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Chapter 11

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Chapter 11

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  1. Chapter 11 Simple Linear Regression and Correlation

  2. Learning Objectives • Use simple linear regression for building empirical models • Estimate the parameters in a linear regression model • Determine if the regression model is an adequate fit to the data • Test statistical hypotheses and construct confidence intervals • Prediction of a future observation • Use simple transformations to achieve a linear regression model • understand the correlation

  3. Relationships between two or more variables Useful for these types of problems Predict a new observation Sometimes a regression model will arise from a theoretical relationship At other times no theoretical knowledge Choice of the model is based on inspection of a scatter diagram Empirical model Regression analysis

  4. Mean of the random variable Y is related to x E[Y|x]=Y|x=0+ 1x 0 and 1 called regression coefficients Appropriate way to generalize this to a probabilistic model Assume that the expected value of Y is a linear function of x Actual value of Yis determined by the mean value function plus a random error term Y=0+ 1x+ Where  called random error term Regression Model

  5. 1 and  • Suppose that the mean and variance of  are 0 and 2 • Slope, 1, can be interpreted as the change in the mean of Y • Height of the line at any value of x is just the expected value of Y for that x • Variability of Y at a particular value of x is determined by the error variance 2 • Implies that there is a distribution of Y-values at each x

  6. Graph of the Variability Y • Distribution of Y for any given value of x • Values of x are fixed, and Y is a random variable with the following mean and variance Mean: Y|x=0+ 1x Variance 2 True regression line x

  7. Values of the intercept, slope and the error variance will not be known Must be estimated from sample data Fitted model is used in prediction of future observations of Y at a particular level of x Simple Linear Regression

  8. Method of Least Squares • True relationship between Y and x is a straight line • Assume n pairs of observations • Estimates of 0 and 1 result in a line that is a “best fit” to the data • Called method of least squares

  9. Least Squares Method • Assuming the n observations in the sample • Sum of the squares of the deviations of the observations from the true regression line • Taking the partial derivatives

  10. Least Squares Method-Cont. • Simplifying • Results are • Fitted or estimated regression line is

  11. Using Special Symbols • Convenient to use special symbols • Numerator • Denominator

  12. Residual Error • Describes the error in the fit of the model to the ith observation yi • Each pair of observations satisfies • Denoted by ei

  13. Estimating 2 • Another unknown parameter,2, the variance of the error term  • Residuals ei are used to obtain an estimate of 2 • Sum of squares of the residuals, often called the error sum of squares • A more convenient computing formula • SST is the total sum of squares

  14. Example • Regression methods were used to analyze the data from a study investigating the relationship between roadway surface temperature (x) and pavement deflection ( y). • Summary quantities were as follows

  15. Questions (a) Calculate the least squares estimates of the slope and intercept. Graph the regression line. (b) Use the equation of the fitted line to predict what pavement deflection would be observed when the surface temperature is 85F. (c) What is the mean pavement deflection when the surface temperature is 90F? (d) What change in mean pavement deflection would be expected for a 1F change in surface temperature?

  16. Solution • Need to have • Hence, the slope and intercept • Regression line

  17. Solution-Cont. • Graph of the regression line

  18. Solution-Cont. • Pavement deflection • Mean pavement deflection • Change in mean pavement deflection

  19. Properties of the Least Estimators • Assumed that the error term  in the model is a random variable • Estimators will be viewed as random variables • Properties of the slope • Properties of the intercept

  20. Analysis of Variance Approach • Used to test for significance of regression • Partitions the total variability in the response variable into two components • First term is called error sum of squares • Second term is called regression sum of squares • Symbolically SST=SSR+SSE • SST is the total corrected sum of squares

  21. Analysis of Variance • SST, SSR, and SSEhas n-1, 1, and n-2 d.o.f, respectively • SSR= β1Sxy and SSE=SST-β1Sxy • Divide by its d.o.f MSR=[SSR/1] and MSE=[SSE/n-2] • Then F=MSR/MSE follows F1,n-2 distribution

  22. Hypothesis Tests for Slope • Adequacy of a linear regression model • Appropriate hypotheses for slope are H0: β1=β1,0 H1: β1#β1,0 • Test Statistic • Follows the F 1,n-2 distribution • Reject H0 if f0>f,1,n-2

  23. Analysis of Variance for Testing Significance of Regression

  24. Example • Consider the data from the previous example on x=roadway surface temperature and y=pavement deflection. • (a) Test for significance of regression using α= 0.05. What conclusions can you draw? • (b) Estimate the standard errors of the slope and intercept.

  25. Solution • Use the steps in hypotheses testing 1) Parameter of interest is slope of the regression line 1 2) 3) 4)  = 0.05 5) The test statistic is 6) Reject H0 if f0 > f,1,18 where f0.05,1,18 = 4.416

  26. Solution 7) Using the results from the previous example • Hence, the test statistic 8) Since 73.95 > 4.416, reject H0 and conclude the model specifies a useful relationship at  = 0.05 • Standard error

  27. Confidence Intervals on the Slope and Intercept • Interested to obtain C.I. estimates of the parameters • Width of these C.I. is a measure of the overall quality of the regression line • 100(1-α)% C.I. on the slope β1 • 100(1-α)% C.I. on the intercept β0

  28. Confidence Interval on the Mean Response • Constructed on the mean response at a specified value of x, say, x0 • Called a C.I. about the regression line • C.I. about the mean responseat the value of x=x0 • Applies only to the interval

  29. Prediction of New Observations • An important application of a regression model • New observation is independent of the observations used to develop the regression model • C.I. for Y|x is inappropriate • Prediction interval on a future observation at the value x0 • Always wider than the C.I. at x0 • Depends on both the error from the fitted model and the error associated with future observations

  30. Example • The first example presented data on roadway surface temperature x and pavement deflection y • Find a 99% confidence interval on each of the following: • (a) Slope • (b) Intercept • (c) Mean deflection when temperature x=85o F • (d) Find a 99% prediction interval on pavement deflection when the temperature is 90oF.

  31. Solution a) Confidence interval on the slope • Critical value t/2,n-2 = t0.005,18 = 2.878 • Hence b) Confidence interval on the intercept

  32. Solution c) 99% confidence interval on  when x=85 F d) 99% prediction interval when x=90 F.

  33. Helpful in checking the errors are approximately normally distributed with constant variance Useful in determining whether additional terms in the model are required Construct normal probability plot of residuals Patterns of residual plots Residual Analysis

  34. Judge the adequacy of a regression model Coefficient of determination Referred as the amount of variability in the data explained by the regression model and SSR is that portion of SST that is explained by the use of the regression model SSE is that portion of SST that is not explained by the use of the regression model Coefficient of Determination(R2)

  35. Inappropriateness of straight-line regression model Scatter diagram Consider the exponential function Y=β0eβ1x transformed to a straight line By a logarithmic transformation ln Y=lnβ0 +β1x + ln  Another intrinsically linear function is Y=β0+β1(1/x)+ By using the reciprocal transformation z=1/x Y=β0 +β1 z + Transformed error terms  are normally distributed Transformation of Data Points

  36. Assumed that x is a mathematical variable and that Y is a random variable Many applications involve situations in which both X and Y are random variables Suppose observations are jointly distributed random variables Measures the strength of linear association between two variables and denoted by  Shows how closely the points in a scatter diagram are spread around the regression line Correlation

  37. Hypothesis Tests • Useful to test the hypotheses H0: =0 H1: #0 • Appropriate test statistic • Follows the t distribution with n-2 degrees of freedom • Reject the null hypothesis if

  38. Next Agenda • Chapters 13 deals with designing and conducting engineering experiments • ANOVA in designing single factor experiments will be emphasized