#### Summary of cases

The following tables show the values for the variances of variables

*s* and

*c* and the values of coefficient

*z* in the expression:

used to generate the data for variable

*i* and individual

*j* in the different cases considered in the simulations -sqrt: square root; note that sqrt(

*z*) is equivalent to

*k* in the main text. Two variables

Detailed list of cases

List of parameter sets used in every simulated case and resulting covariance matrices, eigenvectors and eigenvalues. The expected compositions of eigenvectors were obtained via eigenvector analyses applying R function *eigen* to random samples of size 10^{6}. Note that for four and seven variables cases it was not possible to obtain a constant set of eigenvector coefficients (beyond the first eigenvector) even for such large samples. In any case, The S statistics recognized their equivalence despite differences in eigenvectors’ coefficients (see the S3 and S2 values in the second row and third rows respectively of Figure 3).

In the two variables case we had:

where sqrt is the square root and

*s*
_{1} and

*s*
_{2} had distributions N(0, 0.2), and

*c*, N(0, 0.8). The expected covariance matrix was:

The expected eigenvectors had coefficients (columns):

and the expected eigenvalues were: 1.8 and 0.2.

Compared sample with altered orientation:

where

*s*
_{1} and

*s*
_{2} had distributions N(0, 0.2), and c, N(0, 0.8). The expected covariance matrix was:

The expected eigenvectors had coefficients (columns):

and the expected eigenvalues were: 1.8 and 0.2.

Compared sample with altered shape:

where s

_{1} and s

_{2} had distributions N(0, 0.5), and c, N(0, 0.5). The expected covariance matrix was:

The expected eigenvectors had coefficients (columns):

and the expected eigenvalues were: 1.5 and 0.5.

Compared sample with both orientation and shape altered:

where s

_{1} and s

_{2} had distributions N(0, 0.5), and c, N(0, 0.5). The expected covariance matrix was:

The expected eigenvectors had coefficients (columns):

and the expected eigenvalues were: 1.5 and 0.5.

The expected total variance in all two variable samples was = 2.

In the four variables case, we had:

where

*s*
_{1,}
*s*
_{2,}
*s*
_{3} and

*s*
_{4} had distributions N(0, 0.2), and c, N(0, 0.8). The expected covariance matrix was:

The expected eigenvectors had coefficients (columns):

and the expected eigenvalues were: 3.4, 0.2, 0.2, 0.2.

Compared sample with altered orientation:

where

*s*
_{1,}
*s*
_{2,}
*s*
_{3} and

*s*
_{4} had distributions N(0, 0.2), and c, N(0, 0.8). The expected covariance matrix was:

The expected eigenvectors had coefficients (columns):

and the expected eigenvalues were: 3.4, 0.2, 0.2, 0.2.

Compared sample with altered shape:

where

*s*
_{1,}
*s*
_{2,}
*s*
_{3} and

*s*
_{4} had distributions N(0, 0.4), and c, N(0, 0.6). The expected covariance matrix was:

The expected eigenvectors had coefficients (columns):

and the expected eigenvalues were: 2.8, 0.4, 0.4, 0.4.

Compared sample with both orientation and shape altered:

where

*s*
_{1,}
*s*
_{2,}
*s*
_{3} and

*s*
_{4} had distributions N(0, 0.4), and c, N(0, 0.6). The expected covariance matrix was:

The expected eigenvectors had coefficients (columns):

and the expected eigenvalues were: 2.8, 0.4, 0.4, 0.4.

The expected total variance in all four variable samples was = 4.

In the seven variables case, we had:

where

*s*
_{1,}
*s*
_{2,}
*s*
_{3,}
*s*
_{4,}
*s*
_{5,}
*s*
_{6,} and

*s*
_{7} had distributions N(0, 0.2), and c, N(0, 0.8). The expected covariance matrix was:

The expected eigenvectors had coefficients (columns):

and the expected eigenvalues were: 5.8, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2.

Compared sample with altered orientation:

where

*s*
_{1,}
*s*
_{2,}
*s*
_{3,}
*s*
_{4,}
*s*
_{5,}
*s*
_{6,} and

*s*
_{7} had distributions N(0, 0.2), and c, N(0, 0.8). The expected covariance matrix was:

The expected eigenvectors had coefficients (columns):

and the expected eigenvalues were: 5.8, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2.

Compared sample with altered shape:

where s

_{1,} s

_{2,} s

_{3,} s

_{4,} s

_{5,} s

_{6,} and s

_{7} had distributions N(0, 0.4), and c, N(0, 0.6). The expected covariance matrix was:

The expected eigenvectors had coefficients (columns):

and the expected eigenvalues were: 4.6, 0.4, 0.4, 0.4, 0.4, 0.4, 0.4.

Compared sample with both orientation and shape altered:

where s

_{1,} s

_{2,} s

_{3,} s

_{4,} s

_{5,} s

_{6,} and s

_{7} had distributions N(0, 0.4), and c, N(0, 0.6). The expected covariance matrix was:

The expected eigenvectors had coefficients (columns):

and the expected eigenvalues were: 4.6, 0.4, 0.4, 0.4, 0.4, 0.4, 0.4.

The expected total variance in all seven variable samples was = 7.