Mathematical Sciences: Large-Sample Approximations for Analysis of Qualitative Data

在定性数据分析中使用的大样本近似将被检验。要考虑的问题涉及大量的参数和大量的观测值。将探讨对数线性模型、条件对数线性模型、潜在类模型、规范模型和关联模型。对于最大似然估计的分布、Pearson和似然比卡方统计以及预测的度量，要考虑大样本近似值。在进行分析时，不能假设基础模型是正确的。在大规模调查和人口普查中，统计分析通常是在非常大的样本上进行的。在这类分析中，用于近似观测结果的模型常常涉及大量的未知量，为了应用模型，必须从样本中估计这些未知量。在提出的研究中，对未知数量的估计所期望的变异性和对构建未来观测预测因子的模型值的测量进行了近似。与通常的统计分析相反，即使提出的模型与数据不完全一致，所开发的方法也适用。结果表明，即使近似模型不能完全描述数据，它在统计工作中也是很有价值的。

Large-sample approximations used in the analysis of qualitative data are to be examined. Problems to be considered involve both large numbers of parameters and large numbers of observations. Log-linear models, conditional log-linear models, latent-class models, canonical models, and association models are to be explored. Large-sample approximations are to be considered for distributions of maximum-likelihood estimates, Pearson and likelihood-ratio chi-square statistics, and measures of predictions. Analyses are to be developed without the assumption that the underlying model is correct. In large-scale surveys and censuses, statistical analyses are commonly performed on very large samples. In such analyses, models used to approximate the observations often involve very large numbers of unknown quantities which must be estimated from the sample in order to apply the model. In the proposed research, approximations are developed for the variability expected from the estimates of unknown quantities and for measures of the value of the model for construction of predictors of future observations. In contrast to usual statistical analyses, the methods developed apply even if the proposed model is not entirely consistent with the data. It is shown that approximate models can be quite valuable in statistical work even when they do not fully describe the data.