Optimal and Robust Designs of Step-stress Accelerated Life Testing Experiments for Proportional Hazards Models

Accelerated life testing (ALT) is widely used to obtain reliability information about a product within a limited time frame. The Cox s proportional hazards (PH) model is often utilized for reliability prediction. My master thesis research focuses on designing accelerated life testing experiments for reliability estimation. We consider multiple step-stress ALT plans with censoring. The optimal stress levels and times of changing the stress levels are investigated. We discuss the optimal designs under three optimality criteria. They are D-, A- and Q-optimal designs. We note that the classical designs are optimal only if the model assumed is correct. Due to the nature of prediction made from ALT experimental data, attained under the stress levels higher than the normal condition, extrapolation is encountered. In such case, the assumed model cannot be tested. Therefore, for possible imprecision in the assumed PH model, the method of construction for robust designs is also explored.

Tests of proportional hazards and proportional odds models for grouped survival data

In this paper, we derive score test statistics to discriminate between proportional hazards and proportional odds models for grouped survival data. These models are embedded within a power family transformation in order to obtain the score tests. In simple cases, some small-sample results are obtained for the score statistics using Monte Carlo simulations. Score statistics have distributions well approximated by the chi-squared distribution. Real examples illustrate the proposed tests.

Choosing between Cox proportional hazards and logistic models for interval-censored data via bootstrap

This work develops a new methodology in order to discriminate models for interval-censored data based on bootstrap residual simulation by observing the deviance difference from one model in relation to another, according to Hinde (1992). Generally, this sort of data can generate a large number of tied observations and, in this case, survival time can be regarded as discrete. Therefore, the Cox proportional hazards model for grouped data (Prentice & Gloeckler, 1978) and the logistic model (Lawless, 1982) can befitted by means of generalized linear models. Whitehead (1989) considered censoring to be an indicative variable with a binomial distribution and fitted the Cox proportional hazards model using complementary log-log as a link function. In addition, a logistic model can be fitted using logit as a link function. The proposed methodology arises as an alternative to the score tests developed by Colosimo et al. (2000), where such models can be obtained for discrete binary data as particular cases from the Aranda-Ordaz distribution asymmetric family. These tests are thus developed with a basis on link functions to generate such a fit. The example that motivates this study was the dataset from an experiment carried out on a flax cultivar planted on four substrata susceptible to the pathogen Fusarium oxysoprum. The response variable, which is the time until blighting, was observed in intervals during 52 days. The results were compared with the model fit and the AIC values.

Assessing time-by-covariate interactions in Cox proportional hazards regression models using cubic spline functions

This dissertation develops and explores the methodology for the use of cubic spline functions in assessing time-by-covariate interactions in Cox proportional hazards regression models. These interactions indicate violations of the proportional hazards assumption of the Cox model. Use of cubic spline functions allows for the investigation of the shape of a possible covariate time-dependence without having to specify a particular functional form. Cubic spline functions yield both a graphical method and a formal test for the proportional hazards assumption as well as a test of the nonlinearity of the time-by-covariate interaction. Five existing methods for assessing violations of the proportional hazards assumption are reviewed and applied along with cubic splines to three well known two-sample datasets. An additional dataset with three covariates is used to explore the use of cubic spline functions in a more general setting. ^

Concordance Probability and Discriminatory Power in Proportional Hazards Regression

The concordance probability is used to evaluate the discriminatory power and the predictive accuracy of nonlinear statistical models. We derive an analytic expression for the concordance probability in the Cox proportional hazards model. The proposed estimator is a function of the regression parameters and the covariate distribution only and does not use the observed event and censoring times. For this reason it is asymptotically unbiased, unlike Harrell's c-index based on informative pairs. The asymptotic distribution of the concordance probability estimate is derived using U-statistic theory and the methodology is applied to a predictive model in lung cancer.

Comparison of a traditional and a multilevel Cox Proportional Hazards model

Hierarchically clustered populations are often encountered in public health research, but the traditional methods used in analyzing this type of data are not always adequate. In the case of survival time data, more appropriate methods have only begun to surface in the last couple of decades. Such methods include multilevel statistical techniques which, although more complicated to implement than traditional methods, are more appropriate. ^ One population that is known to exhibit a hierarchical structure is that of patients who utilize the health care system of the Department of Veterans Affairs where patients are grouped not only by hospital, but also by geographic network (VISN). This project analyzes survival time data sets housed at the Houston Veterans Affairs Medical Center Research Department using two different Cox Proportional Hazards regression models, a traditional model and a multilevel model. VISNs that exhibit significantly higher or lower survival rates than the rest are identified separately for each model. ^ In this particular case, although there are differences in the results of the two models, it is not enough to warrant using the more complex multilevel technique. This is shown by the small estimates of variance associated with levels two and three in the multilevel Cox analysis. Much of the differences that are exhibited in identification of VISNs with high or low survival rates is attributable to computer hardware difficulties rather than to any significant improvements in the model. ^

Effects of informative censoring on the hazard function: Application to the proportional hazards regression model

The standard analyses of survival data involve the assumption that survival and censoring are independent. When censoring and survival are related, the phenomenon is known as informative censoring. This paper examines the effects of an informative censoring assumption on the hazard function and the estimated hazard ratio provided by the Cox model.^ The limiting factor in all analyses of informative censoring is the problem of non-identifiability. Non-identifiability implies that it is impossible to distinguish a situation in which censoring and death are independent from one in which there is dependence. However, it is possible that informative censoring occurs. Examination of the literature indicates how others have approached the problem and covers the relevant theoretical background.^ Three models are examined in detail. The first model uses conditionally independent marginal hazards to obtain the unconditional survival function and hazards. The second model is based on the Gumbel Type A method for combining independent marginal distributions into bivariate distributions using a dependency parameter. Finally, a formulation based on a compartmental model is presented and its results described. For the latter two approaches, the resulting hazard is used in the Cox model in a simulation study.^ The unconditional survival distribution formed from the first model involves dependency, but the crude hazard resulting from this unconditional distribution is identical to the marginal hazard, and inferences based on the hazard are valid. The hazard ratios formed from two distributions following the Gumbel Type A model are biased by a factor dependent on the amount of censoring in the two populations and the strength of the dependency of death and censoring in the two populations. The Cox model estimates this biased hazard ratio. In general, the hazard resulting from the compartmental model is not constant, even if the individual marginal hazards are constant, unless censoring is non-informative. The hazard ratio tends to a specific limit.^ Methods of evaluating situations in which informative censoring is present are described, and the relative utility of the three models examined is discussed. ^

Additive Hazards Models with Latent Treatment Effectiveness Lag Time

In many clinical trials to evaluate treatment efficacy, it is believed that there may exist latent treatment effectiveness lag times after which medical procedure or chemical compound would be in full effect. In this article, semiparametric regression models are proposed and studied to estimate the treatment effect accounting for such latent lag times. The new models take advantage of the invariance property of the additive hazards model in marginalizing over random effects, so parameters in the models are easy to be estimated and interpreted, while the flexibility without specifying baseline hazard function is kept. Monte Carlo simulation studies demonstrate the appropriateness of the proposed semiparametric estimation procedure. Data collected in the actual randomized clinical trial, which evaluates the effectiveness of biodegradable carmustine polymers for treatment of recurrent brain tumors, are analyzed.

Modeling multilevel sleep transitional data via Poisson log-linear multilevel models

This paper proposes Poisson log-linear multilevel models to investigate population variability in sleep state transition rates. We specifically propose a Bayesian Poisson regression model that is more flexible, scalable to larger studies, and easily fit than other attempts in the literature. We further use hierarchical random effects to account for pairings of individuals and repeated measures within those individuals, as comparing diseased to non-diseased subjects while minimizing bias is of epidemiologic importance. We estimate essentially non-parametric piecewise constant hazards and smooth them, and allow for time varying covariates and segment of the night comparisons. The Bayesian Poisson regression is justified through a re-derivation of a classical algebraic likelihood equivalence of Poisson regression with a log(time) offset and survival regression assuming piecewise constant hazards. This relationship allows us to synthesize two methods currently used to analyze sleep transition phenomena: stratified multi-state proportional hazards models and log-linear models with GEE for transition counts. An example data set from the Sleep Heart Health Study is analyzed.

A SIMULATION STUDY OF THE SIZE AND POWER OF SOME TWO-SAMPLE TESTS FOR UNEQUALLY CENSORED SURVIVAL DATA (RANK TESTS, LOGRANK, PROPORTIONAL HAZARDS, WILCOXON TESTS, EXPONENTIAL DISTRIBUTION)

Sizes and power of selected two-sample tests of the equality of survival distributions are compared by simulation for small samples from unequally, randomly-censored exponential distributions. The tests investigated include parametric tests (F, Score, Likelihood, Asymptotic), logrank tests (Mantel, Peto-Peto), and Wilcoxon-Type tests (Gehan, Prentice). Equal sized samples, n = 18, 16, 32 with 1000 (size) and 500 (power) simulation trials, are compared for 16 combinations of the censoring proportions 0%, 20%, 40%, and 60%. For n = 8 and 16, the Asymptotic, Peto-Peto, and Wilcoxon tests perform at nominal 5% size expectations, but the F, Score and Mantel tests exceeded 5% size confidence limits for 1/3 of the censoring combinations. For n = 32, all tests showed proper size, with the Peto-Peto test most conservative in the presence of unequal censoring. Powers of all tests are compared for exponential hazard ratios of 1.4 and 2.0. There is little difference in power characteristics of the tests within the classes of tests considered. The Mantel test showed 90% to 95% power efficiency relative to parametric tests. Wilcoxon-type tests have the lowest relative power but are robust to differential censoring patterns. A modified Peto-Peto test shows power comparable to the Mantel test. For n = 32, a specific Weibull-exponential comparison of crossing survival curves suggests that the relative powers of logrank and Wilcoxon-type tests are dependent on the scale parameter of the Weibull distribution. Wilcoxon-type tests appear more powerful than logrank tests in the case of late-crossing and less powerful for early-crossing survival curves. Guidelines for the appropriate selection of two-sample tests are given. ^

Prevalence and prognostic significance of secondary lymphedema following breast cancer

Background The adverse consequences of lymphedema following breast cancer in relation to physical function and quality of life are clear; however, its potential relationship with survival has not been investigated. Our purpose was to determine the prevalence of lymphedema and associated upper-body symptoms at 6 years following breast cancer and to examine the prognostic significance of lymphedema with respect to overall 6-year survival (OS). Methods and Results A population-based sample of Australian women (n=287) diagnosed with invasive, unilateral breast cancer was followed for a median of 6.6 years and prospectively assessed for lymphedema (using bioimpedance spectroscopy [BIS], sum of arm circumferences [SOAC], and self-reported arm swelling), a range of upper-body symptoms, and vital status. OS was measured from date of diagnosis to date of death or last follow-up. Kaplan-Meier methods were used to calculate OS and Cox proportional hazards models quantified the risk associated with lymphedema. Approximately 45% of women had reported at least one moderate to extreme symptom at 6.6 years postdiagnosis, while 34% had shown clinical evidence of lymphedema, and 48% reported arm swelling at least once since baseline assessment. A total of 27 (9.4%) women died during the follow-up period, and lymphedema, diagnosed by BIS or SOAC between 6–18 months postdiagnosis, predicted mortality (BIS: HR=2.5; 95% CI: 0.9, 6.8, p=0.08; SOAC: 3.0; 95% CI: 1.1, 8.7, p=0.04). There was no association (HR=1.2; 95% CI: 0.5, 2.6, p=0.68) between self-reported arm swelling and OS. Conclusions These findings suggest that lymphedema may influence survival following breast cancer treatment and warrant further investigation in other cancer cohorts and explication of a potential underlying biology.

Increased risk of hospitalization for acute lower respiratory tract infection among Australian Indigenous infants 5-23 months of age following pneumococcal vaccination : a cohort study

Background Australian Indigenous children are the only population worldwide to receive the 7-valent pneumococcal conjugate vaccine (7vPCV) at 2, 4, and 6 months of age and the 23-valent pneumococcal polysaccharide vaccine (23vPPV) at 18 months of age. We evaluated this program's effectiveness in reducing the risk of hospitalization for acute lower respiratory tract infection (ALRI) in Northern Territory (NT) Indigenous children aged 5-23 months. Methods We conducted a retrospective cohort study involving all NT Indigenous children born from 1 April 2000 through 31 October 2004. Person-time at-risk after 0, 1, 2, and 3 doses of 7vPCV and after 0 and 1 dose of 23vPPV and the number of ALRI following each dose were used to calculate dose-specific rates of ALRI for children 5-23 months of age. Rates were compared using Cox proportional hazards models, with the number of doses of each vaccine serving as time-dependent covariates. Results There were 5482 children and 8315 child-years at risk, with 2174 episodes of ALRI requiring hospitalization (overall incidence, 261 episodes per 1000 child-years at risk). Elevated risk of ALRI requiring hospitalization was observed after each dose of the 7vPCV vaccine, compared with that for children who received no doses, and an even greater elevation in risk was observed after each dose of the 23vPPV ( adjusted hazard ratio [HR] vs no dose, 1.39; 95% confidence interval [CI], 1.12-1.71;). Risk was highest among children Pp. 002 vaccinated with the 23vPPV who had received < 3 doses of the 7vPCV (adjusted HR, 1.81; 95% CI, 1.32-2.48). Conclusions Our results suggest an increased risk of ALRI requiring hospitalization after pneumococcal vaccination, particularly after receipt of the 23vPPV booster. The use of the 23vPPV booster should be reevaluated.

Effectiveness of 7-valent pneumococcal conjugate vaccine against radiologically diagnosed pneumonia in indigenous infants in Australia

Objective To evaluate the effectiveness of the 7-valent pneumococcal conjugate vaccine (PCV7) in preventing pneumonia, diagnosed radiologically according to World Health Organization (WHO) criteria, among indigenous infants in the Northern Territory of Australia. Methods We conducted a historical cohort study of consecutive indigenous birth cohorts between 1 April 1998 and 28 February 2005. Children were followed up to 18 months of age. The PCV7 programme commenced on 1 June 2001. All chest X-rays taken within 3 days of any hospitalization were assessed. The primary endpoint was a first episode of WHO-defined pneumonia requiring hospitalization. Cox proportional hazards models were used to compare disease incidence. Findings There were 526 pneumonia events among 10 600 children - an incidence of 3.3 per 1000 child-months; 183 episodes (34.8%) occurred before 5 months of age and 247 (47.0%) by 7 months. Of the children studied, 27% had received 3 doses of vaccine by 7 months of age. Hazard ratios for endpoint pneumonia were 1.01 for 1 versus 0 doses; 1.03 for 2 versus 0 doses; and 0.84 for 3 versus 0 doses. Conclusion There was limited evidence that PCV7 reduced the incidence of radiologically confirmed pneumonia among Northern Territory indigenous infants, although there was a non-significant trend towards an effect after receipt of the third dose. These findings might be explained by lack of timely vaccination and/or occurrence of disease at an early age. Additionally, the relative contribution of vaccine-type pneumococcus to severe pneumonia in a setting where multiple other pathogens are prevalent may differ with respect to other settings where vaccine efficacy has been clearly established.