Modeling and scaling of categorical data. (English) Zbl 1407.62192
Rykov, Vladimir V. (ed.) et al., Mathematical and statistical models and methods in reliability. Applications to medicine, finance, and quality control. Invited papers based on the presentation at the 6th international conference (MMR 2009), Moscow, Russia, June 22–26, 2009. Boston, MA: Birkhäuser. Stat. Ind. Technol., 255-264 (2010).
Summary: Estimation and testing of distributions in metric spaces are well known. R. A. Fisher, J. Neyman, W. Cochran, and M. Bartlett achieved essential results on the statistical analysis of categorical data. In the last 40 years, many other statisticians found important results in this field.
Often data sets contain categorical data, e.g., levels of factors or names. There does not exist any ordering or any distance between these categories. At each level there are measured some metric or categorical values. We introduce a new method of scaling based on statistical decisions. For this we define empirical probabilities for the original observations and find a class of distributions in a metric space where these empirical probabilities can be found as approximations for equivalently defined probabilities. With this method we identify probabilities connected with the categorical data and probabilities in metric spaces. Here, we get a mapping from the levels of factors or names into points of a metric space. This mapping yields the scale for the categorical data.
From the statistical point of view we use multivariate statistical methods, we calculate maximum likelihood estimations and compare different approaches for scaling.
For the entire collection see [Zbl 1203.60007].
Often data sets contain categorical data, e.g., levels of factors or names. There does not exist any ordering or any distance between these categories. At each level there are measured some metric or categorical values. We introduce a new method of scaling based on statistical decisions. For this we define empirical probabilities for the original observations and find a class of distributions in a metric space where these empirical probabilities can be found as approximations for equivalently defined probabilities. With this method we identify probabilities connected with the categorical data and probabilities in metric spaces. Here, we get a mapping from the levels of factors or names into points of a metric space. This mapping yields the scale for the categorical data.
From the statistical point of view we use multivariate statistical methods, we calculate maximum likelihood estimations and compare different approaches for scaling.
For the entire collection see [Zbl 1203.60007].
MSC:
| 62H12 | Estimation in multivariate analysis |
| 62C10 | Bayesian problems; characterization of Bayes procedures |
| 62H15 | Hypothesis testing in multivariate analysis |
References:
| [1] | Ahrens, H. and Läuter, J.: Mehrdimensionale Varianzanalyse. Wiley, Akademie-Verlag, Berlin (1981) · Zbl 0467.62061 |
| [2] | Everitt, B.S. and Dunn, G.: Applied Multivariate Data Analysis. Hodder Education, London (2001) · Zbl 1010.62040 |
| [3] | Kendall, M.G. and Stuart, A.: The Advanced Theory of Statistics. Wiley, Griffin & Company, London (1967) · Zbl 0416.62001 |
| [4] | Läuter, H.: Modeling and scaling of categorical data. Preprint, University of Linz (2007) |
| [5] | Voinov, V.G. and Nikulin, M.S.: Unbiased Estimators and Their Applications. Kluwer, Dordecht (1993) · Zbl 0832.62019 · doi:10.1007/978-94-011-1970-2 |
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