ABSTRACT. We investigate non-parametric estimation of a monotone baseline hazard and a decreasing baseline density within the Cox model. Two estimators of a non-decreasing baseline hazard function are proposed. We derive the non-parametric maximum likelihood estimator and consider a Grenander type estimator, defined as the left-hand slope of the greatest convex minorant of the Breslow estimator. We demonstrate that the two estimators are strongly consistent and asymptotically equivalent and derive their common limit distribution at a fixed point. Both estimators of a non-increasing baseline hazard and their asymptotic properties are obtained in a similar manner. Furthermore, we introduce a Grenander type estimator for a non-increasing baseline density, defined as the left-hand slope of the least concave majorant of an estimator of the baseline cumulative distribution function, derived from the Breslow estimator. We show that this estimator is strongly consistent and derive its asymptotic distribution at a fixed point.

Expert elicitation is deployed when data are absent or uninformative and critical decisions must be made. In designing an expert elicitation, most practitioners seek to achieve best practice while balancing practical constraints. The choices made influence the required time and effort investment, the quality of the elicited data, experts' engagement, the defensibility of results, and the acceptability of resulting decisions. This piece outlines some of the common choices practitioners encounter when designing and conducting an elicitation. We discuss the evidence supporting these decisions and identify research gaps. This will hopefully allow practitioners to better navigate the literature, and will inspire the expert judgement research community to conduct well powered, replicable experiments that properly address the research gaps identified.

Illnesses transmitted by food and water cause a major disease burden in the United States despite advancements in food safety, water treatment, and sanitation. We report estimates from a structured expert judgment study using 48 experts who applied Cooke’s classical model of the proportion of disease attributable to 5 major transmission pathways (foodborne, waterborne, person-to-person, animal contact, and environmental) and 6 subpathways (food handler–related, under foodborne; recreational, drinking, and nonrecreational/nondrinking, under waterborne; and presumed person-to-person-associated and presumed animal contact-associated, under environmental). Estimates for 33 pathogens were elicited, including bacteria such as Salmonella enterica, Campylobacter spp., Legionella spp., and Pseudomonas spp.; protozoa such as Acanthamoeba spp., Cyclospora cayetanensis , and Naegleria fowleri ; and viruses such as norovirus, rotavirus, and hepatitis A virus. The results highlight the importance of multiple pathways in the transmission of the included pathogens and can be used to guide prioritization of public health interventions.

Although estimates of regression coefficients depend on other independent variables, there is no assumed dependence relationship between coefficient estimators and the change in value of other variables as in the case of BNs. Nonetheless, this analysis suggests that regression is still more accurate or at least as accurate as BNs for risk estimation for both LRRs and SP tumors.