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

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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.