International
Tables for Crystallography Volume H Powder diffraction Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk © International Union of Crystallography 2018 |
International Tables for Crystallography (2018). Vol. H, ch. 3.8, p. 325
Section 3.8.2.2. Pearson's r coefficient^{a}Department of Chemistry, University of Glasgow, University Avenue, Glasgow, G12 8QQ, UK |
Pearson's r is a parametric linear correlation coefficient widely used in crystallography. It has a similar form to Spearman's test, except that the data values themselves, and not their ranks, are used:where and are the means of intensities taken over the full diffraction pattern. Again, r can lie between −1.0 and +1.0.
Fig. 3.8.1 shows the use of the Pearson and Spearman correlation coefficients (Barr et al., 2004a). In Fig. 3.8.1(a) r = 0.93 and R = 0.68. The high parametric coefficient arises from the perfect match of the two biggest peaks, but the much lower Spearman coefficient acts as a warning that there are unmatched regions in the two patterns. In Fig. 3.8.1(b) the situation is reversed: r = 0.79, whereas R = 0.90, and it can be seen that there is a strong measure of association with the two patterns, although there are some discrepancies in the region 15–35°. In Fig. 3.8.1(c) r = 0.66 and R = 0.22; in this case the Spearman test is again warning of missing match regions. Thus, the use of the two coefficients acts as a valuable balance of their respective properties when processing complete patterns. The Spearman coefficient is also robust in the statistical sense and useful in the case of preferred orientation.
References
Gilmore, C. J., Barr, G. & Paisley, W. (2004). High-throughput powder diffraction. I. A new approach to qualitative and quantitative powder diffraction pattern analysis using full pattern profiles. J. Appl. Cryst. 37, 231–242.Google Scholar