Simple Linear Regression

1. Simple Linear Regression Statistics 700 Week of November 27

2. Example for Illustration • The human body takes in more oxygen when exercising than when it is at rest. To deliver oxygen to the muscles, the heart must beat faster. Heart rate is easy to measure, but measuring oxygen uptake requires elaborate equipment. If oxygen uptake (VO2) can be accurately predicted from heart rate (HR), the predicted values may replace actually measured values for various research purposes. Unfortunately, not all human bodies are the same, so no single prediction equation works for all people. Researchers can, however, measure both HR and VO2 for one person under varying sets of exercise conditions and calculate a regression equation for predicting that person’s oxygen uptake from heart rate. Simple Linear Regression

3. Data From An Individual • Goals in this illustration: • Scatterplot: linear relationship or not? • Obtain the best-fitting line using least-squares. • To test whether the model is significant or not. • To obtain a confidence interval for the regression coefficient. • To obtain predictions. Simple Linear Regression

4. The Scatterplot Simple Linear Regression

5. Simple Linear Regression Model 1. Conditional on X=x, the response variable Y has mean equal to m(x) = a + bx. 2. ais the y-intercept; whileb is the slope of the regression line, which could be interpreted as the change in the mean value per unit change in the independent variable. 3. For each X = x, the conditional distribution of Y is normal with mean m(x) and variance s2. 4. Y1, Y2, …, Yn are independent of each other. Shorthand: Yi = a + bxi + ei with ei IID N(0,s2) Simple Linear Regression

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10. Results of Regression Analysis (using Minitab) Simple Linear Regression

11. Fitted Line on the Scatterplot Simple Linear Regression

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