One assumption of ANOVA is the independence of between-subject groups. This independence is usually established practically, such as based on the biological setup of the experiment. However, how is this independence tested statistically? Moreover, what happens when statistical and practical independence disagree?
Example: let's say the goal is to find out whether animals caged in pairs acquire significantly different weights (highest vs lowest weight for each pair). At first thought, this data can be analyzed with one-way ANOVA (or independent samples t-test, absent other factors) because biologically, each animal is an independent subject. However, these animals are caged in pairs and their weights are indeed highly correlated (r=0.5). I'm not sure what other tests can be run to measure the independence between two continuous variables. Assuming these variables are not statistically independent, does it mean that instead of one-way ANOVA one should run repeated measures ANOVA (or paired samples t-test) despite the practical/biological independence?