# May outliers represent significant values?

Each element of a 56x1 vector represents the functional association between two brain networks. I want to assess which of these 56 values are significant.

One way to deal with this is to use an outlier detection method (the function 'isoutlier' in MatLab 2017 may be useful for that).

Is the use of an outlier detection method the best way to solve this problem? If yes, which one would you suggest? Any other suggestion is appreciated.

It can be assumed that the components of the vector are modeled by indipendent random variables. A null hypothesis may be that the elements follow a normal distribution, and there aren't elements which value is high (or low) enough to consider that element significantly different from the distribution.

Importantly, it should not be assumed that there is a single outlier AND it should not be assumed that outliers should be all positive or all negative. If I would make very rough assumptions, I would expect at least one positive and one negative significant/outlier/extreme value. It should not be assumed that outliers might come from different populations; they are more likely to be completely idiosyncratic.

The 56-elements vector is the following:

15.6102421615702    11.2037663155335    0   2.56486246042907    0   2.95192348899299    25.4261016360899    1.69087452285596    4.58160656448157    6.15352121272156    0   7.15067641442470    22.6569915347745    0   3.11486884384704    0   0.560988297590797   1.72416293971760    2.83706995604589    0.550195573381999   9.37539414273548    8.80726205947249    1.33854718393710    0   8.31660694574501    6.67557053091482    32.0990765988690    19.9783488357544    -3.84674540585432   -1.70069922973462   -5.28546098940327   -1.58616069782078   -2.56805690178644   -18.6007759621189   -0.411482275299029  0   -0.412950001716214  -1.30829388992302   -8.31539662049327   -15.0718661162509   -4.63912678558950   -3.07006832075995   -4.51646951028300   -18.3394207266255   -23.9911274828297   -5.69494580486308   0   -2.03380200815495   -1.64731977997449   0   -0.439186224081322  -4.13238476495763   -0.385501350605197  -5.32409970529450   -10.4842934806172   -1.18223031120631

• Please explain what you mean by "significant." Ordinarily this word has a statistical sense in the context of a hypothesis test: so what is the null hypothesis you have in mind? What can you tell us about this "vector"? Can you assume its components are modeled by independent random variables, for instance? – whuber Mar 30 '17 at 16:15
• Yes,it can be assumed that the components of the vector are modeled by indipendent random variables. A null hypothesis may be that the elements follow a normal distribution, and there aren't elements which value is high (or low) enough to consider that element significantly different from the distribution. Does that make sense? – smndpln Mar 30 '17 at 16:24
• In many situations since an outlier is unusually large or small it can be the most informative data point. If you have many outliers things get complicated and what is important becomes less clear. There can be a masking effect when two outliers are either both extremely large or extremely small. This is especially true when the sample size is small. – Michael R. Chernick Mar 30 '17 at 16:31
• (+1) Yes, that makes sense: many outlier tests are based on precisely those assumptions. They vary in terms of what the likely alternative possibilities are: whether you are looking for single outlier; for multiple outliers; for multiple outliers whose quantity might not be known; and for both high and low outliers or all outliers in just one direction. The more you can tell us concerning such possibilities, the better our recommendations can be. – whuber Mar 30 '17 at 16:48
• I included a note in the question, concerning these possibilities. – smndpln Mar 30 '17 at 16:55