Correlations between cephalometric measurements are the consequence of 2 major components: biologic mechanisms and topographic associations. Topographic, or geometric, relationships arise when the measurements under consideration share common landmarks. Consider, for example, 3 landmarks arranged on a straight line, whose position is not dictated by any biologic factor but is completely random ( Fig 1 ). Although one might assume that the correlation between distances (d1 and d2) should be zero, it is actually statistically significant (and negative; ie, a large d1 relates to a small d2), because point B is shared; any random variation in the horizontal position of point B will affect both measurements in an opposite way. Thus, there can be statistically significant correlations between 2 measurements, even in the absence of biologic factors.

A, Three points, representing anatomic landmarks, lying on a straight line: d1 , distance between points A and B; d2 , distance between points B and C. Distances d1 and d2 originally equal to 100 units. The points were relocated from their original positions to create 100 virtual specimens. Relocation was by random offset from the original position but ensuring that each “point cloud” would be normally distributed with a standard deviation of 10 units. B, Plot of d1 vs d2. The correlation coefficient r for this random sample is –0.52 ( P <0.0001).

The complex nature of topographic associations can be appreciated by observing how they change when the geometry is altered. Figure 2 shows the correlations between the same 2 measurements as point B is relocated at increasing distances from the horizontal line. The correlation changes from negative ( Fig 1 ) to zero ( Fig 2 , A and B ) to positive ( Fig 2 , C and D ). An interesting observation is that the topographic association might be zero, notwithstanding the common landmark shared by the 2 measurements. In our example, this occurs because random variation in the position of the common point B does not affect the measurements in a consistent manner; horizontal variations of point B produce opposite effects on d1 and d2, whereas vertical variations affect both equivalently. A specific geometric arrangement can be created ( Fig 2 , A and B ), where the opposing effects cancel each other, and the correlation is zero.

A, Point B relocated by 100 units upward (therefore, angle ABC is 90°); 100 random virtual specimens constructed by normally distributed, random repositioning of the points around their original position (SD, 10 units). B, Plot of d1 vs d2. The correlation coefficient for this random sample is 0.02 (not significant). C, Point B relocated to 300 units above points A and C. D, Plot of d1 vs d2. The correlation coefficient for this random sample is 0.43 ( P <0.0001).

From the above, it follows that, in addition to the geometric arrangement of the points, another influencing factor is the relative amount of variation along the horizontal and vertical axes. Allowing more variation in point B’s position along the vertical axis in the situation of Figure 2 , A , will produce a positive correlation ( Fig 3 ), whereas a negative correlation will be observed if the point exhibits mostly horizontal variation in position.

A, Same situation as in Figure 2 , A , but now the distribution of point B has double the standard deviation along the vertical axis. B, Plot of d1 vs d2. The correlation coefficient for this random sample is 0.44 ( P <0.0001).

Analogous situations are encountered in cephalometrics, often in more complicated and subtle ways, yet few correlation studies in the orthodontic literature have paid attention to the problem. The classic study of Solow brought the matter into the light, indicating that observed correlations are the net result of topographic and biologic factors. In an attempt to factor out the topographic correlations and thus show the true impact of biologic processes, Solow calculated analytic solutions for various landmark configurations but recognized that “the formulae for the topographical correlations are not sufficiently precise . . . due to the inability . . . to account for dissimilar variation of the reference points in different directions.” Järvinen pointed out the problem when assessing the relationship between cranial base angle and SNA, and Tollaro et al, acknowledging the difficulty, chose to avoid all measurements with explicit geometric associations.

If we accept the difficulty of calculating topographic correlations and removing them to arrive at the true biologic effect, an alternative solution arises from the insight that we could remove the biologic correlations instead. Removal of biologic factors is relatively easy because they are the property of each subject. The landmark configuration of a patient is defined by the underlying biology; to break the link between landmark position and biology, we could create a landmark configuration that is composed of landmarks from different subjects: each cephalometric point selected from a different, randomly chosen, subject of the sample. Such a procedure is a form of randomization. Randomization (permutation or bootstrap) methods provide solutions to problems that cannot be easily solved with analytic, deterministic algorithms. They rely on producing a large number of virtual individuals or samples, computed by random (or pseudo-random) processes.

In this article, a permutation method is proposed to remove biologic associations and create a virtual sample of the same geometric and variational characteristics as the original sample, retaining only the topographic associations. Any difference between the correlations found in the original sample and the constructed sample can be attributed to biologic factors.