Formula For Standard Error Of Multiple Regression
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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 regression with two independent variables in excel interrelationships among all the variables must be taken into account in the weights assigned to the
Multiple Regression Example Problems
variables. The interpretation of the results of a multiple regression analysis is also more complex for the same reason. With two independent variables multiple regression equation example the prediction 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 in the previous chapter except that the w's multiple regression equation with 3 variables 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 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
Standard Error Of The Regression
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 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 120the 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 of the estimate based on a sample Figure 1 shows two regression examples. You can see that standard error of the estimate in Graph A, the points are closer to the line than they are in Graph multiple correlation coefficient formula B. Therefore, the predictions in Graph A are more accurate than in Graph B. Figure 1. Regressions differing in accuracy of prediction. The standard
How To Calculate Multiple Regression By Hand
error of the estimate is 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 http://www.psychstat.missouristate.edu/multibook/mlt06m.html 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, 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 http://onlinestatbook.com/2/regression/accuracy.html 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 is For the data in Table 1, μy = 2.06, SSY = 4.597 and ρ= 0.6268. Therefore, which is the same value computed previously. Similar formulas are used when the standard error of the estimate is computed from a sample rather than a population. The only difference is that the denominator is N-2 rather than N. The reason N-2 is used rather than N-1 is that two parameters (the slope and the intercept) were estimated in order to estimate the sum of squares. Formulas for a sample comparable to the ones for a population are shown below. Please answer the que
i = 1, 2, . . . , n where ui are values of an unobserved error term, u, and. the unknown parameters are constants. http://www.unesco.org/webworld/idams/advguide/Chapt5_2.htm Assumptions The error terms ui are mutually independent and identically distributed, with mean = 0 and http://faculty.cas.usf.edu/mbrannick/regression/Reg2IV.html constant variances E [ui] = 0 V [ui] = This is so, because the observations y1, y2, . . . ,yn are a random sample, they are mutually 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 multiple regression 0, b 1, b 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 standard error of as follows. We can visualize that n 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
ways, that is, using two distinct formulas. Explain the formulas. What happens to b weights if we add new variables to the regression equation that are highly correlated with ones already in the equation? Why do we report beta weights (standardized b weights)? Write a regression equation with beta weights in it. What are the three factors that influence the standard error of the b weight? How is it possible to have a significant R-square and non-significant b weights? Materials The Regression Line With one independent variable, we may write the regression equation as: Where Y is an observed score on the dependent variable, a is the intercept, b is the slope, X is the observed score on the independent variable, and e is an error or residual. We can extend this to any number of independent variables: (3.1) Note that we have k independent variables and a slope for each. We still have one error and one intercept. Again we want to choose the estimates of a and b so as to minimize the sum of squared errors of prediction. The prediction equation is: (3.2) Finding the values of b is tricky for k>2 independent variables, and will be developed after some matrix algebra. It's simpler for k=2 IVs, which we will discuss here. For the one variable case, the calculation of b and a was: For the two variable case: and At this point, you should notice that all the terms from the one variable case appear in the two variable case. In the two variable case, the other X variable also appears in the equation. For example, X2 appears in the equation for b1. Note that terms corresponding to the variance of both X variables occur in the slopes. Also note that a term corresponding to the covariance of X1 and X2 (sum of deviation cross-products) also appears in the formula for the slope. The equation for a with two independent variables is: This equation is a straight-forward generalization of the case for one independent variable. A Numerical Example Suppose we want to predict job performance of Chevy mechanics based on mechanical aptitude test scores and test scores from personality test that measures conscientiousness. Job Perf Mech Apt Consc Y X1 X2 X1*Y X2*Y X1*X2 1 40 25 40 25 1000 2 45 20 90 40 900 1 38 30 38 30 1140 3 50 30 150 90 1500 2 48 28 96 56 1344 3 55 30 165 90 1650 3 53 34 159 102 1802 4 55 36 220 144 1980 4 58 32 232 128 1856 3 40 34 120 102 1360 5 55 38 275 190 2090 3