The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Introductory Statistics Lecture 9 Least Squares Regression 9 28 06 Lecture 9 1 Review Scatter plot Association form direction strength Just graphical not numerical Correlation Direction strength linear Properties Vertical and horizontal lines r 0 Correlation cannot tell the exact relationship 9 28 06 Lecture 9 2 Topics Least Squares Regression Regression lines Equation and interpretation of the line Prediction using the line Correlation and Regression Coefficient of Determination 9 28 06 Lecture 9 3 Age vs Mean Height 9 28 06 Lecture 9 4 To predict mean height at age 32 months 9 28 06 Lecture 9 5 Linear Regression Correlation measures the direction and strength of the linear relationship between two quantitative variables A regression line summarizes the relationship between two variables if the form of the relationship is linear describes how a response variable y changes as an explanatory variable x changes is often used as a mathematical model to predict the value of a response variable y based on a value of an explanatory variable x 9 28 06 Lecture 9 6 Equation of a straight Line A straight line relating y to x has an equation of the form y a bx x explanatory variable y response variable a y intercept b slope of the line 9 28 06 Lecture 9 7 How to fit a line 9 28 06 Lecture 9 8 Error 9 28 06 Lecture 9 9 Least Square Regression Line A line that makes the sum of the squares of the vertical distances of the data points from the line as small as possible Mathematically the line is determined by minimizing y a bx 2 i 9 28 06 i Lecture 9 10 Equation of the Least Squared Regression Line The least squares regression line of y on x is y a bx with slope b r s s y x and intercept a y bx 9 28 06 Lecture 9 11 Interpreting the Regression Line The slope b tells us that along the regression line a change of one standard deviation in x a change of r standard deviations in y a change of 1 unit in x b units in y The point x y is always on line If both x and y are standardized the slope will be r the intercept will be 0 the origin 0 0 is on line why r and b have same sign 9 28 06 Lecture 9 12 Example Age vs Height 84 Height in centimeters 82 y 64 932 0 6348 x 80 78 76 17 5 20 22 5 25 27 5 30 Age in months x 23 5 y 79 85 r 0 9944 s x 3 606 s y 2 302 9 28 06 Lecture 9 b r s s y x a y bx 13 Prediction y a bx is a prediction when the explanatory variable x x What is the average height for a child who is 30 month old How about a 30 year old Do not extrapolate too much for prediction 9 28 06 Lecture 9 14 Correlation and Regression Both for linear relationship between two variables Same sign between b and r r does not depend on which is x and which is y But a regression line does causality 9 28 06 Lecture 9 15 Regression lines depend on x y or y x 9 28 06 Lecture 9 16 Coefficient of Determination r2 The square of the correlation r2 is the proportion of variation in the values of y that is explained by the regression model with x 0 r2 1 The larger r2 the stronger the linear relationship The closer r2 is to 1 the more confident we are in our prediction 9 28 06 Lecture 9 17 Age vs Height r2 0 9888 9 28 06 Lecture 9 18 Age vs Height r2 0 849 9 28 06 Lecture 9 19 Take Home Message Least Squares Regression Regression lines Equation and interpretation of the line Prediction using the line Correlation and Regression Coefficient of Determination 9 28 06 Lecture 9 20