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I have a set of reference wetlands and experimentally manipulated wetlands and I am comparing many water quality variables between them. My thought was to first identify which of the manipulated wetlands fall outside of the natural range for each variable by calculating the mean and standard deviation for the natural wetlands and then seeing which of the experimental wetlands fell more than 2 standard deviations outside of this range, in order to identify wetlands that have water chemistry that falls outside of 95% of natural wetlands.

However, many of my variables are positively skewed. Is there a better way than the 2x the standard deviation to identify upper and lower values that encapsulate 95% of the range?

Note: My question is somewhat similar to this one, but different enough that I think it stands alone (standard deviation to describe variation in positively skewed data)

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    $\begingroup$ May I ask how many reference wetlands you have? And how many measurements you took for a specific water variable at each wetland site? $\endgroup$
    – panda
    Commented Sep 27, 2021 at 21:56
  • $\begingroup$ I have 31 reference wetlands and 21 experimental wetlands, 5 samples from each, and 112 measures of water quality (e.g., levels of lead, zinc, aluminum, ammonia, bicarbonate, pH, dissolved oxygen, ethylbenzene, etc...) $\endgroup$
    – Dugan
    Commented Sep 28, 2021 at 13:42
  • $\begingroup$ So you have 31*5 data points considered to be normal/natural and 5 data points for a site which you would like to know that variable is abnormal or not. How about trying the common-language effect size? janhove.github.io/reporting/2016/11/16/… $\endgroup$
    – panda
    Commented Sep 28, 2021 at 18:28

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No matter how you cut it, it's not often useful to use a symmetric interval to define normal ranges in asymmetric data - assuming of course that those data are filtered free of non-normal observations. Here I mean "normal" in the sense of not abnormal. While we don't often condone discarding outliers, the point is that if you are defining a cut off to define abnormal cases you had best make sure the ranges of normal are in fact calculated from normal values. If you actually know which cases are normal and which are abnormal, then you can look at discrete predictive metrics, like a ROC to define an optimal threshold of abnormal.

That said, it's quite easy to calculate an asymmetric interval representing the range of normal from non-abnormal data, i.e. what one expects with a "central" 95%. Two approaches immediately come to mind: first is to simply calculate the 2.5th and the 97.5th quantile, i.e. which values at which 2.5% and 97.5% of the sample fall below that value. Alternately, you can simply log transform the value, if the resulting distribution has an approximately symmetric distribution, you can treat the distribution as log normal and define empirical normal rules based on the transformed sample.

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  • $\begingroup$ I am a bit confused with the 2nd paragraph, I thought if the data are already normal, the percentile CI is still symmetric $\endgroup$
    – panda
    Commented Sep 27, 2021 at 22:01
  • $\begingroup$ @panda hm reworded that section to speak about ranges of "not abnormal" data so you aren't thinking of symmetrically distributed normal (i.e. Gaussian) data., $\endgroup$
    – AdamO
    Commented Sep 27, 2021 at 22:32
  • $\begingroup$ I agree with this. Indeed, if you want another interval in a positively skewed distribution, that from the minimum to the 95th percentile might serve too. Yet further, all these ideas must be treated circumspectly: if extreme wetlands don't plot as outliers or distinct clumps, they are not that different. It's disturbing how rules of thumb can be taken far too literally, as when Tukey's convention that points more than 1.5 IQR away from the nearer quartile on a boxplot are points to be noted and thought about has morphed in some circles into a rule for outliers that should be deleted! $\endgroup$
    – Nick Cox
    Commented Sep 28, 2021 at 0:17
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    $\begingroup$ @NickCox I had the same thought, or even maybe there's no point even specifying the lower range at all, and we only care when the value is too high. We see this with laboratory analysis all the time, like in asthma, I don't care if the eosinophil is exactly 0, but north of a certain range indicates an eosinophilic asthma and certain treatments won't work. $\endgroup$
    – AdamO
    Commented Sep 28, 2021 at 16:44
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    $\begingroup$ @Dugan you need to define what means "abnormal" and then refer to those observations outside that range as abnormal values. I often see ranges described in terms of a certain quantity above the limit of normal, for instance if a lab has a maximum value of 100, then a value of 500 is 5 times the upper limit of normal. $\endgroup$
    – AdamO
    Commented Sep 28, 2021 at 16:46

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