Reliability factors¶

Reliability factors summarise agreement under a defined observation model. They supplement the pattern, residual, physical model and parameter uncertainties.

Included observations and weights¶

For powder factors, sums run over included points outside excluded intervals with positive standard uncertainty and available finite calculated intensity. Let \(w_i=1/\sigma_i^2\), \(b_i\) be background, \(N\) the included point count and \(P\) the independent free-parameter count.

Powder profile factors¶

\[R_p=\frac{\sum_i|y_{o,i}-y_{c,i}|}{\sum_i|y_{o,i}|},\qquad R_{wp}=\sqrt{\frac{\sum_iw_i(y_{o,i}-y_{c,i})^2}{\sum_iw_i y_{o,i}^2}},\]

\[R_{\exp}=\sqrt{\frac{N-P}{\sum_iw_i y_{o,i}^2}},\qquad \chi^2_{\rm red}=\frac{\sum_iw_i(y_{o,i}-y_{c,i})^2}{N-P}.\]

The engine's factor values are fractions; a displayed percentage is 100 times that fraction. \(R_{wp}/R_{\exp}\) is the square root of reduced χ² under matching sums and degrees of freedom. The name “goodness of fit” is used for both conventions in the literature, so state which is meant.

A fit needs positive degrees of freedom to interpret reduced χ². Nonpositive degrees of freedom or a zero denominator are degenerate cases; the report's numerical guard result is not a statistically meaningful factor.

Background-subtracted factors¶

The conventional background-subtracted factors retain the same residual numerator but replace the normalising observations by net intensity:

\[cR_p=\frac{\sum_i|y_{o,i}-y_{c,i}|}{\sum_i|y_{o,i}-b_i|},\qquad cR_{wp}=\sqrt{\frac{\sum_iw_i(y_{o,i}-y_{c,i})^2}{\sum_iw_i(y_{o,i}-b_i)^2}},\]

\[cR_{\exp}=\sqrt{\frac{N-P}{\sum_iw_i(y_{o,i}-b_i)^2}}.\]

Removing background from the denominator can change the apparent factor substantially. Do not compare a conventional profile Rwp directly with a background-subtracted value while calling them the same definition.

Bragg and F factors¶

Powder phase factors use reflection intensities apportioned from the observed profile according to the calculated contributions:

\[I_{o,h}=\sum_i I_{c,h}(i)\frac{y_{o,i}-b_i}{y_{c,i}-b_i},\qquad R_B=\frac{\sum_h|I_{o,h}-I_{c,h}|}{\sum_h|I_{o,h}|}.\]

\[|F_{c,h}|=\sqrt{m_h|F_h|^2},\qquad |F_{o,h}|=|F_{c,h}|\sqrt{\max(I_{o,h},0)/I_{c,h}},\qquad R_F=\frac{\sum_h\big||F_{o,h}|-|F_{c,h}|\big|}{\sum_h|F_{o,h}|}.\]

The reflection partition is model-dependent, particularly where peaks overlap. These phase factors are not independent directly measured integrated intensities for every powder reflection.

Single-crystal reflection factors¶

The reflection-statistics view also reports amplitude-based and intensity-weighted factors over a declared set \(\mathcal H\):

\[R(F)=\frac{\sum_{h\in\mathcal H}\left|\sqrt{\max(F_{o,h}^2,0)}-\sqrt{\max(F_{c,h}^2,0)}\right|}{\sum_{h\in\mathcal H}\sqrt{\max(F_{o,h}^2,0)}},\]

\[wR(F^2)=\sqrt{\frac{\sum_{h\in\mathcal H}w_h(F_{o,h}^2-F_{c,h}^2)^2}{\sum_{h\in\mathcal H}w_h(F_{o,h}^2)^2}},\qquad w_h=1/u(F_{o,h}^2)^2.\]

r_factor_all and wr_factor_all use all reflections; _gt uses those passing the experiment's observed-reflection threshold; wr_factor_ref uses those selected for fitting. An absent observed threshold means \(I>2u(I)\). An amplitude threshold \(F>n u(F)\) is implemented as \(F_o^2>(n/2)u(F_o^2)\); an intensity threshold uses \(F_o^2>n u(F_o^2)\). The project's optional fit threshold is separate and uses the intensity rule. A reporting threshold does not itself discard weak data from fitting.

Single-crystal reflection exclusions¶

Planned. Single-crystal reflection exclusions are planned. The reporting and fit-threshold rules above describe the current behaviour. See Background and excluded observations for the status alongside powder exclusions.

Per-experiment and pooled summaries¶

For a single-crystal reflection list, the same absolute and weighted formulas operate on observed and calculated \(F^2\) values, giving \(R(F^2)\) and \(wR(F^2)\). There is no powder background partition; the background-subtracted denominators equal the un-subtracted ones. Do not label this intensity-based factor as the amplitude-based \(R(F)\).

Per-experiment factors use that experiment's included observations. Pooled factors combine the underlying sums across experiments; they are not an arithmetic average of each experiment's R factor. Keep the objective weighting and factor definition explicit when interpreting joint fits.

Interpretation¶

There is no universal threshold that proves structural correctness. Changing the data range, uncertainty estimates, background, excluded observations or free set changes the statistic. See assessing a fit.

References¶

McCusker, L. B., Von Dreele, R. B., Cox, D. E., Louër, D. & Scardi, P. (1999). Rietveld refinement guidelines. Journal of Applied Crystallography, 32, 36–50.
Toby, B. H. (2006). R factors in Rietveld analysis: How good is good enough? Powder Diffraction, 21, 67–70.