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Understanding Regression Analysis: Predicting Outcomes with Variables

In regression analysis, we identify response and explanatory variables to predict outcomes. Denoted as y for the response variable and x for the explanatory variable, the regression line provides an equation for predicting y based on x. The y-intercept and slope of this line offer insights into the relationship between the variables. We also measure prediction accuracy through residuals and the method of least squares. This chapter explores these concepts, showcasing examples like using femur length to predict height and interpreting correlation strength in linear relationships.

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Understanding Regression Analysis: Predicting Outcomes with Variables

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  1. Chapter 3Association: Contingency, Correlation, and Regression Section 3.3 Predicting the Outcome of a Variable

  2. Regression Line • The first step of a regression analysis is to identify the response and explanatory variables. • We use y to denote the response variable. • We use x to denote the explanatory variable.

  3. Regression Line: An Equation for Predicting the Response Outcome • The regression line predicts the value for the response variable y as a straight-line function of the value x of the explanatory variable. • Let denote the predicted value of y. The equation for the regression line has the form • In this formula, a denotes the y-intercept and b denotes the slope.

  4. Example: Height Based on Human Remains • Regression Equation: • is the predicted height and is the length of a femur (thighbone), measured in centimeters. • Use the regression equation to predict the height of a person whose femur length was 50 centimeters.

  5. Interpreting the y-Intercept • y-Intercept: • The predicted value for y when • This fact helps in plotting the line • May not have any interpretative value if no observations had x values near 0 It does not make sense for femur length to be 0 cm, so the y-intercept for the equation is not a relevant predicted height.

  6. Interpreting the Slope • Slope: measures the change in the predicted variable (y) for a 1 unit increase in the explanatory variable (x). • Example: A 1 cm increase in femur length results in a 2.4 cm increase in predicted height.

  7. Slope Values: Positive, Negative, Equal to 0 Figure 3.12 Three Regression Lines Showing Positive Association (slope > 0), Negative Association (slope < 0) and No Association (slope = 0). Question Would you expect a positive or negative slope when y = annual income and x = number of years of education?

  8. Residuals Measure the Size of Prediction Errors Residuals measure the size of the prediction errors, the vertical distance between the point and the regression line. • Each observation has a residual • Calculation for each residual: • A large residual indicates an unusual observation. • The smaller the absolute value of a residual, the closer the predicted value is to the actual value, so the better is the prediction.

  9. The Method of Least Squares Yields theRegression Line • Residual sum of squares: • The least squares regression line is the line that minimizes the vertical distance between the points and their predictions, i.e., it minimizes the residual sum of squares. • Note: The sum of the residuals about the regression line will always be zero.

  10. Regression Formulas for y-Intercept and Slope • Slope: • y-Intercept: • Notice that the slope b is directly related to the correlation r, and the y-intercept depends on the slope.

  11. Calculating the slope and y-intercept for the regression line Using the baseball data in Example 9 to illustrate the calculations. The regression line to predict team scoring from batting average is .

  12. The Slope and the Correlation • Correlation: • Describes the strength of the linear association between 2 variables. • Does not change when the units of measurement change. • Does not depend upon which variable is the response and which is the explanatory.

  13. The Slope and the Correlation • Slope: • Numerical value depends on the units used to measure the variables. • Does not tell us whether the association is strong or weak. • The two variables must be identified as response and explanatory variables. • The regression equation can be used to predict values of the response variable for given values of the explanatory variable.

  14. The Squared Correlation • The typical way to interpret is as the proportion of the variation in the y-values that is accounted for by the linear relationship of y with x. • When a strong linear association exists, the regression equation predictions tend to be much better than the predictions using only . • We measure the proportional reduction in error and call it, .

  15. The Squared Correlation • measures the proportion of the variation in the y-values that is accounted for by the linear relationship of y with x. • A correlation of .9 means that • 81% of the variation in the y-values can be explained by the explanatory variable, x.

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