Calculation of reference change values using more than two results is a difficult task: a response

Research into the best estimation of the reference change value (RCV) remains an ongoing issue. We thank Lund et al. ¹ for their critique of our proposal for the calculation of the RCV given variable numbers of samples collected before and after a suspected change in the patient ² with the recognition that with more data the RCV may be smaller rather than larger as was originally proposed. ³ We also agree that the previously described effect of analyte concentration on RCV ⁴ should be included in the equation for multiple samples. That this effect should be included was identified in the text of our previous letter but was not included in the supplied formula.

We would, however, like to suggest a variation to their proposed formulae and table provided by Lund et al. in their critique of our paper. Combining our concepts of multiple samples ² and effect of analyte concentration, ⁴ we obtain the following formulae.

$${\text{RCV}}_{\text{up}}=\{-{\text{CV}}_{\text{T}}^2/{\text{n}}_2^2-[{\text{CV}}_{\text{T}}^4/{\text{n}}_2^2-{\text{CV}}_{\text{T}}^2\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\times ({\text{CV}}_{\text{T}}^2/{\text{n}}_2-1/{\text{Z}}^2)(1/{\text{n}}_1+1/{\text{n}}_2){]}^{0.5}\}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}/({\text{CV}}_{\text{T}}^2/{\text{n}}_2^2-1/{\text{Z}}^2)$$

$${\text{RCV}}_{\text{down}}=\{-{\text{CV}}_{\text{T}}^2/{\text{n}}_2^2+[{\text{CV}}_{\text{T}}^4/{\text{n}}_2^2-{\text{CV}}_{\text{T}}^2\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\times ({\text{CV}}_{\text{T}}^2/{\text{n}}_2-1/{\text{Z}}^2)(1/{\text{n}}_1+1/{\text{n}}_2){]}^{0.5}\}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}/({\text{CV}}_{\text{T}}^2/{\text{n}}_2^2-1/{\text{Z}}^2)$$

In these equations, Z is the probability factor, CV_T is the total result coefficient of variation (including analytical and within-subject biological variation), n₁ is the number of samples taken prior to the change and n₂ is the number taken after the possible change.

These equations differ from those of Lund et al. by interchanging n₂ and n₁. The formulae presented here predict a greater diminution of the RCV in the positive direction if extra samples are taken after the suspected change compared with more samples being collected prior to the suspected change. In their Table 1, with a CV_T of 20%, n₁ = 5, and n₂ = 1 gives an RCV_up of 38.8%. This compares to a comparatively greater RCV_up of 52% if n₁ = 1 and n₂ = 5 according to their table. With a fixed CV_T, the standard deviation of results taken after a possible increase in values (SD_T2) is greater than the SD before the increase (SD_T1) and thus will be reduced to a greater extent by multiple collections. In contrast, the equations we have provided above reverse these values and, we believe, correctly describe this expected effect.

When calculating the RCV in a downwards direction the inverse effect is expected, i.e. that more samples taken prior to the suspected change will have a greater reduction on the RCV than more samples taken after the change.

Although the simplified equation for RCV with multiple samples provided previously ² did not include the effect of analyte concentration, the calculator for probability of a true change referenced in the paper ⁵ did demonstrate the different effects of increasing or decreasing concentrations.

Indeed, this can be seen as a preferred approach to identifying true changes in a patient. The standard approach involves establishing, in advance, the required difference (RCV) for a following sample (or samples) from a previous sample (or samples) to reach a pre-determined level of significance. The outcome is dichotomous (change is significant/not significant) at the pre-determined level of significance. By contrast, the approach envisaged by the calculator is make an assessment after measurement of the following sample(s) to determine the actual probability of the change being due to random effects.

Of course this modelling is based on a simplified scenario where there is a single change in homeostatic set point at a known point in time relative to the sample collections, whereas in reality these assumptions may not be true or known to be true. Additionally, the effect of the reporting interval size also should be included. ⁶

Again we thank the authors for their critique and agree that developing the appropriate calculations for RCV is indeed a difficult task.