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

对定性数据分析中使用的大样本近似方法进行了检验。要考虑的问题既涉及大量参数，也涉及大量观测。将探索对数线性模型、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、皮尔逊和似然比卡方统计量的分布，以及预测的度量，应考虑大样本近似。在进行分析时，不应假设基本模型是正确的。在大规模的调查和普查中，统计分析通常是对非常大的样本进行的。在这种分析中，用于逼近观测值的模型通常涉及非常大量的未知量，必须从样本中估计这些未知量才能应用模型。在拟议的研究中，对未知量估计的预期变异性和构建未来观测的预报器的模型值的度量进行了近似。与通常的统计分析不同，即使拟议的模型与数据不完全一致，所开发的方法也适用。结果表明，即使在不能完全描述数据的情况下，近似模型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.