Evaluation of reproducibility uncertainty in micropipette calibrations for non-nominal volumes through an interlaboratory study

Abstract

The nominal volume of a variable volume micropipette is the maximum volume of the range specified by the micropipette manufacturer. The purpose of the present study is to quantify the reproducibility standard uncertainty for non-nominal volume micropipette calibrations for variable volume micropipettes. An interlaboratory study was implemented for this purpose, using variable volume micropipettes with nominal volumes of 10 μL, 200 μL, and 1000 μL and setting the measurement volumes as 10 %, 50 %, and 100 % of the nominal volumes. In our previous paper (Accredit Qual Assur 19:377, 2014), we quantified the uncertainty due to reproducibility using only data obtained for nominal volume calibrations. From the results obtained, we found that the reproducibility uncertainties quantified for nominal volumes are too small to cover interlaboratory variations for non-nominal volume calibrations. For calibrations of 10 % and 50 % of the nominal volumes, the reproducibility uncertainties must be increased by the factors of 2.7 and 1.4, respectively, for micropipettes with nominal volumes of 10 μL to 1 000 μL, based on the results of this study.

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Appendices

Appendix 1: Approximate uncertainty evaluation

We would like to quantify the reproducibility standard uncertainty independent of the other uncertainty sources. Although we have gathered some information on uncertainty in the interlaboratory study, the evaluated uncertainty sources based on the individual laboratories’ SOPs were not necessarily assessed independently of the reproducibility uncertainty considered in this study. Therefore, apart from the obtained uncertainty assessment information from the participants, we implemented the computation of the standard uncertainty of Vi,j in an approximate manner so carefully that the reproducibility uncertainty is independent of other uncertainty sources. Since the uncertainties other than the reproducibility uncertainty were marginal compared to the interlaboratory variations as mentioned in the main manuscript, only brief explanations are given. Uncertainties are computed basically in accordance with the EURAMET Calibration Guide No. 19 [19] using the information in ISO 8655 Parts 2 [16] and 6 [1].

Uncertainty sources

  1. (1)

    Repeatability uncertainty \(u_{\text{rep}} \left( {V_{i,j} } \right)\)

    This source was evaluated in accordance with the EURAMET Calibration Guide No. 19.

  2. (2)

    Uncertainty relating to the mass scale calibration \(u_{\text{scale}} \left( {V_{i,j} } \right)\)

    This source was evaluated conservatively using the maximum values for “repeatability and linearity” (uncertainty), and “standard uncertainty of measurement” (in the calibration) given in ISO 8655 Part 6. Only for the 200 μL test case, 0.05 mg was used for the both uncertainties instead of 0.2 mg in ISO 8655 Part 6.

  3. (3)

    Uncertainty of the water density \(u_{\text{wat}} \left( {V_{i,j} } \right)\)

    This source was evaluated in accordance with the EURAMET Calibration Guide No. 19 with slight modifications.

  4. (4)

    Uncertainty of the air density \(u_{\text{air}} \left( {V_{i,j} } \right)\)

    This source was evaluated in accordance with the EURAMET Calibration Guide No. 19 with slight modifications. The guide offers the relative standard uncertainty due to using a formula [Eq. (5) in the guide] as 2.4 × 10−4. While that formula was employed, the uncertainty was not used in this study. Although this uncertainty is given for the case where the air temperature, the air pressure, and the relative air humidity are not well specified, we could specify these by their measured values in this study.

  5. (5)

    Uncertainty of the cubic expansion coefficient \(u_{\exp } \left( {V_{i,j} } \right)\)

    This source was evaluated in accordance with the description in the EURAMET Calibration Guide No. 19 that “data from the literature or manufacturer should be used and this would be expected to have an (standard) uncertainty of the order of 5 % to 10 %.” We assumed the relative standard uncertainty of 10 %.

  6. (6)

    Uncertainty of the temperature of the micropipette \(u_{\text{pip}} \left( {V_{i,j} } \right)\)

    The EURAMET Calibration Guide No. 19 offers an approach to determine this uncertainty based on the difference between measured air and water temperatures, which is recommended to be no more than 2 K in the guide. We hence conservatively assumed the uniform distribution with the width of ± 2 K for the uncertainty of \(t_{i,j}^{\text{pip}}\). Although \(t_{i,j}^{\text{pip}}\) was determined to be identical to \(t_{i,j}^{\text{wat}}\), the uncertainty of \(t_{i,j}^{\text{wat}}\) was neglected for \(u_{\text{pip}} \left( {V_{i,j} } \right)\), because it was quite minor.

  7. (7)

    Reproducibility uncertainty \(u_{\rm m}\left(V_i{\rm set}\right)\)\(u_{\rm mod}\left(V_i{\rm set}\right)\)

    For the computation of \(\chi_i^2\) in the Applicability of quantifying reproducibility standard uncertainty for nominal volumes subsection in the Discussion section, the standard uncertainties of reproducibility \(u_{\rm m}\left(V_i^{\rm set}\right)\) in Eq. (5) is employed. For the computation of \(\chi_{\text{mod}, i}^2\) in the Modified reproducibility standard uncertainties for non-nominal volumes subsection in the Discussion section, the standard uncertainties of reproducibility \(u_{\rm mod}\left(V_i^{\rm set}\right)\) in Eq. (6) is employed.

Combination of the standard uncertainties

The combined standard uncertainty of Vi,j, ui,j, is computed by the following equation:

$$\begin{array}{*{20}c} {u_{i,j} = \left[ {\begin{array}{*{20}c} {u_{\text{rep}}^{2} \left( {V_{i,j} } \right) + u_{\text{scale}}^{2} \left( {V_{i,j} } \right) + u_{\text{wat}}^{2} \left( {V_{i,j} } \right) + u_{\text{air}}^{2} \left( {V_{i,j} } \right)} \\ { + u_{\exp }^{2} \left( {V_{i,j} } \right) + u_{\text{pip}}^{2} \left( {V_{i,j} } \right) + u_{\rm m}^{2} \left( {V_{i}^{\text{set}} } \right)} \\ \end{array} } \right]^{1/2} .} \\ \end{array}$$
(18)

It is found that the values are almost the same as the corresponding reproducibility standard uncertainty \(u_{\rm m}\left(V_i{\rm set}\right)\). For the nominal volumes, ui,j was only 12 % larger than \(u_{\rm m}\left(V_i{\rm set}\right)\) at the maximum. Although our assessment of the uncertainties shown in this appendix is approximative, the approximation may not be so crucial, because the reproducibility standard uncertainty is the dominant source of the uncertainty. For the non-nominal volumes, ui,j was 41 % larger than \(u_{\rm m}\left(V_i{\rm set}\right)\) at the maximum. The reproducibility standard uncertainty was still the main source of the assessed standard uncertainty of the calibrated volume.

The modified combined standard uncertainty of Vi,j, \(u_{i, j}^{\bmod }\), is computed by the following equation:

$$\begin{array}{*{20}c} {u_{i, j}^{\bmod } = \left[ {\begin{array}{*{20}c} {u_{\text{rep}}^{2} \left( {V_{i,j} } \right) + u_{\text{scale}}^{2} \left( {V_{i,j} } \right) + u_{\text{wat}}^{2} \left( {V_{i,j} } \right) + u_{\text{air}}^{2} \left( {V_{i,j} } \right)} \\ { + u_{\exp }^{2} \left( {V_{i,j} } \right) + u_{\text{pip}}^{2} \left( {V_{i,j} } \right) + u_{\bmod }^{2} \left( {V_{i}^{\text{set}} } \right)} \\ \end{array} } \right]^{1/2} .} \\ \end{array}$$
(19)

The computed \(u_{i, j}^{\bmod }\) is only 9 % larger than \(u_{\rm mod}\left(V_i{\rm set}\right)\) at the maximum. It can be said that the uncertainties except for the reproducibility uncertainty are negligibly small for the purpose of this study.

Appendix 2: One-way ANOVA in a heteroscedastic condition

We found inhomogeneity in repeatability reported from participants. In other words, repeatability is found to be heteroscedastic in most cases. However, even in the situation with the heteroscedastic variance, the one-way ANOVA in this study is still a way to estimate reproducibility variance unbiasedly. The explanation is given in this appendix.

Instead of the complicated symbols employed in the main manuscript, we define some symbols only for this appendix as follows:

  • a: Number of the levels of the factor in the one-way ANOVA,

  • i: Indication of the level of the factor when Condition i is the i-th level of the factor (i = 1, …, a),

  • n: Number of the repetitions in a condition level,

  • xi,j: measured value in the j-th repetition of Condition i (j = 1, …, n),

  • \(\bar{x}_{\text{i}}\): Average value of xi, 1 to xin,

  • \(\bar{\bar{x}}\): Average value of \(\bar{x}_{ 1}\) to \(\bar{x}_{\text{a}}\).

Moreover, Exp[∙] and Var[∙] denote the operators for the expectation and the variance, respectively.

We assume the following statistical model:

$$\begin{array}{*{20}c} {{x}_{i,j} = \mu + {A}_{i} + {E}_{i,j} ,} \\ \end{array}$$
(20)

where Ai is the effect of the factor for Condition i, and Ei,j is the error for the repeatability.

When homoscedasticity is considered, the variance of Ei,j does not depends on Condition i. We hence define the variances of Ai and Ei,j as follows:

$$Var\left[ {A_{i} } \right] = \sigma_{A}^{2} ,\quad Var\left[ {E_{i,j} } \right] = \sigma_{E}^{2} .$$
(21)

In this case, the unbiased estimator of \(\sigma_A^2\) essentially employed in ISO 5725 Part 2 [9], \(\hat{\sigma }_{A}^{ 2}\), is given as follows:

$$\begin{array}{*{20}c} {\hat{\sigma }_{A}^{2} = \dfrac{1}{a - 1}\mathop \sum \limits_{i = 1}^{a} \left( {\bar{x}_{i} - \bar{\bar{x}} } \right)^{2} - \dfrac{1}{an}\mathop \sum \limits_{i = 1}^{a} \left[ {\dfrac{1}{n - 1}\mathop \sum \limits_{j = 1}^{n} \left( {x_{i,j} - \bar{x}_{i} } \right)^{2} } \right].} \\ \end{array}$$
(22)

When heteroscedasticity is considered, the variance of Ei,j depends on Condition i. We hence define he variances of Ai and Ei,j as follows:

$$Var\left[ {A_{i} } \right] = \sigma_{A}^{2} ,\quad Var\left[ {E_{i,j} } \right] = \sigma_{i}^{2} .$$
(23)

The expectation of the square sum of \(\left( {\bar{x}_{i} - \bar{\bar{x}} } \right)\) is given as follows:

$$\begin{array}{ll} Exp\left[ {\mathop \sum \limits_{i = 1}^{a} \left( {\bar{x}_{i} - \bar{\bar{x}} } \right)^{2} } \right] & = Exp\left[ {\mathop \sum \limits_{i = 1}^{a} \left[ {\left( {\bar{x}_{i} - \mu } \right) + \left( {\mu - \bar{\bar{x}} } \right)} \right]^{2} } \right] \\&= Exp\left[ {\mathop \sum \limits_{i = 1}^{a} \left( {\bar{x}_{i} - \mu } \right)^{2} - 2\mathop \sum \limits_{i = 1}^{a} \left( {\bar{x}_{i} - \mu } \right)\left( {\bar{\bar{x}} - \mu } \right) + a\left( {\bar{\bar{x}} - \mu } \right)^{2} } \right] \\ & = Exp\left[ {\mathop \sum \limits_{i = 1}^{a} \left( {\bar{x}_{i} - \mu } \right)^{2} - a\left( {\bar{\bar{x}} - \mu } \right)^{2} } \right] \\ & = Exp\left[ {\mathop \sum \limits_{i = 1}^{a} \left( {\bar{x}_{i} - \mu } \right)^{2} } \right] - aExp\left[ {\left( {\bar{\bar{x}} - \mu } \right)^{2} } \right]. \\ \end{array}$$
(24)

With

$$\begin{array}{*{20}c} {Exp\left[ {\left( {\bar{x}_{i} - \mu } \right)^{2} } \right] = \sigma_{A}^{2} + \dfrac{{\sigma_{i}^{2} }}{n}} \\ \end{array}$$
(25)

and

$$\begin{array}{ll} Exp\left[ {\left( {\bar{\bar{x}} - \mu } \right)^{2} } \right] & = Exp\left[ {\dfrac{1}{{a^{2} }}\left[ {\mathop \sum \limits_{i = 1}^{a} \left( {\bar{x}_{i} - \mu } \right)} \right]^{2} } \right] \\ &= Exp\left[ {\dfrac{1}{{a^{2} }}\left[ {\mathop \sum \limits_{i = 1}^{a} \left( {\bar{x}_{i} - \mu } \right)^{2} + 2\mathop \sum \limits_{i = 1}^{a} \mathop \sum \limits_{k = i + 1}^{a} \left( {\bar{x}_{i} - \mu } \right)\left( {\bar{x}_{k} - \mu } \right)} \right]} \right] \\ & = Exp\left[ {\dfrac{1}{{a^{2} }}\mathop \sum \limits_{i = 1}^{a} \left( {\bar{x}_{i} - \mu } \right)^{2} } \right] \\&= \dfrac{1}{{a^{2} }}\mathop \sum \limits_{i = 1}^{a} \left( {\sigma_{A}^{2} + \dfrac{{\sigma_{i}^{2} }}{n}} \right) = \dfrac{{\sigma_{A}^{2} }}{a} + \dfrac{1}{{a^{2} }}\mathop \sum \limits_{i = 1}^{a} \dfrac{{\sigma_{i}^{2} }}{n}, \\ \end{array}$$
(26)

the following relation can be yielded:

$$\begin{aligned} & Exp\left[ {\mathop \sum \limits_{i = 1}^{a} \left( {\bar{x}_{i} - \bar{\bar{x}} } \right)^{2} } \right] \\ &\quad = \mathop \sum \limits_{i = 1}^{a} \left( {\sigma_{A}^{2} + \frac{{\sigma_{i}^{2} }}{n}} \right) - a\left[ {\frac{{\sigma_{A}^{2} }}{a} + \frac{1}{{a^{2} }}\mathop \sum \limits_{i = 1}^{a} \frac{{\sigma_{i}^{2} }}{n}} \right] \\ &\quad= \left( {a - 1} \right)\sigma_{A}^{2} + \frac{a - 1}{an}\mathop \sum \limits_{i = 1}^{a} \sigma_{i}^{2} \\ \end{aligned}$$
(27)

Since

$$\begin{array}{*{20}c} {{{Exp}}\left[ {\dfrac{1}{n - 1}\mathop \sum \limits_{j = 1}^{n} \left( {x_{i,j} - \bar{x}_{i} } \right)^{2} } \right] = \sigma_{i}^{2} ,} \\ \end{array}$$
(28)

\(\hat{\sigma }_{A}^{ 2}\) satisfying the following equation is an unbiased estimator of \(\sigma_A^2\):

$$\begin{array}{*{20}c} {\mathop \sum \limits_{i = 1}^{a} \left( {\bar{x}_{i} - \bar{\bar{x}} } \right)^{2} = \left( {a - 1} \right)\hat{\sigma }_{A}^{2} + \dfrac{a - 1}{an}\mathop \sum \limits_{i = 1}^{a} \left[ {\dfrac{1}{n - 1}\mathop \sum \limits_{j = 1}^{n} \left( {x_{i,j} - \bar{x}_{i} } \right)^{2} } \right]} \\ \end{array}$$
(29)

The solution of \(\hat{\sigma }_{A}^{ 2}\) is, in fact, given by Eq. (22). Therefore, it is concluded that even when heteroscedasticity is considered, Eq. (22) gives an unbiased estimator of the variance of the factor in the one-way ANOVA. Although other analysis may be possible, no essential differences can happen as long as we apply an unbiased estimation method, because \(s_{i,j}^{\text{lab}}\) is much larger than \(s_{i,j}^{\text{rep}} /\sqrt {10}\) in this study.

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Shirono, K., Tanaka, H. Evaluation of reproducibility uncertainty in micropipette calibrations for non-nominal volumes through an interlaboratory study. Accred Qual Assur 26, 27–39 (2021). https://doi.org/10.1007/s00769-020-01453-5

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Keywords

  • Piston pipette
  • Micropipette
  • ISO 8655
  • Calibration
  • Non-nominal volume
  • Gravimetric method