A Method Expanding 2 by 2 Contingency Table by Obtaining Tendencies of Boolean Operators: Boolean Monte Carlo Method

Accuracy of medical test and diagnosis are often discussed by 2 by 2 contingency tables. However, it is difficult to apply a 2 by 2 contingency table to multivariate cases because the number of possible categories increases exponentially. The aims of this study is introducing a method by which we can obtain trends of boolean operators and construct a 2 by 2 contingency tables to multivariate cases. In this method, we randomly assigned boolean operators between binary variables and focused on frequencies of boolean operators which could explain outcomes correctly. We determined trends of boolean operators by chi-square test. As an application, we performed this method for a dataset which included patient's age, body mass index (BMI), mean arterial pressure (MAP) and the quantitative measure of diabetes progression ($Y$). We set cutoff to each variable and considered these as binary variables. Interactions of age, BMI and MAP were determined as age $and$ BMI $and$ MAP ($p<0.0001$ for both operators) in estimating $Y$ and sensitivity, specificity, positive and negative predicting value were 0.50, 0.85. 0.59, 0.80 respectively. We may be able to detemine trends of boolean operators and construct a 2 by 2 contingency table in multivariate situation by this method.