Median survival after the onset of dementia is much shorter than has previously been estimated.

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SUMMARY 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.

Right censored survival data collected on a cohort of prevalent cases with constant incidence are length-biased, and may be used to estimate the length-biased (i.e., prevalent-case) survival function. When the incidence rate is constant, so-called stationarity of the incidence, it is more efficient to use this structure for unconditional statistical inference than to carry out an analysis by conditioning on the observed truncation times. It is well known that, due to the informative censoring for prevalent cohort data, the Kaplan-Meier estimator is not the unconditional NPMLE of the length-biased survival function and the asymptotic properties of the NPMLE do not follow from any known result. We present here a detailed derivation of the asymptotic properties of the NPMLE of the length-biased survival function from right censored prevalent cohort survival data with follow-up. In particular, we show that the NPMLE is uniformly strongly consistent, converges weakly to a Gaussian process, and is asymptotically efficient. One important spin-off from these results is that they yield the asymptotic properties of the NPMLE of the incident-case survival function [see Asgharian, M'Lan and Wolfson J. Amer. Statist. Assoc. 97 (2002) 201-209], which is often of prime interest in a prevalent cohort study. Our results generalize those given by Vardi and Zhang [Ann. Statist. 20 (1992) 1022-1039] under multiplicative censoring, which we show arises as a degenerate case in a prevalent cohort setting.

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In a prevalent cohort study with follow-up, the incidence process is not directly observed since only the onset times of prevalent cases can be ascertained. Assessing the "stationarity" of the underlying incidence process can be important for at least three reasons, including an improvement in efficiency when estimating the survivor function. We propose, for the first time, a formal test for stationarity using data from a prevalent cohort study with follow-up. The test makes use of a characterization of stationarity, an extension of this characterization developed in this paper, and of a test for matched pairs of right censored data. We report the results from a power study assuming varying degrees of departure from the null hypothesis of stationarity. The test is also applied to data obtained as part of the Canadian Study of Health and Aging (CSHA) to verify whether the incidence rate of dementia amongst the elderly in Canada has remained constant.

Summary• Ecological and biological processes can change from one state to another once a threshold has been crossed in space or time. Threshold responses to incremental changes in underlying variables can characterize diverse processes from climate change to the desertification of arid lands from overgrazing.• Simultaneously estimating the location of thresholds and associated ecological parameters can be difficult: ecological data are often 'noisy', which can make the identification of the locations of ecological thresholds challenging.• We illustrate this problem using two ecological examples and apply a class of statistical models well-suited to addressing this problem. We first consider the case of estimating allometric relationships between tree diameter and height when the trees have distinctly different growth modes across life-history stages. We next estimate the effects of canopy gaps and dense understory vegetation on tree recruitment in transects that transverse both canopy and gap conditions.• The Bayesian change-point models that we present estimate both threshold locations and the slope or level of ecological quantities of interest, while incorporating uncertainty in the change-point location into these estimates. This class of models is suitable for problems with multiple thresholds and can account for spatial or temporal autocorrelation.

When survival data are collected as part of a prevalent cohort study with follow-up, the recruited cases have already experienced their initiating event, say onset of a disease, and consequently the incidence process is only partially observed. Nevertheless, there are good reasons for interest in certain features of the underlying incidence process, for example whether or not it is stationary. Indeed, the well known relationship between incidence and prevalence, often used by epidemiologists, requires stationarity of the incidence rate for its validity. Also, the statistician can exploit stationarity of the incidence process by improving the efficiency of estimators in a prevalent cohort survival analysis. In addition, whether the incident rate is stationary is often in itself of central importance to medical and other researchers. We present here a necessary and sufficient condition for stationarity of the underlying incidence process, which uses only survival observations, possibly right censored, from a prevalent cohort study with follow-up. This leads to a simple graphical means of checking for the stationarity of the underlying incidence times by comparing the plots of two Kaplan-Meier estimates that are based on partially observed incidence times and follow-up survival data. We use our method to discuss the incidence rate of dementia in Canada between 1971 and 1991.