Biology Reference
In-Depth Information
Statistical Analyses
The first statistical analysis performed was a linear regression between the age and each
one of the 50 morphometric variables studied. The relationship was expressed as
y = a
+
bx
,
where age was represented by the dependent variable (
y
). For those variables which showed a
significant relationship with age, a multiple regression analysis was undertaken with the
expression
y= μ+b
1
x
1
+ …..+b
m
x
m
(Arenas
et al,
1991), were
y
was the dependent variable
(age) and x
1
…………x
m
were the skull variables. Furthermore, the square of multiple
correlation was calculated, which represented the goodness of fit between x
i
, the observed
value of each genetic variable. The better linear predictor, X
i
, is a function of m x
i
and the
expression R
2
=
I
(x
i
- X
i
)
2
/
I
(x
i
- X)
2
, where X is the average of the recorded dependent
variable values. The residual variances were calculated as well; these were the mean square
distances of the points to the regression hyperplanes, R
o
2
/ n , with R
o
2
=
I
(x
i
- X
i
)
2
, where
X
i
was the prediction of x
i
as a function of the regression hyperplane. The number of points
compared is abbreviated as (n) and variance was calculated with the equation var (x
i
) =
i
2
(1
- R
2
), where
i
2
is the variance of the observed distribution of each one of the skull
measurements and R
2
is the square multiple correlation coefficient. The square root of this
value is the typical error of the estimated x
i
value. These analyses were performed with the
statistical programs NCSS and Multicua.
Also, a regression analysis employing distances was carried out. Five different distances
were used (Gower, Absolute value, Mahalanobis, and Minkowski with q exponent equals to 2
and 4; Cuadras & Arenas, 1990). Their mathematical expressions are shown in Table 2.
Table 2. Distances employed in the regression analysis.
Distance
Equation
d
i
2
= 1-s
ij
Gower
p
1
2
d
x
x
Absolute value
ij
ih
jh
h
2
1
d
x
x
)'
C
(
x
x
)
Mahalanobis
ij
i
j
i
j
1
/
q
Minkowski with q exponent
(2 and 4)
p
q
d
x
x
ij
ih
jh
h
1
The coefficient of determination (based on the function of the number of principal
coordinates selected), the residual sum of squares (SSQ =
I
(x
i
- X
i
)
2
, where X
i
is the
prediction of x
i
) and the cross-validation coefficient (C) (calculated with the equation (C =
1/n
I
(x
i
- X
i
)
2
, where X
i
is the prediction of x
i
obtained to eliminate this individual from the
original matrix data) were also estimated.
Additionally for the linear and distance models, we analyzed 24 other regression models
(Table 3) as well as polynomial regressions of third, fourth and fifth order with the variables
which showed significant correlations with the linear regression. Finally, we obtained a
multiple regression with the four variables which presented the highest correlations with the
linear regressions, along with the four variables with the highest correlations with the
Search WWH ::
Custom Search