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

Dear Editor

We read with interest the publication by Jones and Chung, who introduced a concept that reference change values (RCV) decrease when more than two serial results are considered. ¹ Assumptions in this model require knowledge of the characteristics of the serial results before calculation of the RCV, e.g. the results should follow a normal (Gaussian) distribution in steady-state both before and after a possible clinical event. Given n₁ results available before a potential clinical event, the mean of the results is an estimate of the individual’s initial homeostatic set point. After the event, the mean of the n₂ results is an estimate of a (possible) new set point. Jones and Chung ¹ applied a formula for calculating RCV as RCV = Z · CV_T · (1/n₁ + 1/n₂)^½, where CV_T is coefficient of variation of the combined analytical imprecision (CV_A) and within-subject biological variation (CV_I), CV_T = (CV_A² + CV_I ² )^½ and Z is the number of standard deviations (SD) appropriate to the probability desired for detecting differences. ² When n₁ = 1 and n₂ = 1, the formula is RCV = Z · CV_T · (2)^½. This is the commonly used calculation for RCV using two consecutive results. However, we have recently shown that this traditional RCV calculation is incorrect. ³ The original and correct RCV calculation was based on a difference of two results (X2−X1) with use of total standard deviation (SD_T). ⁴ Erroneously, the SD_T was later substituted with CV_T in this RCV formula. ⁵ In this incorrect process, the difference between two results (X2−X1) is transformed from a linear scale to a logarithmic scale when the difference is related to the first result (i.e. [X2−X1]/X1 = X2/X1 − 1, where X2/X1 has a logarithmic scale). In this way, calculations from the formula RCV = Z · CV_T · (2)^½ are not appropriate for this new logarithmic scale based on fractional differences ([X2−X1]/X1). Consequently, when there are increased concentrations over time, this incorrect RCV calculation will produce more false-positive results than theoretically expected and fewer for decreased concentrations. ³ A correct RCV formula with use of CV_T has been stated for two results,^3,6 but a correct RCV formula using more than two results does not exist. We have shown, using simulations, that the RCV concept of Jones and Chung ¹ is only valid if SD_T is used in their formula instead of CV_T (data not shown). In practice, within subject-biological variation data for most measurands are given as CV_I, and, thus, correct RCV calculations using the suggested concept ¹ are not possible.

For practical application of the RCV concept of Jones and Chung, ¹ we have developed a formula for correct RCV for more than two serial results using CV_T. This mathematical RCV formula is generated in a similar way to that described in our previous publication ³ and validated using simulations based on 1 million random numerical results (data not shown). For an increase in results, RCV_up = {−(CV_T) ² /n₁−[(CV_T)⁴/(n₁)²−(CV_T) ²  · ((CV_T) ² /n₁−1/Z²) · (1/n₁ + 1/n₂)]^½}/((CV_T) ² /n₁−1/Z²). For a decrease in results, RCV_down = {−(CV_T) ² /n₁ + [(CV_T)⁴/(n₁)²−(CV_T) ²  · ((CV_T) ² /n₁−1/Z²) · (1/n₁ + 1/n₂)]^½}/((CV_T) ² /n₁−1/Z²). RCV_up and RCV_down define the limits (increase and decrease) for, e.g. 5% false-positive results after the event. In the simulations, we have assumed that the homeostatic set point is identical before and after a clinical event, corresponding to a steady-state situation. However, if, e.g. the mean of results after the event is greater than RCV_up, then the steady-state condition has ended and a (possible) new higher set point is indicated for the patient.

In Table 1, as an example, we have calculated correct RCV_up using CV_T for n₁ and n₂ = 1,2,3,4,5 and when Z = 1.65 and CV_T = 20.0% for increased concentrations (i.e. RCV_up is the upper limit for 5% false-positive results). Analogous RCV results are also listed in Table 1 (in parentheses) using the incorrect formula for RCV (i.e. RCV = Z · CV_T · (1/n₁ + 1/n₂)^½). From Table 1, it is clear that RCV differ significantly when the two different RCV formulae are used. The RCV are symmetrical with use of the incorrect RCV formula (i.e. RCV are the same when n₁ and n₂ are interchanged). It is important to note that RCV results derived using the correct formula are much greater for n₁ = 1 than n₁ = 5 for all values of n₂. The reason is that, when n₁ = 1, there is only one estimate of the initial set point (Y₁ = X1). When n₁ = 5, the initial set point is based on five results and has smaller variation of the mean (Y₅ = [X1 + X2 + X3 + X4 + X5]/5). Related to the new set point (V), RCV is calculated as RCV = (V−Y)/Y. This formula demonstrates the impact from the denominator in fractional differences on the characteristics of the RCV. When Y is small and uncertain (n₁ = 1), RCV will be high and, with further estimates of Y (n₁ = 5), RCV will decrease as shown in Table 1.

Table 1.

RCV as % when CV_T = 20.0% and Z = 1.65 for increased concentrations (i.e. RCV_up define upper limits for 5% false-positive results) using a number of results (n₁) estimating the initial and number results (n₂) estimating the new set point calculated with use of the correct RCV_up formula and the incorrect RCV formula (in parentheses).

	RCVup%
n2\n1	1	2	3	4	5
1	63.1 (46.7)	47.7 (40.4)	42.8 (38.1)	40.3 (36.9)	38.8 (36.1)
2	56.7 (40.4)	40.2 (33.3)	34.7 (30.1)	31.9 (28.6)	30.2 (27.6)
3	54.4 (38.1)	37.3 (30.1)	31.5 (26.9)	28.5 (25.2)	26.7 (24.1)
4	53.2 (36.9)	35.7 (28.6)	29.7 (25.2)	26.6 (23.3)	24.7 (22.1)
5	52.4 (36.1)	34.7 (27.6)	28.6 (24.1)	25.4 (22.1)	23.4 (20.9)

In conclusion, we appreciate some of the ideas advocated in the RCV concept of Jones and Chung, ¹ particularly that of using more results for the estimation of an initial set point. However, we do consider that correct calculation of this RCV concept based on a sequence of serial results is a difficult task.