I am currently reading an article, Analysis of sensory ratings data with cumulative link models. In section 2.4 - example 1, the authors describe an experiment from Bi, 2002*:
... 25 subjects each assess four concordant and four discordant product pairs. The subjects were asked to rate the degree of difference between the sample product pairs on a three point rating scale, where 1 means identical and 3 means different.
The experiment resulted in a 2 x 3 contingency table (please see Table 1 in the paper), and later, to show the application of Pearson $\chi^2$-test statistic, the authors motivated an assumption of non-repeated measures (across six cells of the table). Once the test statistic was calculated (X$^2$ = 6.49), it was assumed to follow a $\chi^2$ distribution with two degrees of freedom. However, I am not sure why the sampling distribution of $X^2$ is assumed to follow $\chi^2_2$ i.e., why are the degrees of freedom two? Could it be possible that the degrees of freedom (DoF) are non-integral in this case?
I have read the related post, 'How to understand degrees of freedom', but was wondering if I could calculate the DoF without relying on any heuristic - I am not even sure what kind of heuristic would be the most accurate in this case.
*Bi, J. (2002). Statistical models for the degree of difference test. Food Quality and Preference, 13:13–37.)
