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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?

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  • $\begingroup$ Correct, scores of animals in the same cage are correlated then the assumption of independence is violated. If your sample size is sufficient, you may want to average the scores of animals in a given cage. The averages will be independent. This will be done implicitly for the between-group variable if you analyze design as repeated measures. $\endgroup$ – David Lane Jan 25 '17 at 20:38
  • $\begingroup$ The purpose is to find if there is a difference between the highest and lowest weight for all the cages, so I don't see how averaging the animals per cage would answer that question. Also, the question would still remain as to how one decides whether the weight groups are statistically dependent. If it's based on the correlation, then is there a particular r beyond which the groups would be treated as dependent, i.e., within-subject instead of between-subject? $\endgroup$ – Sophocles Jan 25 '17 at 22:52
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With animals cage together, there is often a competition effect (the strongest/most aggressive animal takes more food ...) which would lead to a negative correlation, but you reports a positive correlation. So, which species? how was it decided which animals to pair together? If there is large variance among the animals, but similar animals where put together, there could be both a negative correlation due to competition, and a positive correlation due to the similarity (and marginal variance), in your example the last effect winning. So, please also specify how you calculated the correlation.

In short, there will seldom be reasonable to assume independence among animals caged together, but to say much more we need details of your experiment.

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