Lifetime Data Models in Reliability and Survival Analysis

Logistic regression has been applied to numerous investigations that examine the relationship between risk factors and various disease events. Recently, the ability to consider the time element of event occurrences by proportional hazards models has meant that logistic regression has played a less important role in the analysis of survival data. This paper, however, shows that when event times are grouped into intervals, logistic regression can be adapted to the analysis of such data by modeling the interval when an event occurs. Furthermore, it is shown that results from such an adaptation will often lead to parameter estimates close to those obtained by the proportional hazards model in the grouped event time setting. An illustration of the application of logistic regression to survival analysis is based on data from the Framingham Heart Study.

A previous paper (Hutchison, 1988) in this journal has provided an introduction to the basic concepts of survival and event history analysis, originally developed in medical research, econometrics and engineering, and argued the case for their wider application in the social sciences. This paper introduces some further complications that the researcher is likely to meet, and offers some guidelines for handling problems that arise in applying such methods to the highly complex social situations involved.

Great strides in the analysis of survival data using Bayesian methods have been made in the past ten years due to advances in Bayesian computation and the feasibility of such methods. In this chapter, we review Bayesian advances in survival analysis and discuss the various semiparametric modeling techniques that are now commonly used. We review parametric and semiparametric approaches to Bayesian survival analysis, with a focus on proportional hazards models. Reference to other types of models are also given.

PM10 EFFECTS ON PRETERM BIRTH BY VARIOUS STATISTICAL MODELS

The Fundamentals of Survival and Event History Analysis Introduction: What Is Survival and Event History Analysis? Key Concepts and Terminology Censoring and Truncation Mathematical Expression and Relation of Basic Statistical Functions How Do the Survivor, Density and Hazard Function Relate? Why Use Survival and Event History Analysis? Overview of Survival and Event History Models Exercises Using R and Other Computer Programs for Survival and Event History Analysis Introduction: Computer Programs for Survival and Event History Analysis Conducting Serious Data Analysis: Life Lessons Why Use R? Downloading R on Your Personal Computer Add-On Packages Running R Determining and Setting your Working Directory Help and Documentation Importing Data Into R Working With Data: Opening and Accessing Variables from a Data Frame Saving Output as File, Workspace and History and Quitting R Exercises Your First Session: Using the Survival Package and Exploring Data Via Descriptive Statistics and Graphs Your First Session Using the 'Survival' Package In F Loading and Examining the Survival Package and Rcmdrplugin.Survival Plug-In Opening and Examining Data The Surv Object: Packaging the 'Survival Variable' Basic Descriptive Statistics Descriptive Data Exploration with Graphs Exercises Data and Data Reconstruction Introduction: Why Discuss Data and Data Preparation? Sources of Event History Data Single-Episode Data for Single Transition Analyses Multi-Episode Data for Recurrent Event and Frailty Analyses Subject-(Person)-Period Data for Discrete-Time Hazard Models The Counting Process and Episode Splitting A Note on Dates Exercises Non-Parametric Methods: Estimating and Comparing Survival Curves Using the Kaplan-Meier Estimator Introduction The Kaplan-Meier Estimator Producing Kaplan-Meier Estimates Plotting the Kaplan-Meier Survival Curve Testing Differences Between Two Groups Using Survdiff Stratifying the Analysis by a Covariate Exercises The Cox Proportional-Hazards Regression Introduction: Why is The Cox Model So Popular? The Cox Regression Model Estimating and Interpreting The Cox Model with Fixed Covariates The Cox Regression Model with Time-Varying Covariates Exercises Parametric Models Introduction: What are Parametric Models and Why Use Them? Proportional Hazards (Ph) Versus Accelerated Failure Time (Aft) Models The Path to Choosing a Model Estimating and Interpreting Parametric Survival Models Exponential and Piecewise Constant Exponential Model Weibull Model Log-Logistic and Log-Normal Models Additional Parametric Models Finding the Best Fitting Parametric Model Exercises Model Building and Diagnostics Introduction Model Building and Selection of Covariates Assessing the Overall Goodness of Fit of Your Model What is Residual Analysis? Testing Overall Model Adequacy: Cox-Snell Residuals Testing the Proportional Hazards Assumption: Schoenfeld Residuals Checking For Influential Observations: Score Residuals (Dfbeta Statistics) Assessing Nonlinearity: Martingale Residual and Component-Plus-Residual Plots Exercises Correlated and Discrete-Time Survival Data: Frailty, Recurrent Events and Discrete-Time Models Introduction Shared Frailty: Modeling Recurrent Events and Clustering In Groups Other Frailty Models: Unshared, Nested, Joint and Additive Models Estimating Frailty Models in R Example of Frailty Model Estimation and Interpretation Discrete-Time and Count Models Exercises Multiple Events and Entire Histories: Competing Risk, Multistate Models and Sequence Analysis Introduction Competing Risk Models Multistate Models Sequence Analysis: Modeling Entire Histories Exercises Appendix : Datasets Used in this Book

Cox’s Proportional Hazards (CPH) model is quite likely the most popular modeling technique in survival analysis. While the CPH model is able to represent relationships between a collection of risks and their common effect, Bayesian networks have become an attractive alternative with far broader applications. Our paper focuses on a Bayesian network interpretation of the CPH model. We provide a method of encoding knowledge from existing CPH models in the process of knowledge engineering for Bayesian networks. We compare the accuracy of the resulting Bayesian network to the CPH model, Kaplan-Meier estimate, and Bayesian network learned from data using the EM algorithm. Bayesian networks constructed from CPH model lead to much higher accuracy than other approaches, especially when the number of data records is very small.

A Bayesian network interpretation of the Cox's proportional hazard model

In time-varying covariate analysis of clinical survival data, it is often of interest to estimate the proportion of treatment effect (PTE), along with its confidence intervals, explained by a surrogate marker. The conventional procedure for such an analysis fits data into two working models separately to estimate the treatment effects before and after adjustment of the covariate. The construction of confidence intervals for the PTE under the conventional procedure lacks support by standard statistical software such as SAS, and could be very computationally demanding even after the support is available in the future. To overcome this problem, we propose a new procedure to simplify the computation. Under the new procedure, the treatment effects before and after adjustment of the covariate are simultaneously estimated from a single model. More important than saving computational effort, the new procedure can also be effectively applied to multiple-covariate models for the decomposition of overall treatment effect and for the comparison of PTE among several surrogate markers. The new procedure is applied to the motivating data example from the LIFE study, and demonstrates flexibility that the conventional procedure currently lacks.

The modeling of time to event data is an important topic with many applications in diverse areas. The collective of methods to analyze such data are called survival analysis, event history analysis or duration analysis. Survival analysis is widely applicable because the definition of an ’event’ can be manifold and examples include death, graduation, purchase or bankruptcy. Hence, application areas range from medicine and sociology to marketing and economics. In this paper, we review the theoretical basics of survival analysis including estimators for survival and hazard functions. We discuss the Cox Proportional Hazard Model in detail and also approaches for testing the proportional hazard (PH) assumption. Furthermore, we discuss stratified Cox models for cases when the PH assumption does not hold. Our discussion is complemented with a worked example using the statistical programming language R to enable the practical application of the methodology.

A survival analysis on a data set of 295 early breast cancer patients is performed in this study. A new proportional hazards model, hypertabastic model was applied in the survival analysis. We assume a proportional hazards model, and select two sets of risk factors for death and metastasis for breast cancer patients respectively by using standard variable selection methods. To evaluate the performance of the new model and compare it with other popular distributions, Cox, Weibull and log-logistic models were fitted to the data besides the hypertabastic model. Result shows that the hypertabastic proportional hazards model outperformed all the comparison models and provided the best fit for the breast cancer data. In addition, we observed that the gene expression variable, wound response signature, combined with other clinical variables, can provide an effective model to predict the overall survival and hazard rate for breast cancer patients.

1. Probability Models. 1.1 Discrete Probability Models. 1.2 Conditional Probability and Independence. 1.3 Random Variables. 1.4 Expectation. 1.5 The Variance. 1.6 Covariance and Correlation. 2. Special Discrete Distributions. 2.1 The Binomial Distribution. 2.2 The Hypergeometric Distribution. 2.3 The Geometric and Negative Binomial Distributions. 2.4 The Poisson Distribution. 3. Continuous Random Variables. 4.1 Continuous RV's and Their Distributions. 4.2 Expected Values and Variances. 4.3 Transformations of Random Variables. 4.4Joint Densities. 4 Special Continuous Distributions. 4.1 The Normal Distribution. 4.2 The Gamma Distribution. 5. Conditional Distributions. 5.1 The Discrete Case. 5.2 Conditional Expectations for the Discrete Case. 5.3 Conditional Densities and Expectations for Continuous RV's. 6. Limit Laws. 6.1 Moment Generating Functions. 6.2 Convergence in Probability and in Distribution. 6.3 The Central Limit Theorem. 6.4 The Delta-Method. 7. Estimation. 7.1 Point Estimation. 7.2 The Method of Moments. 7.3 Maximum Likelihood. 7.4 Consistency. 7.5 The OMEGA-Method. 7.6 Confidence Intervals. 7.7 Fisher Information, The Cramer-Rao Bound, and Asymptotic Normality of MLE's. 7.8 Sufficiency. 8. Testing Hypotheses. 8.1 Introduction. 8.2 The Neyman-Pearson Lemma. 8.3 The Likelihood Ratio Test. 8.4 The p-Value and the Relationship Between Tests of Hypotheses and Confidence Intervals. 9. The Multivariate Normal, Chi-square, t, and F-Distributions. 9.1 The Multivariate Normal Distribution. 9.2 The Central and Noncentral Chi-Square Distributions. 9.3 Student's t-Distribution. 9.4 The F-Distribution. 10.3 Nonparametric Statistics. 10.1 The Wilcoxon Test and Estimator. 10.2 One Sample Methods. 10.3 The Kolmogorov-Smirnov Tests. 11. Linear Models. 11.1 The Principle of Least Squares. 11.2 Linear Models. 11.3 F-Tests for H0. 11.4 Two-Way Analysis of Variance. 12. Frequency Data. 12.1 Logistic Regression. 12.2 Two-Way Frequency Tables. 12.3 Chi-Square Goodness of Fit Tests. 13. Miscellaneous Topics. 13.1 Survival Analysis. 13.2 Bootstrapping. 13.3 Bayesian Statistics. 13.4 Sampling.

Background: A diuretic strategy is needed that is superior to current clinical care in the management of cardiorenal syndrome. Current HF guidelines do not provide any standard protocol for diuretic dosing. Aim: To determine if a protocolized diuretic treatment strategy (ProDiuS), compared to usual care (UC), results in improved decongestion, clinical outcomes, and health-related quality of life (HRQOL), while preserving renal function in hospitalized patients with cardiorenal syndrome. Materials and Methods: This trial was a prospective randomized single-blind trial of participants hospitalized for cardiorenal syndrome from November 1, 2013 to July 9, 2015. Participants were randomized to ProDiuS vs. UC and followed daily in hospital, and at 1-month and 3-month follow-up. ProDiuS was a stepped diuretic strategy targeting daily urine output of 3-5 L/day. UC was care at the discretion of treating providers. The primary outcome, change in body weight (kg) from randomization to 96 hours (day 4), was compared between the ProDiuS and UC groups using the t-test. Data analysis for secondary outcomes between the two groups were conducted using the t-test or Wilcoxon rank sum test depending on whether data was skewed for continuous variables, as well as linear regression modeling. For ordinal variables or proportions, data were analyzed using the exact chi-square test and logistic regression modeling. For mortality, time to death was analyzed as time-to-event data, with censoring at the time of death, date of last follow-up, or the end of the study (3 month follow-up), using Kaplan-Meier curves and log-rank tests, and Cox proportional hazards models to adjust for continuous and discrete covariates in the survival analysis. Results: The study did not enroll the prespecified number of subjects due to slow recruitment. Out of 786 prescreened patients, 19 participants were included in the trial. There were no significant differences in baseline characteristics. Mean age was 68.7±7.3 years and 72.2% were male. There was borderline higher change in body weight from baseline to day 4 or discharge in ProDiuS vs. UC (-6.12 vs. -2.07 kg; p=0.05). Net negative fluid balance, length of hospitalization, HF rehospitalizations, mortality, acute kidney injury, adverse outcomes, and HRQOL scores were similar between groups. Conclusion: Due to small sample size, firm conclusions cannot be drawn. However, these findings suggest that ProDiuS results in similar clinical and HRQOL outcomes as UC in HF patients treated at a large tertiary medical center in the short term. This trial was conducted in an advanced HF population on specialized HF services, which may have attenuated the efficacy of ProDiuS vs. UC. Further studies with larger sample size and more diverse HF populations are needed to determine the efficacy and safety of ProDiuS. Several lessons learned in attempting to design a trial of protocolized diuretic strategy in the cardiorenal population are discussed.

The decline in caries prevalence, the increases in the level of fluoride exposure, and the lack of placebo control subjects have complicated caries clinical trials in recent times. There has been a substantial increase in the numbers of subjects required for the detection of statistically significant differences between dental products, and hence, the cost of these trials has grown enormously. This study uses a new statistical approach to the analysis of the data from these trials with the ultimate aim of providing a more sensitive method of analysis. The new approach uses survival analysis, where the outcome measure is the survival time of an individual tooth surface. It exploits recent developments in the analysis of clustered survival data where survival times within the same cluster or subject are correlated. To illustrate, the new method of analysis was used for the North Wales, UK, caries clinical trial. It is concluded that survival analysis uses most of the data available in a caries clinical trial, an outcome measure that is easily understood and may lead to a more sensitive method of analysis.

Background: The PROCLAIMSM registry (www.proclaimregistry.com), created in 2011, is the largest collection of IL-2 treated patients in the US and provides real-time insights into sequencing or combining IL-2 with new and old therapies and how this may affect patient outcomes. Previously, we reported a median overall survival (mOS) of 20 months with a median follow-up of 37 months in metastatic melanoma (mM) patients treated with high dose IL-2 (HD IL-2) between 2007 and 2012 from a retrospective cohort. These findings led to the hypothesis that improved mOS may have been a result of subsequent salvage therapies, including checkpoint inhibitors. Purpose: To report on the analysis of survival data from 26 sites. Methods: Patients must have received at least one dose of HD IL-2 for this analysis. Those that received checkpoint therapy prior to HD IL-2 were excluded. Statistics and survival analysis on prospectively entered patients were performed on datasets as of March 16th, 2015. Results: The median overall survival (mOS) for the 236 patients was 18.4 months with a median follow-up of 21.7 months. Patients were stratified into three groups; HD IL-2 only (n=123), HD IL-2 followed by ipilimumab (Post ipi, n=78), and HD IL-2 followed by PD-1 inhibitors (Post αPD-1, n=35). There were four patients in the Post αPD-1 group that received anti-PD-L1 treatment. Patients in the HD IL-2 only, Post ipi, and Post αPD-1 groups achieved a mOS of 14, 15.7, and 28.7 months, respectively. The estimated 12-month survival rates were 56%, 64%, and 97%, respectively. There were 10/78 (13%) and 3/35 (8.6%) post therapy treatment-related incidences of autoimmune events in the Post Ipi and Post αPD-1 groups, respectively. No treatment related deaths were reported. Conclusions: This is the first report of clinical data relating to HD IL-2 use followed by checkpoint blockade of the PD-1 pathway. Treatment with anti-PD-1 after initial therapy with HD IL-2 had significantly prolonged survival compared to patients treated with ipilimumab. Moreover, improved survival was not observed in patients treated with follow-on ipilimumab compared to patients treated only with HD IL-2. Anti-PD-1 therapy after HD IL-2, appears to be safe, is therapeutically active. These data support the concept of investigating IL-2 therapy in combination or sequence with newly developed immune checkpoint inhibitors. Citation Format: Michael K.K. Wong, Michael A. Morse, David F. McDermott, Joseph I. Clark, Howard L. Kaufman, Gregory A. Daniels, Hong Hua, Sandra Aung. Overall survival of metastatic melanoma patients treated with HD IL-2 followed by immune checkpoint blockade of the CTLA-4 or the PD-1 pathways: Analysis of data on the current use of HD IL-2. [abstract]. In: Proceedings of the CRI-CIMT-EATI-AACR Inaugural International Cancer Immunotherapy Conference: Translating Science into Survival; September 16-19, 2015; New York, NY. Philadelphia (PA): AACR; Cancer Immunol Res 2016;4(1 Suppl):Abstract nr B140.

Clinicians often wish to use data from clinical trials or hospital databases to study disease natural history. Of particular interest are estimated survival and prognostic factors. In this context, it may be appropriate to measure survival from diagnosis or some other time origin, possibly prior to study entry. We describe the application of methods for truncated survival data, and compare these with the standard product limit estimator and proportional hazards models in the measurement of survival from entry. Theoretical considerations suggest that analysis of survival from entry may under- or overestimate the survival distribution of interest, depending on the shape of the true underlying hazard. Analogous results hold for the coefficients from a proportional hazards model. We illustrate our findings with data from a multicenter clinical trial and a hospital database.