Regression & Prediction 4/15/2019 Regression & Prediction

4/15/2019 Linear Regression Finding the best fitting straight line for a set of data This line is represented by the equation Y = bX + a, where a & b are fixed constants

Least Squares Regression 4/15/2019 Least Squares Regression Most commonly used regression line Makes the sum of the squared errors as small as possible Regression Line Equation Y=bX+a Y(hat) is the predicted value of Y given a certain X b is the slope a is the y-intercept

4/15/2019 Slope (b) Indicates by how much Y will change when X is increased by one point

Relationship of “b” to “r” 4/15/2019 Relationship of “b” to “r” When “r” is positive, “b” is positive When “r” is zero, “b” is zero When “r” is negative, “b” is negative

4/15/2019 Y-Intercept (a) a = MY - bMX Value of Y when X is equal to 0

How Good is Our Prediction - Standard Error of Estimate 4/15/2019 How Good is Our Prediction - Standard Error of Estimate Provides a measure of how accurately the regression equation predicts the Y values Gives a measure of the standard distance between a regression line and the actual data points Analogous to standard deviation.

Standard Error of Estimate - Formula 4/15/2019 Standard Error of Estimate - Formula

Another Formula for Standard Error of Estimate 4/15/2019 Another Formula for Standard Error of Estimate SSerror=(1-r2)SSY By using this formula for SSerror the standard error of estimate can also be computed as

4/15/2019 Example Exam 1 (X) Final Grades (Y) 62 74 73 93 88 68 82 79 85 91 77 72 94 96 65 61 92 98 95 A professor claims that the scores on the 1st exam provide an excellent indication of how students will perform throughout the term

Exam 1 (x) Final Grades (Y) X2 Y2 XY 62 74 3844 5476 4588 73 93 5329 8649 6789 88 68 7744 4624 5984 82 79 6724 6241 6478 85 91 7225 8281 7735 77 72 5929 5184 5544 94 96 8836 9216 9024 65 61 4225 3721 3965 92 8464 8372 6068 7905 98 95 9604 9025 9310 4/15/2019

4/15/2019 Calculate SP

4/15/2019 Calculate SSx And the SSx

4/15/2019 Calculate SSY And the SSy

4/15/2019 Calculate r

Calculate Regression Equation 4/15/2019 Calculate Regression Equation Y’ = bX+a a = MY - bMX

Standard Error of Estimate 4/15/2019 Exam 1 (x) Final Grades (Y) Predicted Final (Y-Y’) (Y-Y')2 62 74 70.35 3.65 13.3225 73 93 77.61 -15.39 236.8521 88 68 87.51 -19.51 380.6401 82 79 83.55 -4.55 20.7025 85 91 85.53 5.47 29.9209 77 72 80.25 -8.25 68.0625 94 96 91.47 4.53 20.5209 65 61 72.33 -11.33 128.3689 92 89.49 2.51 6.3001 78.27 3.73 13.9129 7.47 55.8009 98 95 94.11 0.89 0.7921 We know now that our predictions for y based on x will not be perfect, however we can calculate approximately how far off our predictions will be. That is, we need is is called the standard error of the estimate. Our regression equation is: Y’ = 0.66X+29.43 We can do this by finding the predicted value for each X value and then subtracting Y-hat from Y and squaring the deviation, and then dividing it by its degree of freedom to obtain a measure of variance and taking the square root

Standard Error of Estimate

Multiple Regression: Some Additional Points 4/15/2019 Multiple Regression: Some Additional Points If b1>b2, does that mean that X1 is a better predictor than X2?