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双标图与对应分析的拟合优度AI 解读

The present paper examines proportional goodness of fit to variables recorded on individuals, the variances and covariances of the variables, and the form and distances between individuals. No single plot displays all three optimally in the sense of least squares. However, even aspects which are non‐optimally fitted by biplots and Benzécri plots often closely preserve the optimal fit. This is shown by means of a preservation‐of‐fit function which depends on the type of…

回归与位移效应的再随机化推断：计算可行的方法AI 解读

An idea due to Quade (1973) is used to develop a method that reduces the computational load for rerandomization inferences in certain situations, and, for experiments carried out with a sufficiently small reference set of randomizations, makes computations entirely feasible. Consider an experimental design for N units for which an experimental plan E is to be chosen randomly from a set of K plans. Plan E will assign treatment dosage dE,u to unit u with postulated additive…

矩阵的加性与乘性模型最小二乘逼近AI 解读

Summary Reduced rank approximation of matrices by Householder–Young methods is shown to be equivalent to fitting of a bilinear (multiplicative) model and to projection onto an optimally chosen subspace. The relation to linear approximation is made evident. Least squares fitting of a mixed linear (additive) and bilinear (multiplicative) model is proved to be a two-stage process: (1) fit the linear part of the model, then take residuals, and (2) fit the bilinear part to the…

双因素方差分析中交互作用的同时统计推断AI 解读

Three statistics for testing additivity and making detailed inferences on interactions in a two-way setup are considered. The first method uses the well-known sum of squares for testing and yields Scheffé bounds for all contrasts; the second introduces a maximum characteristic root statistic and Roy bounds, but is restricted to product-type contrasts; the last is limited to tetrad differences and its statistic is the maximum of all such differences. Bounds on percentage…

双因素方差分析中交互作用的同时统计推断AI 解读

Three statistics for testing additivity and making detailed inferences on interactions in a two-way setup are considered. The first method uses the well-known sum of squares for testing and yields Scheffé bounds for all contrasts; the second introduces a maximum characteristic root statistic and Roy bounds, but is restricted to product-type contrasts; the last is limited to tetrad differences and its statistic is the maximum of all such differences. Bounds on percentage…

乘积型交互对比的同时置信区间AI 解读

Summary The set of product-type interaction contrasts, which contains the subsets of interaction residuals, tetrad contrasts, double-dichotomy contrasts and pooled-tetrad contrasts, is discussed. In the Normal case with equal and known variances, simultaneous confidence intervals for a given set of such contrasts can be constructed in several ways. Five such methods (the Scheffé method, the Tukey method, the Dunn method, a method based on Roy's maximum-root statistic and a…

矩阵的双标图展示及其在主成分分析中的应用AI 解读

Any matrix of rank two can be displayed as a biplot which consists of a vector for each row and a vector for each column, chosen so that any element of the matrix is exactly the inner product of the vectors corresponding to its row and to its column. If a matrix is of higher rank, one may display it approximately by a biplot of a matrix of rank two which approximates the original matrix. The biplot provides a useful tool of data analysis and allows the visual appraisal of…

摘要暂缺

This paper presents procedures for multivariate analysis of variance which simultaneously test for significance all variables together as well as all subsets of variables, or the differences between all samples as well as between the samples of any subgroup. These procedures ensure a predetermined probability of making no type I error on any of the tests and are coherent in preserving the implication relations which exist between hypotheses on different groups of samples.…