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. ^

Distribution of quality adjusted life expectancy: A systems approach

Statistical methods are developed which assess survival data for two attributes; (1) prolongation of life, (2) quality of life. Health state transition probabilities correspond to prolongation of life and are modeled as a discrete-time semi-Markov process. Imbedded within the sojourn time of a particular health state are the quality of life transitions. They reflect events which differentiate perceptions of pain and suffering over a fixed time period. Quality of life transition probabilities are derived from the assumptions of a simple Markov process. These probabilities depend on the health state currently occupied and the next health state to which a transition is made. Utilizing the two forms of attributes the model has the capability to estimate the distribution of expected quality adjusted life years (in addition to the distribution of expected survival times). The expected quality of life can also be estimated within the health state sojourn time making more flexible the assessment of utility preferences. The methods are demonstrated on a subset of follow-up data from the Beta Blocker Heart Attack Trial (BHAT). This model contains the structure necessary to make inferences when assessing a general survival problem with a two dimensional outcome. ^

Selection bias and the cross-validation of regression models for prediction

Strategies are compared for the development of a linear regression model with stochastic (multivariate normal) regressor variables and the subsequent assessment of its predictive ability. Bias and mean squared error of four estimators of predictive performance are evaluated in simulated samples of 32 population correlation matrices. Models including all of the available predictors are compared with those obtained using selected subsets. The subset selection procedures investigated include two stopping rules, C$\sb{\rm p}$ and S$\sb{\rm p}$, each combined with an 'all possible subsets' or 'forward selection' of variables. The estimators of performance utilized include parametric (MSEP$\sb{\rm m}$) and non-parametric (PRESS) assessments in the entire sample, and two data splitting estimates restricted to a random or balanced (Snee's DUPLEX) 'validation' half sample. The simulations were performed as a designed experiment, with population correlation matrices representing a broad range of data structures.^ The techniques examined for subset selection do not generally result in improved predictions relative to the full model. Approaches using 'forward selection' result in slightly smaller prediction errors and less biased estimators of predictive accuracy than 'all possible subsets' approaches but no differences are detected between the performances of C$\sb{\rm p}$ and S$\sb{\rm p}$. In every case, prediction errors of models obtained by subset selection in either of the half splits exceed those obtained using all predictors and the entire sample.^ Only the random split estimator is conditionally (on $\\beta$) unbiased, however MSEP$\sb{\rm m}$ is unbiased on average and PRESS is nearly so in unselected (fixed form) models. When subset selection techniques are used, MSEP$\sb{\rm m}$ and PRESS always underestimate prediction errors, by as much as 27 percent (on average) in small samples. Despite their bias, the mean squared errors (MSE) of these estimators are at least 30 percent less than that of the unbiased random split estimator. The DUPLEX split estimator suffers from large MSE as well as bias, and seems of little value within the context of stochastic regressor variables.^ To maximize predictive accuracy while retaining a reliable estimate of that accuracy, it is recommended that the entire sample be used for model development, and a leave-one-out statistic (e.g. PRESS) be used for assessment. ^

A multiplicative model for standardizing vital rates without using external standards

Standardization is a common method for adjusting confounding factors when comparing two or more exposure category to assess excess risk. Arbitrary choice of standard population in standardization introduces selection bias due to healthy worker effect. Small sample in specific groups also poses problems in estimating relative risk and the statistical significance is problematic. As an alternative, statistical models were proposed to overcome such limitations and find adjusted rates. In this dissertation, a multiplicative model is considered to address the issues related to standardized index namely: Standardized Mortality Ratio (SMR) and Comparative Mortality Factor (CMF). The model provides an alternative to conventional standardized technique. Maximum likelihood estimates of parameters of the model are used to construct an index similar to the SMR for estimating relative risk of exposure groups under comparison. Parametric Bootstrap resampling method is used to evaluate the goodness of fit of the model, behavior of estimated parameters and variability in relative risk on generated sample. The model provides an alternative to both direct and indirect standardization method. ^

Bayesian inference and its application to meta-analysis of epidemiologic data

In Part One, the foundations of Bayesian inference are reviewed, and the technicalities of the Bayesian method are illustrated. Part Two applies the Bayesian meta-analysis program, the Confidence Profile Method (CPM), to clinical trial data and evaluates the merits of using Bayesian meta-analysis for overviews of clinical trials.^ The Bayesian method of meta-analysis produced similar results to the classical results because of the large sample size, along with the input of a non-preferential prior probability distribution. These results were anticipated through explanations in Part One of the mechanics of the Bayesian approach. ^

Mitral valve prolapse and its relationship with stroke

The purpose of this study was to elucidate the relationship between mitral valve prolapse and stroke. A population-based historical cohort investigation was conducted among residents of Olmsted County, Minnesota who had an initial echocardiographic diagnosis of mitral valve prolapse from 1975 through 1989. This cohort (N = 1085) was followed for stroke outcomes using the resources of an operational medical record linkage system. There was an overall two-fold increase in the incidence of stroke among individuals with mitral valve prolapse relative to a standard population (standardized morbidity ratio = 2.12, 95% confidence limits = 1.33-3.21). When the data were partitioned by duration of follow-up from the diagnosis of mitral valve prolapse, or by the calendar years at echocardiographic diagnosis, respectively, the association between mitral valve prolapse and stroke was not modified. Mitral valve prolapse subjects 85 years and older were at highest increased risk of developing strokes relative to the general population (standardized morbidity ratio = 5.47, 95% confidence limits = 2.20-11.24). Coronary heart disease, atrial fibrillation, diabetes mellitus and hypertension, were unlikely to have confounded the association between mitral valve prolapse and stroke.^ The cumulative risk of first stroke among individuals initially diagnosed with mitral valve prolapse age 15 to 64 years, given survival to 15.2 years of follow-up, was 4.0%. The cumulative risk of first stroke among individuals initially diagnosed with mitral valve prolapse age 65 to 74 years, given survival to 11.2 years of follow-up, was 13.2%. The cumulative risk of first stroke among individuals initially diagnosed with mitral valve prolapse age 75 years and older, given survival to 6.7 years of follow-up, was 30.6%.^ Among individuals with mitral valve prolapse, age, diabetes, and atrial fibrillation were associated with an increased risk of stroke. Atrial fibrillation was associated with a four-fold rate of stroke and diabetes associated with a seven-fold rate of stroke.^ Findings from this research support the hypothesis that mitral valvular heart prolapse is linked with a stroke sequela. ^

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 effect of life events on blood pressure and body mass index in children

In Conroe, Texas, 492 students ages 5 to 15 participated in a screening examination for cardiovascular risk factor study. Among 492 students, 141 elementary and junior high students participated in the present sub-study to investigate the effect of the number of recent life events on blood pressure and on body mass index. Using the elementary and junior high school Coddington scales, life events occurring in the past 12 months were measured for students ages 9 to 14 years, no significant differences in life events were observed by age and sex. The number of life events was not related to blood pressure but was positively correlated to body mass index in children and adolescents. ^

A Bayesian approach to estimating the regression coefficients of a multinomial logit model

A Bayesian approach to estimation of the regression coefficients of a multinominal logit model with ordinal scale response categories is presented. A Monte Carlo method is used to construct the posterior distribution of the link function. The link function is treated as an arbitrary scalar function. Then the Gauss-Markov theorem is used to determine a function of the link which produces a random vector of coefficients. The posterior distribution of the random vector of coefficients is used to estimate the regression coefficients. The method described is referred to as a Bayesian generalized least square (BGLS) analysis. Two cases involving multinominal logit models are described. Case I involves a cumulative logit model and Case II involves a proportional-odds model. All inferences about the coefficients for both cases are described in terms of the posterior distribution of the regression coefficients. The results from the BGLS method are compared to maximum likelihood estimates of the regression coefficients. The BGLS method avoids the nonlinear problems encountered when estimating the regression coefficients of a generalized linear model. The method is not complex or computationally intensive. The BGLS method offers several advantages over Bayesian approaches. ^

Modeling the survival of T-cell lymphocytes using compound branching processes

Radiotherapy has been a method of choice in cancer treatment for a number of years. Mathematical modeling is an important tool in studying the survival behavior of any cell as well as its radiosensitivity. One particular cell under investigation is the normal T-cell, the radiosensitivity of which may be indicative to the patient's tolerance to radiation doses.^ The model derived is a compound branching process with a random initial population of T-cells that is assumed to have compound distribution. T-cells in any generation are assumed to double or die at random lengths of time. This population is assumed to undergo a random number of generations within a period of time. The model is then used to obtain an estimate for the survival probability of T-cells for the data under investigation. This estimate is derived iteratively by applying the likelihood principle. Further assessment of the validity of the model is performed by simulating a number of subjects under this model.^ This study shows that there is a great deal of variation in T-cells survival from one individual to another. These variations can be observed under normal conditions as well as under radiotherapy. The findings are in agreement with a recent study and show that genetic diversity plays a role in determining the survival of T-cells. ^

Prospective run theory for multinomial outcomes

The focus of this study was to generalize the theory of runs to multinomial outcomes using the generating function approach. Detailed discussion is provided for determining the probability distributions for all runs of length i in a sequence of n trials for the binomial and trinomial cases. The generalization to multinomial case is also presented. Application to data for patients from a long term disability care facility is presented to illustrate the use of Run Theory in determining the probability of a dominant state of treatment associated with a patient during his/her hospitalization. ^

Maximal exercise hemodynamics and risk of mortality in apparently healthy men and women

The use of exercise electrocardiography (ECG) to detect latent coronary heart disease (CHD) is discouraged in apparently healthy populations because of low sensitivity. These recommendations however, are based on the efficacy of evaluation of ischemia (ST segment changes) with little regard for other measures of cardiac function that are available during exertion. The purpose of this investigation was to determine the association of maximal exercise hemodynamic responses with risk of mortality due to all-causes, cardiovascular disease (CVD), and coronary heart disease (CHD) in apparently healthy individuals. Study participants were 20,387 men (mean age = 42.2 years) and 6,234 women (mean age = 41.9 years) patients of a preventive medicine center in Dallas, TX examined between 1971 and 1989. During an average of 8.1 years of follow-up, there were 348 deaths in men and 66 deaths in women. In men, age-adjusted all-cause death rates (per 10,000 person years) across quartiles of maximal systolic blood pressure (SBP) (low to high) were: 18.2, 16.2, 23.8, and 24.6 (p for trend $<$0.001). Corresponding rates for maximal heart rate were: 28.9, 15.9, 18.4, and 15.1 (p trend $<$0.001). After adjustment for confounding variables including age, resting systolic pressure, serum cholesterol and glucose, body mass index, smoking status, physical fitness and family history of CVD, risks (and 95% confidence interval (CI)) of all-cause mortality for quartiles of maximal SBP, relative to the lowest quartile, were: 0.96 (0.70-1.33), 1.36 (1.01-1.85), and 1.37 (0.98-1.92) for quartiles 2-4 respectively. Similar risks for maximal heart rate were: 0.61 (0.44-0.85), 0.69 (0.51-0.93), and 0.60 (0.41-0.87). No associations were noted between maximal exercise rate-pressure product mortality. Similar results were seen for risk of CVD and CHD death. In women, similar trends in age-adjusted all-cause and CVD death rates across maximal SBP and heart rate categories were observed. Sensitivity of the exercise test in predicting mortality was enhanced when ECG results were evaluated together with maximal exercise SBP or heart rate with a concomitant decrease in specificity. Positive predictive values were not improved. The efficacy of the exercise test in predicting mortality in apparently healthy men and women was not enhanced by using maximal exercise hemodynamic responses. These results suggest that an exaggerated systolic blood pressure or an attenuated heart rate response to maximal exercise are risk factors for mortality in apparently healthy individuals. ^

Genetic epidemiologic methods in risk determination in Li-Fraumeni syndrome

Li-Fraumeni syndrome (LFS) is characterized by a variety of neoplasms occurring at a young age with an apparent autosomal dominant transmission. Individuals in pedigrees with LFS have high incidence of second malignancies. Recently LFS has been found to be associated with germline mutations of a tumor-suppressor gene, p53. Because LFS is rare and indeed not a clear-cut disease, it is not known whether all cases of LFS are attributable to p53 germline mutations and how p53 plays in cancer occurrence in such cancer syndrome families. In the present study, DNAs from constitutive cells of two-hundred and thirty-three family members from ten extended pedigrees were screened for p53 mutations. Six out of the ten LFS families had germline mutations at the p53 locus, including point and deletion mutations. In these six families, 55 out of 146 members were carriers of p53 mutations. Except one, all mutations occurred in exons 5 to 8 (i.e., the "hot spot" region) of the p53 gene. The age-specific penetrance of cancer was estimated after the genotype for each family member at risk was determined. The penetrance was 0.15, 0.29, 0.35, 0.77, and 0.91 by 20, 30, 40, 50 and 60 year-old, respectively, in male carriers; 0.19, 0.44, 0.76, and 0.90 by 20, 30, 40, and 50 year-old, respectively, in female carriers. These results indicated that one cannot escape from tumorigenesis if one inherits a p53 mutant allele; at least ninety percent of p53 carriers will develop cancer by the age of 60. To evaluate the possible bias due to the unexamined blood-relatives in LFS families, I performed a simulation analysis in which a p53 genotype was assigned to each unexamined person based on his cancer status and liability to cancer. The results showed that the penetrance estimates were not biased by the unexamined relatives. I also determined the sex, site, and age-specific penetrance of breast cancer in female carriers and lung cancer in male carriers. The penetrance of breast cancer in female carriers was 0.81 by age 45; the penetrance of lung cancer in male carriers was 0.78 by age 60, indicating that p53 play a key role for tumorigenesis in common cancers. ^

Natural history of placebo treatment in adults acquiring travelers' diarrhea in Mexico

The natural history of placebo treated travelers' diarrhea and the prognostic factors of recovery from diarrhea were evaluated using 9 groups of placebo treated subjects from 9 clinical trial studies conducted since 1975, for use as a historical control in the future clinical trial of antidiarrheal agents. All of these studies were done by the same group of investigators in one site (Guadalajara, Mexico). The studies are similar in terms of population, measured parameters, microbiologic identification of enteropathogens and definitions of parameters. The studies had two different durations of followup. In some studies, subjects were followed for two days, and in some they were followed for five days.^ Using definitions established by the Infectious Diseases society of America and the Food and Drug Administration, the following efficacy parameters were evaluated: Time to last unformed stool (TLUS), number of unformed stools post-initiation of placebo treatment for five consecutive days of followup, microbiologic cure, and improvement of diarrhea. Among the groups that were followed for five days, the mean TLUS ranged from 59.1 to 83.5 hours. Fifty percent to 78% had diarrhea lasting more than 48 hours and 25% had diarrhea more than five days. The mean number of unformed stools passed on the first day post-initiation of therapy ranged from 3.6 to 5.8 and, for the fifth day ranged from 0.5 to 1.5. By the end of followup, diarrhea improved in 82.6% to 90% of the subjects. Subjects with enterotoxigenic E. coli had 21.6% to 90.0% microbiologic cure; and subjects with shigella species experienced 14.3% to 60.0% microbiologic cure.^ In evaluating the prognostic factors of recovery from diarrhea (primary efficacy parameter in evaluating the efficacy of antidiarrheal agents against travelers' diarrhea). The subjects from five studies were pooled and the Cox proportional hazard model was used to evaluate the predictors of prolonged diarrhea. After adjusting for design characteristics of each trial, fever with a rate ratio (RR) of 0.40, presence of invasive pathogens with a RR of 0.41, presence of severe abdominal pain and cramps with a RR of 0.50, number of watery stools more than five with a RR of 0.60, and presence of non-invasive pathogens with a RR of 0.84 predicted a longer duration of diarrhea. Severe vomiting with a RR of 2.53 predicted a shorter duration of diarrhea. The number of soft stools, presence of fecal leukocytes, presence of nausea, and duration of diarrhea before enrollment were not associated with duration of diarrhea. ^

A Bayesian analysis for spatial processes with application to mapping

In geographical epidemiology, maps of disease rates and disease risk provide a spatial perspective for researching disease etiology. For rare diseases or when the population base is small, the rate and risk estimates may be unstable. Empirical Bayesian (EB) methods have been used to spatially smooth the estimates by permitting an area estimate to "borrow strength" from its neighbors. Such EB methods include the use of a Gamma model, of a James-Stein estimator, and of a conditional autoregressive (CAR) process. A fully Bayesian analysis of the CAR process is proposed. One advantage of this fully Bayesian analysis is that it can be implemented simply by using repeated sampling from the posterior densities. Use of a Markov chain Monte Carlo technique such as Gibbs sampler was not necessary. Direct resampling from the posterior densities provides exact small sample inferences instead of the approximate asymptotic analyses of maximum likelihood methods (Clayton & Kaldor, 1987). Further, the proposed CAR model provides for covariates to be included in the model. A simulation demonstrates the effect of sample size on the fully Bayesian analysis of the CAR process. The methods are applied to lip cancer data from Scotland, and the results are compared. ^