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Subject: Re: Statistical problem
From: "Gerard Y. Conan" <[log in to unmask]>
Reply-To:Academic forum on fisheries ecology and related topics <[log in to unmask]>
Date:Wed, 18 Sep 1996 18:04:57 -0300

text/plain (100 lines)

This seems a simple problem, but in fact it is not a simple problem if you
want to give it some thoughts. 

What you really want to do is compare two samples for which you use the
allometric model W=a.L power b. This relationship is not linear, you use a
transformation to linearize and tentatively achieve homoscedascity. You
already know that your two samples do not come from the same population, you
wish to check whether the model you are using fits the two differents
populations equally well with the same parameters. Your goal is to use only
one model and the same parameters for simplification. There are measurement
errors on both variables, and the relationship between the two variables is
probably specific to each individual.

Given this you may of course agree to use accepted procedures and for
instance simplify by assuming that no error is made on measurements of
length in order to have a predictive rather than a functional regression.
You may also accept that comparing the tramsformed data rather than the
original is acceptable.

Using standard packages without giving some thoughts to the nature of the
problem may be misleading. Actually the true problem seems to be "may I use
the same relationship over and over again, knowing that the populations keep
on changing", this is not the standard statistics problem of comparing
sample regressions to check if they come from the same population
regression. If you were taking large enough samples, you would probably gain
enough power to detect that your samples come from different populations
even if the differences between populations were very small. May be what you
want to do is check whether you can use the same preset relationship over
and over again for the purpose of consistency and accept the risk of a given
error on your predictions of weight from a known size, eventually estimate
the range of variation of your estimates from the actual value in 95% of
cases, and this will involve both precision and bias calculations.

On the other hand, if you wish to do what "everybody else does" just assume
that the linear transformation is alright, that the regressions are
predictive regressions (no errors on L), that all animals in a sample obey
to the same law but with an added random variability component which obeys
to the same probability distribution for all individuals. Allow that the
"law" may vary between samples i.e. that not all samples may obey to the
same "law". If you accept that logic, then pick up the latest edition of a
most respected book of statistics such as Snedecor and Cochran, Statistical
Methods, Iowa State University Press, Ames Iowa, USA. check chapter 14.6
Comparison of regression lines. You will find that all you require are the
following statistics for each sample: sumX, sumXX,sumXY,sumYY,sumY, and N.
Everything else is quite straightforward and can be calculated by hand, no
need for a package as long as you have a table of one tailed and two tailed
F and eventually a chi square table. This is a comparison of regression
lines by ANOVA, it is not an ANOVA, which would be less powerfull, and would
indeed require the same Ln L values for each sample, it is not an ANCOVA
sticto sensu, which would allow you to compare the Ln of weights, once you
have removed the linear trend associated with different ln L.

Best wishes.  

At 12:41 PM 9/18/96 +0200, you wrote:
>Dear Fish-folks,
>I want to compare linear regressions (for example ln length - ln weigth)
>with respect
>to differences in slope and intercept. We have found several
>complicate formulae in statistical textbooks. However, I know that one
could use
>also the ANOVA-approach - but how? I have intensively studied the manual of our
>statistical software (Statistika), and have found that linear regression and
>ANOVA are basically identical. Thus, it may be simple if we have similar
>each year. But in practice, the fish lengths (x-values) are measured
>independently each year,
>that means the range of fish lengths and weights may vary. My idea was to group
>the weigths of fish as dependent variables, and to use the lengths as
>and then to run an ANCOVA-procedure. But I could not identify whether this is
>correct. Has anyone some experience with this topic?
>Many thanks
>Yours sincerely
>Dr. Thomas Mehner
>Institute of Hydrobiology
>University of Technology
>D-01062 Dresden
>Tel.: (++49)(351) 463 2018
>Fax:  (++49)(351) 463 7108
>internet: [log in to unmask]
Gerard Y. Conan, M.S., Ph.D.
Research Scientist, Department of Fisheries and Oceans Canada
North Atlantic Fisheries Center, St John's, Newfoundland
Phone 1 709 772 3898, FAX 1 709 772 4105
Home Fax and Phone 1 709 739 4639

Adjunct Professor, Department of Mathematics and Statistics, 
Université de Moncton. E.mail [log in to unmask] 

Have a nice day ! 

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