1 A General Class of Models for Recurrent Events Edsel A. Pena University of South Carolina at Columbia [ Research support from.

Slides:



Advertisements
Similar presentations
COMPUTER INTENSIVE AND RE-RANDOMIZATION TESTS IN CLINICAL TRIALS Thomas Hammerstrom, Ph.D. USFDA, Division of Biometrics The opinions expressed are those.
Advertisements

Agency for Healthcare Research and Quality (AHRQ)
Topic: Several Approaches to Modeling Recurrent Event Data Presenter: Yu Wang.
1 Non-Linear and Smooth Regression Non-linear parametric models: There is a known functional form y=  x,  derived from known theory or from previous.
1 Goodness-of-Fit Tests with Censored Data Edsel A. Pena Statistics Department University of South Carolina Columbia, SC [ Research.
3.2 OLS Fitted Values and Residuals -after obtaining OLS estimates, we can then obtain fitted or predicted values for y: -given our actual and predicted.
The General Linear Model Or, What the Hell’s Going on During Estimation?
HSRP 734: Advanced Statistical Methods July 24, 2008.
Biostat/Stat 576 Chapter 6 Selected Topics on Recurrent Event Data Analysis.
Cox Model With Intermitten and Error-Prone Covariate Observation Yury Gubman PhD thesis in Statistics Supervisors: Prof. David Zucker, Prof. Orly Manor.
Visual Recognition Tutorial
Goodness of Fit of a Joint Model for Event Time and Nonignorable Missing Longitudinal Quality of Life Data – A Study by Sneh Gulati* *with Jean-Francois.
Nonparametric Estimation with Recurrent Event Data Edsel A. Pena Department of Statistics University of South Carolina Research.
Nonparametric Estimation with Recurrent Event Data Edsel A. Pena Department of Statistics University of South Carolina Research.
Nonparametric Estimation with Recurrent Event Data Edsel A. Pena Department of Statistics University of South Carolina Research.
Nonparametric Estimation with Recurrent Event Data Edsel A. Pena Department of Statistics, USC Research supported by NIH and NSF Grants Based on joint.
Development of Empirical Models From Process Data
NACC National Alzheimer’s Coordinating Center Time Dependent Exposure in Case-Control Studies Roger Higdon, PhD Senior Biostatistician NACC, University.
Role and Place of Statistical Data Analysis and very simple applications Simplified diagram of scientific research When you know the system: Estimation.
Biostatistics in Research Practice: Non-parametric tests Dr Victoria Allgar.
On a dynamic approach to the analysis of multivariate failure time data Odd Aalen Section of Medical Statistics, University of Oslo, Norway.
EVIDENCE BASED MEDICINE
Cohort Studies Hanna E. Bloomfield, MD, MPH Professor of Medicine Associate Chief of Staff, Research Minneapolis VA Medical Center.
Chapter 14 Inferential Data Analysis
Survival Analysis A Brief Introduction Survival Function, Hazard Function In many medical studies, the primary endpoint is time until an event.
Survival Analysis: From Square One to Square Two
Exponential Distribution & Poisson Process
1 A General Class of Models for Recurrent Events Edsel A. Pena University of South Carolina at Columbia [ Research support from.
HSTAT1101: 27. oktober 2004 Odd Aalen
Simple Linear Regression
Essentials of survival analysis How to practice evidence based oncology European School of Oncology July 2004 Antwerp, Belgium Dr. Iztok Hozo Professor.
1 Performance Evaluation of Computer Networks: Part II Objectives r Simulation Modeling r Classification of Simulation Modeling r Discrete-Event Simulation.
Dr Laura Bonnett Department of Biostatistics. UNDERSTANDING SURVIVAL ANALYSIS.
On ranking in survival analysis: Bounds on the concordance index
Software Reliability SEG3202 N. El Kadri.
Chap 20-1 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chapter 20 Sampling: Additional Topics in Sampling Statistics for Business.
1 Data Analysis © 2005 Germaine G Cornelissen-Guillaume. Copying and distribution of this document is permitted in any medium, provided this notice is.
Introduction to Research
Various topics Petter Mostad Overview Epidemiology Study types / data types Econometrics Time series data More about sampling –Estimation.
Borgan and Henderson:. Event History Methodology
HSRP 734: Advanced Statistical Methods July 17, 2008.
Construct-Centered Design (CCD) What is CCD? Adaptation of aspects of learning-goals-driven design (Krajcik, McNeill, & Reiser, 2007) and evidence- centered.
HSRP 734: Advanced Statistical Methods July 31, 2008.
PROBABILITY AND STATISTICS FOR ENGINEERING Hossein Sameti Department of Computer Engineering Sharif University of Technology Principles of Parameter Estimation.
1 Data Mining, Data Perturbation and Degrees of Freedom of Projection Regression T.C. Lin * appear in JRSS series C.
Survival Analysis 1 Always be contented, be grateful, be understanding and be compassionate.
Threshold Regression Models Mei-Ling Ting Lee, University of Maryland, College Park
Parametric Conditional Frailty Models for Recurrent Cardiovascular Events in the LIPID Study Dr Jisheng Cui Deakin University, Melbourne.
1 EXAKT SKF Phase 1, Session 2 Principles. 2 The CBM Decision supported by EXAKT Given the condition today, the asset mgr. takes one of three decisions:
Fall 2002Biostat Statistical Inference - Proportions One sample Confidence intervals Hypothesis tests Two Sample Confidence intervals Hypothesis.
N318b Winter 2002 Nursing Statistics Specific statistical tests Chi-square (  2 ) Lecture 7.
Empirical Likelihood for Right Censored and Left Truncated data Jingyu (Julia) Luan University of Kentucky, Johns Hopkins University March 30, 2004.
Assessing Estimability of Latent Class Models Using a Bayesian Estimation Approach Elizabeth S. Garrett Scott L. Zeger Johns Hopkins University Departments.
1 Using dynamic path analysis to estimate direct and indirect effects of treatment and other fixed covariates in the presence of an internal time-dependent.
Love does not come by demanding from others, but it is a self initiation. Survival Analysis.
STA347 - week 91 Random Vectors and Matrices A random vector is a vector whose elements are random variables. The collective behavior of a p x 1 random.
1 Borgan and Henderson: Event History Methodology Lancaster, September 2006 Session 6.1: Recurrent event data Intensity processes and rate functions Robust.
Stats Term Test 4 Solutions. c) d) An alternative solution is to use the probability mass function and.
Using Prior Scores to Evaluate Bias in Value-Added Models Raj Chetty, Stanford University and NBER John N. Friedman, Brown University and NBER Jonah Rockoff,
1 Ka-fu Wong University of Hong Kong A Brief Review of Probability, Statistics, and Regression for Forecasting.
Clinicaloptions.com/hepatitis NAFLD and NASH Prevalence in US Cohort Slideset on: Williams CD, Stengel J, Asike MI, et al. Prevalence of nonalcoholic fatty.
STA248 week 121 Bootstrap Test for Pairs of Means of a Non-Normal Population – small samples Suppose X 1, …, X n are iid from some distribution independent.
Analysis of Mismeasured Data David Yanez Department of Biostatistics University of Washington July 5, 2005 Biost/Stat 579.
Carolinas Medical Center, Charlotte, NC Website:
Survival Analysis: From Square One to Square Two Yin Bun Cheung, Ph.D. Paul Yip, Ph.D. Readings.
Jeffrey E. Korte, PhD BMTRY 747: Foundations of Epidemiology II
Cox Regression Model Under Dependent Truncation
Parametric Methods Berlin Chen, 2005 References:
Learning From Observed Data
Biomarkers as Endpoints
Presentation transcript:

1 A General Class of Models for Recurrent Events Edsel A. Pena University of South Carolina at Columbia [ Research support from NIH, NSF ENAR Talk, Arlington, VA, 3/18/02

2 Recurrent Phenomena Public Health and Medical Settings hospitalization of a subject with a chronic disease, e.g., end stage renal disease drug/alcohol abuse of a subject headaches tumor occurrence cyclic movements in the small bowel during fasting state depression episodes of epileptic seizures Prevalent in other areas (reliability, economics, etc.) as well.

3 A Pictorial Representation: One Subject An observable covariate vector: X(s) = (X 1 (s), X 2 (s), …, X q (s)) t Z Unobserved Frailty s 0  End of observation period Observed events Unobserved Event An intervention is performed just after each event T1T1 T2T2 T3T3 T4T4 S1S1 S2S2 S3S3 S4S4  -S 4

4 Features in Recurrent Event Modeling Intervention effects after each event occurrence. Effects of accumulating event occurrences on the subject. Could be a weakening or an strengthening effect. Effects of possibly time-dependent covariates. Possible associations of event occurrences for a subject. A possibly random observation period per subject. Number of observable events per subject is random and is informative on stochastic mechanism generating events. Informative right-censoring mechanism for the inter- event time that covers end of observation period.

5 Random Entities: One Subject X(s) = covariate vector, possibly time-dependent T 1, T 2, T 3, … = the inter-event or gap times S 1, S 2, S 3, … = the calendar times of occurrences F + = { : 0 < s} = filtration including info about interventions, covariate, etc. in [0, s] Z = unobserved frailty variable N + (s) = number of events observed on or before calendar time s Y + (s) = indicator of whether the subject is still at risk just before calendar time s

6 A General Class of Models {A + (s|Z): s > 0} is a predictable non-decreasing process such that given Z and with respect to the filtration F + : is a square-integrable zero-mean (local) martingale. As in previous works (Aalen, Gill, Andersen and Gill, Nielsen, et al, others) we assume

7 Modeling the Intensity Process [Pena and Hollander, to appear] Specify, possibly in a dynamic fashion, a predictable, observable process {E(s): 0 < s <  }, called the effective age process, satisfying E(0) = e 0 > 0; E(s) > 0 for every s; On [S k-1, S k ), E(s) is monotone and differentiable with a nonnegative derivative.

8 Specification of the Intensity Process

9 Model Components 0 (.) = an unknown baseline hazard rate function, possibly parametrically specified. E(s) = effective age of the subject at calendar time s. Idea is that a performed intervention changes the effective age of subject acting on the baseline hazard rate.  (.;  ) = a +function on {0,1,2,…} of known form with  (0;  ) = 1 and with unknown parameter . Encodes effect of accumulating event occurrences on the subject.  (.) = positive link function containing the effect of subject covariates.  is unknown. Z = unobservable frailty variable, which when integrated out, induces associations among the inter-event times.

10 Illustration: Effective Age Process “Possible Intervention Effects” s 0  Calendar Time Effective Age, E(s) No improvement Perfect intervention Some improvement Complications

11 Special Cases of the Class of Models IID “Renewal” Model without frailties: Considered by Gill (‘81 AS), Wang and Chang (‘99, JASA), Pena, Strawderman and Hollander (‘01, JASA). IID “Renewal” Model with frailties: Considered by Wang and Chang (‘99), PSH (‘01).

12 Generality and Flexibility Extended Cox PH Model: Considered by Prentice, Williams, and Petersen (PWP) (‘81); Lawless (‘87), Aalen and Husebye (‘91). Also by PWP (‘81), Brown and Proschan (‘83) and Lawless (‘87) called a “minimal repair model” in the reliability literature.

13 A generalized Gail, Santner and Brown (‘80) tumor occurrence model and Jelinski and Moranda (‘72) software reliability model: Let I 1, I 2, I 3, … be IND Ber[p(s)] rvs and Let  k = min{j >  k-1 : I j = 1}. If we generalize the BP (‘83) and Block, Borges and Savits (‘85) minimal repair model. Also considered in Presnell, Hollander and Sethuraman (‘94, ‘97).

14 Other Models In Class Dorado, Hollander and Sethuraman (‘97), Kijima (‘89), Baxter, Kijima and Tortorella (‘96), Stadje and Zuckerman (‘91), and Last and Szekli (‘98): Two simple forms for the  function:  initial measure of “defectiveness” or event “proneness.”

15 Relevance Flexibility and generality of this class of models will allow better modeling of observed phenomena, and allow testing of specific/special models using this larger class. Question: Is this relevant in biostatistical modeling?? Question: Is this relevant in biostatistical modeling?? Answer: Answer: The fact that it contains models currently being used indicates the model’s importance. However However, a “paradigm shift” is needed in the data gathering since the model requires the assessment of the effective age. But, this could be provided by the medical or public health experts after each intervention.

16 Most often it is the case of “A Data in Search of a Model;” but, sometimes* as in this case, it is “A Model in Search of a Data!” *A modern example of such a situation is that which led to the 1919 Eddington expedition. On the Model’s Immediate Applicability

17 Some Issues on Inference Need to take into account the sum-quota data accrual scheme which leads to an informative random number of events and informative right-censoring (cf., PSH (‘01) in renewal model).

18 Example: Variances of EDF, PLE, and GPLE For GPLE in “Renewal (IID) Model” [PSH ‘01, JASA]:

19 Identifiability: Model without Frailty If for each (   the support of E(T 1 ) contains [0,  ], and if  satisfies then the statistical model is identifiable.

20 Other Statistical Issues Parameter Estimation, especially when baseline hazard is non-parametrically specified. In progress! Testing and Group Comparisons. Model Validation and Diagnostics.