Kronecker product, Loewner ordering, lower bound principle, monotonicity,

For frequency counts, the situation of extra zeros often arises in biomedical applications. This is demonstrated with count data from a dental epidemiological study in Belo Horizonte (the Belo Horizonte caries prevention study) which evaluated various programmes for reducing caries. Extra zeros, however, violate the variance±mean relationship of the Poisson error structure. This extra-Poisson variation can easily be explained by a special mixture model, the zero-in¯ated Poisson (ZIP) model. On the basis of the ZIP model, a graphical device is presented which not only summarizes the mixing distribution but also provides visual information about the overall mean. This device can be exploited to evaluate and compare various groups. Ways are discussed to include covariates and to develop an extension of the conventional Poisson regression. Finally, a method to evaluate intervention effects on the basis of the ZIP regression model is described and applied to the data of the Belo Horizonte caries prevention study.

Maximum likelihood estimation, curvature, monotonicity, algorithms, Newton-Raphson algorithm,

Background Low nurse staffing levels are associated with adverse patient outcomes from hospital care, but the causal relationship is unclear. Limited capacity to observe patients has been hypothesised as a causal mechanism. Objectives This study determines whether or not adverse outcomes are more likely to occur after patients experience low nurse staffing levels, and whether or not missed vital signs observations mediate any relationship. Design Retrospective longitudinal observational study. Multilevel/hierarchical mixed-effects regression models were used to explore the association between registered nurse (RN) and health-care assistant (HCA) staffing levels and outcomes, controlling for ward and patient factors. Setting and participants A total of 138,133 admissions to 32 general adult wards of an acute hospital from 2012 to 2015. Main outcomes Death in hospital, adverse event (death, cardiac arrest or unplanned intensive care unit admission), length of stay and missed vital signs observations. Data sources Patient administration system, cardiac arrest database, eRoster, temporary staff bookings and the Vitalpac system (System C Healthcare Ltd, Maidstone, Kent; formerly The Learning Clinic Limited) for observations. Results Over the first 5 days of stay, each additional hour of RN care was associated with a 3% reduction in the hazard of death [hazard ratio (HR) 0.97, 95% confidence interval (CI) 0.94 to 1.0]. Days on which the HCA staffing level fell below the mean were associated with an increased hazard of death (HR 1.04, 95% CI 1.02 to 1.07), but the hazard of death increased as cumulative staffing exposures varied from the mean in either direction. Higher levels of temporary staffing were associated with increased mortality. Adverse events and length of stay were reduced with higher RN staffing. Overall, 16% of observations were missed. Higher RN staffing was associated with fewer missed observations in high-acuity patients (incidence rate ratio 0.98, 95% CI 0.97 to 0.99), whereas the overall rate of missed observations was related to overall care hours (RN + HCA) but not to skill mix. The relationship between low RN staffing and mortality was mediated by missed observations, but other relationships between staffing and mortality were not. Changing average skill mix and staffing levels to the levels planned by the Trust, involving an increase of 0.32 RN hours per patient day (HPPD) and a similar decrease in HCA HPPD, would be associated with reduced mortality, an increase in staffing costs of £28 per patient and a saving of £0.52 per patient per hospital stay, after accounting for the value of reduced stays. Limitations This was an observational study in a single site. Evidence of cause is not definitive. Variation in staffing could be influenced by variation in the assessed need for staff. Our economic analysis did not consider quality or length of life. Conclusions Higher RN staffing levels are associated with lower mortality, and this study provides evidence of a causal mechanism. There may be several causal pathways and the absolute rate of missed observations cannot be used to guide staffing decisions. Increases in nursing skill mix may be cost-effective for improving patient safety. Future work More evidence is required to validate approaches to setting staffing levels. Other aspects of missed nursing care should be explored using objective data. The implications of findings about both costs and temporary staffing need further exploration. Trial registration This study is registered as ISRCTN17930973. Funding This project was funded by the National Institute for Health Research (NIHR) Health Services and Delivery Research programme and will be published in full in Health Services and Delivery Research; Vol. 6, No. 38. See the NIHR Journals Library website for further project information.

Meta-analysis of rare event studies has recently become a subject of controversy and debate. We will argue and demonstrate in this paper that the occurrence of zero events in clinical trials or cohort studies, even if zeros occur in both arms (the case of a double-zero trial), is less problematic, at least from a statistical perspective, if the available statistical tools are applied in the appropriate way. In particular, it is neither necessary nor advisable to exclude studies with zero events from the meta-analysis. In terms of statistical tools, we will focus here on Mantel-Haenszel techniques, mixed Poisson regression and related regression models.