# Find significantly differences between multi-variant groups

I have 11 groups, each consist of about 200 observations, and each observation has 2 variables (which are actually the first two PLS components if it makes any difference). Each group is a substance with increasing amounts of contaminant. The first group has 0% contamination, the second group has 1% contamination.... the 11th group consists of 10% contamination. When I plot all the groups, base on the two variables I get the following:

My data is organized in pandas dataframe and looks like this:

         score1     score2
H01     -0.153515   0.013216
H01     -0.149235   0.007089
H01     -0.152169   0.006924
H01     -0.160296   0.008324
H01     -0.158890   0.017218
H01     -0.160165   0.011170


I would like to know which group is significantly different from which group and which one is not. I know that if you have only one variable, you can use Tukey’s test and then, each group will get a label, where similar groups will get similar labels. However, this does not apply to multi-variant cases, such as in my case where I have 2 variables. I tried MANOVA but it only tells me if there is at least one group which is different but does not tell me which one is different from which. Is there a statistical test I can apply that take into consideration both variables simultaneously in order to look for significant differences between groups? I prefer the solution to be in python libraries if possible, but any direction for a solution would be appreciated.

UPDATE: I run MANOVA using python- statsmodels.multivariate.manova as suggested, for H12 and H14 and I got weird-look results:

                    Value    Num DF Den DF  F Value     Pr > F
Wilks' lambda   0.0321726   2   354     5324.57     6.6844e-265
Pillai's trace  0.967827    2   354     5324.57     6.6844e-265
Hotelling-Lawley trace  30.0823     2   354     5324.57     6.6844e-265
Roy's greatest root     30.0823     2   354     5324.57     6.6844e-265


If I understand correctly, all methods have the same extremely low P value (6.6844e-265), which mean that the groups are significantly different, but as can be seen in the plot, H12 and H14 are practically on top of each other. Am I interpreting the MANOVA table correctly?

• There are methods to perform pairwise comparison in MANOVA, for example, the statement MANOVA in SAS PROC GLM can do it. But I have no idea on python. – user158565 Nov 24 '18 at 20:30
• I'm not sure I fully understand your comment. Pairwise means two pairs of variables? Do you suggest that I use MANOVA? because from my understanding MANOVA will just indicate a difference and not where is the difference – user88484 Nov 24 '18 at 20:49
• You have 11 groups. MANOVA can tell you there are differences among 11 groups on that two response variables. Pairwise comparison means to compare between two specified groups on that 2 response variables. – user158565 Nov 24 '18 at 20:52
• So you suggest that I do some kind of iteration between each pair of groups? and perform MANOVA for each pair – user88484 Nov 25 '18 at 8:16