Valid analytical performance specifications for combined analytical bias and imprecision for the use of common reference intervals

Abstract

Background

Many clinical decisions are based on comparison of patient results with reference intervals. Therefore, an estimation of the analytical performance specifications for the quality that would be required to allow sharing common reference intervals is needed. The International Federation of Clinical Chemistry (IFCC) recommended a minimum of 120 reference individuals to establish reference intervals. This number implies a certain level of quality, which could then be used for defining analytical performance specifications as the maximum combination of analytical bias and imprecision required for sharing common reference intervals, the aim of this investigation.

Methods

Two methods were investigated for defining the maximum combination of analytical bias and imprecision that would give the same quality of common reference intervals as the IFCC recommendation. Method 1 is based on a formula for the combination of analytical bias and imprecision and Method 2 is based on the Microsoft Excel formula NORMINV including the fractional probability of reference individuals outside each limit and the Gaussian variables of mean and standard deviation. The combinations of normalized bias and imprecision are illustrated for both methods. The formulae are identical for Gaussian and log-Gaussian distributions.

Results

Method 2 gives the correct results with a constant percentage of 4.4% for all combinations of bias and imprecision.

Conclusion

The Microsoft Excel formula NORMINV is useful for the estimation of analytical performance specifications for both Gaussian and log-Gaussian distributions of reference intervals.

Introduction

Analytical performance specifications (APS) for the maximum allowable analytical bias and imprecision, and combinations of these, that were appropriate for use of common reference intervals across laboratories were proposed by Gowans et al. ¹ This model was based on the recommendation for the establishment of population-based reference intervals of the International Federation of Clinical Chemistry (IFCC). ² The aim of the proposal was to define the APS for the analytical quality that would be required to allow sharing common reference intervals among homogeneous populations. The concept was that a single high-quality laboratory should establish the reference interval from a large population of apparently healthy reference individuals, where the term ‘reference interval’ is considered as the combined within- and between-subject biological variation, s_B, excluding the analytical imprecision.

This interval could then be used by other laboratories with similar populations, if they could attain or surpass these APS. This would ensure that all laboratories could use a reference interval which was within the IFCC-recommended 90% confidence interval (CI) for the upper and lower reference limits, which is based on 120 individuals.

The derivation of the 90% CI for both the upper and lower reference limits according to the IFCC recommendation was based on the formula, CI = ± (2.81·s_B)/N^½, where s_B is the standard deviation of a reference interval with a Gaussian distribution and N is the number of reference individuals (minimum 120). ² This is equal to 0.25·s_B, which was determined as the maximum allowable numerical bias when imprecision was negligible. ¹ Thus, the limits for the 95% reference interval are ±1.96·s_B ± 0.25·s_B (± 1.71·s_B to ±2.21·s_B), yielding a maximum of 4.4% of the reference individuals outside both of the reference limits. The basic concept was to determine the combination of analytical bias and imprecision that would keep the maximum percentage of reference individuals outside each reference limit to no more than that obtained when the IFCC recommendation was applied. The purpose of this investigation was to find a method for the combination of analytical bias and imprecision according to the IFCC recommendation for parametric reference intervals based on the minimum useful number of reference individuals that would keep the maximum percentage of reference individuals outside each reference limit to the theoretically correct 4.4% for both reference distributions: ‘Original scale’ (Gaussian distribution) and ‘Logarithmic scale’ (ln-Gaussian distribution).

Methods

Two methods were examined for the combination of analytical bias and imprecision that would keep the maximum percentage of reference individuals outside each reference limit to 4.4%. All calculations were performed using Microsoft Excel 2013.

Method 1

A formula suggested for the maximum combination of analytical bias and imprecision (s_A), which was assumed to give a constant maximum of false-positive of 4.4% was presented in 2001 ³

$$\text{1}.\text{96}\cdot {\left({\text{s}}_{\text{B}}{\hspace{0.17em}}^2+{\text{s}}_{\text{A}}{\hspace{0.17em}}^2\right)}^{1/2}+\left|\text{bias}\right|=\left(\text{1}.\text{96}+0.\text{25}\right)\cdot {\text{s}}_{\text{B}}$$

|bias| is the numerical estimate of bias, in order to keep the symmetry of the Gaussian distribution.

Method 2

A formula for the probability, P, that the percentage of combined |bias| and s_A that would keep the maximum percentage of reference individuals outside each reference limit to 0.044 (4.4%). This function can be calculated using Microsoft Excel 2013 with the function for Gaussian distributions with normalized values

$$\text{NORMINV}\left(probability;mean;standard\hspace{0.5em}deviation\right)=\text{NORMINV}\left(0.044;0.0;{[1+{\{s_A/s_B\}}^{\text{2}}]}^{1/2}\right)$$

where the standard deviation is the combined and normalized standard deviation equal to [1 + {s_A/s_B} ² ]^½.

Results

The functions of the two methods for combined bias and imprecision are shown in Figure 1 with normalized imprecision as function of normalized bias.

Figure 1.

Analytical performance specifications for reference intervals with constant 4.4% of apparently healthy reference individuals outside both reference limits for each combination of bias and imprecision (method 2, blue graph). Relationships between analytical bias (abscissa) and imprecision (ordinate) are shown normalized, that is, both are divided by the biological standard deviation of the Gaussian reference interval, s_B (excluding the analytical imprecision). For Gaussian distributions, bias and imprecision are measured in concentration units and for ln-Gaussian distributions in the corresponding natural logarithmic units. The black lines illustrates the example with serum magnesium treated as ln-Gaussian with σ_B = 0.070, and the analytical imprecision, CV_A = 0.027 (2.7%), then σ_A = {ln[(0.027)² + 1]}^½ = 0.027 and σ_A/σ_B = 0.027/0.0700 = 0.386 (ordinate) and the corresponding bias, bias/σ_B is ±0.127 (abscissa) equal to 0.127·0.0700 = 0.00889 which transforms to fractional bias by exp(0.00889)−1 = 0.00893 or 0.89%. The dotted purple graph (method 1) represents the function according to the formula. ³

Method 1

The dotted purple graph in Figure 1 represents the function according to the formula. ³ The maximum s_A/s_B is 0.52 for bias equal to zero and with the percentage outside the reference limit 4.1%.

Method 2

The solid blue graph in Figure 1 represents the function according to the Gaussian distributions: NORMINV(probability;mean;standard deviation). The maximum s_A/s_B is 0.56 for bias equal to zero and with the percentage outside the reference limit 4.4%, and therefore this is the correct method.

Discussion

The shapes of the curves are rather close, but Method 2, with the formula NORMINV(probability;mean;standard deviation), is correct with the result of 4.4%, and thus the blue graph in Figure 1 is correct, fulfilling the original purpose of keeping the maximum percentage of healthy individuals outside the reference limits constant. ¹

However, importantly, most population-based reference intervals are skew and calculations should more correctly be done after transformation to logarithms, often described as natural logarithmic (ln-Gaussian) distributions. This means that the logarithmic data can also be described as in Figure 1, and the logarithmic standard deviation (σ) can be related to the coefficient of variation by the formula CV = (exp{σ ² }−1)^½ from Fokkema et al. ⁴ and the transformed formula σ = {ln[(CV) ²  + 1]}^½ from Lund et al. ⁵ This is demonstrated in the following example.

Example

One example of this new valid approach can be constructed from data generated in the Nordic Reference Interval Project 2000 ⁶ for serum magnesium with reference interval 0.71–0.94 mmol/L and ln-Gaussian interval –0.342 to –0.061. This project on the derivation of common reference intervals was based on 3000 apparently healthy individuals; however, the analytical imprecision was not documented, so a CV_A of 0.015 (1.5%) has been assumed for this example; in consequence, the standard deviation, including imprecision, is 0.281/3.92 = 0.0716, and with subtraction of imprecision is 0.0700, which gives the maximum |bias| = 0.0700· 0.25 = 0.0175 and exp(0.0175)−1 ∼ 1.77%, and maximum imprecision σ_A = 0.0700· 0.56 = 0.0392 and CV_A = (exp(0.0392²)−1)^½ ∼ 3.92% (the difference is negligible for small values, i.e. when CV_A < 10%). If the CV_A of a laboratory is 0.027 (2.7%), then σ_A = {ln[(0.027)² + 1]}^½ = 0.027 and σ_A/σ_B = 0.027/0.0700 = 0.386. Then, the allowable ln-bias is ± 0.127 calculated or read from Figure 1 and, when multiplied with σ_B, equal to 0.127· 0.0700 = 0.00889 which transforms to fractional bias by exp(0.00889)−1 = 0.00893 or 0.89%.