Multiple Regression Standard Error Of The Regression Coefficient
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Define "regression coefficient" Define "beta weight" Explain what R is and how it is related to r Explain why a regression weight is called a "partial slope" Explain why the sum of squares explained in a multiple
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regression model is usually less than the sum of the sums of squares in standard error of multiple regression coefficient formula simple regression Define R2 in terms of proportion explained Test R2 for significance Test the difference between a complete and reduced
Multiple Regression Standard Error Formula
model for significance State the assumptions of multiple regression and specify which aspects of the analysis require assumptions In simple linear regression, a criterion variable is predicted from one predictor variable. In multiple regression, the multiple regression example criterion is predicted by two or more variables. For example, in the SAT case study, you might want to predict a student's university grade point average on the basis of their High-School GPA (HSGPA) and their total SAT score (verbal + math). The basic idea is to find a linear combination of HSGPA and SAT that best predicts University GPA (UGPA). That is, the problem is to find the multiple regression equation values of b1 and b2 in the equation shown below that give the best predictions of UGPA. As in the case of simple linear regression, we define the best predictions as the predictions that minimize the squared errors of prediction. UGPA' = b1HSGPA + b2SAT + A where UGPA' is the predicted value of University GPA and A is a constant. For these data, the best prediction equation is shown below: UGPA' = 0.541 x HSGPA + 0.008 x SAT + 0.540 In other words, to compute the prediction of a student's University GPA, you add up (a) their High-School GPA multiplied by 0.541, (b) their SAT multiplied by 0.008, and (c) 0.540. Table 1 shows the data and predictions for the first five students in the dataset. Table 1. Data and Predictions. HSGPA SAT UGPA' 3.45 1232 3.38 2.78 1070 2.89 2.52 1086 2.76 3.67 1287 3.55 3.24 1130 3.19 The values of b (b1 and b2) are sometimes called "regression coefficients" and sometimes called "regression weights." These two terms are synonymous. The multiple correlation (R) is equal to the correlation between the predicted scores and the actual scores. In this example, it is the correlation between UGPA' and UGPA, which turns out to be 0.7
it comes to determining how well a linear model fits the data. However, I've stated previously that R-squared is overrated. Is there
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a different goodness-of-fit statistic that can be more helpful? You bet! Today, standard error of regression coefficient I’ll highlight a sorely underappreciated regression statistic: S, or the standard error of the regression. S provides
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important information that R-squared does not. What is the Standard Error of the Regression (S)? S becomes smaller when the data points are closer to the line. In the regression http://onlinestatbook.com/2/regression/multiple_regression.html output for Minitab statistical software, you can find S in the Summary of Model section, right next to R-squared. Both statistics provide an overall measure of how well the model fits the data. S is known both as the standard error of the regression and as the standard error of the estimate. S represents the average distance that the http://blog.minitab.com/blog/adventures-in-statistics/regression-analysis-how-to-interpret-s-the-standard-error-of-the-regression observed values fall from the regression line. Conveniently, it tells you how wrong the regression model is on average using the units of the response variable. Smaller values are better because it indicates that the observations are closer to the fitted line. The fitted line plot shown above is from my post where I use BMI to predict body fat percentage. S is 3.53399, which tells us that the average distance of the data points from the fitted line is about 3.5% body fat. Unlike R-squared, you can use the standard error of the regression to assess the precision of the predictions. Approximately 95% of the observations should fall within plus/minus 2*standard error of the regression from the regression line, which is also a quick approximation of a 95% prediction interval. For the BMI example, about 95% of the observations should fall within plus/minus 7% of the fitted line, which is a close match for the prediction interval. Why I Like the Standard Error of the Regression (S) In many cases, I prefer the
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 http://www.psychstat.missouristate.edu/multibook/mlt06m.html the weights assigned to the variables. The interpretation of the results of a multiple regression analysis is also more complex for the same reason. With two independent variables 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 have been replaced with b's and the X'i has been replaced with a Y'i. The "b" values are called multiple regression 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 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 regression standard error 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 20 162 120 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 obbe down. Please try the request again. Your cache administrator is webmaster. Generated Fri, 21 Oct 2016 01:04:36 GMT by s_wx1011 (squid/3.5.20)