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Stat 1301 More on Regression

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Outline of Lecture 1. Regression Effect and Regression Fallacy 2. Regression Line as Least Squares Line 3. Extrapolation 4. Multiple Regression

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1. Regression Effect and Regression Fallacy

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Hypothetical Grades for the First 2 Tests in a Class of STAT 1301 Hypothetical Grades for the First 2 Tests in a Class of STAT 1301 AVG x = 75SD x = 10 (Test 1) AVG y = 75SD y = 10 (Test 2) r = 0.7 Test - Retest Situation

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Predict the score on Test 2 for a student whose Test 1 score was... Predict the score on Test 2 for a student whose Test 1 score was... (a) 95 (a) 95 (b) 60 Regression Line: Y =.7X + 22.5 ^

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l Test-retest situation: - Bottom group on Test 1 does better on Test 2 - Top group on Test 1 falls back on Test 2 The Regression Fallacy l attributing the regression effect to something besides natural spread around the line. The Regression Effect

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Regression Effect - Explanation Students scoring 95 on Test 1 3 categories (a) Students who will average 95 for the course (a) Students who will average 95 for the course (b) Great students having a bad day (c) “Pretty good” students having a good day - There are more students in category (c) than in (b) - Thus, we expect the “average” performance for those who scored 95 on Test 1 to drop

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Regression Effect - Examples 4-yr-olds with IQ’s of 120 typically have adult IQ’s around 110. 4-yr-olds with IQ’s of 70 typically have adult IQ’s around 85. Of major league baseball teams with winning records, typically 2/3 win fewer games the next year.

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Note: n The regression effect does not explain a change in averages n If r > 0: if X is above AVGx, then the predicted Y must be above AVGy if X is below AVGx, then the predicted Y must be below AVGy if X is below AVGx, then the predicted Y must be below AVGy

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2. Regression Line as Least Squares Line

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What line is “closest” to the points ?

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The regression line has smallest RMS size of deviations from points to the line.

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The regression line is also called the least squares line.

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3. Extrapolation l Predicting beyond the range of predictor variables

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3. Extrapolation l Predicting beyond the range of predictor variables 6 NOT a good idea

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4. Multiple Regression Using more than one independent variable to predict dependent variable. Using more than one independent variable to predict dependent variable.Example: PredictY = son’s height UsingX 1 = father’s height X 2 = mother’s height

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4. Multiple Regression Using more than one independent variable to predict dependent variable. Using more than one independent variable to predict dependent variable.Example: PredictY = son’s height UsingX 1 = father’s height X 2 = mother’s height Equation:Y = m 1 X 1 + m 2 X 2 + b

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