Improved likelihood ratio tests for complete contingency tables
Improved likelihood ratio tests for complete contingency tables
复制标题
改进完整列联表的似然比检验
DOI：
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发表时间：
1976
期刊：
影响因子：
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通讯作者：
David A. Williams
中科院分区：
文献类型：
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作者：
David A. Williams
Lawley（1956）描述了如何通过将-2 log A检验统计量乘以一个选择的乘数来改进渐近似然比检验，使得修改后的统计量的零分布更好地近似于其渐近x2分布。本文将这种方法应用于完全列联表假设的渐近似然比检验。改进的测试来自封闭形式的最大似然估计的假设。复合零假设相对于复合备择假设的渐近似然比检验通常可以通过比较-2 log A与x2分布来获得，其中A是两个假设下的最大似然比。Lawley（1956）指出，可以通过将-2 log A乘以选定的比例因子来改进此类检验，以便所得统计量的零分布具有与X2相同的矩，忽略n-2阶量，其中n是样本的大小。当统计量2 log A可以表示为观测值的显式函数时，乘数最容易通过直接计算其期望值来找到，直到n-1阶。这些改进的似然比检验在多变量分析领域得到了广泛的应用。近年来，在发展使用对数线性模型分析多维列联表的方法方面取得了相当大的进展。Plackett（1974）对此作了全面的评述。在这些模型中，细胞频率可以被视为独立的泊松变量，其对数期望在主效应和相互作用参数的分层集合中是线性的。用于检验给定模型拟合优度的-2 log A统计量采用S = 2 EX log（X/u）的形式，其中X是单元频率，u是其期望值的最大似然估计值。在实践中，当X = 0时，X log（Xl，u）可以被零代替。根据Nelder & Wedderburn（1972），S将被称为模型的偏差。为了检验具有偏差S1的给定模型与具有附加估计参数和偏差S2的备选模型，似然比检验统计量为S S1-S2
SUMMARY Lawley (1956) describes how asymptotic likelihood ratio tests can in general be improved by multiplying the -2 log A test statistic by a multiplier chosen so that the null distribution of the modified statistic is better approximated by its asymptotic x2 distribution. This paper applies this technique to asymptotic likelihood ratio tests of hypotheses concerning complete contingency tables. Improved tests are derived for hypotheses with closed form maximum likelihood estimators. An asymptotic likelihood ratio test of a composite null hypothesis against a composite alternative can in general be obtained by comparing -2 log A with a x2 distribution, where A is the ratio of maximized likelihoods under the two hypotheses. Lawley (1956) showed that such tests can be improved by multiplying -2 log A by a scale factor chosen so that the null distribution of the resulting statistic has the same moments as X2 ignoring quantities of order n-2, where n is the size of the sample. When the statistic - 2 log A can be expressed as an explicit function of the observations the multiplier is most easily found by calculating directly its expectation as far as terms of order n-1. These improved likelihood ratio tests have been most used in the area of multivariate analysis. In recent years considerable progress has been made in developing methods for analyzing multidimensional contingency tables using log linear models. A comprehensive review is given by Plackett (1 974). In these models the cell frequencies may be regarded as independent Poisson variables whose log expectations are linear in a hierarchical set of main effect and interaction parameters. The -2 log A statistic for testing the goodness of fit of a given model takes the form S = 2EX log (X/u), where the X are the cell frequencies and the ,u are the maximum likelihood estimators of their expectations. In practice X log (X/,u) can be replaced by zero when X = 0. Following Nelder & Wedderburn (1972), S will be termed the deviance of the model. To test a given model with deviance S1 against an alternative with additional estimated parameters and deviance S2, the likelihood ratio test statistic iS S1-S2