Greater than 30% outliers in small dataset - what to do? Standard test? Test with outliers removed? Robust statistics?

I have a small-sample dataset representing observations from a longitudinal study. My principal interest is in 'change scores' across three parameters (A, B, C). This requires simple paired t-tests. However, applying the median absolute deviation rule, I've found that the change scores for each parameter contains a large number of outliers (30-45%).

This represents a substantial amount of data relative to the full sample, and thus my concern. I have several questions I'd appreciate any comment on:

• Is there a rule for when removal of data from outliers is too much (i.e., where outliers represent too great a proportion of the full dataset)?
• How should I proceed with analysis? A regular test with all data? A test with data removed? Or a robust t-test using trimmed means and winsorized variance?

Example figures:

Histogram of Parameter A

Parameter A vs Parameter B

• this 30%-45% number, do you obtain it by flagging an observation as outlier whenever it is flagged as outlier on either one of the the three parameters? Aug 23, 2018 at 8:50
• Yes. For example, the change score for parameter A has 30% outliers, parameter B has 40%, and parameter C 35%. Aug 23, 2018 at 10:47
• so if you had only parameter A to apply the mad rule on, you would have 30% outliers? Is this what you mean? Also, can you post the result of doing length(unique(x))/length(x)) when, again, x is just the first parameter ('parameter A')? [length(unique()) counts the number of different values, so length(unique({1,1,1,2})) is 2] Aug 23, 2018 at 12:17
• Yes. If only looking at the change scores for parameter A, I would obtain 30% outliers following the MAD rule. The result of your function is 1: all change score observations are different. Aug 24, 2018 at 2:41
• ok. These 30% outliers, when you plot them (histogram of parameters individually or --even better-- considering plots of A vs B and A vs C and B vs C) do they form a cohesive group (are all or a large proportion of the outliers bundled together?) Aug 24, 2018 at 5:50