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In this paper, a novel decision-making trial and evaluation laboratory (DEMATEL) theory with a shrinkage coefficient of indirect relation matrix is proposed, and a useful validity index, called Liu’s validity index, is also proposed for evaluating the performance of any DEMATEL model. If the shrinkage coefficient of an indirect relation matrix is equal to 1, then this new theory is identical to the traditional theory; in other words, it is a generalization of the traditional theory. Furthermore, the indirect relation is always considerably greater than the direct one in traditional DEMATEL theory, which is unreasonable and unfair because it overemphasizes the influence of the indirect relation. We prove in this paper that if the shrinkage coefficient is equal to 0.5, then the indirect relation is less than its direct relation. Because the shrinkage coefficient belongs to [0.5, 1], according to Liu’s validity index, we can find a more appropriate shrinkage coefficient to obtain a more efficient DEMATEL method. Some crucial properties of this new theory are discussed, and a simple example is provided to illustrate the advantages of the proposed theory.

In this paper, I point out that both Joe’s and Ding’s strength statistics can only be used for testing the pair-wise independence, and I propose a novel G-square based strength statistic, called Liu’s generalized correlation coefficient, it can be used to detect and compare the strength of not only the pair-wise independence but also the mutual independence of any multivariate variables. Furthermore, I proved that only Liu’s generalized correlation coefficient is strictly increasing on its number of variables, it is more sensitive and useful than Cramer’s V coefficient, in other words, Liu generalized correlation coefficient is not only the G-square based strength statistic, but also an improved statistic for detecting and comparing the strengths of deferent associations of any two or more sets of multivariate variables, moreover, this new strength statistic can also be tested by G2.

The most important issue in DEMATEL theory is how to obtain a reliable initial direct relation matrix with order n, the traditional theory obtains it by using the pair-wise comparison method, in which, each respondent must answer n(n-1) times pair-wise comparisons of all of the direct influences, if n is a large number, the work of pair-wise comparing is becoming hard, time-consuming, and unreliable. In this paper, for overcoming above drawbacks, we replace the pair-wise comparison method with Liu's ordering theory to find the initial direct relation matrix. This new method without pair-wise comparing can be used for any order n, a simple example was also provided in this paper to illustrate the advantages of the proposed theory.