[R] Use of geometric mean for geochemical concentrations [RESOLVED]

[Richard O'Keefe]

Suppose we measure the concentration of urea in four samples.
Suppose the results are 1, 2, 3, and 4 μmol·L−1,
How should we summarise this?

If you collect a predetermined volume V of water and measure the
number of moles M(V) of urea then calculated
C(V) = M(V)/V, and then you did that four times, then pooling the four
samples gives you
(M(V1)+M(V2)+M(V3)+M(V4))/(V1+V2+V3+V4) =
(V1*C(V1)+V2*C(V2)+V3*C(V3)+V4*C(V4))/(V1+V2+V3+V4)
and this just *is* the concentration of urea in the pooled samples.
If the samples are all the same volume, this reduces to the arithmetic mean.
In this case, 2.5 μmol·L−1,

If you predetermine a number of moles M of urea and collect water
until the volume V(M) contains M moles,
then calculated C(M) = M/V(M), and then you did that four times,
pooling the samples gives you
(M1+M2+M3+M4)/(M1/C(M1) + M2/C(M2) + M3/C(M3) + M4/C(M4))
and this just *is* the concentration of urea in the pooled samples.
If the samples all contain the same amount of urea, this reduces to
the geometric mean.
In this case, 1.92 μmol·L−1,

Really, it's not a statistical question but a semantic one: what
physical reality do the data reflect and what combinations of the
numbers make sense in/of the real world?

On Thu, 25 Jan 2024 at 06:25, Rich Shepard <rshepard using appl-ecosys.com> wrote:
>
> On Mon, 22 Jan 2024, Rich Shepard wrote:
>
> > As an aquatic ecologist I see regulators apply the geometric mean to
> > geochemical concentrations rather than using the arithmetic mean. I want to
> > know whether the geometric mean of a set of chemical concentrations (e.g.,
> > in mg/L) is an appropriate representation of the expected value. If not, I
> > want to explain this to non-technical decision-makers; if so, I want to
> > understand why my assumption is wrong.
>
> Many of you provided excellent comments, and so did a couple of folks on
> StackExchange. Rather than responding to individual posts I've waited until
> the thread petered out to provide an overall response.
>
> I've two points to make: one on mean calculations and the second on the
> context I didn't sufficiently provide when I posted my question.
>
> Responses confirmed that the appropriate model for calculating means depends
> on the data set and the question(s) the data are to answer. So the summary
> answer to my question (as stated) is: it depends. :-) Thank you.
>
> What prompted my thread-starting message is that I work in the realm of
> environmental regulation compliance, including the Clean Water Act and the
> Endangered Species Act. There is one state environmental regulator that
> provides state-wide point source storm water discharges under a General
> permit for smaller industrial activities. The permit monitoring requirements
> are 4 samples per year, one each quarter for a small set of water chemical
> and physical constituents (really!) and the reporting requirements are to
> use the geometric mean to summarize the four data points. I have my clients
> calculate an arithmetic mean in addition. (For the record, if you have an
> Agriculture Department General Storm Water Discharge Permit for a point
> source such as a livestock feed lot you need only a single sample (after the
> rains start) to comply with the permit. Feh!
>
> Germane to Bert's comments about all the wrong ways to treat
> non-detected/censored water chemical analyses, I discovered Dennis Helsel by
> his 2005 article in Environmental Science & Technology (Oct. 16th). Bought
> his book when it was published in 2012 and have used survival analyses on
> censored data ever since. (Also presented a Continuing Legal Education talk
> in 2016 with a nice thank-you email from a state district judge who
> attended.)
>
> I greatly appreciate all your comments and apologize for not better
> explaining the context of my question when I posted my first message.
>
> Regards,
>
> Rich
>
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