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Unit 4

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Unit 4

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  1. Unit 4 Residuals, Residual Plots, & Influential points

  2. Residuals (error) - • The vertical deviation between the observations & the LSRL • the sum of the residuals is always zero • error = observed - expected

  3. Residual plot • A scatterplot of the (x, residual) pairs. • Residuals can be graphed against other statistics besides x • Purpose is to tell if a linear association exist between the x & y variables

  4. 60 64 68 Consider a population of adult women. Let’s examine the relationship between their height and weight. Weight Height

  5. Residuals Weight 60 64 68 Height Suppose we now take a random sample from our population of women.

  6. Residual plot • A scatterplot of the (x, residual) pairs. • Residuals can be graphed against other statistics besides x • Purpose is to tell if the model is an appropriate fit between the x & y variables • If no pattern exists between the points in the residual plot, then the model is appropriate.

  7. Model is appropriate Model is NOT appropriate

  8. Residuals x Age Range of Motion 35 154 24 142 40 137 31 133 28 122 25 126 26 135 16 135 14 108 20 120 21 127 30 122 One measure of the success of knee surgery is post-surgical range of motion for the knee joint following a knee dislocation. Is there a linear relationship between age & range of motion? Sketch a residual plot. Since there is no pattern in the residual plot, there is a linear relationship between age and range of motion

  9. Residuals Age Range of Motion 35 154 24 142 40 137 31 133 28 122 25 126 26 135 16 135 14 108 20 120 21 127 30 122 Plot the residuals against the y-hats. How does this residual plot compare to the previous one?

  10. Residuals Residuals x Residual plots are the same no matter if plotted against x or y-hat.

  11. Coefficient of determination- • r2 • the proportion of variation in y that can be attributed to a approximate linear relationship between x & y • remains the same no matter which variable is labeled x

  12. Age Range of Motion 35 154 24 142 40 137 31 133 28 122 25 126 26 135 16 135 14 108 20 120 21 127 30 122 Let’s examine r2. Suppose you were going to predict a future y but you didn’tknow the x-value. Your best guess would be the overall mean of the existing y’s. Total sum of the squared deviations - Total variation SStotal = 1564.917

  13. Age Range of Motion 35 154 24 142 40 137 31 133 28 122 25 126 26 135 16 135 14 108 20 120 21 127 30 122 Now suppose you were going to predict a future y but you DO know the x-value. Your best guess would be the point on the LSRL for that x-value (y-hat). Sum of the squared residuals using the LSRL. SSResid = 1085.735

  14. SSTotal = 1564.917 Age Range of Motion 35 154 24 142 40 137 31 133 28 122 25 126 26 135 16 135 14 108 20 120 21 127 30 122 SSResid = 1085.735 By what percent did the sum of the squared error go down when you went from just an “overall mean” model to the “regression on x” model? This is r2 – the amount of the variation in the y-values that is explained by the x-values.

  15. Age Range of Motion 35 154 24 142 40 137 31 133 28 122 25 126 26 135 16 135 14 108 20 120 21 127 30 122 How well does age predict the range of motion after knee surgery? 30.6% of the variation in range of motion after knee surgery can be explained by the approximate linear regression of age and range of motion.

  16. Interpretation of r2 r2% of the variation in y can be explained by the approximate linear regression of x & y.

  17. Computer-generated regression analysis of knee surgery data: Predictor Coef Stdev T P Constant 107.58 11.12 9.67 0.000 Age 0.8710 0.4146 2.10 0.062 s = 10.42 R-sq = 30.6% R-sq(adj) = 23.7% Be sure to convert r2 to decimal beforetaking the square root! NEVER use adjusted r2! What is the equation of the LSRL? Find the slope & y-intercept. What are the correlation coefficient and the coefficient of determination?

  18. In a regression setting, an outlier is a data point with a large residual Outlier –

  19. Influential point- • A point that influences where the LSRL is located • If removed, it will significantly change the slope of the LSRL

  20. Racket Resonance Acceleration (Hz) (m/sec/sec) 1 105 36.0 2 106 35.0 3 110 34.5 4 111 36.8 5 112 37.0 6 113 34.0 7 113 34.2 8 114 33.8 9 114 35.0 10 119 35.0 11 120 33.6 12 121 34.2 13 126 36.2 14 189 30.0 One factor in the development of tennis elbow is the impact-induced vibration of the racket and arm at ball contact. Sketch a scatterplot of these data. Calculate the LSRL & correlation coefficient. Does there appear to be an influential point? If so, remove it and then calculate the new LSRL & correlation coefficient.

  21. (189,30) could be influential. Remove & recalculate LSRL

  22. (189,30) was influential since it moved the LSRL

  23. Which of these measures are resistant? • LSRL • Correlation coefficient • Coefficient of determination NONE – all are affected by outliers