Supplementary Material
Remodelling of biological parameters during human aging: evidence for complex regulation in longevity and in
type 2 diabetes
Journal name: AGE
Liana Spazzafumo1*, Fabiola Olivieri
2,3*
, Angela Marie Abbatecola4, Gastone Castellani, Daniela Monti8, Rosamaria
Lisa5, Roberta Galeazzi4, Cristina Sirolla1, Roberto Testa6, Rita Ostan7, Maria Scurti 7, Calogero Caruso10, Sonya
Vasto10, Rosanna Vescovini9, Giulia Ogliari11, Daniela Mari11, Fabrizia Lattanzio4, Claudio Franceschi7
Corresponding author:
Dr.ssa Liana Spazzafumo
Biostatistical Center
Polo Scientifico Tecnologico INRCA
Via Birarelli, 8
tel.+39 071 8004105
fax +39071 206791
l.spazzafumo@inrca.it
Models
Factor structure analysis: the method of functional and structural relationships
This method is based on a generalization of the mathematical way to express physical laws, where variables are related
by causal effect (for example, the perfect gases law). The method has been developed for two and n variables and we
will briefly describe these approaches in the linear case.
Two mathematical variables, X and Y, are linearly related if
Y   0  1 X and in this case X and Y are
functionally related. In general, we do not observe X and Y, while the observed values of the associated random
variables  and  are defined by:
 i  X i   ij i  1,2,, n
 i  Yi   ij
The estimation of 0 and 1, as well as the existence of the error terms in the X and Y variables poses a problem that is
similar but quite different from the classical regression due to the presence of an error term:
   0  1  (  1 )
This relation is called a structural relation between the observed random variables  and  and is the result of the
functional relation between X and Y.
The generalization to n variables X0, X1,….Xn, after adding the dummy variable X0=1 is:
k

j 0
j
Xj 0
along with the following associated random variables:
 ji  X ji   ji
i  1,2,, n, j  1,2,, n
This problem can be reformulated as the determination of an hyperplane in k dimensions and the distance of a point 1i,
2i,…. ki from the hyperplane is:
k
 
j 0
j
ji
 k

  j 2 


 j 0

k
Using specific algebraic manipulations we formulated the following system to be solved as:
k
c 
j 0
ij
j


n
l
l  1,2, k
with
1  1
r12
r13
r12
12
r23
r13
r23
1  3



r1k
r2 k
r3k
 r1k
 r2 k
 r3k  0

 1k
Where
j 

nS j
2
is the Sj the standard deviation of the i and the rij their correlations.
We then solved the second equation for  and hence find the ’s from the first.