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. ^

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. ^

Demographic analysis of Certified Industrial Hygienists (CIH) with the use of survival analysis techniques

Objective. The goal of this study is to characterize the current workforce of CIHs, the lengths of professional practice careers of the past and current CIHs.^ Methods. This is a secondary data analysis of data compiled from all of the nearly 50 annual roster listings of the American Board of Industrial Hygiene (ABIH) for Certified Industrial Hygienists active in each year since 1960. Survival analysis was performed as a technique to measure the primary outcome of interest. The technique which was involved in this study was the Kaplan-Meier method for estimating the survival function.^ Study subjects: The population to be studied is all Certified Industrial Hygienists (CIHs). A CIH is defined by the ABIH as an individual who has achieved the minimum requirements for education, working experience and through examination, has demonstrated a minimum level of knowledge and competency in the prevention of occupational illnesses. ^ Results. A Cox-proportional hazards model analysis was performed by different start-time cohorts of CIHs. In this model we chose cohort 1 as the reference cohort. The estimated relative risk of the event (defined as retirement, or absent from 5 consecutive years of listing) occurred for CIHs for cohorts 2,3,4,5 relative to cohort 1 is 0.385, 0.214, 0.234, 0.299 relatively. The result show that cohort 2 (CIHs issued from 1970-1980) has the lowest hazard ratio which indicates the lowest retirement rate.^ Conclusion. The manpower of CIHs (still actively practicing up to the end of 2009) increased tremendously starting in 1980 and grew into a plateau in recent decades. This indicates that the supply and demand of the profession may have reached equilibrium. More demographic information and variables are needed to actually predict the future number of CIHs needed. ^

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. ^

The number and timing of time -dependent covariate measurements for the Cox regression model

The problem of analyzing data with updated measurements in the time-dependent proportional hazards model arises frequently in practice. One available option is to reduce the number of intervals (or updated measurements) to be included in the Cox regression model. We empirically investigated the bias of the estimator of the time-dependent covariate while varying the effect of failure rate, sample size, true values of the parameters and the number of intervals. We also evaluated how often a time-dependent covariate needs to be collected and assessed the effect of sample size and failure rate on the power of testing a time-dependent effect.^ A time-dependent proportional hazards model with two binary covariates was considered. The time axis was partitioned into k intervals. The baseline hazard was assumed to be 1 so that the failure times were exponentially distributed in the ith interval. A type II censoring model was adopted to characterize the failure rate. The factors of interest were sample size (500, 1000), type II censoring with failure rates of 0.05, 0.10, and 0.20, and three values for each of the non-time-dependent and time-dependent covariates (1/4,1/2,3/4).^ The mean of the bias of the estimator of the coefficient of the time-dependent covariate decreased as sample size and number of intervals increased whereas the mean of the bias increased as failure rate and true values of the covariates increased. The mean of the bias of the estimator of the coefficient was smallest when all of the updated measurements were used in the model compared with two models that used selected measurements of the time-dependent covariate. For the model that included all the measurements, the coverage rates of the estimator of the coefficient of the time-dependent covariate was in most cases 90% or more except when the failure rate was high (0.20). The power associated with testing a time-dependent effect was highest when all of the measurements of the time-dependent covariate were used. An example from the Systolic Hypertension in the Elderly Program Cooperative Research Group is presented. ^

The critical role and regulation of transcription factor FoxM1 in gastric cancer development and progression

The mammalian Forkhead Box (Fox) transcription factor (FoxM1) is implicated in tumorgenesis. However, the role and regulation of FoxM1 in gastric cancer remain unknown.^ I examined FoxM1 expression in 86 cases of primary gastric cancer and 57 normal gastric tissue specimens. I found weak expression of FoxM1 protein in normal gastric mucosa, whereas I observed strong staining for FoxM1 in tumor-cell nuclei in various gastric tumors and lymph node metastases. The aberrant FoxM1 expression is associated with VEGF expression and increased angiogenesis in human gastric cancer. A Cox proportional hazards model revealed that FoxM1 expression was an independent prognostic factor in multivariate analysis. Furthermore, overexpression of FoxM1 by gene transfer significantly promoted the growth and metastasis of gastric cancer cells in orthotopic mouse models, whereas knockdown of FoxM1 expression by small interfering RNA did the opposite. Next, I observed that alteration of tumor growth and metastasis by elevated FoxM1 expression was directly correlated with alteration of VEGF expression and angiogenesis. In addition, promotion of gastric tumorigenesis by FoxM1 directly and significantly correlated with transactivation of vascular endothelial growth factor (VEGF) expression and elevation of angiogenesis. ^ To further investigate the underlying mechanisms that result in FoxM1 overexpression in gastric cancer, I investigated FoxM1 and Krüppel-like factor 4 (KLF4) expressions in primary gastric cancer and normal gastric tissue specimens. Concomitance of increased expression of FoxM1 protein and decreased expression of KLF4 protein was evident in human gastric cancer. Enforced KLF4 expression suppressed FoxM1 protein expression. Moreover, a region within the proximal FoxM1 promoter was identified to have KLF4-binding sites. Finally, I found an increased FoxM1 expression in gastric mucosa of villin-Cre -directed tissue specific Klf4-null mice.^ In summary, I offered both clinical and mechanistic evidence that dysregulated expression of FoxM1 play an important role in gastric cancer development and progression, while KLF4 mediates negative regulation of FoxM1 expression and its loss significantly contributes to FoxM1 dysregulation. ^

The combined effect of pravastatin and aspirin on clinical cardiovascular events

The 3-hydroxy-3methylglutaryl coenzyme A (HMG-CoA) reductase inhibitors, or statins, can achieve significant reductions in plasma low-density lipoprotein (LDL)-cholesterol levels. Experimental and clinical evidence now shows that some statins interfere with formation of atherosclerotic lesions independent of their hypolipidemic properties. Vulnerable plaque rupture can result in thrombus formation and artery occlusion; this plaque deterioration is responsible for most acute coronary syndromes, including myocardial infarction (MI), unstable angina, and coronary death, as well as coronary heart diseaseequivalent non-hemorrhagic stroke. Inhibition of HMG-CoA reductase has potential pleiotropic effects other than lipid-lowering, as statins block mevalonic acid production, a precursor to cholesterol and numerous other metabolites. Statins' beneficial effects on clinical events may also thus involve nonlipid-related mechanisms that modify endothelial function, inflammatory responses, plaque stability, and thrombus formation. Aspirin, routinely prescribed to post-MI patients as adjunct therapy, may potentiate statins beneficial effects, as aspirin does not compete metabolically with statins but acts similarly on atherosclerotic lesions. Common functions of both medications include inhibition of platelet activity and aggregation, reduction in atherosclerotic plaque macrophage cell count, and prevention of atherosclerotic vessel endothelial dysfunction. The Cholesterol and Recurrent Events (CARE) trial provides an ideal population in which to examine the combined effects of pravastatin and aspirin. Lipid levels, intermediate outcomes, are examined by pravastatin and aspirin status, and differences between the two pravastatin groups are found. A modified Cox proportional-hazards model with aspirin as a time-dependent covariate was used to determine the effect of aspirin and pravastatin on the clinical cardiovascular composite endpoint of coronary heart disease death, recurrent MI or stroke. Among those assigned to pravastatin, use of aspirin reduced the composite primary endpoint by 35%; this result was similar by gender, race, and diabetic status. Older patients demonstrated a nonsignificant 21% reduction in the primary outcome, whereas the younger had a significant reduction of 43% in the composite primary outcome. Secondary outcomes examined include coronary artery bypass graft (38% reduction), nonsurgical bypass, peripheral vascular disease, and unstable angina. Pravastatin and aspirin in a post-MI population was found to be a beneficial combination that seems to work through lipid and nonlipid, anti-inflammatory mechanisms. ^

Extension of k-ratio methods to rank-based analyses

An extension of k-ratio multiple comparison methods to rank-based analyses is described. The new method is analogous to the Duncan-Godbold approximate k-ratio procedure for unequal sample sizes or correlated means. The close parallel of the new methods to the Duncan-Godbold approach is shown by demonstrating that they are based upon different parameterizations as starting points.^ A semi-parametric basis for the new methods is shown by starting from the Cox proportional hazards model, using Wald statistics. From there the log-rank and Gehan-Breslow-Wilcoxon methods may be seen as score statistic based methods.^ Simulations and analysis of a published data set are used to show the performance of the new methods. ^

Effects of breastfeeding and naturally acquired immunity on infection with {\it Giardia lamblia}: A study in a cohort of Mexican infants

Giardia lamblia is one of the most common causes of gastrointestinal tract infection among young children worldwide. Yet host protection against this parasite and the effect of infection with Giardia on infant growth are poorly understood. It was hypothesized that among young children, protection against infection with Giardia is afforded by breastfeeding and previous infection with the parasite and further, that infection with Giardia decreases growth velocity. From 4/88 to 4/90, 197 infants in a poor area of Mexico City were followed from 0 to 18 months of age, with stool specimens, symptoms and feeding status data collected weekly. A total of 6,031 stool specimens were tested for Giardia antigen by enzyme-linked immunosorbent assay. There were 1.0 Giardia infections per child-year; 25% were symptomatic and 54% lasted more than 1 month; 94 infants had 1, and 33 had 2 or more infections. Breastfeeding status was coded and analyzed for each child-week of follow up. 91% of study infants were breastfed from birth, 57% at 6 months and 38% at 12 months of age. Rate ratios for non-breastfeeding adjusted for confounding factors were calculated from stratified analyses and the Cox proportional hazards model. Not breastfeeding was a significant risk factor for first infection with Giardia vs. any breastfeeding (adjusted RR = 1.8; 1.1, 2.8) at all ages; a dose response was demonstrated by degree of breastfeeding. The adjusted rate ratio for non-breastfeeding vs. partial breastfeeding was 1.6 (1.03, 2.6) and for non-breastfeeding vs. complete breastfeeding was 4.7 (1.4, 15.9). Among Giardia infected infants, breastfeeding did not protect against diarrheal symptoms or shorten the duration of carriage. First and repeat infections with Giardia did not differ in duration or the percent symptomatic. The analysis of growth and Giardia infection was inconclusive but suggested that a history of Giardia infection might be associated with decreased weight velocity, while an immediate chronic infection might be associated with increased weight velocity. In summary, these data indicate that breastfeeding protects infants against infection with Giardia; provide no evidence of protection against repeat infections resulting from a prior infection and suggest but do not establish that a history of Giardia infection might be associated with decreased growth in young children. ^

Joint effects of pulmonary function and aerobic power on survival in a cohort of healthy adults

Objective measurements of physical fitness and pulmonary function are related individually to long-term survival, both in healthy people and in those who are ill. These factors are furthermore known to be related to one another physiologically in people with pulmonary disease, because advanced pulmonary disease causes ventilatory limitation to exercise. Healthy people do not have ventilatory limitation to exercise, but rather have ventilatory reserve. The relationship between pulmonary function and exercise performance in healthy people is minimal. Exercise performance has been shown to modify the effect of pulmonary function on mortality in people with chronic obstructive pulmonary disease, but the relationship between these factors in healthy people has not been studied and is not known. The purpose of this study is to quantify the joint effects of pulmonary function and exercise performance as these bear on mortality in a cohort of healthy adults. This investigation is an historical cohort study over 20 years of follow-up of 29,624 adults who had complete preventive medicine, spirometry and treadmill stress examinations at the Cooper Clinic in Dallas, Texas.^ In 20 years of follow-up, there were 738 evaluable deaths. Forced expiratory volume in one second (FEV$\sb1$) percent of predicted, treadmill time in minutes percent of predicted, age, gender, body mass index, baseline smoking status, serum glucose and serum total cholesterol were all significant, independent predictors of mortality risk. There were no frank interactions, although age had an important increasing effect on the risk associated with smoking when other covariates were controlled for in a proportional-hazards model. There was no confounding effect of exercise performance on pulmonary function. In agreement with the pertinent literature on independent effects, each unit increase in FEV$\sb1$ percent predicted was associated with about eight tenths of a percent reduction in adjusted mortality rate. The concept of physiologic reserve is useful in interpretation of the findings. Since pulmonary function does not limit exercise tolerance in healthy adults, it is reasonable to expect that exercise tolerance would not modify the effect of pulmonary function on mortality. Epidemiologic techniques are useful for elucidating physiological correlates of mortality risk. ^

Early identification of symptomatic aortic stenosis using echocardiographic data

Background: Heart failure (CHF) is the most frequent and prognostically severe symptom of aortic stenosis (AS), and the most common indication for surgery. The mainstay of treatment for AS is aortic valve replacement (AVR), and the main indication for an AVR is development of symptomatic disease. ACC/AHA guidelines define severe AS as an aortic valve area (AVA) ≤1cm², but there is little data correlating echocardiogram AVA with the onset of symptomatic CHF. We evaluated the risk of developing CHF with progressively decreasing echocardiographic AVA. We also compared echocardiographic AVA with Jet velocity (V2) and indexed AVA (AVAI) to assess the best predictor of development of symptomatic CHF.^ Methods and Results: This retrospective cohort study evaluated 518 patients with asymptomatic moderate or severe AS from a single community based cardiology practice. A total of 925 echocardiograms were performed over an 11-year period. Each echocardiogram was correlated with concurrent clinical assessments while the investigator was blinded to the echocardiogram severity of AS. The Cox Proportional hazards model was used to analyze the relationship between AVA and the development of CHF. The median age of patients at entry was 76.1 years, with 54% males. A total of 116 patients (21.8%) developed new onset CHF during follow-up. Compared to patients with AVA >1.0cm², patients with lower AVA had an exponentially increasing risk of developing CHF for each 0.2cm² decrement in AVA, becoming statistically significant only at an AVA less than 0.8 cm². Also, compared to V2 and AVAI, AVA added more information to assessing risk for development of CHF (p=0.041). ^ Conclusion: In patients with normal or mildly impaired LVEF, the risk of CHF rises exponentially with decreasing valve area and becomes statistically significant after AVA falls below 0.8cm². AVA is a better predictor of CHF when compared to V2 or AVAI.^

Genetic variants in the mammalian target of rapamycin (mTOR) signaling pathway as predictors of survival and clinical response in women with ovarian cancer

Background. The mTOR pathway is commonly altered in human tumors and promotes cell survival and proliferation. Preliminary evidence suggests this pathway's involvement in chemoresistance to platinum and taxanes, first line therapy for epithelial ovarian cancer. A pathway-based approach was used to identify individual germline single nucleotide polymorphisms (SNPs) and cumulative effects of multiple genetic variants in mTOR pathway genes and their association with clinical outcome in women with ovarian cancer. ^ Methods. The case-series was restricted to 319 non-Hispanic white women with high grade ovarian cancer treated with surgery and platinum-based chemotherapy. 135 SNPs in 20 representative genes in the mTOR pathway were genotyped. Hazard ratios (HRs) for death and Odds ratios (ORs) for failure to respond to primary therapy were estimated for each SNP using the multivariate Cox proportional hazards model and multivariate logistic regression model, respectively, while adjusting for age, stage, histology and treatment sequence. A survival tree analysis of SNPs with a statistically significant association (p<0.05) was performed to identify higher order gene-gene interactions and their association with overall survival. ^ Results. There was no statistically significant difference in survival by tumor histology or treatment regimen. The median survival for the cohort was 48.3 months. Seven SNPs were significantly associated with decreased survival. Compared to those with no unfavorable genotypes, the HR for death increased significantly with the increasing number of unfavorable genotypes and women in the highest risk category had HR of 4.06 (95% CI 2.29–7.21). The survival tree analysis also identified patients with different survival patterns based on their genetic profiles. 13 SNPs on five different genes were found to be significantly associated with a treatment response, defined as no evidence of disease after completion of primary therapy. Rare homozygous genotype of SNP rs6973428 showed a 5.5-fold increased risk compared to the wild type carrying genotypes. In the cumulative effect analysis, the highest risk group (individuals with ≥8 unfavorable genotypes) was significantly less likely to respond to chemotherapy (OR=8.40, 95% CI 3.10–22.75) compared to the low risk group (≤4 unfavorable genotypes). ^ Conclusions. A pathway-based approach can demonstrate cumulative effects of multiple genetic variants on clinical response to chemotherapy and survival. Therapy targeting the mTOR pathway may modify outcome in select patients.^

Risk factors of pneumonia in traumatic spinal-cord-injured patients

Pneumonia is a well-documented and common respiratory infection in patients with acute traumatic spinal cord injuries, and may recur during the course of acute care. Using data from the North American Clinical Trials Network (NACTN) for Spinal Cord Injury, the incidence, timing, and recurrence of pneumonia were analyzed. The two main objectives were (1) to investigate the time and potential risk factors for the first occurrence of pneumonia using the Cox Proportional Hazards model, and (2) to investigate pneumonia recurrence and its risk factors using a Counting Process model that is a generalization of the Cox Proportional Hazards model. The results from survival analysis suggested that surgery, intubation, American Spinal Injury Association (ASIA) grade, direct admission to a NACTN site and age (older than 65 or not) were significant risks for first event of pneumonia and multiple events of pneumonia. The significance of this research is that it has the potential to identify patients at the time of admission who are at high risk for the incidence and recurrence of pneumonia. Knowledge and the time of occurrence of pneumonias are important factors for the development of prevention strategies and may also provide some insights into the selection of emerging therapies that compromise the immune system. ^

Adaptive clinical trial design for targeted therapy development

The development of targeted therapy involve many challenges. Our study will address some of the key issues involved in biomarker identification and clinical trial design. In our study, we propose two biomarker selection methods, and then apply them in two different clinical trial designs for targeted therapy development. In particular, we propose a Bayesian two-step lasso procedure for biomarker selection in the proportional hazards model in Chapter 2. In the first step of this strategy, we use the Bayesian group lasso to identify the important marker groups, wherein each group contains the main effect of a single marker and its interactions with treatments. In the second step, we zoom in to select each individual marker and the interactions between markers and treatments in order to identify prognostic or predictive markers using the Bayesian adaptive lasso. In Chapter 3, we propose a Bayesian two-stage adaptive design for targeted therapy development while implementing the variable selection method given in Chapter 2. In Chapter 4, we proposed an alternate frequentist adaptive randomization strategy for situations where a large number of biomarkers need to be incorporated in the study design. We also propose a new adaptive randomization rule, which takes into account the variations associated with the point estimates of survival times. In all of our designs, we seek to identify the key markers that are either prognostic or predictive with respect to treatment. We are going to use extensive simulation to evaluate the operating characteristics of our methods.^