Abstract
For multi-way contingency tables with nominal categories, this paper proposes three kinds of proportional reduction in error measures, which describe the relative decrease in the probability of making an error in predicting the value of one variable when the values of the other variables are known, as opposed to when they are not known. The measures have forms of arithmetic, geometric and harmonic means. An example is shown.
Highlights
Consider an R × C contingency table with both nominal categories of the explanatory variable X and the response variable Y
For multi-way contingency tables with nominal categories, this paper proposes three kinds of proportional reduction in error measures, which describe the relative decrease in the probability of making an error in predicting the value of one variable when the values of the other variables are known, as opposed to when they are not known
Goodman and Kruskal (1954) proposed two kinds of measures, i.e., (1) the measure which describes the proportional reduction in variation (PRV) in predicting the Y category obtained when the X category is known, as opposed to when the X category is not known, and (2) the measure which describes the proportional reduction in error (PRE) in predicting it
Summary
Consider an R × C contingency table with both nominal categories of the explanatory variable X and the response variable Y. Consider the situation where the explanatory and response variables are not defined In this case, the following measure λ is given: pimi +. See Goodman and Kruskal (1954) This indicates the PRE in predicting the category of either variable as between knowing and not knowing the category of the other variable. For a two-way contingency table with both nominal categories, Yamamoto and Tomizawa (2010) proposed new PRE measures, say Λ, expressed as the arithmetic, geometric and harmonic means of λB and λA. For a two-way contingency table with nominal-ordinal categories, Yamamoto, Nozaki and Tomizawa (2011) proposed a PRE measure the detail is omitted.
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