Example prostate cancer data n 97 p 8 used in many

Info icon This preview shows pages 61–63. Sign up to view the full content.

Example: Prostate cancer data ( n = 97 , p = 8) used in many text-books. To study relationship between Y = the level of prostate-specific antigen ( lpsa ) and a num- ber of clinical measures in men who were about to receive a radical prostatectomy : X 1 = log cancer volume ( lcavol ) X 2 = log prostate weight ( lweight ) X 3 = age X 4 = log of the amount of benign prostatic hyperplasia( lbph ), X 5 = seminal vesicle invasion ( svi , binary) X 6 = log of capsular penetration ( lcp ), X 7 = Gleason score ( gleason , ordered categorical) X 8 = percent of Gleason scores 4 or 5 ( pgg45 ) PAGE 61
Image of page 61

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

2.10 Robust Regression c circlecopyrt HYON-JUNG KIM, 2017 Table 1: Coefficient estimates OLS LASSO intercept 0.669 0.355 lcavol 0.587 0.516 lweight 0.454 0.345 age -0.020 lbph 0.107 0.051 svi 0.766 0.567 lcp -0.105 gleason 0.045 pgg45 0.005 0.002 0 0.2 0.4 0.6 0.8 1 -0.2 0 0.2 0.4 0.6 0.8 1 lcavol lweight age lbph svi lcp gleason pgg45 Coe ffi cients normalized bardbl ˆ β λ bardbl 1 Note: The dashed vertical line depicts the minimum BIC value 2.10 Robust Regression Recall that the ordinary least squares estimates for linear regression are optimal when all of the regression assumptions are valid. When the errors are nonnormal, or there are big PAGE 62
Image of page 62
2.10 Robust Regression c circlecopyrt HYON-JUNG KIM, 2017 outliers, least squares regression can perform poorly. Robust regression is not overly affected by violations of assumptions by the underlying data-generating process. Robust loss functions 1. Least absolute deviance (LAD) with L 1 -loss: ρ ( e ) = | e | 2. Huber’s loss function: convex and differentiable ρ k ( e ) = 1 2 e 2 , for | e | ≤ k k | e | − 1 2 k 2 , for | e | > k where c is a user-defined tuning constant that affects robustness and efficiency of the method. 3. Tukey’s biweight loss fucntion ρ k ( e ) = min braceleftBig 1 , 1 (1 ( e/k ) 2 ) 3 bracerightBig 4. Least trimmed squares (LTS) h summationdisplay j =1 ( r 2 ) ( i ) where ( r 2 ) ( i ) are the order statistics of the squared residuals. LTS estimator chooses the regression coefficients minimize the sum of the smallest h of the squared residuals. ï 4 ï 2 0 2 4 0 1 2 3 4 L 1 -loss ï 4 ï 2 0 2 4 0 1 2 3 4 5 Huber loss ï 4 ï 2 0 2 4 0 1 2 3 4 Tukey loss PAGE 63
Image of page 63
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern