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Uncertainty of measurement and heteroscedasticity by xavier fuentes-arderiu

 
Uncertainty of measurement and heteroscedasticity

Xavier Fuentes-Arderiu, Bernardino Gonz�lez-de-la-Presa, Juan Cuadros-Mu�oz
Servei de Bioqu�mica Cl�nica
Ciutat Sanit�ria i Universit�ria de Bellvitge
08907 L'Hospitalet de Llobregat
Catalonia, Spain

Fax +34 93 260 75 46
xfa@csub.scs.es


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Uncertainty of measurement (hereafter referred to as uncertainty) is a parameter, associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand (that is to say the measured quantity) [1]; in other words, uncertainty is  numerical information that complements a result of measurement, indicating the magnitude of the doubt about this result.

The international scientific and standardization bodies recommend that the uncertainty of patients' results obtained in clinical laboratories should be known [2, 3]; the rationale for this recommendation is that full interpretation of the value of a quantity obtained by measurement requires also evaluation of the doubt attached to its value. The common opinion of these bodies is that clinical laboratories should supply information about the uncertainty of their results of measurement, when applicable.

The uncertainty that should be written together with a clinical laboratory result is the so called expanded uncertainty, obtained by the positive squared root of the sum of the variances, corresponding to different sources of uncertainty affecting the measurement process �that is to say the combined uncertainty�multiplied by a coverage factor [1, 4, 5]. Among these causes, day-to-day imprecision is generally responsible for an important part of uncertainty.

With regard to day-to-day imprecision, the phenomenon called heteroscedasticity should be taken into account: day-to-day metrological variance depends on the value of the measurand (the opposite phenomenon is called homoscedasticity). In some cases of heteroscedasticity, in spite of variance differences with the measurand value, the coefficient of variation reminds constant; in these cases, the calculation of the variance due to day-to day imprecision is easy to carry out (knowing the measured value and the constant coefficient of variation). But, when heteroscedasticity is present and the coefficient of variation also depends on the value of the measurand, to know the day-to-day imprecision it is necessary (i) to know the mathematical or graphical relationship between variance and measurand value, called variance function, or (ii) to know the mathematical or graphical relationship between coefficient of variation and measurand value, called the imprecision profile.

Variance functions may be estimated using by using the maximum approximate conditional likelihood method [6-7]. The equation used in this method is s2 = (b1 + b2 c)J, where s2 is the variance of replicate measurements, c the values of the different concentrations and b1,  b2 and J are three parameters defining the function. There is no underlying physical or chemical law for this function.

We have applied the maximum approximate conditional likelihood method using the Sadler et al. program [6] to repeated results of several measurement procedures of different quantities used in our laboratory. For each quantity, variances and coefficients of variation have been estimated with 20 replicated results, one per day, over 20 working days, in aliquots (stored at -20 oC) of seven serum pools with values representing the entire measurement range (Table 1).

Quantity

Serum  pool

Mean value

CV

S�Bilirubin; subst.c. [mmol/L]

1

4.7

10.0

(Hitachi 747/Jendrassik-Gr�f/Roche Diagnostics)

2

29.5

4.3

 

3

83.5

1.6

 

4

119.7

1.4

 

5

198.2

1.4

 

6

265.3

1.9

 

7

331.8

1.9

 

 

 

 

S�Ferritin; mass.c. [mg/L]

1

21.3

9.5

(Hitachi 917/Tina-Quant reagents/Roche Diagnostics)

2

53.4

5.9

 

3

89.1

4.5

 

4

136.7

4.1

 

5

215.1

3.0

 

6

326.5

2.5

 

7

503.3

2.6

 

 

 

 

S�Triiodothyronine; subst.c. [pmol/L]

1

0.67

18.1

(Elecsys 2010/Roche Diagnostics)

2

1.64

9.5

 

3

2.51

7.1

 

4

4.34

5.1

 

5

5.41

4.6

 

6

7.51

4.2

 

7

9.26

5.3

S = serum; subst.c. = substance concentration; mass c. = mass concentration.

The graphical outputs of the program show coefficients of variation approximately constant for several measurement procedures, such as those related to cholesterol, glucose and protein. However, other measurement procedures have clearly different coefficient of variation for each value of the measured quantity as can be apprecated in Figure 1 (the shared zone is the 95 % confidence interval).









 

 

 




Coming back to the uncertainty estimation, when a measurement procedure has a behavior such as represented in Fig. 1, the Sadler et al. program [6] allow us to predict, within the measurement range, the variance corresponding to the measurand value, and this variance may be used to estimate the uncertainty.

Therefore, clinical laboratories using the maximum approximate conditional likelihood method may know its imprecision profiles and variance functions in order to estimate appropriately its uncertainties.

References

  1. International Organization for Standardization, International Electrotechenical Commission, International Organization of Legal Metrology, International Bureau of Weights and Measures. Guide to the expression of uncertainty in measurement. Geneva: ISO, 1993.

  2. International Union of Pure and Applied Chemistry, International Federation of Clinical Chemistry. Compendium of Terminology and Nomenclature of Properties in Clinical Laboratory Sciences (Recommendations 1995). [Prepared for publication by J.C. Rigg, S.S. Brown, R. Dybkaer, H. Olesen]. Oxford: Blackwell Science; 1995.

  3. International Organization for Standardization. Quality management in the medical laboratory. ISO/DIS 15189. Geneva: ISO; 2000.

  4. Taylor BN, Kuyatt CE. National Institute of Standards and Technology. Guidelines for evaluating and expressing the uncertainty of NIST measurement results. NIST Technical Note 1297, 1994 Edition.

  5. Eurachem. Quantifying uncertainty in analytical measurement. London: Eurachem, British Standards Institute; 1995.

  6. Sadler WA, Smith MH, Legge HM. A method for direct estimation of imprecision profiles, with reference to immunoassay data. Clin Chem 1988;34:1058-61.

  7. Sadler WA, Smith MH. Use and abuse of imprecision profiles: some pitfalls illustrated by computing and plotting confidence intervals. Clin Chem 1990;36:1346-50.

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