Statistical bias in least-squares refinement

Statistical fluctuations in counting rates etc., as well as defects in the structural model, can introduce bias in the estimation of parameters by least-squares refinements. Of the residuaIs in common use, only unweighted R₂ = Σ(I_o - I_c)² is free from statistical bias. Order-of-magnitude estimates of the bias can be derived, but it seems better to avoid it by adjusting the weights. To the second order, refinement of R₂ is unbiased if the intensity used in calculating the usual weights is not I_o but ⅓ (I_o + 2I_c). There seems to be no simple method of avoiding bias in R₁.