A generalization of the Thurstone method for multiple choice and incomplete paired comparisons

A ranking method based on paired comparisons is proposed. The object's characteristics are considered as random variables and the observers judge about their differences. The differences are classified. More than two classes are allowed. Assuming Gauss distributed latent random variables we set up the likelihood function and estimate the parameters by the maximum likelihood method. The rank of the objects is the order of the expectations. We analyse the log-likelihood function and provide reasonable conditions for the existence of the maximum value and the uniqueness of the maximizer. Some illustrative examples are also presented. The method can be applied in case of incomplete comparisons as well. It allows constructing confidence intervals for the probabilities and testing the hypothesis that there are no significant differences between the expectations.