The STARANISO Server
Anisotropy of the Diffraction Limit
The problem here is that the PDB does not compute the correct value of the completeness for anisotropic data, for the simple reason that it does not currently capture any of the information required to determine the anisotropic cut-off.  This is something that we are currently discussing with them.
The correct anisotropic completeness values are in the table 'Merging statistics table for observed data extracted from the final MRFANA log file' under 'Compl. Ellip.' (i.e. the ellipsoidal completeness). STARANISO defines completeness in the conventional way, i.e. the fraction of reflections inside some data-dependent cut-off surface that were actually measured.  The cut-off surface is defined such that it encloses the set of reflections for which some metric of statistical significance (such as the mean I/σ(I) or CC(1/2)) exceeds a significance threshold set by the user.
The main difference in STARANISO lies in the method of determining the cut-off surface: for data assumed to be isotropic it is done in the standard way by computing the metric in spherical bins so that the cut-off surface is a sphere.  For data assumed to be anisotropic (i.e. the default) it is done by computing the 'moving-average' of the metric within a sphere of predefined radius in reciprocal space centred on each reflection in turn.  For anisotropic data CC(1/2) is not suitable as a metric so only the mean I/σ(I) is used.  Then an ellipsoid is fitted by least squares to the resulting cut-off surface points and the ellipsoid becomes the cut-off surface only for the purpose of estimating the completeness.  Note that there's no reason for the true cut-off surface to be an ellipsoid: an ellipsoid represents only one example of anisotropy (the only constraint on the shape of the cut-off surface is that it is locally smooth and it has at least the point symmetry of the Laue class).
In the absence of the definition of the cut-off ellipsoid (i.e. its semi-axis lengths and directions), the PDB simply assume that anisotropic data are isotropic and use a spherical surface of radius equal to the diffraction limit to compute the completeness.  This means that reflections with statistically non-significant values of the mean I/σ(I) that lie outside the ellipsoid but still inside the limiting sphere are included in the count of statistically-significant reflections, which will underestimate the completeness.
Whether or not the additional anisotropic data that would be rejected by a spherical cut-off will affect map interpretation will obviously depend on whether those data enhance the resolution of the map in a useful direction (e.g. it may help to resolve atoms whose relative positions lie approximately in that direction, but it is unlikely to help with resolving atom pairs in the low-resolution directions).  The magnitude of this effect will depend on the ellipsoidal completeness, not the spherical completeness value currently computed by the PDB.  In any case including that data surely cannot do any harm!
You should quote the fitted ellipsoid dimensions as given under 'Diffraction limits & eigenvectors of ellipsoid fitted to diffraction cut-off surface'.  It's important to understand that the ellipsoid is not the same as the cut-off surface: it's only an approximation to it.
The 'lowest limit' refers to the reflection with the lowest d* that was cut by STARANISO, i.e. how deeply STARANISO's cut-off surface cut into the input data.
The 'worst' and 'best' limits refer to the reflections which lie on the cut-off surface and which have the lowest and highest d* after STARANISO's cut-off, though not necessarily cut by STARANISO since the cut-off may already have been applied to the input data.
The 'worst' and 'best' directions of the cut-off surface are not mutually perpendicular because there's no reason why they should be!  The data are not cut off by the ellipsoid; rather the ellipsoid is the best fit to the empirically-determined cut-off surface, which therefore may in places go inside or outside the ellipsoid. So the actual cut-offs are not necessarly exactly at the axes of the ellipsoid: they may vary in direction depending on how good is the fit (an exception is that for consistency we apply a final spherical cut-off at the longest axis of the ellipsoid).
The anisotropy of the intensity values is well-described by an ellipsoid (i.e. the 'quadratic form' of the anisotropic B-factor expression); however the anisotropy of <I/σ(I)> is not so simply described because σ(I) depends strongly on the redundancies and these may be completely arbitrary since they are determined by the user-selected collection strategy.  For example the cut-off surface could in principle (though admittedly very unlikely in practice!) be shoe-box shaped, in which case the 'worst' direction would be along the shortest edge of the shoe box and the 'best' direction would be along the body diagonal (i.e. from the centre to a corner).  These directions are clearly not perpendicular.
The only constraints on the shape of the cut-off surface are that it is locally smooth and it has at least the symmetry of the Laue group, i.e. it could be a shoe box with sides of different lengths only in triclinic, monoclinic or orthorhombic or space groups.  For example in cubic space groups it would have to have at least the symmetry of a cube (but again note that the worst and best directions in a cube are not perpendicular).
The problem is that it's impossible to describe an arbitrary surface with only a few numbers! - though of course it's easy to visualise it as a 3-D plot using the WebGL tool.  So we use the approximately-fitted ellipsoid to provide a rough definition of the shape of the cut-off surface using the minimum number of numbers (i.e. not more than 6).
I would say that the justification is that the distribution of statistical significance, however it is measured, is clearly systematic, i.e. there are obviously regions of reciprocal space where the significance of the data is systematically higher than in other regions.  The problem is to delineate the significant region while making minimal assumptions about its shape (one need assume only that the surface is locally smooth and that its symmetry is the same as that of the Laue class, which are very mild assumptions and rather obvious).  Removal of the data in the non-significant region is justified because they would only contribute to the noise in the refinement and map calculation, and not contribute any useful signal.
While I would agree that while it would be hard to demonstrate any benefit of an anisotropic cut-off in cases of mild anisotropy, it would be even harder to demonstrate that any harm comes of it! (and where would one draw the line between mild and strong anisotropy?).
Deciding what constitutes a significant anisotropy effect is harder than it might seem because the effects do not depend only on the anisotropy ratio or the variation in B factor with direction. They depend also strongly on the resolution. Thus quite small anisotropic ratios and B factor variations can have a big effect at high resolution through the exponent term in the Debye-Waller factor.
Note that the cut-off surface in the cubic case may also be anisotropic, because we are not assuming that the surface is an ellipsoid, only that it has cubic symmetry.  If we assumed that the surface is really an ellipsoid it would indeed be constrained by cubic symmetry to be a sphere.  A cube has different properties when measured along its edges compared with its diagonals, and indeed along any arbitrary direction, so is anisotropic by definition.  The sphere is the only shape which is isotropic but a crystal cannot be built from spheres, so all crystals, including cubic ones, can be anisotropic.
STARANISO does indeed perform three different corrections: the cut-off described above, the anisotropically-corrected prior intensity in the French-Wilson treatment, and the anisotropic correction of the resulting Fs.
We thought long and hard about this and decided that it doesn't make sense to average data where the contributions to the mean intensity from the structure-dependent terms (i.e. the contributions from the different levels of structure) differ.  Therefore for anisotropic data we continue to average the statistics in spherical shells, but adjust the widths of the shells to contain equal numbers of reflections (otherwise the statistics for the outer shells containing small numbers of reflections become meaningless).
Tickle, I.J., Flensburg, C., Keller, P., Paciorek, W., Sharff, A.,
Vonrhein, C., Bricogne, G. (2018). STARANISO (http://staraniso.globalphasing.org/cgi-bin/staraniso.cgi).
Cambridge, United Kingdom: Global Phasing Ltd.