Multiple Regression Error Variance
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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 multiple regression formula explained in a multiple regression model is usually less than the sum of multiple regression example the sums of squares in simple regression Define R2 in terms of proportion explained Test R2 for significance Test the multiple regression spss difference between a complete and reduced 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
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predicted from one predictor variable. In multiple regression, the 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 variance of error term in regression predicts University GPA (UGPA). That is, the problem is to find the 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 corre
not going to use total because it's just the sum of snatch and clean. Data The heaviest weights (in kg) that men who weigh more than multiple regression analysis 105 kg were able to lift are given in the table. Data Dictionary Age
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The age the competitor will be on their birthday in 2004. Body The weight (kg) of the competitor Snatch The
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maximum weight (kg) lifted during the three attempts at a snatch lift Clean The maximum weight (kg) lifted during the three attempts at a clean and jerk lift Total The total weight (kg) http://onlinestatbook.com/2/regression/multiple_regression.html lifted by the competitor Age Body Snatch Clean Total 26 163.0 210.0 262.5 472.5 30 140.7 205.0 250.0 455.0 22 161.3 207.5 240.0 447.5 27 118.4 200.0 240.0 440.0 23 125.1 195.0 242.5 437.5 31 140.4 190.0 240.0 430.0 32 158.9 192.5 237.5 430.0 22 136.9 202.5 225.0 427.5 32 145.3 187.5 232.5 420.0 27 124.3 190.0 225.0 415.0 20 142.7 185.0 220.0 405.0 29 127.7 170.0 215.0 https://people.richland.edu/james/ictcm/2004/multiple.html 385.0 23 134.3 160.0 210.0 370.0 18 137.7 155.0 192.5 347.5 Regression Model If there are k predictor variables, then the regression equation model is y = β0 + β1x1 + β2x2 + ... + βkxk + ε. The x1, x2, ..., xk represent the k predictor variables. Those parameters are the same as before, β0 is the y-intercept or constant, β1 is the coefficient on the first predictor variable, β2 is the coefficient on the second predictor variable, and so on. ε is the error term or the residual that can't be explained by the model. Those parameters are estimated by b0, b1, b2, ..., bk. This gives us a regression equation used for prediction of y = b0 + b1x1 + b2x2 + ...+ bkxk. Basically, everything we did with simple linear regression will just be extended to involve k predictor variables instead of just one. Regression Analysis Explained Round 1: All Predictor Variables Included Minitab was used to perform the regression analysis. This is not really something you want to try by hand. Response Variable: clean Predictor Variables: age, body, snatch Regression Equation The regression equation isclean = 32.9 + 1.03 age + 0.106 body + 0.
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