International
Tables for
Crystallography
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 3.8, p. 325

Section 3.8.2.2. Pearson's r coefficient

C. J. Gilmore,a G. Barra and W. Donga*

aDepartment of Chemistry, University of Glasgow, University Avenue, Glasgow, G12 8QQ, UK
Correspondence e-mail:  chris@chem.gla.ac.uk

3.8.2.2. Pearson's r coefficient

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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:[{r_{ij}} = {{\sum\limits_{k = 1}^n {\left({{x_k} - \overline x } \right)} \left({{y_k} - \overline y } \right)} \over {{{\left [{\sum\limits_{k = 1}^n {{{\left({{x_k} - \,\overline x } \right)}^2}} \sum\limits_{k = 1}^n {{{\left({{y_k} - \,\overline y } \right)}^2}} } \right]}^{1/2}}}}, \eqno(3.8.2)]where [\overline x] and [\overline y] 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[link] shows the use of the Pearson and Spearman correlation coefficients (Barr et al., 2004a[link]). In Fig. 3.8.1[link](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[link](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[link](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.

[Figure 3.8.1]

Figure 3.8.1 | top | pdf |

The use of the Pearson (r) and Spearman (R) correlation coefficients to quantitatively match powder patterns: (a) r = 0.93, R = 0.68; (b) r = 0.79, R = 0.90; (c) r = 0.66, R = 0.22.

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








































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