060 -0.760 0.578 4.00 3.75 2.485 1.265 1.600 5.00 2.25 2.910 -0.660 0.436 Sum 15.00 10.30 10.30 0.000 2.791 The last column shows that the sum of the squared errors of prediction is 2.791. Therefore, the standard error of the estimate is There is a version of the formula for the standard error in terms of Pearson's correlation: where ρ is the population value of Pearson's correlation and SSY
Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site http://stats.stackexchange.com/questions/27916/standard-errors-for-multiple-regression-coefficients About Us Learn more about Stack Overflow the company Business Learn more about http://people.duke.edu/~rnau/regnotes.htm 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 analysis, data mining, and data visualization. Join them; it only takes a minute: Sign up Here's how it works: Anybody multiple regression 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 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 multiple regression equation 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 deviation of a beta parameter is gotten by taking the corresponding term in $(X^TX)^{-1}$ multiplying it by the sample estimate of the residual variance and then taking the square root. This is not a very simple calculation but any software package will compute it for you and provide it in the output. Example On page 134 of Draper and Smith (referenced in
1: descriptive analysis · Beer sales vs. price, part 2: fitting a simple model · Beer sales vs. price, part 3: transformations of variables · Beer sales vs. price, part 4: additional predictors · NC natural gas consumption vs. temperature What to look for in regression output What's a good value for R-squared? What's the bottom line? How to compare models Testing the assumptions of linear regression Additional notes on regression analysis Stepwise and all-possible-regressions Excel file with simple regression formulas Excel file with regression formulas in matrix form If you are a PC Excel user, you must check this out: RegressIt: free Excel add-in for linear regression and multivariate data analysis Additional notes on linear regression analysis To include or not to include the CONSTANT? Interpreting STANDARD ERRORS, "t" STATISTICS, and SIGNIFICANCE LEVELS of coefficients Interpreting the F-RATIO Interpreting measures of multicollinearity: CORRELATIONS AMONG COEFFICIENT ESTIMATES and VARIANCE INFLATION FACTORS Interpreting CONFIDENCE INTERVALS TYPES of confidence intervals Dealing with OUTLIERS Caution: MISSING VALUES may cause variations in SAMPLE SIZE MULTIPLICATIVE regression models and the LOGARITHM transformation To include or not to include the CONSTANT? Most multiple regression models include a constant term (i.e., an "intercept"), since this ensures that the model will be unbiased--i.e., the mean of the residuals will be exactly zero. (The coefficients in a regression model are estimated by least squares--i.e., minimizing the mean squared error. Now, the mean squared error is equal to the variance of the errors plus the square of their mean: this is a mathematical identity. Changing the value of the constant in the model changes the mean of the errors but doesn't affect the variance. Hence, if the sum of squared errors is to be minimized, the constant must be chosen such that the mean of the errors is zero.) In a simple regression model, the