Section 3.2.3.2. Concluding remarks

Proper tests of statistical significance of this or that aspect of a least-squares plane can be made if the plane has been based on a proper weight matrix as discussed in Section 3.2.3; if it can be agreed that the random errors of observation are normally distributed; and if an agreeable test (null) hypothesis can be formulated. For example, one may ask for the probability that a degree of fit of the observed positions to the least-squares plane at least as poor as the fit that was found might occur if the atoms in truth lie precisely on a plane. The test answers this question: a table of probabilities displayed as a function of and provides the answer. Here is just our minimized and is the number of degrees of freedom for the problem of the plane (erroneously cited in at least one widely used crystallographic system of programs as ). There will not usually be any reason to believe that the atoms are exactly coplanar in any case; nevertheless, this test may well give a satisfying indication of whether or not the atoms are, in the investigator's judgment, essentially coplanar. It must be emphasized that as calculated in Section 3.2.3 will include proper allowance for uncertainties in the d and orientation of the plane with greater reliability than the estimates of Section 3.2.2, which are based on nominally arbitrary weights. Both, however, will allow for the large variations in d and tilt that can arise in either case if n is small. Some of the earlier, less complete discussions of this problem have been mentioned in Section 3.2.2.

Among the problems not considered here are ones of fitting more than one plane to a set of observed positions, e.g. of two planes fitted to three sets of atoms associated, respectively, with the first plane, the second plane, and both planes, and of the angle between the two planes. For the atoms common to both planes there will be a fundamental point of difference between existing programs (in which, in effect, the positions of the atoms in common are inconsistently adjusted to one position on the first plane and, in general, a different position on the second) and what we would advocate as the proper procedure of requiring the adjusted positions of such atoms to lie on the line of intersection of the two planes. As to the dihedral angle there is a difficulty, noted by WMC (1973, p. 2705), that the usual formulation of in terms of the cosine of the dihedral angle reduces to 0/0 at . However, this variance is obviously well defined if the plane normals and their covariances are well defined. The essential difficulty lies with the ambiguity in the direction of the line of intersection of the planes in the limit of zero dihedral angle. For the torsion angle about a line defined by two atoms, there should be no such difficulty. It seems likely that for the two-plane problem proposed above, the issue that decides whether the dihedral angle will behave like the standard dihedral angle or, instead, like the torsion angle, will be found to be whether or not two or more atoms are common to both planes.

All that we have tried to bring out about the covariances of derived quantities involving the plane requires that the covariances of the experimental atom positions (reduced in our formulations to Cartesian coordinates) be included. However, such covariances of derived quantities are often not available in practice, and are usually left unused even if they are. The need to use the covariances, not just the variances, has been obvious from the beginning. It has been emphasized in another context by Schomaker & Marsh (1983) and much more strongly and generally by Waser (1973), whose pleading seems to have been generally ignored, by now, for about thirty years.