A comparison of two approaches for power and sample size calculations in logistic regression models

Whittemore (1981) proposed an approach for calculating the sample size needed to test hypotheses with specified significance and power against a given alternative for logistic regression with small response probability. Based on the distribution of covariate, which could be either discrete or continuous, this approach first provides a simple closed-form approximation to the asymptotic covariance matrix of the maximum likelihood estimates, and then uses it to calculate the sample size needed to test a hypothesis about the parameter. Self et al. (1992) described a general approach for power and sample size calculations within the framework of generalized linear models, which include logistic regression as a special case. Their approach is based on an approximation to the distribution of the likelihood ratio statistic. Unlike the Whittemore approach, their approach is not limited to situations of small response probability. However, it is restricted to models with a finite number of covariate configurations. This study compares these two approaches to see how accurate they would be for the calculations of power and sample size in logistic regression models with various response probabilities and covariate distributions. The results indicate that the Whittemore approach has a slight advantage in achieving the nominal power only for one case with small response probability. It is outperformed for all other cases with larger response probabilities. In general, the approach proposed in Self et al. (1992) is recommended for all values of the response probability. However, its extension for logistic regression models with an infinite number of covariate configurations involves an arbitrary decision for categorization and leads to a discrete approximation. As shown in this paper, the examined discrete approximations appear to be sufficiently accurate for practical purpose.     

中国经济总量及要素增长的基本趋势分析

研究的目的在于回答中国经济增长及生产要素的趋势到底是什么样的？其结论对中国经济的长期预测具有重要意义。虽然所研究的与现有的收敛性分析及影响因素分析不同，但从长期看，中国经济增长及生产要素有着自身的时间特性，各种影响因素是经济增长短期波动的原因。通过分析发现：中国经济（GDP）增长、资本存量服从指数时间模型，而劳动力增长则服从增长曲线时间模型。据此，并对中国GDP、资本存量及劳动力的增长速度进行了区间估计。     

Project Portfolio Selection: A Newsvendor Approach

We consider the problem of balancing the penalties associated with budgetary slack (being underbudget) and cost overruns in the project portfolio selection problem by addressing randomness in project costs and making individual project budgets decision variables. Setting the budget for a single project is shown to be analogous to the newsvendor problem. For related versions of the project portfolio selection problem we provide optimal and heuristic procedures. Numerical experiments are used to test the procedures and provide managerial guidelines. We show project budgets should be set so that each project in the portfolio has the same probability of running over budget, it is better to have a larger number of projects with less than ideal funding compared to a smaller number of projects with ideal funding, and substantial opportunities to select more projects with a higher expected profit are available if an aggregate portfolio budget is used.     

Eyes on the account size: Interactions between attention and budget in consumer choice

The context surrounding a consumer decision, such as one’s overall budget available for purchases, can exert a strong effect on the subjective value of a product. Across three eye-tracking studies, we explore the attentional processes through which budget size influences consumers’ purchasing behavior. Higher budgets increased and sped up purchasing even when items were affordable at all budget sizes. Moreover, attention interacted with budget size to promote purchasing at higher budgets. Finally, individual differences in the magnitude of the budget effect related to attentional patterns: those whose decisions depended more on budget exhibited more budget-price transitions and less variability in search patterns compared to those whose decisions were less dependent on budget. These findings indicate that attention moderates the effect of budgets on purchasing decisions, allowing low budgets to serve as self-control devices and large budgets to generate impulse purchases.