Multivariate Kolmogorov-Smirnov for two samples

Question

Let’s say we have data of some fruits and we want to consider different statistics for each fruit (there are various samples for each statistic). For example:

Watermelon:

• Sugar: 50, 57, 36…
• Water: 101, 143, 128…
• …

Melon:

• Sugar: …
• Water: …
• …

(…More fruits…)

We want to know if, based on this statistics, fruits are the same. I thought about using Kolmogorov-Smirnov test in order to check it. So we have a CDF for each statistic (sugar, water, etc.) for each fruit. If we want to consider all statistics for each fruit, it would be nice to use the multivariate Kolmogorov-Smirnov test for two samples and compare every pair of fruits.

The problem is that I cannot find information about the multivariate test with two samples and I read somewhere that the multivariate test is not that good. Could you tell me a different test I could use? Or the Kolmogorov-Smirnov is a good idea (where can I read about the multivariate for two-samples)?

Answer 1

There is a generalization of the KS statistic for multivariate distributions called the energy distance, described in Szekely and Rizzo (2013). When two joint distributions are equal, the energy distance between them is zero. It is possible to use a test statistic formed from the energy distance to test the hypothesis that two distributions are the same. There is also a generalization of ANOVA that uses the energy distance called distance components (DISCO) which can be used to tests whether multiple distributions are equal to each other.

Answer 2

I think that unless you have many observations for each (fruit, statistic) combination it doesn't make sense to compare distributions. E.g. in ANOVA in theory they could compare distributions of response under different treatments, but instead they typically compare just treatment means. I would suggest you try MANOVA to compare your fruits to each other. An example is provided here.