Uncertainty calculation for calibrators of the IFCC HbA1c standardization network

Abstract

Within the last decade, the IFCC HbA1c standardization network has established the metrologically highest reference measurement procedure for HbA1c testing. Based on this procedure, reference calibrators are produced which in turn provide the starting point for the standardization of the manufactures routine HbA1c assays. According to the IVD directive, the uncertainty of the reference calibrators must be calculated and reported together with their assigned values. Within this article, we elaborate the uncertainty calculation according to GUM (guide to the expression of uncertainty in measurement) in detail. Finally, the results are validated by a simulation study.

Appendices

Appendix A Variance components

The calculation of the uncertainty of the mean of a set of measurements, gathered in a specific design, requires a careful consideration of the respective design structure. The simplest case of such a design is the random-effects model, which can be written as

$$Y_{ij} = \mu + a_i + \varepsilon _{ij} ,\quad i = i, \ldots ,I,\quad j = 1, \ldots ,J,$$

with Y _ij denoted as the measured result obtained e.g., in laboratory i, and repetition j, μ denotes the overall mean, a _i the laboratory-specific effect with mean zero and between-laboratory variance $\sigma _a^2$ and $\varepsilon _{ij}$ the repetition effect, with mean zero and within-laboratory variance $\sigma _\varepsilon ^2$. As long as no statistical tests are performed, no distribution assumptions need to be made. Note that the variance of an individual measurement is split up into the two variance sources, i.e., ${\rm Var}(Y_{ij}) = \sigma _a^2 + \sigma _\varepsilon ^2$.

In our particular case, we are interested in the calculation of the variance of the mean $\bar Y = \frac{1}{{I \cdot J}}\sum\limits_{i,j} {Y_{ij}}$.

However measurements from the same laboratory are no longer independent, as ${\rm Cor}(Y_{ij} ,Y_{ik}) = \sigma _a^2$, therefore the variance of the mean is not simply the sum of the individual variances, divided by the number of observations, but becomes ${\rm Var}(\bar Y) = \frac{1}{I}\sigma _a^2 + \frac{1}{{I \cdot J}}\sigma _\varepsilon ^2$. We note that the between-laboratory variation is only reduced by a factor of 1/I, instead of 1/IJ. The between-laboratory and within-laboratory variances can be estimated by ANOVA estimation [16]. Define

$$\displaylines{ {\rm MSA} = \frac{J}{{I - 1}}\sum\limits_i {(Y_{i.} - \bar Y)^2},\cr {\rm MSE} = \frac{1}{{I(J - 1)}}\sum\limits_{i,j} {(Y_{ij} - \bar Y_{i.})^2}.}$$

An estimator of the between-laboratory variance is given by $\hat \sigma _a^2 = \frac{{{\rm MSA} - {\rm MSE}}}{{IJ}}$ and $\hat \sigma _\varepsilon ^2 = {\rm MSE}$.

The interested reader is referred to [16], for the uncertainty calculation in more complicated models, as well as for the estimation of the variance components in cases of unbalanced data.

Appendix B Transformation of uncertainty, correlation and bias from calibrators

The derivation of Eq. (1.3) is based on the following ideas: A (non)-linear calibration curve, derived from I calibrators with assigned values c _i and measured signals s _i , may be approximated between two points $(c_i^* ,s_i^*),(c_{i + 1}^* ,s_{i + 1}^*)$ by a straight line $f(a,b,x) = a + bx$. The assigned values c _i have uncertainties u(c _i ), maximum bias $\delta(c_i)$ and correlation ${\rm Cor}(c_i ,c_{i + 1}) = \rho _{i,i + 1}$.

A value c ^*, lying within $c_i^* \le c^* < c_{i + 1}^*$, might be approximated by $\hat c = \frac{{s^* - \hat a}}{{\hat b}}$, the value read from the straight line, with $\hat b = \frac{{s_{i + 1}^* - s_i^*}}{{c_{i + 1}^* - c_i^*}}$ and $\hat a = s_i^* - \hat b \cdot c_i^*$. An estimator of $\hat c$ is therefore given by

$$\hat c = g(c_i^* ,c_{i + 1}^*) = (c_{i + 1}^* - c_i^*) \cdot \frac{{s^* - s_i^*}}{{s_{i + 1}^* - s_i^*}} + c_i^* .$$

Defining by $s = \frac{{s^* - s_i^*}}{{s_{i + 1}^* - s_i^*}} = \frac{{c^* - c_i^*}}{{c_{i + 1}^* - c_i^*}}$, we obtain immediately $\frac{{\partial g}}{{\partial c_{i + 1}^*}} = s,\quad \frac{{\partial g}}{{\partial c_i^*}} = 1 - s$ and applying Eq. (1.1) the uncertainty component due to the uncertainty of the calibrator value of the read value c, is given by

$$\displaylines{ u_{{\rm cal}} (c^*) = \dot s^2 \cdot u^2 (c_{i + 1}^*) + (1 - \dot s)^2 \cdot u^2 (c_i^*)\cr +\ \rho _{i,i + 1} \cdot s \cdot (1 - s) \cdot u(c_i^*) \cdot u(c_{i + 1}^*).}$$

The maximum bias of the value $\hat c$, can be calculated by regarding $\hat c - \hat c^t = g(c_i^* ,c_{i + 1}^*) - g(c_i^{*t} ,c_{i + 1}^{*t})$, where $g(c_i^{*t} ,c_{i + 1}^{*t})$ denotes the straight line between the true calibrator values. One can show that

$$\displaylines{ \hat c - \hat c^t = s \cdot d(c_{i + 1}^*) + (1 - s) \cdot d(c_i^*) \le s \cdot \delta (c_{i + 1}^*)\cr +\ (1 - s) \cdot \delta (c_i^*),}$$

where d(c _i ) denotes the unknown difference between the assigned value of the calibrator and its true value and $\delta (c_i)$ the known maximum bias. Therefore, the maximum bias of the read value is approximated by

$$ \delta (c^*) = s \cdot \delta (c_{i + 1}^*) + (1 - s) \cdot \delta (c_i^*).$$