David B. Wolfson

A reevaluation of the duration of survival after the onset of dementia.

BACKGROUND Dementia shortens life expectancy; estimates of median survival after the onset of dementia have ranged from 5 to 9.3 years. Previous studies of people with existing dementia, however, may have underestimated the deleterious effects of dementia on survival by failing to consider persons with rapidly progressive illness who died before they could be included in a study (referred to as length bias). METHODS We used data from the Canadian Study of Health and Aging to estimate survival from the onset of symptoms of dementia; the estimate was adjusted for length bias. A random sample of 10,263 subjects 65 years old or older from throughout Canada was screened for cognitive impairment. For those with dementia, we ascertained the date of onset and conducted follow-up for five years. RESULTS We analyzed data on 821 subjects, of whom 396 had probable Alzheimers disease, 252 had possible Alzheimers disease, and 173 had vascular dementia. For the group as a whole, the unadjusted median survival was 6.6 years (95 percent confidence interval, 6.2 to 7.1). After adjustment for length bias, the estimated median survival was 3.3 years (95 percent confidence interval, 2.7 to 4.0). The median survival was 3.1 years for subjects with probable Alzheimers disease, 3.5 years for subjects with possible Alzheimers disease, and 3.3 years for subjects with vascular dementia. CONCLUSIONS Median survival after the onset of dementia is much shorter than has previously been estimated.

Length-Biased Sampling With Right Censoring: An Unconditional Approach

When survival data arise from prevalent cases ascertained through a cross-sectional study, it is well known that the survivor function corresponding to these data is length biased and different from the survivor function derived from incident cases. Length-biased data have been treated both unconditionally and conditionally in the literature. In the latter case, where length bias is viewed as being induced by random left truncation of the survival times, the truncating distribution is assumed to be unknown. Conditioning on the observed truncation times hence causes very little loss of information. In many instances, however, it can be supposed that the truncating distribution is uniform, and it has been pointed out that under these circumstances, an unconditional analysis will be more informative. There are no results in the current literature that give the asymptotic properties of the unconditional nonparametric maximum likelihood estimator (NPMLE) of the unbiased survivor function in the presence of censoring. This article fills that gap by giving this NPMLE and its accompanying asymptotic properties when the data are purely length biased. An example of survival with dementia is presented in which the conditional and unconditional estimators are compared.

Sample Size Calculations for Binomial Proportions via Highest Posterior Density Intervals

Three different Bayesian approaches to sample size calculations based on highest posterior density (HPD) intervals are discussed and illustrated in the context of a binomial experiment. The preposterior marginal distribution of the data is used to find the sample size needed to attain an expected HPD coverage probability for a given fixed interval length. Alternatively, one can find the sample size required to attain an expected HPD interval length for a fixed coverage. These two criteria can lead to different sample size requirements. In addition to averaging, a worst possible outcome scenario is also considered. The results presented here provide an exact solution to a problem recently addressed in the literature.

Some comments on Bayesian sample size determination

SUMMARY Several criteria for Bayesian sample size determination have recently been proposed. Criteria based on highest posterior density (HPD) intervals from the exact posterior distribution in general lead to smaller sample sizes than those based on non-HPD intervals and/or normal approximations to the exact density. The economies are variable, however, and depend both on the prior inputs and the desired posterior accuracy and coverage probability. In our reply we review several properties of sample size methods and discuss the importance of these properties in the context of a binomial experiment. A general algorithm for Bayesian sample size determination that is useful for more complex sampling situations based on Monte Carlo simulations is briefly described.

Covariate Bias Induced by Length-Biased Sampling of Failure Times

Although many authors have proposed different approaches to the analysis of length-biased survival data, a number of issues have not been fully addressed. The most important among these issues is perhaps that regarding inclusion of covariates into the analysis of length-biased lifetime data collected through cross-sectional sampling of a population. One aspect of this problem, which appears to have been neglected in the literature, concerns the effect of length bias on the sampling distribution of the covariates. In most regression analyses, it is conventional to condition on the observed covariate values; however, certain covariate values could be preferentially selected into the sample, being linked to the long-term survivors, who themselves are favored by the sampling mechanism. This observation raises two questions: (1) Does the conditional analysis of covariates lead to biased estimators of regression coefficients?; and (2) does inference through the joint l likelihood of covariates and failure times yield more efficient estimators of the regression parameters? We present a joint likelihood approach and study the large-sample behavior of the resulting maximum likelihood estimators (MLEs). We find that these MLEs are more efficient than their conditional counterparts even though the two MLEs are asymptotically equal. Our results are illustrated using data on survival with dementia, collected as part of the Canadian Study of Health and Aging.

The latent period of multiple sclerosis: a critical review.

It is a widely held belief that multiple sclerosis is a disease with a long latent period that is preceded by heightened susceptibility before adolescence. There has, however, been little research focused on either the estimation of the latent period or determination of the susceptibility period. In this article, we present a critical assessment of the relevant literature on migrant studies, cluster studies, the Faroe Islands “epidemic,” a sibling study, and novel statistical approaches as they pertain to the pre-onset natural history of multiple sclerosis. We also discuss the roles of the latent and susceptibility periods in the design, analysis, and interpretation of epidemiologic studies. (Epidemiology 1993;4:464–470)

Estimation in multi-path change-point problems

The current literature deals with the change-point problem only in the context of the obser¬vation of a single sequence. In this paper, inference will be based on the observation of TV sequences of random variables, each sequence containing one change-point. This extension allows the effective use of bootstrap and empirical Bayes methods, both of which are not feasible in the single-path context. Two classes of these “multi-path” change-point problems are considered. If the change-point is assumed to occur at the the same position in each sequence, then the terminology “fixed-tau multi-path change-point” will be used. In other cases, one may expect the change-point to occur at random positions in each sequence, according to some distribution, a “random-tau multi-path change-point” problem. Examples and simulations are given.

National General Practice Study of Epilepsy and Epileptic Seizures: Objectives and Study Methodology of the Largest Reported Prospective Cohort Study of Epilepsy

Most available data on the prognosis of epileptic seizures come from hospital-based clinics in which patients with chronic or severe disease are over-represented. The National General Practice Study of Epilepsy and Epileptic Seizures is a large prospective community-based study of people with newly diagnosed seizures which aims to address questions related to the early prognosis of epilepsy. 275 general practitioners throughout the United Kingdom have registered a total of 1,195 patients. In this paper we discuss the background to the study and the methodology used.

Bayesian sample size determination for binomial proportions

This paper presents several new results on Bayesian sample size deter- mination for estimating binomial proportions, and provides a comprehensive com- parative overview of the subject. We investigate the binomial sample size problem using generalized versions of the Average Length and Average Coverage Criteria, the Median Length and Median Coverage Criteria, as well as the Worst Outcome Criterion and its modied version. We compare sample sizes derived from highest posterior density and equal-tailed credible intervals. In some cases, we derive, for the rst time, closed form sample size formulae, and where this is not possible, we describe various numerical approaches. These range in complexity from Monte Carlo simulations to more sophisticated curve tting techniques, third order an- alytic approximations, and exact, but more computationally-intensive, methods. We compare the accuracy and eciency of the dieren t computational methods for each of the criteria and make recommendations about which methods are preferred. Finally, we consider, again for the rst time, issues surrounding prior robustness on the choice of sample size. Examples are given throughout the text.

Interval-based versus decision theoretic criteria for the choice of sample size

SUMMARY Several different criteria for Bayesian sample size determination have recently been proposed. Bayesian approaches are natural, since at the planning stage of an experiment one is forced to consider prior notions about unknown parameter values that may affect the choice of a final sample size. For this, all the methods consider a prior distribution over the unknown parameters. Differences between the methods have been driven by the type of inferences that will be made, e.g. hypothesis testing or interval estimation, the latter based on posterior means and variances or highest posterior density regions. A more fundamental question, however, is whether to introduce formally a loss or utility function to aid in choosing the sample size. In this paper, we discuss the advantages and disadvantages of taking a fully decision theoretic approach versus one of the simpler approaches, which only implicitly consider utilities in balancing increased precision against the increased costs associated with larger sample sizes. Throughout, we emphasize the practical aspects of sample size estimation, raising issues that would face the consumer of statistics in selecting a sample size in a given experiment.