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I have three groups A, B and C and I want to prove that similarity between group B and group A is significantly higher than between group B and group C.

How to prove it?

Simple example:

I have three groups of people from three countries Albania (A), Belarus (B) and China (C). I have information about their weights. It occurs that the average weight of Belarussians is the highest, Albanians are thinner, Chinese are the thinnest. I can check using ANOVA or a non-parametrical test that there are significant differences among groups.

Additionally I may check that Belarussians are significantly fatter than Albanians and significantly fatter than Chinese. The question is how can I prove that the weight of Albanians is significantly more similar to Belarussians than the weight of Chinese?

Of course with just one attribute I can check if Albanians are significantly fatter than Chinese and – if so – it means that they are significantly closer to Belarussians. But is it the only way? And what if it occurs that there are no significant differences between A and C?

Now, let’s say that I have two attributes: weight and height. It can be checked that there are significant differences between groups when taking into account both attributes. The question is how can I prove with this attributes that Belarussians and Albanians are more similar to each other than Chinese. This time there are two axes and each person is just a point in two dimensional space so there is no easy answer that A is between B and C as it was in the case of one attribute. I calculate average values of attributes for every group and check Euclidean distances between these three “averaged persons” and it occurs that the distance between B and A is smaller than between B and C. But how to check the significance of this result?

Like in the sample picture - how to prove that group A is significantly more similar to B than C to B? Three groups in 2D space - is B significantly more similar to A than C?

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First, as I was just saying elsewhere, you can't expect to be able to prove anything.

Second, you can't use significance testing to show that two things are similar (or more similar to each other than either is to something else). Think about how significance testing works: you try to show that two things are different by assuming that they're the same and then showing that this assumption implies the results you obtained were unlikely. The problem with trying to do the opposite, by assuming that two things are different, is that this assumption gets you basically nothing, far less than what would be needed to compute a relevant probability. In simple terms, if you knew that $x = 0$, that would tell you a lot about $x$, but if you only knew that $x ≠ 0$, $x$ could still be any positive or negative number.

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