Some general comparative points on Chao’s and Zelterman’s estimators of population size

Two simple and frequently used capture–recapture estimates of the population size are compared: Chao's lower-bound estimate and Zelterman's estimate allowing for contaminated distributions. In the Poisson case it is shown that if there are only counts of ones and twos, the estimator of Zelterman is always bounded above by Chao's estimator. If counts larger than two exist, the estimator of Zelterman is becoming larger than that of Chao's, if only the ratio of the frequencies of counts of twos and ones is small enough. A similar analysis is provided for the binomial case. For a two-component mixture of Poisson distributions the asymptotic bias of both estimators is derived and it is shown that the Zelterman estimator can experience large overestimation bias. A modified Zelterman estimator is suggested and also the bias-corrected version of Chao's estimator is considered. All four estimators are compared in a simulation study.

The sequential analysis of repeated binary responses: a score test for the case of three time points

In this paper a robust method is developed for the analysis of data consisting of repeated binary observations taken at up to three fixed time points on each subject. The primary objective is to compare outcomes at the last time point, using earlier observations to predict this for subjects with incomplete records. A score test is derived. The method is developed for application to sequential clinical trials, as at interim analyses there will be many incomplete records occurring in non-informative patterns. Motivation for the methodology comes from experience with clinical trials in stroke and head injury, and data from one such trial is used to illustrate the approach. Extensions to more than three time points and to allow for stratification are discussed. Copyright © 2005 John Wiley & Sons, Ltd.

Some general comparative points on Chao's and Zelterman's estimators of the population size

Two simple and frequently used capture–recapture estimates of the population size are compared: Chao's lower-bound estimate and Zelterman's estimate allowing for contaminated distributions. In the Poisson case it is shown that if there are only counts of ones and twos, the estimator of Zelterman is always bounded above by Chao's estimator. If counts larger than two exist, the estimator of Zelterman is becoming larger than that of Chao's, if only the ratio of the frequencies of counts of twos and ones is small enough. A similar analysis is provided for the binomial case. For a two-component mixture of Poisson distributions the asymptotic bias of both estimators is derived and it is shown that the Zelterman estimator can experience large overestimation bias. A modified Zelterman estimator is suggested and also the bias-corrected version of Chao's estimator is considered. All four estimators are compared in a simulation study.

Turning on the alarm: the neural mechanisms of the transition from innocuous to painful sensation

The experience of pain occurs when the level of a stimulus is sufficient to elicit a marked affective response, putatively to warn the organism of potential danger and motivate appropriate behavioral responses. Understanding the biological mechanisms of the transition from innocuous to painful levels of sensation is essential to understanding pain perception as well as clinical conditions characterized by abnormal relationships between stimulation and pain response. Thus, the primary objective of this study was to characterize the neural response associated with this transition and the correspondence between that response and subjective reports of pain. Towards this goal, this study examined BOLD response profiles across a range of temperatures spanning the pain threshold. 14 healthy adults underwent functional magnetic resonance imaging (fMRI) while a range of thermal stimuli (44-49oC) were applied. BOLD responses showed a sigmoidal profile along the range of temperatures in a network of brain regions including insula and mid- cingulate, as well as a number of regions associated with motor responses including ventral lateral nuclei of the thalamus, globus pallidus and premotor cortex. A sigmoid function fit to the BOLD responses in these regions explained up to 85% of the variance in individual pain ratings, and yielded an estimate of the temperature of steepest transition from non-painful to painful heat that was nearly identical to that generated by subjective ratings. These results demonstrate a precise characterization of the relationship between objective levels of stimulation, resulting neural activation, and subjective experience of pain and provide direct evidence for a neural mechanism supporting the nonlinear transition from innocuous to painful levels along the sensory continuum.