R help - Jan 2024 - Use of geometric mean for geochemical concentrations

A statistical question, not specific to R.

I'm asking for a pointer for a source of definitive descriptions of what
types of data are best summarized by the arithmetic, geometric, and harmonic
means.

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.

TIA,

Rich

better posted on r-sig-ecology? -- or maybe even stack exchange?

Cheers,
Bert

On Mon, Jan 22, 2024 at 7:45?AM Rich Shepard <rshepard at appl-ecosys.com>
wrote:
> A statistical question, not specific to R.
>
> I'm asking for a pointer for a source of definitive descriptions of
what
> types of data are best summarized by the arithmetic, geometric, and
> harmonic
> means.
>
> 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.
>
> TIA,
>
> Rich
>
> ______________________________________________
> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
> http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>
	[[alternative HTML version deleted]]

>>>>> Rich Shepard 
>>>>>     on Mon, 22 Jan 2024 07:45:31 -0800 (PST) writes:
    > A statistical question, not specific to R.  I'm asking for
    > a pointer for a source of definitive descriptions of what
    > types of data are best summarized by the arithmetic,
    > geometric, and harmonic means.

In spite of  off-topic:

I think it is a good question, not really only about
geo-chemistry, but about statistics in applied sciences (and
engineering for that matter).

Something I sure good applied statisticians in the 1980's and
1990's would all know the answer of :

To use the geometric mean instead of the arithmetic mean
is basically  *equivalent* to  first log-transform the data
and then work with that transformed data:
Not just for computing average, but for more relevant modelling,
inference, etc.

John W Tukey (and several other of the grands of the time)
had the log transform among the  "First aid transformations":

If the data for a continuous variable must all be positive it is
also typically the case that the distribution is considerably
skewed to the right.
In such a case behave as a good human who sees another human in
health distress: apply First Aid -- do the things you learned to
do quickly without too much thought, because things must happen
fast ---to hopefully save the other's life.

Here: Do log transform all such variables with further ado,
and only afterwards start your (exploratory and more) data analysis.

Now,  mean(log(y)) = log(geometricmean(y)), 
where mean() is the arithmetic mean as in R
{mathematically; on the computer you need all.equal(), not '==' !!}

I.e., according to Tukey and all the other experienced applied
statisticians of the past, the geometric mean is the "best thing" 
to do for such positive right-skewed data   in the same sense
that the log-transform is the best "a priori" transformation for
such data -- with the one advantage even that you need to fiddle
with zeroes when log-transforming, whereas the geometric mean
works already for zeroes.

Martin


    > 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.

    > TIA,

    > Rich

    > ______________________________________________
    > R-help at r-project.org mailing list -- To UNSUBSCRIBE and
    > more, see https://stat.ethz.ch/mailman/listinfo/r-help
    > PLEASE do read the posting guide
    > http://www.R-project.org/posting-guide.html and provide
    > commented, minimal, self-contained, reproducible code.

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