Three methods A, B and C are used to measure a continuous variable x for each subject in a study. There are two types of subjects in the study: type1 and type2.
Difference between type
xA1 - xA2 (difference in means of type1 and type2 as measured by method A)
xB1 - xB2
xC1 - xC2
Difference between method
xA - xB (difference in means between the measurements by A and B)
xB - xC
xC - xA
Now, how to statistically compare the difference between type and difference between method?
Bonus question: How to check if the method choice affects the group difference and also quantify it?
One toy example would be an experiment where weights of subjects (male and female) are measured using three different scales. How to statistically compare inter-scale difference with the inter-gender difference?
One approach I am thinking is explained here -> http://stats.stackexchange.com/questions/77269/statistical-comparison-of-2-independent-cohens-dsStatistical comparison of 2 independent Cohen's ds But both differences explained in the link are type1 - type2 kind and I am not sure if it is applicable to compare method1 - method2 vs. type1 - type2 differences. Also, I am wondering if there is an alternative way of comparing without calculating Cohen's d.