Least Squares Regression

  If we have two variables X and Y, we often would like to model the relation as a line  Draw a line through the scatter diagram  We want to find the line that “best” describes the linear relationship … the regression line Best Fit

 ● We want to use a linear model ● Linear models can be written in several different (equivalent) ways  y = m x + b  y – y 1 = m ( x – x 1 )  y = b 1 x + b 0 ● Because the slope and the intercept both are important to analyze, we will use y = b 1 x + b 0 Linear Model

 ● One difference between math and stat is that statistics assumes that the measurements are not exact, that there is an error or residual ● The formula for the residual is always Residual = Observed – Predicted ● This relationship is not just for this chapter … it is the general way of defining error in statistics Residuals

  What the residual is on the scatter diagram Residual The model line The x value of interest The observed value y The residual The predicted value y

 ● We want to minimize the residuals, but we need to define what this means ● We use the method of least-squares  We consider a possible linear mode  We calculate the residual for each point  We add up the squares of the residuals ● The line that has the smallest is called the least-squares regression line Least-Squares Regression Line

  The equation for the least-squares regression line is given by y = b 1 x + b 0  b 1 is the slope of the least-squares regression line  b 0 is the y-intercept of the least-squares regression line Least Squares Regression Line

 ● Finding the values of b 1 and b 0, by hand, is a very tedious process ● You should use calculator for this ● Finding the coefficients b 1 and b 0 is only the first step of a regression analysis  We need to interpret the slope b 1  We need to interpret the y-intercept b 0 Tough Stuff

  Interpreting the slope b 1  The slope is sometimes defined as  The slope is also sometimes defined as  The slope relates changes in y to changes in x Slope

 The speed of a golf club and the distance the ball went were measured for a linear relationship. Draw a scatter pot of the data, and the least- squares regression line. Write down the equation. Example

 Club Head Speed (mph) Distance (yards) Data Using your equation, estimate how far a golf ball would travel if it was hit at 104 mph. Interpret the slope of the regression equation

Sample Problem  The heights and weights of 11 men between the ages of 21 and 26 were measured. The data are presented in the table below.  A. Create at a scatter diagram to confirm that an approximately linear relationship exists between x and y.  B. Find the least squares regression line, treating height, x, as the explanatory variable and weight, y, as the response variable – two ways, 1. by hand and 2. use calculator  c. Interpret the slope and intercept, if appropriate.  d. Use the regression line to predict the weight of a man who is 73 inches tall?

If the least-squares regression line is used to make predictions based on values of the explanatory variable that are much larger or much smaller than the observed values, we say the researcher is working outside the scope of the model. Never use a least-squares regression line to make predictions outside the scope of the model because we can’t be sure the linear relation continues to exist © 2010 Pearson Prentice Hall. All rights reserved