There was strong disagreement among members of the Subcommittee over the question of whether 'observations' used in a refinement should be net integrated intensities `I`, values of |`F`|^{2} or values of |`F`|, or indeed whether it makes a difference. The critical factors in the transformation from peak and background scan intensities through net integrated intensities to |`F`|^{2} concern the application of correction terms associated with absorption, the Lorentz-polarisation factor, thermal diffuse scattering * etc*., whereas the change from |`F`|^{2} to |`F`| concerns the square-root function. The extraction of the square root is a non-linear operation that has the potential of introducing a bias proportional to the variance of |`F`|^{2} (Wilson, 1976b, 1979), with the additional problem of determining what to do for the very weak reflections where statistical fluctuations in the peak and background measurements may cause the net intensity to be negative. The widely used formula for the s.u. of |`F`|, `u`(|`F`|) = ` u`(|`F`|^{2})/2|`F`|, may be appropriate for strong reflections, but must be modified for weak reflections in order to prevent `u`(|`F`|) from becoming infinite at |`F`| = 0. French & Wilson (1978) and Gonschorek (1985) have proposed methods for obtaining |`F`| and `u`(|`F`|) from |`F`|^{2} and ` u`(|`F`|^{2}).

The partial derivative of the calculated quantity ` C`_{j} = |`F`|^{n} = (`A`^{2} + `B`^{2})^{n/2} with respect to the variable `v`_{r} is

(18) . . `C`_{j}/`v`_{r} = `n`|`F`|^{n - 2}{`A`(`A`/`v`_{r}) + ` B`(`B`/`v`_{r})}.

If the calculated structure factor is zero, i.e. `A` = `B` = 0, `C`_{j}/`v`_{r} = 0 for ` n` > 2, and undefined for `n` < 2. The contributions of the `j`th observation to the normal-equations matrix and vector are ` w`_{j}(`C`_{j}/`v`_{r})(`C`_{j}/`v`_{s}) and ` w`_{j}(`O`_{j} - `C`_{j})(`C`_{j}/`v`_{r}), respectively. If the weight is chosen according to ` w`(|`F`_{j}|) = 4|`F`_{j}|^{2}`w`(|` F`_{j}|^{2}), then the contribution to the matrix is exactly the same, and the contribution to the vector nearly the same, for refinements on |`F`| and |`F`|^{2}. In many cases, omission of weak reflections has a negligible effect on the results (see recommendation 6), and the two kinds of refinement are then nearly identical. For this reason, the |`F`| versus |`F`|^{2} controversy is often considered to be irrelevant.

**The arguments in favor of refinement on | F|** are based on a mathematical analysis by Prince & Nicholson (1985). They observe that different reflections have different leverage which is a quantity that measures the influence of an individual reflection on the fit. It is proportional to the contributions to the matrix and vector described above. Because |

**The arguments in favor of refinement on | F|^{2}** are based on the bias introduced by extracting the square root (Wilson 1976b, 1979), and on the undesirable discontinuity of the partial derivatives of |

**The arguments in favor of refinement on I** are an extension of those for refinement on |

© 1989, 1995 International Union of Crystallography

Updated 23rd Sept. 1996