No abstract

Extraordinary amounts of data are being produced in many branches of science. Proven statistical methods are no longer applicable with extraordinary large data sets due to computational limitations. A critical step in big data analysis is data reduction. Existing investigations in the context of linear regression focus on subsampling-based methods. However, not only is this approach prone to sampling errors, it also leads to a covariance matrix of the estimators that is typically bounded from below by a term that is of the order of the inverse of the subdata size. We propose a novel approach, termed information-based optimal subdata selection (IBOSS). Compared to leading existing subdata methods, the IBOSS approach has the following advantages: (i) it is significantly faster; (ii) it is suitable for distributed parallel computing; (iii) the variances of the slope parameter estimators converge to 0 as the full data size increases even if the subdata size is fixed, i.e., the convergence rate depends on the full data size; (iv) data analysis for IBOSS subdata is straightforward and the sampling distribution of an IBOSS estimator is easy to assess. Theoretical results and extensive simulations demonstrate that the IBOSS approach is superior to subsampling-based methods, sometimes by orders of magnitude. The advantages of the new approach are also illustrated through analysis of real data.

Deriving optimal designs for nonlinear models is challenging in general. Although some recent results allow us to focus on a simple subclass of designs for most problems, deriving a specific optimal design mainly depends on algorithmic approaches. There is need of a general and efficient algorithm which is more broadly applicable than the current state of the art methods. We present a new algorithm that can be used to find optimal designs with respect to a broad class of optimality criteria, when the model parameters or functions thereof are of interest, and for both locally optimal and multistage design strategies. We prove convergence to the desired optimal design, and show that the new algorithm outperforms the best available algorithm in various examples.

Deriving optimal designs for nonlinear models is in general challenging. One crucial step is to determine the number of support points needed. Current tools handle this on a case-by-case basis. Each combination of model, optimality criterion and objective requires its own proof. The celebrated de la Garza Phenomenon states that under a (p − 1)th-degree polynomial regression model, any optimal design can be based on at most p design points, the minimum number of support points such that all parameters are estimable. Does this conclusion also hold for nonlinear models? If the answer is yes, it would be relatively easy to derive any optimal design, analytically or numerically. In this paper, a novel approach is developed to address this question. Using this new approach, it can be easily shown that the de la Garza phenomenon exists for many commonly studied nonlinear models, such as the Emax model, exponential model, three-and four-parameter log-linear models, Emax-PK1 model, as well as many classical polynomial regression models. The proposed approach unifies and extends many well-known results in the optimal design literature. It has four advantages over current tools: (i) it can be applied to many forms of nonlinear models; to continuous or discrete data; to data with homogeneous or non-homogeneous errors; (ii) it can be applied to any design region; (iii) it can be applied to multiple-stage optimal design; and (iv) it can be easily implemented.

We propose a new approach for identifying the support points of a locally optimal design when the model is a nonlinear model. In contrast to the commonly used geometric approach, we use an approach based on algebraic tools. Considerations are restricted to models with two parameters, and the general results are applied to often used special cases, including logistic, probit, double exponential and double reciprocal models for binary data, a loglinear Poisson regression model for count data, and the Michaelis-Menten model. The approach, which is also of value for multi-stage experiments, works both with constrained and unconstrained design regions and is relatively easy to implement.

We extend the approach in [Ann. Statist. 38 (2010Statist. 38 ( ) 2499Statist. 38 ( -2524 for identifying locally optimal designs for nonlinear models. Conceptually the extension is relatively simple, but the consequences in terms of applications are profound. As we will demonstrate, we can obtain results for locally optimal designs under many optimality criteria and for a larger class of models than has been done hitherto. In many cases the results lead to optimal designs with the minimal number of support points.

Teachers are playing crucial roles in the implementation of formative assessment, which has been widely recognised as a valuable strategy in enhancing students' learning outcomes. However, systematic analysis on factors that might facilitate or hinder teachers' intentions and implementations regarding formative assessment is scarce. This review covers 52 eligible studies and identifies factors, which have been categorised into personal and contextual factors, that influence teachers' intentions and implementations regarding formative assessment. The results of this review may benefit researchers, school leaders, and policy makers when they aspire to facilitate the implementation of formative assessment.

Generalized linear models with group effects are commonly used in scientific studies. However, there appear to be no results for selecting optimal designs. In this paper, we identify the structure of locally optimal designs, provide a general strategy to determine the design points and the corresponding weights for optimal designs, and present theoretical results for the special case of D-optimality. The results can be applied to many commonly studied models, including the logistic, probit, and loglinear models. The design region can be restricted or unrestricted, and the results can also be applied for a multi-stage approach.