1. Simple Linear Regression 1

2. 1. Introduction Response (out come or dependent) variable (Y): height of the wife Predictor (explanatory or independent) variable (X): height of the husband Example: George Bush :1.81m Laura Bush: ? David Beckham: 1.83m Victoria Beckham: 1.68m Brad Pitt: 1.83m Angelina Jolie: 1.70m ● To predict height of the wife in a couple, based on the husband’s height 2

3. Regression analysis: ● The earliest form of linear regression was the method of least squares, which was published by Legendre in 1805, and by Gauss in 1809. ● The method was extended by Francis Galton in the 19th century to describe a biological phenomenon. ● This work was extended by Karl Pearson and Udny Yule to a more general statistical context around 20th century. ● regression analysis is a statistical methodology to estimate the relationship of a response variable to a set of predictor variable. History: ● when there is just one predictor variable, we will use simple linear regression. When there are two or more predictor variables, we use multiple linear regression. ● when it is not clear which variable represents a response and which is a predictor, correlation analysis is used to study the strength of the relationship 3

4. A probabilistic model We denote the n observed values of the predictor variable x as We denote the corresponding observed values of the response variable Y as 4

5. Notations of the simple linear Regression - Observed value of the random variable Y i depends on x i - random error with unknown mean of Y i Unknown Slope True Regression Line Unknown Intercept 5

6. 6 Simple Linear Regression Model y = dependent variable x = independent variable  0 = intercept  1 = slope of the line = error variable x y 00 Run Rise   = Rise/Run  0 and  1 are unknown, therefore, are estimated from the data.

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8. Linear function of the predictor variable Have a common variance, Same for all values of x. Normally distributed 4 BASIC ASSUMPTIONS – for statistical inference Independent 8

9. Comments: 1.Linear not in x But in the parameters and 2. Predictor variable is not set as predetermined fixed values, is random along with Y. The model can be considered as a conditional model Example: linear, logx = x* Example: Height and Weight of the children. Height (X) – given Weight (Y) – predict Conditional expectation of Y given X = x 9

10. 2. Fitting the Simple Linear Regression Model 2.1 Least Squares (LS) Fit 10

11. Example 10.1 (Tires Tread Wear vs. Mileage: Scatter Plot. From: Statistics and Data Analysis; Tamhane and Dunlop; Prentice Hall. ) 11

12. 12 Estimating the Coefficients The estimates are determined by drawing a sample from the population of interest, calculating sample statistics. producing a straight line that cuts into the data.           The question is: Which straight line fits best? x y

13. 13 3 3 For the least squares regression method, the best line is the one that minimizes the sum of squared vertical differences between the points and the line.     4 1 1 4 (1,2) 2 2 (2,4) (3,1.5) (4,3.2) 2.5 The smaller the sum of squared differences the better the fit of the line to the data.

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15. The “best” fitting straight line in the sense of minimizing Q: LS estimate One way to find the LS estimate and Setting these partial derivatives equal to zero and simplifying, we get the normal equations: 15

16. Solve the equations and we get 16

17. To simplify, we introduce The resulting equation is known as the least squares regression line, which is an estimate of the true regression line. 17

18. Example 10.2 (Tire Tread vs. Mileage: LS Line Fit) Find the equation of the line for the tire tread wear data from Table10.1,we have and n=9.From these we calculate 18

19. The slope and intercept estimates are Therefore, the equation of the LS line is Conclusion: there is a loss of 7.281 mils in the tire groove depth for every 1000 miles of driving. Given a particular We can find This means that the mean groove depth for all tires driven for 25,000miles is estimated to be 178.62 miles. 19

20. 2.2 Goodness of Fit of the LS Line Coefficient of Determination and Correlation The residuals: are used to evaluate the goodness of fit of the LS line. 20

21. We define: Note: total sum of squares (SST) Regression sum of squares (SSR) Error sum of squares (SSE) is called the coefficient of determination 21

22. Example 10.3 (Tire Tread Wear vs. Mileage: Coefficient of Determination and Correlation For the tire tread wear data, calculate using the result s from example 10.2 We have Next calculate Therefore The Pearson correlation is where the sign of r follows from the sign of since 95.3% of the variation in tread wear is accounted for by linear regression on mileage, the relationship between the two is strongly linear with a negative slope. 22

23. The Maximum Likelihood Estimators (MLE) Consider the linear model: where is drawn from a normal population with mean 0 and standard deviation σ, the likelihood function for Y is: Thus, the log-likelihood for the data is:

24. The MLE Estimators Solving We obtain the MLEs of the three unknown model parameters The MLEs of the model parameters a and b are the same as the LSEs – both unbiased The MLE of the error variance, however, is biased:

25. 2.3 An Unbiased Estimator of   An unbiased estimate of is given by Example 10.4(Tire Tread Wear Vs. Mileage: Estimate of Find the estimate of for the tread wear data using the results from Example 10.3 We have SSE=2351.3 and n-2=7,therefore Which has 7 d.f. The estimate of is miles. 25

26. Under the normal error assumption * Point estimators: * Sampling distributions of and : For mathematical derivations, please refer to the Tamhane and Dunlop text book, P331. 3. Statistical Inference on   and   26

27. * Pivotal Quantities (P.Q.’s): * Confidence Intervals (CI’s): 27 Statistical Inference on   and  , Con’t

28. * Hypothesis tests: -- Test statistics: -- At the significance level, we reject in favor of if and only if (iff) -- The first test is used to show whether there is a linear relationship between x and y Statistical Inference on   and  , Con’t 28

29. Analysis of Variance (ANOVA), Con’t Mean Square: -- a sum of squares divided by its d.f. 29

30. Analysis of Variance (ANOVA) ANOVA Table Example: Source of Variation (Source) Sum of Squares (SS) Degrees of Freedom (d.f.) Mean Square (MS) F Regression Error SSR SSE 1 n - 2 Total SST n - 1 Source SS d.f. MS F Regression Error 50,887.20 2531.53 1 7 50,887.20 361.25 140.71 Total53,418.73 8 30

31. 31 4. Finance Application: Market Model One of the most important applications of linear regression is the market model. It is assumed that rate of return on a stock (R) is linearly related to the rate of return on the overall market. R =  0 +  1 R m +  Rate of return on a particular stockRate of return on some major stock index The beta coefficient measures how sensitive the stock’s rate of return is to changes in the level of the overall market.

32. 32 Example The market model Estimate the market model for Nortel, a stock traded in the Toronto Stock Exchange. Data consisted of monthly percentage return for Nortel and monthly percentage return for all the stocks. This is a measure of the stock’s market related risk. In this sample, for each 1% increase in the TSE return, the average increase in Nortel’s return is.8877%. This is a measure of the total risk embedded in the Nortel stock, that is market-related. Specifically, 31.37% of the variation in Nortel’s return are explained by the variation in the TSE’s returns.

33. 5. Regression Diagnostics 5.1 Checking for Model Assumptions Checking for Linearity Checking for Constant Variance Checking for Normality Checking for Independence 33

34.  Checking for Linearity Xi =Mileage Y=β 0 + β 1 x Yi =Groove Depth ^ ^ ^ ^ Y=β 0 + β 1 x Yi =fitted value ^ ei =residual Residual = ei = Yi- Yi iXiYi ^ Yiei 10394.33360.6433.69 24329.50331.51-2.01 38291.00302.39-11.39 412255.17273.27-18.10 516229.33244.15-14.82 620204.83215.02-10.19 724179.00185.90-6.90 828163.83156.787.05 932150.33127.6622.67 34

35.  Checking for Normality 35

36.  Checking for Constant Variance Var(Y) is not constant. A sample residual plots when Var(Y) is constant. 36

37.  Checking for Independence Does not apply for Simple Linear Regression Model Only apply for time series data 37

38. 5.2 Checking for Outliers & Influential Observations  What is OUTLIER  Why checking for outliers is important  Mathematical definition  How to deal with them 38

39. 5.2-A. Intro Recall Box and Whiskers Plot (Chapter 4 of T&D) Where (mild) OUTLIER is defined as any observations that lies outside of Q1-(1.5*IQR) and Q3+(1.5*IQR) (Interquartile range, IQR = Q3 − Q1) (Extreme) OUTLIER as that lies outside of Q1-(3*IQR) and Q3+(3*IQR) Observation "far away" from the rest of the data 39

40. 5.2-B. Why are outliers a problem? May indicate a sample peculiarity or a data entry error or other problem ; Regression coefficients estimated that minimize the Sum of Squares for Error (SSE) are very sensitive to outliers >>Bias or distortion of estimates; Any statistical test based on sample means and variances can be distorted In the presence of outliers >>Distortion of p- values; Faulty conclusions. Example: ( Estimators not sensitive to outliers are said to be robust ) Sorted DataMedianMeanVariance95% CI for mean Real Data 1 3 5 9 1256.020.6[0.45, 11.55] Data with Error 1 3 5 9 120527.62676.8[-36.630,91.83] 40

41. 5.2-C. Mathematical Definition  Outlier The standardized residual is given by If |e i *|>2, then the corresponding observation may be regarded an outlier. Example: (Tire Tread Wear vs. Mileage) STUDENTIZED RESIDUAL: a type of standardized residual calculated with the current observation deleted from the analysis. The LS fit can be excessively influenced by observation that is not necessarily an outlier as defined above. i123456789 ei*ei*2.25-0.12-0.66-1.02-0.83-0.57-0.400.431.51 41

42. 5.2-C. Mathematical Definition  Influential Observation Observation with extreme x-value, y-value, or both. On average h ii is (k+1)/n, regard any h ii >2(k+1)/n as high leverage; If x i deviates greatly from mean x, then h ii is large; Standardized residual will be large for a high leverage observation; Influence can be thought of as the product of leverage and outlierness. Example: (Observation is influential/ high leverage, but not an outlier) eg.1 with without eg.2 scatter plot residual plot 42

43. 5.2-C. SAS code of the tire example SAS code Data cars; Input x y; Datalines; 0 394.33 4 329.50 8 291.00 12 255.17 16 229.33 20 204.83 24 179.00 28 163.83 32150.33 ; Run; proc reg data=cars; model y=x; output out=resid rstudent=r h=lev cookd=cd dffits=dffit; Run; 43

44. 5.2-C. SAS code of the tire example SAS code – continued proc print data=resid; where abs(r)>=2 or lev>(4/9) or cd>(4/9) or abs(dffit)>(2*sqrt(1/9)); run; 44

45. 5.2-C. SAS output of the tire example SAS output 45

46. 5.2-D. How to deal with Outliers & Influential Observations Investigate (Data errors? Rare events? Can be corrected?) Ways to accommodate outliers  Non Parametric Methods (robust to outliers)  Data Transformations  Deletion (or report model results both with and without the outliers or influential observations to see how much they change) 46

47. 5.3 Data Transformations Reason To achieve linearity To achieve homogeneity of variance To achieve normality or symmetry about the regression equation 47

48. Types of Transformation Linearzing Transformation transformation of a response variable, or predicted variable, or both, which produces an approximate linear relationship between variables. Variance Stabilizing Transformation make transformation if the constant variance assumption is violated 48

49. Linearizing Transformation Use mathematical operation, e.g. square root, power, log, exponential, etc. Only one variable needs to be transformed in the simple linear regression. Which one? Predictor or Response? Why? 49

50. XiYi ^ log Yi ^ Y = ^ exp (logYi ) Ei 0394.335.926374.6419.69 4329.505.807332.58-3.08 8291.005.688295.24-4.24 12255.175.569262.09-6.92 16229.335.450232.67-3.34 20204.835.331206.54-1.71 24179.005.211183.36-4.36 28163.835.092162.771.06 32150.334.973144.505.83 e.g. We take a log transformation on Y =  exp (-  x) log Y = log  -  x 50

51. Variance Stabilizing Transformation Delta method : Two terms Taylor-series approximations Var( h(Y)) ≈ [h  2 g 2 (  where  Var(Y) = g 2  Y) =   set [h’(  ] 2 g 2 (  2. h’(  = 3. h  =  h(y) = e.g. Var(Y) = c 2  2, where c > 0, g  = c  ↔ g(y) = cy h(y) =  Therefore it is the logarithmic transformation 51

52. 6. Correlation Analysis Pearson Product Moment Correlation: a measurement of how closely two variables share a linear relationship. Useful when it is not possible to determine which variable is the predictor and which is the response. Health vs wealth. Which is predictor? Which is response? 52

53. Statistical Inference on the Correlation Coefficient ρ We can derive a test on the correlation coefficient in the same way that we have been doing in class. Assumptions X, Y are from the bivariate normal distribution Start with point estimator r: sample correlation coefficient: estimator of the population correlation coefficient ρ Get the pivotal quantity The distribution of r is quite complicated T 0 : test statistic for ρ = 0 Do we know everything about the p.q.? Yes: T 0 ~ t n-2 under H 0 : ρ=0 53

54. Bivariate Normal Distribution pdf: Properties μ 1, μ 2 means for X, Y σ 1 2, σ 2 2 variances for X, Y ρ the correlation coeff between X, Y 54

55. Derivation of T 0 Therefore, we can use t as a statistic for testing against the null hypothesis H 0 : β 1 =0 Equivalently, we can test against H 0 : ρ=0 55 Are these equivalent? Substitute After a few steps we can see that these two t-tests are indeed equivalent

56. Exact Statistical Inference on ρ Test H 0 : ρ=0, H a : ρ≠0 Test statistic: Reject H 0 iff Example A researcher wants to determine if two test instruments give similar results. The two test instruments are administered to a sample of 15 students. The correlation coefficient between the two sets of scores is found to be 0.7. Is this correlation statistically significant at the.01 level? H 0 : ρ=0, H a : ρ≠0 for α =.01, 3.534 = t 0 > t 13,.005 = 3.012 ▲ Reject H 0 56

57. Approximate Statistical Inference on ρ There is no exact method of testing ρ vs an arbitrary ρ 0 Distribution of R is very complicated T 0 ~ t n-2 only when ρ = 0 To test ρ vs an arbitrary ρ 0 one can use Fisher’s transformation Therefore, let 57

58. Approximate Statistical Inference on ρ Test : Sample estimate: Z test statistic: CI for ρ: We reject H 0 if |z 0 | > z α/2 58

59. Approximate Statistical Inference on ρ using SAS Code: Output: 59

60. Pitfalls of Regression and Correlation Analysis Correlation and causation Ticks cause good health Coincidental data Sun spots and republicans Lurking variables Church, suicide, population Restricted range Local, global linearity 60

61. 7. Simple Linear Regression in Matrix Form Linear regression model The normal equations: vector of response X matrix vector of errors vector of parameters  Using summa- tions Using matrix notation: →

62. Matrix Form of Multiple Regression by LS or y = X  +  in short LS criterion is: The LS solutions are:

63. Summary Linear regression analysis The Least squares (LS) estimates:   and   Correlation Coefficient ρ and r Probabilistic model for Linear regression: Confidence Interval & Prediction interval Outliers? Influential Observations? Data Transformations? Correlation Analysis Model Assumptions 63

64. Least Squares (LS) Fit Sample correlation coefficient r Statistical inference on ß 0 & ß 1 Prediction Interval Model Assumptions Correlation Analysis Linearity Constant Variance Normality Independence 64

65. 65 Homework 5. Due Thursday, 9/24/2015 Please solve the following two questions using SAS

66. Homework 5. Continued 66