Modeling and estimating a threshold effect: An application to improving cardiac surgery practices

Estimating thresholds when a threshold effect exists has important applications in biomedical research. However, models/methods commonly used in the biomedical literature may lead to a biased estimate. For patients undergoing coronary artery bypass grafting (CABG), it is thought that exposure to low oxygen delivery (DO2) contributes to an increased risk of avoidable acute kidney injury. This research is motivated by estimating the threshold of nadir DO2 for CABG patients to help develop an evidence-based guideline for improving cardiac surgery practices. We review several models (sudden-jump model, broken-stick model, and the constrained broken-stick model) that can be adopted to estimate the threshold and discuss modeling assumptions, scientific plausibility, and implications in estimating the threshold. Under each model, various estimation methods are studied and compared. In particular, under a constrained broken-stick model, a modified two-step Newton-Raphson algorithm is introduced. Through comprehensive simulation studies and an application to data on CABG patients from the University of Michigan, we show that the constrained broken-stick model is flexible, more robust, and able to incorporate scientific knowledge to improve efficiency. The two-step Newton-Raphson algorithm has good computational performances relative to existing methods.

Significance of crinoid preservation: Clare Shale Formation (Upper Carboniferous), Fisherstreet Bay, Doolin, County Clare, Ireland

Some fossils, such as crinoid stems, are not widely appreciated by collectors and researchers, yet can provide unique data regarding taphonomy and palaeoecology. A long crinoid pluricolumnal showing a distinctive pattern of preservation was collected from the Clare Shale Formation (Upper Carboniferous) at Fisherstreet Bay, Doolin, County Clare, western Ireland. The specimen is partly disarticulated and represents the mesistele to mesistele/dististele transition; attachment was by unbranched long, slender radices; pluricolumnal heteromorphic; fragments of pluricolumnal are of multiples of a unit length. This specimen, cladid? sp. indet., slumped to the seafloor after death and started to disarticulate as ligaments rotted. By reference to the broken stick model, the pattern of disarticulation suggests that the noditaxis of the heteromorphic stem was N212.