1. Simple Linear Regression 1. review of least squares procedure 2. inference for least squares lines
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3. The Model House size House Cost Most lots sell for $25,000 Building a house costs about $75 per square foot. House cost = 25000 + 75(Size) The model has a deterministic and a probabilistic components
4. The Model House cost = 25000 + 75(Size) House size House Cost Most lots sell for $25,000 However, house cost vary even among same size houses! Since cost behave unpredictably, we add a random component.
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7. The Least Squares (Regression) Line A good line is one that minimizes the sum of squared differences between the points and the line.
8. The Least Squares (Regression) Line 3 3 4 4 (1,2) 2 2 (2,4) (3,1.5) Sum of squared differences = (2 - 1) 2 + (4 - 2) 2 + (1.5 - 3) 2 + (4,3.2) (3.2 - 4) 2 = 6.89 2.5 Let us compare two lines The second line is horizontal The smaller the sum of squared differences the better the fit of the line to the data. 1 1 Sum of squared differences = (2 -2.5) 2 + (4 - 2.5) 2 + (1.5 - 2.5) 2 + (3.2 - 2.5) 2 = 3.99
9. The Estimated Coefficients To calculate the estimates of the slope and intercept of the least squares line , use the formulas: The regression equation that estimates the equation of the first order linear model is: Alternate formula for the slope b 1
14. Interpreting the Linear Regression -Equation This is the slope of the line. For each additional mile on the odometer, the price decreases by an average of $0.0623 The intercept is b 0 = $17067. 0 No data Do not interpret the intercept as the “ Price of cars that have not been driven” 17067
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16. The Normality of From the first three assumptions we have: y is normally distributed with mean E(y) = 0 + 1 x, and a constant standard deviation x 1 x 2 x 3 The standard deviation remains constant, but the mean value changes with x 0 + 1 x 1 0 + 1 x 2 0 + 1 x 3 E(y|x 2 ) E(y|x 3 ) E(y|x 1 )
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28. Coefficient of determination x 1 x 2 y 1 y 2 Two data points (x 1 ,y 1 ) and (x 2 ,y 2 ) of a certain sample are shown. Total variation in y = Variation explained by the regression line + Unexplained variation (error) Variation in y = SSR + SSE y
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43. For each residual we calculate the standard deviation as follows: A Partial list of Standard residuals Residual Analysis Standardized residual ‘i’ = Residual ‘i’ Standard deviation
44. It seems the residual are normally distributed with mean zero Residual Analysis
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48. Patterns in the appearance of the residuals over time indicates that autocorrelation exists. + + + + + + + + + + + + + + + + + + + + + + + + + Time Residual Residual Time + + + Note the runs of positive residuals, replaced by runs of negative residuals Note the oscillating behavior of the residuals around zero. 0 0 Non Independence of Error Variables
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50. + + + + + + + + + + + + + + + + The outlier causes a shift in the regression line … but, some outliers may be very influential An outlier An influential observation + + + + + + + + + + +
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