Error Prediction 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 interrelationships among all the variables must be taken into account in multiple regression prediction in r the weights assigned to the variables. The interpretation of the results of a multiple regression analysis is also multiple regression prediction interval more complex for the same reason. With two independent variables the prediction of Y is expressed by the following equation: Y'i = b0 + b1X1i + multiple regression prediction interval excel 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 regression
Multiple Regression Prediction Equation
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 PSY645 prediction interval multiple regression formula 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 obtained as a tthe 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 prediction interval for multiple regression calculator in Graph A, the points are closer to the line than they are in Graph
Error Prediction Linear Regression
B. Therefore, the predictions in Graph A are more accurate than in Graph B. Figure 1. Regressions differing in accuracy of prediction. The standard
Logistic Regression Prediction
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 ques
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 regression model is usually less than the sum of http://onlinestatbook.com/2/regression/multiple_regression.html the sums of squares in simple regression Define R2 in terms of proportion explained Test R2 for significance Test the 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 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 multiple regression 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 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 multiple regression prediction 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 0.79. That is, R = 0.79. Note that R will never be negative since if there are negative correlations between the predictor variables and the criterion, the regression weights will be negative so that the correlation between the predicted and actual scores will be positive. Interpretation of Regression Coefficients A regression coefficient in mul