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AMS 572 Group #2 1

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Outline Jinmiao Fu—Introduction and History Ning Ma—Establish and Fitting of the model Ruoyu Zhou—Multiple Regression Model in Matrix Notation Dawei Xu and Yuan Shang—Statistical Inference for Multiple Regression Yu Mu—Regression Diagnostics Chen Wang and Tianyu Lu—Topics in Regression Modeling Tian Feng—Variable Selection Methods Hua Mo—Chapter Summary and modern application 4

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Introduction Multiple linear regression attempts to model the relationship between two or more explanatory variables and a response variable by fitting a linear equation to observed data. Every value of the independent variable x is associated with a value of the dependent variable 5

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Example: The relationship between an adult’s health and his/her daily eating amount of wheat, vegetable and meat. 6

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History 7

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Correlation coefficient Method of moments Pearson's system of continuous curves. Karl Pearson (1857–1936) Lawyer, Germanist, eugenicist, mathematician and statistician Chi distance, P-value Statistical hypothesis testing theory, statistical decision theory. Pearson's chi-square test, Principal component analysis. 8

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Sir Francis Galton FRS (16 February 1822 – 17 January 1911) Anthropology and polymathy Doctoral students Karl Pearson In the late 1860s, Galton conceived the standard deviation. He created the statistical concept of correlation and also discovered the properties of the bivariate normal distribution and its relationship to regression analysis 9

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Galton invented the use of the regression line (Bulmer 2003, p. 184), and was the first to describe and explain the common phenomenon of regression toward the mean, which he first observed in his experiments on the size of the seeds of successive generations of sweet peas. 10

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The publication by his cousin Charles Darwin of The Origin of Species in 1859 was an event that changed Galton's life. He came to be gripped by the work, especially the first chapter on "Variation under Domestication" concerning the breeding of domestic animals. 11

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Adrien-Marie Legendre (18 September 1752 – 10 January 1833) was a French mathematician. He made important contributions to statistics, number theory, abstract algebra and mathematical analysis. He developed the least squares method, which has broad application in linear regression, signal processing, statistics, and curve fitting. 12

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Johann Carl Friedrich Gauss (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics. 13

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Gauss, who was 23 at the time, heard about the problem and tackled it. After three months of intense work, he predicted a position for Ceres in December 1801—just about a year after its first sighting—and this turned out to be accurate within a half-degree. In the process, he so streamlined the cumbersome mathematics of 18th century orbital prediction that his work—published a few years later as Theory of Celestial Movement—remains a cornerstone of astronomical computation. 14

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It introduced the Gaussian gravitational constant, and contained an influential treatment of the method of least squares, a procedure used in all sciences to this day to minimize the impact of measurement error. Gauss was able to prove the method in 1809 under the assumption of normally distributed errors (see Gauss–Markov theorem; see also Gaussian). The method had been described earlier by Adrien-Marie Legendre in 1805, but Gauss claimed that he had been using it since

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Sir Ronald Aylmer Fisher FRS (17 February 1890 – 29 July 1962) was an English statistician, evolutionary biologist, eugenicist and geneticist. He was described by Anders Hald as "a genius who almost single-handedly created the foundations for modern statistical science," and Richard Dawkins described him as "the greatest of Darwin's successors". 16

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In addition to "analysis of variance", Fisher invented the technique of maximum likelihood and originated the concepts of sufficiency, ancillarity, Fisher's linear discriminator and Fisher information. 17

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Probabilistic Model : the observed value of the random variable(r.v.) unknown model parameters depends on fixed predictor values n is the number of observations. ~ N (0, ) i.i.d,i=1,2,3,…,n 19

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Fitting the model LS provides estimates of the unknown model parameters, which minimizes Q (j=1,2, …,k) 20

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Tire tread wear vs. mileage (example11.1 in textbook) Mileage (in 1000 miles) Groove Depth (in mils) The table gives the measurements on the groove of one tire after every 4000 miles. Our Goal: to build a model to find the relation between the mileage and groove depth of the tire. 21

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Data example ； Input mile depth ； Sqmile=mile*mile ； Datalines; ; run; Proc reg data=example; Model Depth= mile sqmile; Run; SAS code----fitting the model 22

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Depth= mile+0.172sqmile 23

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Goodness of Fit of the Model Residuals are the fitted values total sum of squares (SST): regression sum of squares (SSR): An overall measure of the goodness of fit Error sum of squares (SSE): 24

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1. Transform the Formulas to Matrix Notation 26

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2. Example 11.2 (Tire Wear Data: Quadratic Fit Using Hand Calculations) 30

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Statistical Inference for Multiple Regression Determine which predictor variables have statistically significant effects We test the hypotheses: If we can’t reject H 0j, then x j is not a significant predictor of y. 35

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Review statistical inference for Simple Linear Regression Statistical Inference on 36

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Statistical Inference on What about Multiple Regression? The steps are similar 37

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Statistical Inference on What’s V jj ? Why ? 1. Mean Recall from simple linear regression, the least squares estimators for the regression parameters and are unbiased. Here, of least squares estimators is also unbiased. 38

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Statistical Inference on 2.Variance Constant Variance assumption: – 39

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Statistical Inference on Let V jj be the j th diagonal of the matrix 40

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Statistical Inference on 41

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Statistical Inference on 42

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Statistical Inference on Therefore, 43

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Statistical Inference on Derivation of confidence interval of The 100(1-α)% confidence interval for is 44

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Statistical Inference on Rejects H 0j if 45

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Prediction of Future Observation Having fitted a multiple regression model, suppose we wish to predict the future value of Y for a specified vector of predictor variables x*=(x 0 *,x 1 *,…,x k *) One way is to estimate E(Y*) by a confidence interval(CI). 46

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Prediction of Future Observation 47

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F-Test for Consider: Here is the overall null hypothesis, which states that none of the variables are related to. The alternative one shows at least one is related. 48

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How to Build a F-Test…… The test statistic F=MSR/MSE follows F- distribution with k and n-(k+1) d.f. The α -level test rejects if recall that MSE(error mean square) with n-(k+1) degrees of freedom. 49

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The relation between F and r F can be written as a function of r. By using the formula: F can be as: We see that F is an increasing function of r ² and test the significance of it. 50

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Analysis of Variance (ANOVA) The relation between SST, SSR and SSE: where they are respectively equals to: The corresponding degrees of freedom(d.f.) is: 51

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ANOVA Table for Multiple Regression Source of Variation (source) Sum of Squares (SS) Degrees of Freedom (d.f.) Mean Square (MS) F Regression Error SSR SSE k n-(k+1) TotalSSTn-1 This table gives us a clear view of analysis of variance of Multiple Regression. 52

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Extra Sum of Squares Method for Testing Subsets of Parameters Before, we consider the full model with k parameters. Now we consider the partial model: while the rest m coefficients are set to zero. And we could test these m coefficients to check out the significance: 53

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Building F-test by Using Extra Sum of Squares Method Let and be the regression and error sums of squares for the partial model. Since SST Is fixed regardless of the particular model, so: then, we have: The α-level F-test rejects null hypothesis if 54

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Remarks on the F-test The numerator d.f. is m which is the number of coefficients set to zero. While the denominator d.f. is n-(k+1) which is the error d.f. for the full model. The MSE in the denominator is the normalizing factor, which is an estimate of σ² for the full model. If the ratio is large, we reject. 55

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Links between ANOVA and Extra Sum of Squares Method Let m=1 and m=k respectively, we have: From above we can derive: Hence, the F-ratio equals: with k and n-(k+1) d.f. 56

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5 Regression Diagnostics 5.1 Checking the Model Assumptions Plots of the residuals against individual predictor variables: check for linearity A plot of the residuals against fitted values: check for constant variance A normal plot of the residuals: check for normality 58

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A run chart of the residuals: check if the random errors are auto correlated. Plots of the residuals against any omitted predictor variables: check if any of the omitted predictor variables should be included in the model. 59

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Example: Plots of the residuals against individual predictor variables 60

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SAS code 61

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Example: plot of the residuals against fitted values 62

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SAS code 63

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Example: normal plot of the residuals 64

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SAS code 65

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5.2 Checking for Outliers and Influential Observations Standardized residuals Large values indicate outlier observation. Hat matrix If the Hat matrix diagonal, then ith observation is influential. 66

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Example: graphical exploration of outliers 67

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Example: leverage plot 68

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5.3 Data transformation Transformations of the variables(both y and the x’s) are often necessary to satisfy the assumptions of linearity, normality, and constant error variance. Many seemingly nonlinear models can be written in the multiple linear regression model form after making a suitable transformation. For example, after transformation: or 69

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Multicollinearity Multicollinearity occurs when two or more predictors in the model are correlated and provide redundant information about the response. Example of multicollinear predictors are height and weight of a person, years of education and income, and assessed value and square footage of a home. Consequences of high multicollinearity: a. Increased standard error of estimates of the β ’s b. Often confused and misled results. 71

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Detecting Multicollinearity Easy way: compute correlations between all pairs of predictors. If some r are close to 1 or -1, remove one of the two correlated predictors from the model. Equal to 1 Correlations X1X1 colinear X2X2 independent X3X3 X2X2 72

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Detecting Multicollinearity Another way: calculate the variance inflation factors for each predictor xj: where is the coefficient of determination of the model that includes all predictors except the jth predictor. If VIFj≥10, then there is a problem of multicollinearity. 73

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Muticollinearity-Example See Example11.5 on Page 416, Response is the heat of cement on a per gram basis (y) and predictors are tricalcium aluminate(x1), tricalcium silicate(x2), tetracalcium alumino ferrite(x3) and dicalcium silicate(x4). 74

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Muticollinearity-Example Estimated parameters in first order model: ˆy = x x x x 4. F = with p−value below Individual t−statistics and p−values: 2.08 (0.071), 0.7 (0.501) and 0.14 (0.896), (0.844). Note that sign on β 4 is opposite of what is expected. And very high F would suggest more than just one significant predictor. 75

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Muticollinearity-Example Correlations Correlations were r13 = , r24 = Also the VIF were all greater than 10. So there is a multicollinearity problem in such model and we need to choose the optimal algorithm to help us select the variables necessary. 76

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Muticollinearity-Subsets Selection Algorithms for Selecting Subsets – All possible subsets Only feasible with small number of potential predictors (maybe 10 or less) Then can use one or more of possible numerical criteria to find overall best – Leaps and bounds method Identifies best subsets for each value of p Requires fewer variables than observations Can be quite effective for medium-sized data sets Advantage to have several slightly different models to compare 77

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Muticollinearity-Subsets Selectioin – Forward stepwise regression Start with no predictors – First include predictor with highest correlation with response – In subsequent steps add predictors with highest partial correlation with response controlling for variables already in equations – Stop when numerical criterion signals maximum (minimum) – Sometimes eliminate variables when t value gets too small Only possible method for very large predictor pools Local optimization at each step, no guarantee of finding overall optimum – Backward elimination Start with all predictors in equation – Remove predictor with smallest t value – Continue until numerical criterion signals maximum (minimum) Often produces different final model than forward stepwise method 78

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Muticollinearity-Best Subsets Criteria Numerical Criteria for Choosing Best Subsets – No single generally accepted criterion Should not be followed too mindlessly – Most common criteria combine measures of with add penalties for increasing complexity (number of predictors) – Coefficient of determination Ordinary multiple R-square Always increases with increasing number of predictors, so not very good for comparing models with different numbers of predictors – Adjusted R-Square Will decrease if increase in R-Square with increasing p is small 79

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Muticollinearity-Best Subsets Criteria – Residual mean square (MSEp) Equivalent to adjusted r-square except look for minimum Minimum occurs when added variable doesn't decrease error sum of squares enough to offset loss of error degree of freedom – Mallows' Cp statistic Should be about equal to p and look for small values near p Need to estimate overall error variance – PRESS statistic The one associated with the minimum value of PRESSp is chosen Intuitively easier to grasp than the Cp-criterion. 80

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Muticollinearity-Forward Stepwise First include predictor with highest correlation with response >FIN=4 81

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Muticollinearity-Forward Stepwise In subsequent steps add predictors with highest partial correlation with response controlling for variables already in equations. (if Fi>FIN=4, enter the Xi and Fi

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Muticollinearity-Forward Stepwise

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Muticollinearity-Forward Stepwise Summarize the stepwise algorithms Therefore our “Best Model” should only include x 1 and x 2, which is y= x x 2 84

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Muticollinearity-Forward Stepwise Check the significance of the model and individual parameter again. We find p value are all small and each VIF is far less than

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Muticollinearity-Best Subsets Also we can stop when numerical criterion signals maximum (minimum) and sometimes eliminate variables when t value gets too small. 86

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Muticollinearity-Best Subsets The largest R squared value is associated with the full model. The best subset which minimizes the Cp-criterion includes x 1,x 2 The subset which maximizes Adjusted R squared or equivalently minimizes MSEp is x 1,x 2,x 4. And the Adjusted R squared increases only from to by the addition of x 4 to the model already containing x 1 and x 2. Thus the simpler model chosen by the Cp-criterion is preferred, which the fitted model is y= x x 2 87

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Polynomial model Polynomial models are useful in situations where the analyst knows that curvilinear effects are present in the true response function. We can do this with more than one explanatory variable using Polynomial regression model: 88

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Multicollinearity-Polynomial Models Multicollinearity is a problem in polynomial regression (with terms of second and higher order): x and x 2 tend to be highly correlated. A special solution in polynomial models is to use zi = xi − ¯xi instead of just xi. That is, first subtract each predictor from its mean and then use the deviations in the model. 89

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Multicollinearity – Polynomial model Example: x = 2, 3, 4, 5, 6 and x 2 = 4, 9, 16, 25, 36. As x increases, so does x 2. rx,x 2 = = 4 then z = −2,−1, 0, 1, 2 and z 2 = 4, 1, 0, 1, 4. Thus, z and z 2 are no longer correlated. rz,z 2 = 0. We can get the estimates of the β’s from the estimates of the γ ’s. Since 90

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Dummy Predictor Variable The dummy variable is a simple and useful method of introducing into a regression analysis information contained in variables that are not conventionally measured on a numerical scale, e.g., race, gender, region, etc. 91

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Dummy Predictor Variable The categories of an ordinal variable could be assigned suitable numerical scores. A nominal variable with c≥2 categories can be coded using c – 1 indicator variables, X1,…,Xc-1, called dummy variables. Xi=1, for ith category and 0 otherwise X1=,…,=Xc-1=0, for the c th category 92

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Dummy Predictor Variable If y is a worker’s salary and Di = 1 if a non-smoker Di = 0 if a smoker We can model this in the following way: 93

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Dummy Predictor Variable Equally we could have used the dummy variable in a model with other explanatory variables. In addition to the dummy variable we could also add years of experience (x), to give: For smoker For non-smoker 94

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Dummy Predictor Variable Non-smoker Smoker α α+β y x 95

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Dummy Predictor Variable We can also add the interaction to between smoking and experience with respect to their effects on salary. For non-smoker For smoker 96

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Dummy Predictor Variable Non-smoker Smoker α α+β y x 97

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Standardized Regression Coefficients We typically wants to compare predictors in terms of the magnitudes of their effects on response variable. We use standardized regression coefficients to judge the effects of predictors with different units 98

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Standardized Regression Coefficients They are the LS parameter estimates obtained by running a regression on standardized variables, defined as follows: Where and are sample SD’s of and 99

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Standardized Regression Coefficients Let And The magnitudes of can be directly compared to judge the relative effects of on y. 100

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Standardized Regression Coefficients Since, the constant can be dropped from the model. Let be the vector of the and be the matrix of 101

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Standardized Regression Coefficients So we can get This method of computing is numerically more stable than computing directly, because all entries of R and r are between -1 and

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Standardized Regression Coefficients Example (Given in page 424) From the calculation, we can obtain that And sample standard deviations of x1,x2 and are Then we have Note that,although.Thus x1 has a larger effect than x2 on y. 103

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Standardized Regression Coefficients We can also use the matrix method to compute standardized regression coefficients. First we compute the correlation matrix between x1,x2 and y Then we have Next calculate Hence Which is as same result as before 104

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How to decide their salaries? Lionel Messi 10,000,000 EURO/yr Carles Puyol 5,000,000 EURO/yr Attacker Defender 5 years 11 years more than 20 goals per year less than 1 goals per year 106

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How to select variables? 1) Stepwise Regression 2)Best Subset Regression 107

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Stepwise Regression Partial F-test Partial Correlation Coefficients How to do it by SAS? Drawbacks 108

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Partial F-test (p-1)-Variable Model: p-Variable Model: 109

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How to do the test? We reject in favor of at level α if 110

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Another way to interpret the test: test statistics: We reject at level α if 111

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Partial Correlation Coeffientients test statistics: *Add to the regression equation that includes only if is large enough. 112

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How to do it by SAS? (EX9 Continuity of Ex5) No.X1X2X3X4Y The table shows data on the heat evolved in calories during the hardening of cement on a per gram basis (y) along with the percentages of four ingredients: tricalcium aluminate (x1), tricalcium silicate (x2), tetracalcium alumino ferrite (x3), and dicalcium silicate (x4). 113

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SAS Code data example1; input x1 x2 x3 x4 y; datalines; ; Run; proc reg data=example1; model y= x1 x2 x3 x4 /selection=stepwise; run; 114

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SAS output 115

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SAS output 116

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Interpretation At the first step, x4 is chosen into the equation as it has the largest correlation with y among the 4 predictors; At the second step, we choose x1 into the equation for it has the highest partial correlation with y controlling for x4; At the third step, since is greater than, x2 is chosen into the equation rather than x3. 117

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At the 4th step, we removed x4 from the model since its partial F-statistics is too small. From Ex11.5, we know that x4 is highly correlated with x2. Note that in Step4, the R-Square is , which is slightly higher that , the R-Square of Step 2. It indicates that even x4 is the best predictor of y, the pair (x1,x2) is a better predictor than the predictor (x1,x4). Interpretation 118

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Drawbacks The final model is not guaranteed to be optimal in any specified case. It yields a single final model while in practice there are often several equally good model. 119

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Best Subset Regression Comparison to Stepwise Method Optimality Criteria How to do it by SAS? 120

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In best subsets regression, a subset of variables is chosen from that optimizes a well- defined objective criterion. The best regression algorithm permits determination of a specified number of best subsets from which the choice of the final model can be made by the investigator. Comparison to Stepwise Regression 121

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Optimality Criteria 122

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Optimality Criteria Standardized mean square error of prediction: involves unknown parameters such as ‘s, so minimize a sample estimate of. Mallows’ : 123

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It practice, we use the because of its ease of computation and its ability to judge the predictive power of a model. Optimality Criteria 124

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How to do it by SAS?(Ex11.9) proc reg data=example1; model y= x1 x2 x3 x4 /selection=adjrsq mse cp; run; 125

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SAS output 126

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The best subset which minimizes the is x1, x2 which is the same model selected using stepwise regression in the former example. The subset which maximizes is x1, x2, x4. However, increases only from to by the addition of x4 to the model which already contains x1 and x2. Thus, the model chosen by the is preferred. Interpretation 127

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Multiple Regression Model Fitting the MLR Model MLR Model in Matrix Notation are unknown parameters Model (Extension of Simple Regression): Least squares method : Goodness of fit of the model: 129

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Statistical Inference for Multiple Regression Regression Diagnostics Hypotheses: Test statistic: At least one Statistical Inference on vs. Hypotheses:vs. Test statistic: Residual Analysis Data Transformation 130

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Compare the full model: the partial model: Hypotheses: vs. Test statistic: RejectH 0 when The General Hypothesis Test: Estimating and Predicting Future Observations: Let and Test statistic: CI for the estimated mean * : PI for the estimated Y * : 131

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Topics in regression modeling Multicollinearity Polynomial Regression Dummy Predictor Variables Logistic egression Model Variable Selection Methods Stepwise Regression: Stepwise Regression Algorithm Best Subsets Regression Strategy for building a MLR model partial F-test partial Correlation Coefficient 132

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Application of the MLR model Linear regression is widely used in biological, chemistry, finance and social sciences to describe possible relationships between variables. It ranks as one of the most important tools used in these disciplines. 133

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Multiple linear regression Chemistry heredity Financial market biology Housing price 134

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Broadly speaking, an asset pricing model can be expressed as: Example Where, and k denote the expected return on asset i, the kth risk factor and the number of risk factors, respectively. denotes the specific return on asset i. 135

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The equation can also be expressed in the matrix notation: is called the factor loading 136

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What’s the most important factors? Interest rate Inflation rate Employment rate Rate of return on the market portfolio Government policies GDP 137

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Method Step 1: Find the efficient factors (EM algorithms, maximum likelihood) Step 2: Fit the model and estimate the factor loading (Multiple linear regression) 138

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According to the multiple linear regression and run data on SAS, we can get the factor loading and the coefficient of multiple determination We can ensure the factors that mostly effect the return in term of SAS output and then build the appropriate multiple factor models We can use the model to predict the future return and make a good choice! 139

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