# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

4 Pages

### appndx_b

Course: ETD 02112000, Fall 2009
School: Virginia Tech
Rating:

Word Count: 1022

#### Document Preview

B Appendix Estimation of Parameters and Baseline Hazard Function in Proportional Hazards Model B.1 Estimation of the Coefficient Among the proposed methods for estimating the parameters , those of marginal and partial likelihood are the most commonly used in practice. Since the partial likelihood method developed by Cox (1972) is considered to be the most general of the existing estimation techniques and will be...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Virginia >> Virginia Tech >> ETD 02112000

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
B Appendix Estimation of Parameters and Baseline Hazard Function in Proportional Hazards Model B.1 Estimation of the Coefficient Among the proposed methods for estimating the parameters , those of marginal and partial likelihood are the most commonly used in practice. Since the partial likelihood method developed by Cox (1972) is considered to be the most general of the existing estimation techniques and will be used in the data analysis involved in the present study, further details on its derivation are given in this section. Consider the following two scenarios. In scenario 1 an item fails at 100 hours. In scenario 2 an item fails at 100 hours given that it has survived for 99 hours. In the first scenario we use the failure time density to evaluate the corresponding probability function. In the second, we need to use the hazard function to evaluate the conditional probability. Now let us consider the event that item i fails from the survivor or risk set R( j ) and R( j ) = {k: tk j } in which: tk is the failure time for item k and 1 < 2 < ... < n denotes the ordered failure times of the n pipes. To evaluate the corresponding probability, we first evaluate the probability that one item fails in the survivor set R( j ) which is given by: P[one item fails at t from R( j )] = P[item k fails at time j from R( j )] = = kR(t) P[ j t k < j + t | t k j ] h k (t )t k kR ( j ) Consider the probability, P[item `i' fails at time j given that it belongs to R( j ) | one item fails in R( j )] in which R( j ) ={k: tk j } which in turn can be evaluated as P[item k R( j ) fails] P[one item fails in R( j )] = = P[ j t k < j + t] P[t k t] P[one item fails in R( j )] h i ( j ) kR( j ) h k ( j ) By regarding the selection of an item from the risk set at its observed failure time as an independent event, a joint probability statement more appropriately called the likelihood function is written as: 108 L(; t 1 , t 2 ,..., t n ) = i =1 n kR(t i ) h i (t ) h k (t) By applying the proportional hazards model structure, h(t;z) = h 0 (t) (z ; ) , the likelihood function becomes n n (B.1.1) h 0 ( j ) exp(z i ) exp( z i ) L() = = h ( ) exp(z k ) j=1 exp( z k ) j=1 0 j kR( j ) kR( j ) Several methods have also been proposed for dealing with the problem of ties in failure times [Cox and Oaks (1984), Kalbfleisch and Prentice (1980)]. In the approximate likelihood function in the case of ties as proposed by Breslow (1974), the partial likelihood function is written as: n exp(s i ) L() = m i i =1 exp(z j ) jR ( j ) where mi is the number of failures at i and s i is the vector sum of the covariates of the mi items. The conditional log likelihood obtained from equation (B.1.1) is given by (B.1.2) n n n = l L() = z i - log exp( z k ) i i =1 i =1 kR ( j ) i =1 In order to obtain maximum likelihood estimates the of first and second derivatives of li are first calculated. If zir denotes the value of the rth component of the explanatory variable z on the ith subject, then we have: (B.1.3) z exp( T z ) li kR (i ) = z ir - lr exp( T z k ) kR ( i ) kr k = z ir - A ir () Also: li kR (i ) = z ir - r s exp( T z k ) kR ( i ) 2 z kr exp( T z k ) + A ir ()A is () = -C irs () For all risk sets i = 1, ... , k, we obtain from equation (B.1.3) the score function Ur( ) for the rth component: 109 li k = z ir - A ir () r i =1 i =1 The information matrix I( ) of negative second derivatives has elements: U r () = k ( ) (B.1.4) I rs () = C irs () i =1 k (B.1.5) Maximum likelihood estimates of can be obtained by iterative use of equation (B.1.4) and (B.1.5). The Newton-Raphson iterative algorithm can be applied to obtain the maximum likelihood estimates of . B.2 Estimation of the Baseline Hazard Function By referring to equation (5.1.5), it is seen that we still need to estimate the baseline hazard function h0(t). From equation (5.1.2), it is seen that the baseline hazard function represents the hazard rate that a piece of equipment would experience if all the values of the covariates are equal to zero. We can say that the baseline hazard function is the hazard rate when the covariates have no influence on the failure pattern. The baseline hazard function can be estimated by first obtaining a cumulative baseline hazard function. First, compute the cumulative hazard function as follows. From the given data we know the observed failure times, j . Then for a fixed value of t, H0(t) is estimated by (B.2.1) dj ^ H 0 (t ) = ^ < t exp( z k ) j kR ( j ) where, j : observed and ordered failure times dj : number...

Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

Virginia Tech - ETD - 10022000
Chapter 2 Basic dusty plasma principles2.1 Deby shieldingA fundamental characteristic of the behavior of the plasma is its ability to shield out electric potentials that are applied to it. Suppose an electric field is applied into a plasma by putt
Virginia Tech - ETD - 10022000
Chapter 1 IntroductionDusty plasmas are low-temperature multispecies ionized gases including electrons, ions, and negatively (or positively) charged dust grains typically micrometer or submicrometer size. In such circumstances, the dust particles ca
Virginia Tech - ETD - 10172003
CHAPTER TWO DESIGN METHODS AND PROCESS This chapter reviews the basic concepts of design problems, methods, and processes. It presents the pertinent research studies on site analysis and discusses its position in the design process. The chapter also
Virginia Tech - ETD - 100699
2.0 Research Objectives and Thesis Organization2.1Problem DefinitionQuantitative feedback of error is a prerequisite for any attempt to interactively modify or match a surface patch. The degree to which a designer is successful in modeling a pa
Virginia Tech - ETD - 05062002
Chapter 6Comparison of Analysis with FLAC Analysis and Experimental Test6.1 Introduction This chapter compares the results formulated for the tube with an apron attached and constant cross-sectional area with a FLAC analysis performed by another
Virginia Tech - ESM - 12793
SF =Ultimate Load Design Load orSF =The Load or Stress that should cause failure The Maximum Load or Stress that will be permittedWe may need to consider the Safety Factor for multiple parameters: Safety Factor on Yield Stress Safety Factor
Virginia Tech - ESM - 12803
5.4 Anisotropic Materials Isotropic: properties the same in all directions. Homogeneous: properties the same from point to point. &quot;An&quot; Greek, Latin for `not', `without', `lacking' Anisotropic &quot;not isotropic&quot;Figure 5.14 Anisotropic materials: (a)
Virginia Tech - ESM - 12812
Name: _ (last, first)ESM 2204 Sample Final Exam December xx, xxxxClosed book, closed notes, and one 8.5&quot; 11&quot; formula sheet allowed. Please make sure your test includes 7 multiple choice problems and one work-out problem. Show all work on this te
Virginia Tech - ESM - 12816
COURSE: ESM 2304 Dynamics of Particles and Rigid Bodies, Spring Semester, 2008 TEXTBOOK: Engineering Mechanics: Dynamics, Sixth Edition (2007), by Meriam and Kraige (Note: You cannot use any previous edition.) INSTRUCTOR: L. Glenn Kraige OFFICE: 324
Virginia Tech - MATH - 1224
RECITATION 9 ^ 1. Consider the trajectory r(t) = (cos t + t sin t)^ + (sin t - t cos t)^ + t2 k. i j 1 r (t) |r (t)|^ (a) By differentiating r(t) and then normalizing to have length one, find the unit tangent vector T : ^ T =^ (b) By differentiat
Virginia Tech - Y - 05
Math5524- Homework7- Orthogonalization - Konat e Problem 1: Consider in E = R4 the classical inner productn&lt; x, y &gt;=n=1xi yi ;and the two vectors u1 = (1, 4, 0, 2)T ;u2 = (2, -2, 1, 3)T .1.1: Find the inner product of u1 and u2 . 1.2: Fin
Virginia Tech - MATH - 5726
Template for the SOR MethodThis is a more extensive description of the template for the SOR method posted as template.m. Please note that a &quot;typewriter&quot; font is used for variable names as they appear in the Matlab code (like Mat, u), while the usua
Virginia Tech - MATH - 5725
Math 5415 Homework for Chapters 20+: Interest Rate ModelingProblem J. Consider the Vasiek model for the short rate r(t): (22.28). c a) Let y(t) = eat r(t), work out dy(t) and simplify. By integrating both sides, obtain an expression for r(t). (You
Virginia Tech - MATH - 5726
Organization of efd.mThe explicit finite difference method for the heat equation is carried out by the script efd.m. It assumes that the following variables and function m-files are defined and produce the desired values. 1. a,b are the upper and l
Virginia Tech - MATH - 5726
Growth Conditions and Risk-Neutral PricingWe said in class that in order for a solution u(x, ) of the heat equation to be given by the integral formula (FS) we needed to know that, for each time interval 0 &lt; &lt; T , The solution satises a growth of
Virginia Tech - MATH - 5726
Math 5726 Homework Set #1Problem A. Verify that each of the following is a solution of the heat equation: 1. u(x, ) = (4 )-1/2 e-x 2. u(x, ) = e-c2 2/4.sin(cx), for any constant c.3. u(x, ) = erf( 2x ).(The error function erf(x
Virginia Tech - Y - 05
Math5524- Homework6- Vectors spaces - Konat e Problem 1: Consider the matrix 1 2 3 1 0 1 2 A= 2 3 2 4 1 1.1: determine the linear application f the matrix A is associated with. 1.2: determine a basis of the null space N (A) of A and its dimension. 1.
Virginia Tech - Y - 05
Homework3 - LU factorization Problem 1: Consider a gaussian elimination performed on a 3 by 3 matrix which produced the following elimination coefficients and rules: step1: row 1 is multiplied by -2 and then added to row 2; step2: row 1 is multiplied
Virginia Tech - AOE - 2104
AOE 2104: Homework Assignment 1 1. Consider the low-speed ight of the Space Shuttle as it is nearing a landing. If the air pressure and temperature of the shuttle are 1.2 atm and 300 K, respectively, what are the density and specic volume? Solution:
Virginia Tech - MECH - 533
1. Consider a reference frame that undergoes an angular displacement from Fa to Fb about the 3&quot; axis through angle , followed by an angular displacement from Fb to Fc about the 2&quot; axis through angle . Note that is NOT the classical nutation angle. a
Virginia Tech - AOE - 4140
Chapter 3 KinematicsAs noted in the Introduction, the study of dynamics can be decomposed into the study of kinematics and kinetics. For the translational motion of a particle of mass m, this decomposition amounts to expressing Newton's second law,
Virginia Tech - V - 16
Journal of Technology EducationVol. 16 No. 2, Spring 2005Coming to Terms with Engineering Design as ContentTheodore Lewis With the publication of standards for teaching, learning, and the inculcation of technological literacy (International Tech
Virginia Tech - V - 18
Journal of Technology EducationVol. 18 No. 2, Spring 2007Identifying the Paradigm of Design Faculty in Undergraduate Technology Teacher Education in the United StatesScott A. Warner, Laura Morford Erli, Chad W. Johnson, and Scott W. Greiner Davi
Virginia Tech - V - 17
Journal of Technology EducationVol. 17 No. 2, Spring 2006Book Reviewde Vries, M. J. (2005). Teaching about technology: An introduction to the philosophy of technology for non-philosophers. Dordrecht, The Netherlands: Springer. \$129.00 (hardcover
Virginia Tech - V - 11
Journal of Technology EducationVol. 11 No. 1, Fall 1999Talking Technology: Language and Literacy in the Primary School Examined Through Childrens Encounters with MechanismsEric Parkinson The Role of Language and Dedicated Terminology Language pl
Virginia Tech - V - 17
Journal of Technology EducationVol. 17 No. 2, Spring 2006Productivism and the Product Paradigm in Technological EducationLeo Elshof &quot;To keep every cog and wheel is the first precaution of intelligent tinkering&quot; American ecologist Aldo Leopold (1
Virginia Tech - V - 6
Journal of Technology EducationVol. 6 No. 1, Fall 1994Diderot, the Mechanical Arts, and the Encyclopdie: In Search of the Heritage of Technology EducationJohn R. Pannabecker1 In a recent symposium on critical issues in technology education, Walt
Virginia Tech - ETD - 12202002
terms would improve the model fit. Additional regression diagnostics determined that no individual points were outliers or had undue influence on the model shape. Together, v and b had a significant effect on root growth, explaining 33% and 61% of th
Virginia Tech - ETD - 11292001
Appendix A Math Pretest and Posttest961.Assuming that the pattern continues, what are the next 3 terms? 39, 34, 29, _, _, _ A B C D 14, 19, 24 24, 19, 14 28, 24, 20 29, 24, 195.Use the pattern of designs below. What is the total number of r
Virginia Tech - ETD - 08192002
Appendix A Definition of Terms93Definition of Molecular and Biochemical Terms [Taken directly from Glossary of Griffiths et al, 1999] Antigen: A molecule that is recognized by antibody (immunoglobulin) molecules. Generally, multiple antibody mole
Virginia Tech - ETD - 040799
Appendix A Finite Element AnalysisDue to the complexity of the geometry of the test cavity, no analytical solution is available. The finite element method was used to numerically approximate the acoustic behavior of the test cavity. The analysis w