Biology Reference
In-Depth Information
To use the dot product to calculate the angle between two vectors of size variables, we
would first estimate the regression coefficients for each one, then normalize the vectors to
unit length (meaning that the square root of their summed squared coefficients equals one).
To calculate the angle for shape variables, we would similarly first estimate the regression
coefficients for each such component, such as regression coefficients for coordinates
obtained by GPA or partial warp scores. We then calculate the dot product by multiplying
the allometric coefficient of one species by the allometric coefficient of that same variable in
the other, then multiply the coefficient of the next variable in one species by the allometric
coefficient of that same variable in another, and so for all coefficients. Finally, we would
sum all those products. This gives the correlation between the vectors (R v ). Because a corre-
lation is a cosine of an angle, we can also write the equation for the dot product as:
A
B
5 j A jj B j cos
θ
(11.6)
is the magnitude (length) of A, which is calculated by (A 1 2
A 2 2
...A P 2 ) 1/2 and
where jAj
1
1
is the length of B, calculated by (B 1 2
B 2 2
...B P 2 ) 1/2 , and
jBj
θ
similarly,
1
1
is the angle
between them.
If A and B are unit vectors, the two lengths
jAj
and
jBj
are both one, so, to find the
angle between the two vectors we solve for
θ
by:
θ 5
arccos
ð
A
B
Þ=ððj
A
jj
B
jÞÞ
(11.7)
When two vectors are parallel, the angle between them is 0 and the vector correlation
between them is 1.0; in contrast, when two vectors point in exactly the opposite direction
(which is termed being anti-parallel), the angle between them is 180 and the vector corre-
lation between them is
1.0. The angle between perpendicular (orthogonal) vectors is 90 ,
and the correlation between them is 0.0.
2
Testing the Statistical Significance of the Angle
Once we have computed an angle between two regression vectors, we are left with the
question of whether it is statistically significant. Rather than attempt to find an analytic test
of significance, we can rely on a bootstrap or permutation procedure (see Chapter 8 for an
overview of resampling methods and bootstrapping, and Chapter 9 for a more detailed dis-
cussion of permutation tests). Using bootstrapping, we can determine a confidence interval
for the range of angles between regression vectors that can be produced by random varia-
tion within each group. At issue is whether the uncertainty of our estimate of each vector
(due to sampling) is so large that we cannot reject the null hypothesis of no difference.
To estimate the range of angles within each species, we estimate the residuals from the
regression of shape on the independent variable. Each individual gives a multidimensional
set of residuals that describe the deviation of that individual from its expected shape. We
then form a pair of bootstrap sets for each group that will be used to calculate the angle
between the vectors. These pairs are constructed by resampling the residuals (with replace-
ment) and randomly assigning them to expected values of shape (derived from the original
regression model) at the values of size observed in the original data. This procedure pre-
serves the covariance structure among variables and is a multivariate extension of the stan-
dard approach to estimation of uncertainties of regression slopes by resampling.
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