Three ideal observer models for rule learning in simple languages

Children learning the inflections of their native language show the ability to generalize beyond the perceptual particulars of the examples they are exposed to. The phenomenon of “rule learning”—quick learning of abstract regularities from exposure to a limited set of stimuli—has become an important model system for understanding generalization in infancy. Experiments with adults and children have revealed differences in performance across domains and types of rules. To understand the representational and inferential assumptions necessary to capture this broad set of results, we introduce three ideal observer models for rule learning. Each model builds on the next, allowing us to test the consequences of individual assumptions. Model 1 learns a single rule, Model 2 learns a single rule from noisy input, and Model 3 learns multiple rules from noisy input. These models capture a wide range of experimental results—including several that have been used to argue for domain-specificity or limits on the kinds of generalizations learners can make—suggesting that these ideal observers may be a useful baseline for future work on rule learning.     

Sample size planning with the cost constraint for testing superiority and equivalence of two independent groups

The allocation of sufficient participants into different experimental groups for various research purposes under given constraints is an important practical problem faced by researchers. We address the problem of sample size determination between two independent groups for unequal and/or unknown variances when both the power and the differential cost are taken into consideration. We apply the well‐known Welch approximate test to derive various sample size allocation ratios by minimizing the total cost or, equivalently, maximizing the statistical power. Two types of hypotheses including superiority/non‐inferiority and equivalence of two means are each considered in the process of sample size planning. A simulation study is carried out and the proposed method is validated in terms of Type I error rate and statistical power. As a result, the simulation study reveals that the proposed sample size formulas are very satisfactory under various variances and sample size allocation ratios. Finally, a flowchart, tables, and figures of several sample size allocations are presented for practical reference.     

New heterogeneous test statistics for the unbalanced fixed‐effect nested design

When the underlying variances are unknown or/and unequal, using the conventional F test is problematic in the two‐factor hierarchical data structure. Prompted by the approximate test statistics (Welch and Alexander–Govern methods), the authors develop four new heterogeneous test statistics to test factor A and factor B nested within A for the unbalanced fixed‐effect two‐stage nested design under variance heterogeneity. The actual significance levels and statistical power of the test statistics were compared in a simulation study. The results show that the proposed procedures maintain better Type I error rate control and have greater statistical power than those obtained by the conventional F test in various conditions. Therefore, the proposed test statistics are recommended in terms of robustness and easy implementation.     

The First Asian Ecumenical Confession of Faith: The So‐Called Twelve Articles of Faith of Many Asian Protestant Churches

The Twelve Articles of Faith, written by Western Presbyterian and Reformed missionaries in India around 1900, have arguably been the most important ecumenical confession of faith of many Asian Protestant churches (Indian, Korean and Chinese Protestant churches and beyond). The articles by and large adopt the spirit and content of the Westminster Confession and Catechisms, and have some apparently Calvinistic elements. But they also have some new Arminian, universalistic and ecumenical elements that include the unique historical and theological implications of the epochal Western missions in Asia of the 19th and the early 20th centuries: both Christ's atonement for all people and a non‐predestinarian order of salvation. These elements, however, seem to entail a weak, individualistic ecclesiology along with a Biblicist or fundamentalist interpretation of the Bible. Nonetheless, having influenced millions of Asian Christians, the articles have been and continue to be an important ecumenical confession of faith, and in this way can contribute to uniting Asian‐and world‐Christian churches.     

Saving for a Rainy Day? Comparative Optimism About Disability in Old Age

This study examined young (n = 40) and middle‐aged (n = 30) adults' susceptibility to comparative optimism and comparative pessimism regarding disability in old age and their willingness to save for long‐term care. Participants rated their risk of diverse levels of disability in old age, compared to another similar person, and indicated the amount of money they would be willing to save for future long‐term care. While middle‐aged participants showed the same level of comparative optimism for diverse disability levels, younger participants showed increasing levels of comparative optimism with increasing disability. Participants' comparative optimism levels and age both predicted their intentions to save. The findings are discussed in terms of theories of judgment and behavioral decision making.