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.

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. ^

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. ^

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. ^

The prognostic significance of quality of life following breast cancer [Conference Abstract]

Introduction: Evidence suggests a positive association between quality of life (QOL). and overall survival(OS). among metastatic breast cancer (BC). patients, although the relationship in early-stage BC is unclear. This work examines the association between QOL and OS following a diagnosis of early-stage BC. ----- Methods: A population-based sample of Queensland women (n=287). with early-stage, invasive, unilateral BC, were prospectively observed for a median of 6.6 years. QOL was assessed at six and 18 months post-diagnosis using the Functional Assessment of Cancer Therapy, Breast FACT-B+4. questionnaire. Raw scores for the FACT-B+4 scales were computed and individuals were categorised according to whether QOL declined, remained stable or improved over time. OS was measured from the date of diagnosis to the date of death or was censored at the date of last follow-up. Risk ratios (RR) and 95% confidence intervals (CI). for the association between QOL and OS were obtained using Cox proportional hazards survival models adjusted for confounding characteristics. ----- Results: A total of 27 (9.4%). women died during the follow-up period. Three baseline QOL scales (emotional, general and overall QOL) were significantly associated with OS, with RRs ranging between 0.89 95% CI: 0.81, 0.98; P=0.01. and 0.98 (95% CI: 0.96, 0.99; P=0.03),indicating a 2%-11% reduced risk of death for every one unit increase in QOL. When QOL was categorised according to changes between six and 18 months post-diagnosis, analyses showed that for those who experienced declines in functional and physical QOL, risk of death increased by two- (95% CI: 1.43, 12.52; P<0.01) and four-fold (95% CI: 1.15, 7.19; P=0.02), respectively. Conclusions: This work indicates that specific QOL scales at six months post-diagnosis, and changes in certain QOL scales over the subsequent 12-month period (as measured by the FACT-B+4), are associated with overall survival in women with early-stage breast cancer.

Patterns, correlates, and prognostic significance of quality of life following breast cancer

Objective To describe quality of life (QOL) over a 12-month period among women with breast cancer, consider the association between QOL and overall survival (OS), and explore characteristics associated with QOL declines. Methods A population-based sample of Australian women (n=287) with invasive, unilateral breast cancer (Stage I+), was observed prospectively for a median of 6.6 years. QOL was assessed at six, 12 and 18 months post-diagnosis, using the Functional Assessment of Cancer Therapy, Breast (FACT-B+4) questionnaire. Raw scores for the FACT-B+4 and subscales were computed and individuals were categorized according to whether QOL declined, remained stable or improved between six and 18 months. Kaplan-Meier and Cox proportional hazards survival methods were used to estimate OS and its associations with QOL. Logistic regression models identified factors associated with QOL decline. Results Within FACT-B+4 sub-scales, between 10% and 23% of women showed declines in QOL. Following adjustment for established prognostic factors, emotional wellbeing and FACT-B+4 scores at six months post-diagnosis were associated with OS (p<0.05). Declines in physical (p<0.01) or functional (p=0.02) well-being between six and 18 months post-diagnosis were also associated significantly with OS. Receiving multiple forms of adjuvant treatment, a perception of not handling stress well and reporting one or more other major life events at six months post-diagnosis were factors associated with declines in QOL in multivariable analyses. Conclusions Interventions targeted at preventing QOL declines may ultimately improve quantity as well as quality of life following breast cancer.

Identification of factors contributing to recurrence of venous leg ulcers

Background and Significance Venous leg ulcers are a significant cause of chronic ill-health for 1–3% of those aged over 60 years, increasing in incidence with age. The condition is difficult and costly to heal, consuming 1–2.5% of total health budgets in developed countries and up to 50% of community nursing time. Unfortunately after healing, there is a recurrence rate of 60 to 70%, frequently within the first 12 months after heaing. Although some risk factors associated with higher recurrence rates have been identified (e.g. prolonged ulcer duration, deep vein thrombosis), in general there is limited evidence on treatments to effectively prevent recurrence. Patients are generally advised to undertake activities which aim to improve the impaired venous return (e.g. compression therapy, leg elevation, exercise). However, only compression therapy has some evidence to support its effectiveness in prevention and problems with adherence to this strategy are well documented. Aim The aim of this research was to identify factors associated with recurrence by determining relationships between recurrence and demographic factors, health, physical activity, psychosocial factors and self-care activities to prevent recurrence. Methods Two studies were undertaken: a retrospective study of participants diagnosed with a venous leg ulcer which healed 12 to 36 months prior to the study (n=122); and a prospective longitudinal study of participants recruited as their ulcer healed and data collected for 12 months following healing (n=80). Data were collected from medical records on demographics, medical history and ulcer history and treatments; and from self-report questionnaires on physical activity, nutrition, psychosocial measures, ulcer history, compression and other self-care activities. Follow-up data for the prospective study were collected every three months for 12 months after healing. For the retrospective study, a logistic regression model determined the independent influences of variables on recurrence. For the prospective study, median time to recurrence was calculated using the Kaplan-Meier method and a Cox proportional-hazards regression model was used to adjust for potential confounders and determine effects of preventive strategies and psychosocial factors on recurrence. Results In total, 68% of participants in the retrospective study and 44% of participants in the prospective study suffered a recurrence. After mutual adjustment for all variables in multivariable regression models, leg elevation, compression therapy, self efficacy and physical activity were found to be consistently related to recurrence in both studies. In the retrospective study, leg elevation, wearing Class 2 or 3 compression hosiery, the level of physical activity, cardiac disease and self efficacy scores remained significantly associated (p<0.05) with recurrence. The model was significant (p <0.001); with a R2 equivalent of 0.62. Examination of relationships between psychosocial factors and adherence to wearing compression hosiery found wearing compression hosiery was significantly positively associated with participants’ knowledge of the cause of their condition (p=0.002), higher self-efficacy scores (p=0.026) and lower depression scores (p=0.009). Analysis of data from the prospective study found there were 35 recurrences (44%) in the 12 months following healing and median time to recurrence was 27 weeks. After adjustment for potential confounders, a Cox proportional hazards regression model found that at least an hour/day of leg elevation, six or more days/week in Class 2 (20–25mmHg) or 3 (30–40mmHg) compression hosiery, higher social support scale scores and higher General Self-Efficacy scores remained significantly associated (p<0.05) with a lower risk of recurrence, while male gender and a history of DVT remained significant risk factors for recurrence. Overall the model was significant (p <0.001); with an R2 equivalent 0.72. Conclusions The high rates of recurrence found in the studies highlight the urgent need for further information in this area to support development of effective strategies for prevention. Overall, results indicate leg elevation, physical activity, compression hosiery and strategies to improve self-efficacy are likely to prevent recurrence. In addition, optimal management of depression and strategies to improve patient knowledge and self-efficacy may positively influence adherence to compression therapy. This research provides important information for development of strategies to prevent recurrence of venous leg ulcers, with the potential to improve health and decrease health care costs in this population.

Relationships between preventive activities, psychosocial factors and recurrence of venous leg ulcers : a prospective study

Aim To identify relationships between preventive activities, psychosocial factors and leg ulcer recurrence in patients with chronic venous leg ulcers. Background Chronic venous leg ulcers are slow to heal and frequently recur, resulting in years of suffering and intensive use of health care resources. Methods A prospective longitudinal study was undertaken with a sample of 80 patients with a venous leg ulcer recruited when their ulcer healed. Data were collected from 2006–2009 from medical records on demographics, medical history and ulcer history; and from self-report questionnaires on physical activity, nutrition, preventive activities and psychosocial measures. Follow-up data were collected via questionnaires every three months for 12 months after healing. Median time to recurrence was calculated using the Kaplan-Meier method. A Cox proportional-hazards regression model was used to adjust for potential confounders and determine effects of preventive strategies and psychosocial factors on recurrence. Results: There were 35 recurrences in a sample of 80 participants. Median time to recurrence was 27 weeks. After adjustment for potential confounders, a Cox proportional hazards regression model found that at least an hour/day of leg elevation, six or more days/week in Class 2 (20–25mmHg) or 3 (30–40mmHg) compression hosiery, higher social support scale scores and higher General Self-Efficacy scores remained significantly associated (p<0.05) with a lower risk of recurrence, while male gender and a history of DVT remained significant risk factors for recurrence. Conclusion Results indicate that leg elevation, compression hosiery, high levels of self-efficacy and strong social support will help prevent recurrence.