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Abstract Magic squares have been extremely useful and popular in combinatorics and statistics. One generalization of magic squares ismagic rectangleswhich are useful for designing experiments in statistics. A necessary and sufficient condition for the existence of magic rectangles restricts the number of rows and columns to be either both odd or both even. In this paper, we generalize magic rectangles to even by oddnearly magic rectangles. We also prove necessary and sufficient conditions for the existence of a nearly magic rectangle, and construct one for each parameter set for which they exist. 

Alternative design and analysis methods for screening experiments based on locating arrays are presented. The number of runs in a locating array grows logarithmically based on the number of factors, providing efficient methods for screening complex engineered systems, especially those with large numbers of categorical factors having different numbers of levels. Our analysis method focuses on levels of factors in the identification of important main effects and two-way interactions. We demonstrate the validity of our design and analysis methods on both well-studied and synthetic data sets and investigate both statistical and combinatorial properties of locating arrays that appear to be related to their screening capability. 

While generalized linear mixed models are useful, optimal design questions for such models are challenging due to complexity of the information matrices. For longitudinal data, after comparing three approximations for the information matrices, we propose an approximation based on the penalized quasi-likelihood method.We evaluate this approximation for logistic mixed models with time as the single predictor variable. Assuming that the experimenter controls at which time observations are to be made, the approximation is used to identify locally optimal designs based on the commonly used A- and D-optimality criteria. The method can also be used for models with random block effects. Locally optimal designs found by a Particle Swarm Optimization algorithm are presented and discussed. As an illustration, optimal designs are derived for a study on self-reported disability in olderwomen. Finally,we also study the robustness of the locally optimal designs to mis-specification of the covariance matrix for the random effects. 

Subdata selection from big data is an active area of research that facilitates inferences based on big data with limited computational expense. For linear regression models, the optimal design-inspired Information-Based Optimal Subdata Selection (IBOSS) method is a computationally efficient method for selecting subdata that has excellent statistical properties. But the method can only be used if the subdata size, k, is at last twice the number of regression variables, p. In addition, even when $$k\ge 2p$$, under the assumption of effect sparsity, one can expect to obtain subdata with better statistical properties by trying to focus on active variables. Inspired by recent efforts to extend the IBOSS method to situations with a large number of variables p, we introduce a method called Combining Lasso And Subdata Selection (CLASS) that, as shown, improves on other proposed methods in terms of variable selection and building a predictive model based on subdata when the full data size n is very large and the number of variables p is large. In terms of computational expense, CLASS is more expensive than recent competitors for moderately large values of n, but the roles reverse under effect sparsity for extremely large values of n.