Reliability Factors

EasyDiffraction reports several complementary measures of agreement between observed and calculated diffraction data. They are ratios internally; values shown with a percent sign are multiplied by 100.

For the equations below, \(y_i^{\mathrm{obs}}\) and \(y_i^{\mathrm{calc}}\) are the observed and calculated intensities, \(\sigma_i\) is the standard uncertainty of the observed intensity, and

\[ w_i = \frac{1}{\sigma_i^2} \]

is its inverse-variance weight. The sum runs over the \(N\) data points included in the reported value.

R-factor (Rf)

The unweighted profile R-factor is the absolute difference between observed and calculated intensities, normalized by the total absolute observed intensity:

\[ R_f = \frac{\sum_i \left|y_i^{\mathrm{obs}}-y_i^{\mathrm{calc}}\right|} {\sum_i \left|y_i^{\mathrm{obs}}\right|}. \]

Lower values indicate closer agreement. This factor does not use the measurement uncertainties, so every data point contributes according to the magnitude of its absolute residual.

Squared-residual R-factor (Rf²)

The value labelled Rf² in the fit summary is the unweighted root-squared-residual ratio:

\[ R_{f^2} = \left[ \frac{\sum_i \left(y_i^{\mathrm{obs}}-y_i^{\mathrm{calc}}\right)^2} {\sum_i \left(y_i^{\mathrm{obs}}\right)^2} \right]^{1/2}. \]

Despite the compact Rf² label, this is not the algebraic square of \(R_f\). The label indicates that squared intensities and residuals are used before taking the square root.

Weighted R-factor (wR)

The weighted R-factor is the root ratio of weighted squared residuals:

\[ wR = \left[ \frac{\sum_i w_i \left(y_i^{\mathrm{obs}}-y_i^{\mathrm{calc}}\right)^2} {\sum_i w_i \left(y_i^{\mathrm{obs}}\right)^2} \right]^{1/2}, \qquad w_i = \frac{1}{\sigma_i^2}. \]

Consequently, a point with a smaller standard uncertainty has more influence than a less precise point. EasyDiffraction expects standard uncertainties as input and converts them to inverse-variance weights; it does not use \(\sigma_i\) itself as the weight.

Chi-square and reduced chi-square

The uncertainty-weighted sum of squared residuals is

\[ \chi^2 = \sum_i \left( \frac{y_i^{\mathrm{obs}}-y_i^{\mathrm{calc}}}{\sigma_i} \right)^2 = \sum_i w_i \left(y_i^{\mathrm{obs}}-y_i^{\mathrm{calc}}\right)^2. \]

If \(p\) free parameters were fitted, the number of degrees of freedom is \(\nu=N-p\), and the reported goodness-of-fit is the reduced chi-square:

\[ \chi_\nu^2 = \frac{\chi^2}{\nu}. \]

A value near 1 means that the size of the residuals is consistent with the stated standard uncertainties. A much larger value can indicate a poor model or underestimated uncertainties; a much smaller value can indicate overestimated uncertainties or an over-flexible model.

For a joint fit, EasyDiffraction also multiplies each experiment's squared normalized residuals by its normalized joint-fit weight. Those experiment weights are normalized so that their sum equals the number of experiments.

Expected weighted profile R-factor

For powder fits, the expected weighted profile R-factor is

\[ wR_{\mathrm{expected}} = \left[ \frac{\nu} {\sum_i w_i \left(y_i^{\mathrm{obs}}\right)^2} \right]^{1/2}. \]

It is the weighted profile R-factor expected when \(\chi_\nu^2=1\). Therefore, \(wR / wR_{\mathrm{expected}} = \sqrt{\chi_\nu^2}\) when the same data points and weights are used for both values.

Bragg R-factor (BR)

When observed and calculated structure-factor magnitudes are available, EasyDiffraction can report the Bragg R-factor:

\[ BR = \frac{\sum_h \left|F_h^{\mathrm{obs}}-F_h^{\mathrm{calc}}\right|} {\sum_h F_h^{\mathrm{obs}}}. \]

Here \(h\) indexes reflections and \(F_h\) is a structure-factor magnitude. Lower values indicate closer agreement between observed and calculated reflection amplitudes.

Names and data subsets

The fit summary uses the short labels Rf, Rf², wR, and BR. Saved deterministic fit results also expose IUCr-style names:

Saved result	Definition and scope
R_factor_all	\(R_f\) for all included observations
wR_factor_all	\(wR\) for all included observations
R_factor_gt	\(R_f\) for observations satisfying \(y_i^{\mathrm{obs}}>3\sigma_i\)
wR_factor_gt	\(wR\) for observations satisfying \(y_i^{\mathrm{obs}}>3\sigma_i\)
prof_R_factor	\(R_f\) for all included powder-profile points
prof_wR_factor	\(wR\) for all included powder-profile points
prof_wR_expected	\(wR_{\mathrm{expected}}\) for all included powder-profile points
reduced_chi_square	\(\chi_\nu^2\) for the fitted residual vector

Only finite observations with finite calculated values and positive, finite standard uncertainties are included in saved deterministic statistics. A metric is unavailable when its denominator is zero or when it does not apply to the fitted data.