Inference for Least S quares L ines

1. Inference for Least Squares Lines Lecture Unit 8.1 - 8.3

2. Warm-Up The least squares slope b1 is an estimate of the true slope of the line that relates global average temperature to CO2. Since b1 = 0.0103 is very close to 0, could it be that the true slope is NOT significantly different from 0 (that is, atmospheric CO2 gives very little information about global average temperature)?

3. The Estimated Coefficients Bivariate data: (x1 , y1 ), (x2 , y2 ), (x3 , y3 ), … , (xn , yn ) To calculate the estimates of the slope and intercept of the least squares line , use the formulas: The least squares prediction equation that estimates the mean value of y for a particular value of x is:

4. The Model • The first order linear model y = response variable x = explanatory variable b0 = y-intercept b1 = slope of the line e = error variable b0 and b1 are unknown populationparameters, therefore are estimated from the data. y Rise b1 = Rise/Run Run b0 x

5. The Least Squares (Regression) Line A good line is one that minimizes the sum of squared differences between the points and the line.

6. The Simple Linear Regression Line Example: • A car dealer wants to find the relationship between the odometer reading and the selling price of used cars. • A random sample of 100 cars is selected, and the data recorded. • Find the regression line. Explanatoryvariable x Responsevariable y

7. The Simple Linear Regression Line • Solving by hand: Calculate a number of statistics where n = 100.

8. The Simple Linear Regression Line

9. Interpreting the Least Squares Line 17248.73 No data 0 The intercept is b0 = $17248.73. This is the slope of the line. For each additional mile on the odometer, the price decreases by an average of $0.0669 Do not interpret the intercept as the “Price of cars that have not been driven”

10. Error Variable: Required Conditions • The error e is a critical part of the regression model. • Four requirements involving the distribution of e must be satisfied. • The distribution of the e’s can be described by a normal model • The mean of e is zero: E(e) = 0. • The standard deviation of e is se for all values of x. • The errors associated with different values of y are all independent.

11. E(y|x3) b0 + b1x3 E(y|x2) b0 + b1x2 E(y|x1) b0 + b1x1 The Normality of  The distribution of the e’s can be described by a normal model The mean of e is zero: E(e) = 0. The standard deviation of e is se for all values of x. The standard deviation remains constant, m3 m2 but the mean value changes with x m1 From the first three assumptions we have: y is normally distributed with mean E(y) = b0 + b1x, and a constant standard deviation se x1 x2 x3

12. Assessing the Model • The least squares method will produces a regression line whether or not there is a linear relationship between x and y. • Consequently, it is important to assess how well the linear model fits the data. • Several methods are used to assess the model. All are based on the sum of squares for errors, SSE.

13. A shortcut formula Sum of Squares for Errors: SSE • This is the sum of the squared differences between the observed y’s and the predicted y’s. • It can serve as a measure of how well the line fits the data. SSE is defined by

14. Standard Error of Estimate • The mean error is equal to zero. • If the standard deviation e of the residuals is small the errors tend to be close to zero (close to the mean error). Then, the model fits the data well. • Therefore we can use e as a measure of the suitability of using a linear model. • An estimator of e is given by se

15. Standard Error of Estimate,Example • Example: • Calculate the standard error of estimate for the previous example and describe what it tells you about the model fit. • Solution

16. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Testing the slope • When no linear relationship exists between two variables, the regression line should be horizontal. q q Linear relationship. Linear relationship. Linear relationship. Linear relationship. No linear relationship. Different inputs (x) yield the same output (y). Different inputs (x) yield different outputs (y). The slope is not equal to zero The slope is equal to zero

17. The standard error of b1. Testing the Slope • We can draw inference about 1 from b1 by testing H0: 1 = 0 H1: 1 = 0 (or < 0,or > 0) • The test statistic is • If the error variable is normally distributed, the statistic is Student t distribution with d.f. = n-2. where

18. Testing the Slope,Example • Example • Test to determine whether there is enough evidence to infer that there is a linear relationship between the car auction price and the odometer reading for all three-year-old Tauruses in the previous example.

19. Testing the Slope, Example Bivariate data: (x1 , y1 ), (x2 , y2 ), (x3 , y3 ), … , (x100 , y100 ) H0: 1 = 0 HA: 1 0 • Solving by hand • To compute “t” we need the values of b1 and sb1. • P-value = 2P(t98 > |-13.44|) ~ 0

20. Testing the Slope (Example) • Using the computer H0: 1 = 0 HA: 1 0 Reject H0:1=0. There is strong evidence to infer that the odometer reading affects the auction selling price.

21. Confidence Interval for the the Slope, Example Bivariate data: (x1 , y1 ), (x2 , y2 ), (x3 , y3 ), … , (x100 , y100 )

22. Confidence Interval for the Slope (Example) • Using the computer 95% confidence interval for the slope 1

23. Warm-Up (cont.) Results of hypothesis test H0 : 1 = 0 HA : 1  0 Reject H0 Confidence interval for 1

24. Coefficient of determination Reduction in prediction error when use x: TSS-SSE = SSR

25. Explained in part by Remains, in part, unexplained Coefficient of determination Reduction in prediction error when use x:TSS-SSE = SSR or TSS = SSR + SSE The regression model SSR Overall variability in y TSS The error SSE

26. y Coefficient of determination: graphically y2 Two data points (x1,y1) and (x2,y2) of a certain sample are shown. Variation in y = SSR + SSE (TSS) y1 x1 x2 + Unexplained variation (error) Total variation in y = Variation explained by the regression line

27. Coefficient of determination • R2 (=r2 ) measures the proportion of the variation in y that is explained by the variation in x. • r2 takes on any value between zero and one since the correlation r satisfies -1r1). r2 = 1: Perfect match between the line and the data points. r2 = 0: There are no linear relationship between x and y.

28. Coefficient of Determination,Example • Example • Find the coefficient of determination for the used car price –odometer example. What does this statistic tell you about the model? • Solution • Solving by hand;

29. Coefficient of Determination • Using the computerFrom the regression output we have 64.8% of the variation in the auction selling price is explained by the variation in odometer reading. The rest (35.2%) remains unexplained by this model.

30. Using the Regression Equation • We can use the least squares line to predict the values of y. • To make a prediction we use • Point prediction, and • Interval prediction

31. A point prediction Point Prediction • Example • Predict the selling price of a three-year-old Taurus with 40,000 miles on the odometer. • It is predicted that a 40,000 miles car would sell for $14,574. • How close is this prediction to the real price?

32. The prediction interval Interval Estimates • Two intervals can be used to discover how closely the predicted value will match the true value of y. • Prediction interval – predicts y for a given value of x (price prediction for a specific car with 40,000 miles on odometer) • Confidence interval – estimates the average y for a given x (estimate the average price of all cars with 40,000 miles on odometer). • The confidence interval

33. Interval Estimates,Example • Example - continued • Provide an interval estimate for the bidding price on a Ford Taurus with 40,000 miles on the odometer. • Two types of predictions are required: • A prediction for a specific car • An estimate for the average price per car

34. Interval Estimates,Example • Solution • A 95% predictioninterval provides the price estimate for a single car: t.025,98

35. Interval Estimates,Example • Solution – continued • A 95% confidence interval provides the estimate of the mean price per car for all Ford Taurus’s with 40,000 miles on the odometer. • The confidence interval (95%) =

36. The effect of the given x on the length of the interval • As xmoves away from x the interval becomes longer. That is, the shortest interval is found at x.

37. The effect of the given x on the length of the interval • As xmoves away from x the interval becomes longer. That is, the shortest interval is found at x. • As xmoves away from x the interval becomes longer. That is, the shortest interval is found at x.

38. The effect of the given x on the length of the interval • As xmoves away from x the interval becomes longer. That is, the shortest interval is found at x. • As xmoves away from x the interval becomes longer. That is, the shortest interval is found at x.

39. Regression Diagnostics - I • The three conditions required for the validity of the regression analysis are: • the error variable is normally distributed. • the error variance is constant for all values of x. • The errors are independent of each other. • How can we diagnose violations of these conditions?

40. Residual Analysis • Examining the residuals (or standardized residuals), help detect violations of the required conditions. • Example – continued: • Nonnormality. • Use Excel to obtain the standardized residual histogram. • Examine the histogram and look for a bell shaped. diagram with a mean close to zero.

41. Standardized residual ‘i’ = Residual ‘i’ Standard deviation Residual Analysis A Partial list of Standard residuals For each residual we calculate the standard deviation as follows:

42. Residual Analysis It seems the residual are normally distributed with mean zero

43. ^ y + + + + + + + + + + + + + + + + + + + + + + + ^ The spread increases with y Heteroscedasticity • When the requirement of a constant variance is violated we have a condition of heteroscedasticity. • Diagnose heteroscedasticity by plotting the residual against the predicted y. Residual + + + + + + + + + + + + + ^ + + + y + + + + + + + +

44. Homoscedasticity • When the requirement of a constant variance is not violated we have a condition of homoscedasticity. • Example - continued

45. Non Independence of Error Variables • A time series is constituted if data were collected over time. • Examining the residuals over time, no pattern should be observed if the errors are independent. • When a pattern is detected, the errors are said to be autocorrelated. • Autocorrelation can be detected by graphing the residuals against time.

46. Non Independence of Error Variables Patterns in the appearance of the residuals over time indicates that autocorrelation exists. Residual Residual + + + + + + + + + + + + + + 0 0 + Time Time + + + + + + + + + + + + + Note the oscillating behavior of the residuals around zero. Note the runs of positive residuals, replaced by runs of negative residuals

47. Outliers • An outlier is an observation that is unusually small or large. • Several possibilities need to be investigated when an outlier is observed: • There was an error in recording the value. • The point does not belong in the sample. • The observation is valid. • Identify outliers from the scatter diagram. • It is customaryto suspect an observation is an outlier if its |standard residual| > 2

48. + + + + + + + + + + + An influential observation An outlier + + … but, some outliers may be very influential + + + + + + + + + + + + + + The outlier causes a shift in the regression line

49. Procedure for Regression Diagnostics • Develop a model that has a theoretical basis. • Gather data for the two variables in the model. • Draw the scatter diagram to determine whether a linear model appears to be appropriate. • Determine the regression equation. • Check the required conditions for the errors. • Check the existence of outliers and influential observations • Assess the model fit. • If the model fits the data, use the regression equation.