-------- Original Message --------
Subject: Re: correcting for body size
Date: Sun, 20 May 2012 09:46:09 -0400
From: GREGORY CAMPBELL <g.v.campb...@btopenworld.com>
Reply-To: GREGORY CAMPBELL <g.v.campb...@btopenworld.com>
To: morphmet@morphometrics.org <morphmet@morphometrics.org>
Dear Saad: It's perfectly possible for a fitted line to account for a
lot of the co-variance of two variables (and therefore have a high
r-squared value), but not be a good explanation of that co-variance:
-the fitted line can tend to under-estimate large and small values in
the range while over-estimating for the values near the average, or the
other way around (non-linearity);
- the precision of the estimate can be very good for part of the range,
but poor elsewhere (heteroscedasticity)
-the arithmetic used to fit the line doesn't work properly because the
points are not distributed in a bivariate Gaussian-normal way
(non-normality).
All these errors of explanation (and others I haven't remembered) are
caused by bias in estimation across the range. So they show up in plots
of the residuals (difference between true and line-estimated value): the
residuals show some systematic variation when plotted against the
estimating (ordinal) variable, or they are not normally distributed.
Since there is no causal relationship between body size and eye area
(big bodies do not cause big eyes), it makes less sense to use a Type I
regression (the standard type, designed to produce good estimation
formulae, using ordinary least-squares fitting). Better to use a Type II
regression (which generates formulae for lines using equal emphasis on
both variables).
There are two types of Type II regressions available. Major-axis
regression works if the uncertainty (the difference between the value
measured and the true value of a variable) is the same for all your
variables. Reduced major-axis regression (also called standardised major
axis regression) works if the uncertainty is not likely to be the same
(the structures being measured are qualitatively different, the measured
values are different orders of magnitude, or the structures are measured
in different units). Reduced-major-axis is probably the safer option
here (readers please comment).
If the residuals show some bias, carrying out the regression again after
log-transforming both dimensions (to either their base-10 logs or base-e
ln-values) often removes the biases. It also makes it easier to compare
the allometric relationship between the dimensions. Since slopes in
log-transformed data represent the exponent of their relationship, it
can be used to study the scaling of the variables. Since your eye
dimension is an area, you would expect it to increase as the square of
your body size, so the slope of the log-transformed dimension should be
exactly two. If the slope is more than two, your flies are investing
proportionally more in growing eyes than bodies, and less than two they
are investing proportionally less in growing eyes than bodies. You can
test whether a slope is statistically different from an expected value
using a t-test, or if it lies within the 95% confidence interval (lots
of statistical packages provide this).
Bivariate allometry with Type II regression is not standard in most of
the common statistics packages. I use the paleotological freeware, PAST
(it's easy to load data from Access, Excel, and Minitab).
If you do use log-transformed data, you still have to look for
systematic bias by examining your residuals (non-linear allometry lurks
in some animal growth).
A more complete discussion of regression: see Warton et al 2006
(Biological Reviews 81: 259).
A more complete discussion of allometry: start with Gould 1966
(Biological Reviews 41: 587).
Greg Campbell
The Naive Chemist
*From:* morphmet <morphmet_modera...@morphometrics.org>
*To:* morphmet <morphmet@morphometrics.org>
*Sent:* Wednesday, 16 May 2012, 19:14
*Subject:* Re: correcting for body size
> -------- Original Message --------
> Subject: correcting for body size
> Date: Tue, 15 May 2012 11:03:28 -0400
> From: Saad Arif <arifs...@gmail.com <mailto:arifs...@gmail.com>>
> To: morphmet@morphometrics.org <mailto:morphmet@morphometrics.org>
>
> Hello all,
>
> I have linear measurements for eye area and body size in Drosophila. I
> want to get residuals for eye area by regressing it on body size. Both
> variables appear to have a straight-line relationship. my question is:
> do i need to square my body size measurement before i regress it to
> eye area even if they are both seemingly linearly related?
>
> Any response would be appreciated!
>
> Thanks in advance.
>
> Saad
>
>
>
>
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