A Weibull-model-based approach is examined to handle under- and overdispersed discrete data in a hierarchical framework. This methodology was first introduced by Nakagawa and Osaki (1975, IEEE Transactions on Reliability, 24, 300–301), and later examined for under- and overdispersion by Klakattawi et al. (2018, Entropy, 20, 142) in the univariate case. Extensions to hierarchical approaches with under- and overdispersion were left unnoted, even though they can be obtained in a simple manner. This is of particular interest when analysing clustered/longitudinal data structures, where the underlying correlation structure is often more complex compared to cross-sectional studies. In this article, a random-effects extension of the Weibull-count model is proposed and applied to two motivating case studies, originating from the clinical and sociological research fields. A goodness-of-fit evaluation of the model is provided through a comparison of some well-known count models, that is, the negative binomial, Conway–Maxwell–Poisson and double Poisson models. Empirical results show that the proposed extension flexibly fits the data, more specifically, for heavy-tailed, zero-inflated, overdispersed and correlated count data. Discrete left-skewed time-to-event data structures are also flexibly modelled using the approach, with the ability to derive direct interpretations on the median scale, provided the complementary log–log link is used. Finally, a large simulated set of data is created to examine other characteristics such as computational ease and orthogonality properties of the model, with the conclusion that the approach behaves best for highly overdispersed cases.

Objective: To examine if personal and comparative optimism, perceived effectiveness, and moralization of vaccination predict people's decision to get vaccinated. Methods: We measured self-reported vaccination decisions in a five-wave longitudinal study (N ≍ 5,000/wave) in Belgium over a six months period (December 2020–May 2021) during the COVID-19 pandemic. Among the predictors were demographic factors, personal and comparative optimism for three aspects of COVID-19 (infection, severe disease, good outcome), perceived effectiveness of vaccination, and the extent to which vaccination is being viewed in prosocial terms (altruism, civic spirit) versus as instrumental in one's self-interest (common sense, concern about one's health). Results: The actual availability of vaccines changed people's outlook on vaccination. Marked differences emerged in vaccination decision between linguistic-cultural regions (Flemish Region, Walloon Region, Brussels Capital Region). Personal and comparative optimism predicted vaccination decisions to different extents depending on participants’ age and on whether the optimism was for infection, severe disease, or a good outcome. In older participants, vaccination decision was mostly predicted by personal optimism; in younger participants, it was mostly predicted by comparative optimism. Moralizing vaccination predicted a lower likelihood of a positive vaccination decision, that is, higher vaccine hesitancy or refusal, particularly in older participants. Conclusions: Assessments of risk perception serving to inform vaccination campaigns should differentiate between expectations concerning the risk of infection and expectations concerning the outcome of an infection. Public health messages should address comparative optimism, particularly when targeting younger populations. Contrary to popular belief, moralizing vaccination may reduce the willingness to get vaccinated.

Purpose: This cross-sectional study investigates the association between retinal vessel complexity and age and studies the effects of cardiovascular health determinants. Methods: Retinal vessel complexity was assessed by calculating the boxcounting fractal dimension (D f ) from digital fundus photographs of 850 subjects (3-97 years). All photographs were labelled as 'non-pathological' by the treating ophthalmologist. Results: Statistical models showed a significantly decreasing relationship between age and D f (linear: R-squared = 0.1897, p < 0.0001; quadratic: Rsquared = 0.2343, p < 0.0001; cubic: R-squared = 0.2721, p < 0.0001), with the cubic regression model offering the best compromise between accuracy and model simplicity. Multivariate cubic regression showed that age, spherical equivalent and smoking behaviour have an effect (p < 0.0001) on D f . A significantly increasing effect of the number of pack-years on D f was observed (effect: 0.0004, p = 0.0017), as well as a significantly decreasing effect of years since tobacco abstinence (effect: −0.0149, p < 0.0001). Conclusion:We propose using a cubic trend with age, refractive error and smoking behaviour when interpreting retinal vessel complexity.