Auditors who perform audit sampling are often interested in obtaining evidence for or against the hypothesis that the misstatement in a population of items is lower than a critical limit, the so-called performance materiality. Here, we propose to perform this hypothesis test using a Bayesian approach that involves the use of an impartial prior distribution, assigning equal prior probabilities to the competing interval hypotheses. Firstly, we argue that the impartial prior distribution is sensible for auditors because it is easy to justify, interpret, and explain. Secondly, we show that Bayes factors computed using this prior distribution have desirable statistical properties. Finally, we compare these Bayes factors with traditional p-values in an audit sampling context and elaborate on the merits of the impartial Bayesian hypothesis test.
We establish upper bounds for the weak and strong error resulting from a perturbation of the noise driving the stochastic Burgers equation, where we assume the noise to be additive and of trace class and the initial value to be sufficiently regular. More specifically, let Qi, i∈{1,2}, be positive trace class operators on L2(0,1) and for i∈{1,2} let Xi:[0,T]×[0,1]×Ω→R be the solution to the stochastic Burgers equation driven by a Qi-Brownian motion WQi:[0,T]×Ω→L2(0,1) on the temporal-spatial domain [0,T]×[0,1]. We prove a weak error bound of the form|E[φ(X1(T,⋅))]−E[φ(X2(T,⋅))]|≲‖(−A)−1−(Q1−Q2)‖L1(L2), where φ:L2(0,1)→R is a sufficiently smooth function, ‖⋅‖L1(L2) is the trace class norm for operators on L2(0,1), and A:D(A)⊆L2(0,1)→L2(0,1) is the one-dimensional Dirichlet Laplacian that represents the leading term in the Burgers equation. In addition, we prove a strong error bound of the formE∫01|X1(T,z)−X2(T,z)|2dz≲‖(−A)−12−|Q112−Q212|‖L2(L2)2, where ‖⋅‖L2(L2) is the Hilbert–Schmidt norm for operators on L2(0,1). In particular, our results provide upper bounds for the weak and strong error arising when approximating the trace class noise by finite-dimensional noise; the rates we obtain reflect the general philosophy that the weak convergence rate should be twice the strong rate.
Classroom discussions on controversial issues are widely regarded as an essential part of democratic citizenship education, yet there is limited knowledge about how teachers can be supported in facilitating them. This study evaluates a professional development program (PDP) aimed at supporting in-service teachers in guiding classroom discussions on controversial issues. The program involved peer-coaching sessions where teachers experimented with discussion strategies and reflected on their experiences. The study adopted a mixed-methods design, combining a quasi-experimental study (N = 54) to assess effects on teachers’ self-efficacy regarding an open and safe classroom climate, co-construction, and content quality, with stimulated recall interviews (N = 8) to explore how teachers experienced the PDP and applied it in their classroom practice. The results showed that participation in the PDP was associated with significantly greater growth in teachers’ self-efficacy for supporting an open and safe classroom climate and co-construction, compared to a control group. No significant effects were found for teacher self-efficacy with respect to promoting content quality. The qualitative findings mirrored these results and provided further insight into teachers’ perceptions of how the PDP supported their approaches to classroom discussions, while also highlighting persistent challenges, particularly in balancing neutrality with normative guidance during discussions. The findings suggest that practice-oriented professional development, combined with structured opportunities for collaborative reflection, can strengthen teachers’ confidence in guiding such discussions and support teachers in reflecting on and further developing their approaches to classroom discussions on controversial issues.
Relatively little is known about the specific characteristics of the spelling abilities of children with developmental language disorder (DLD). This concerns knowledge about spelling patterns and the item characteristics contributing to spelling performance. It has been found that vowel duration spelling in multisyllabic words is challenging for typically developing (TD) children and that this performance is influenced by implicit cues, including semantic familiarity, orthographic frequency, and morphology. We investigated outcomes on long and short vowel duration spelling in Dutch and the potential impact of implicit cues in Dutch Grade 5 children with DLD (n = 74). We did so by comparing spelling dictation outcomes of this group of children to that of Grade 3 TD children (n = 47) and Grade 5 TD children (n = 63). The dictation task consisted of items differing in semantic familiarity, orthographic frequency, and morphology. We found that children with DLD made more errors on long and short vowel duration patterns than both TD peers and younger TD children, indicating severe difficulties. The influence of implicit cues was similar for children with DLD and younger TD children, though results varied slightly per pattern: In the consonant doubling pattern, morphology affected the spelling performance of younger TD children, but not that of children with DLD, whereas morphology affected the performance of all groups in vowel degemination. Semantic familiarity also affected the spelling accuracy of children with and without DLD, while frequency did not. The contributions of semantic familiarity and morphology can help shape spelling instruction.
The 16th Acromegaly Consensus Conference in September 2024 updated recommendations on diagnosis and treatment of acromegaly comorbidities. Since the 2020 acromegaly comorbidity management guideline was published, new evidence has emerged on novel and known comorbidities and new treatment approaches. Forty-three experts in the management of acromegaly reviewed the current literature and assessed changes in clinical practice standards and management. Current outcome goals were considered and updated, with a focus on the impact of current and emerging treatments of these comorbidities. Participants assessed factors that determine pharmacological choices, as well as use of specific agents in the management of the most relevant acromegaly comorbidities. We present consensus recommendations highlighting optimization of evidence-based acromegaly comorbidities management.
