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Clinical Chemistry 52: 895-896, 2006; 10.1373/clinchem.2006.067991
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(Clinical Chemistry. 2006;52:895-896.)
© 2006 American Association for Clinical Chemistry, Inc.

Letters to the Editor

How Reliable Are Critical Error Calculations?

Wojciech Gernand

Department of, Laboratory Diagnostics, Medical University of Lublin, ul. Chodzki 1, 20-093 Lublin, Poland, E-mail gernand{at}wp.pl


To the Editor:

The critical systematic error ({Delta}SEC) and the critical random error ({Delta}REC) are indicators used in quality-control planning (1). {Delta}SEC is calculated with the following formula:

Formula 1(1)
where TEA is the total error allowable, B is the method bias, s is the standard deviation of the sample, and z is a factor specifying the one-tailed significance. {Delta}REC is calculated with the formula:

Formula 2(2)
where z2 is a factor specifying the two-tailed significance.

{Delta}SEC and {Delta}REC allow the selection of appropriate quality-control procedures through power function graphs (2).

In practice, critical errors are only estimators of the true {Delta}SEC and {Delta}REC values, which are unknown. These estimators are calculated based on a limited number of results, which are assumed to be representative of the total population of results that could be obtained by stable performance measurement methods in the same material. The reliability of these estimations determines the practical value of decisions about an appropriate quality-control strategy.

Confidence intervals (CIs) should be used to express the reliability of an estimated statistic (3). In the above formulas, TEA, z, and z2 are constant values. B and s are the difference and standard deviation, respectively, with individual CIs that can be determined by traditional parametric statistical methods. However, obtaining CIs for {Delta}SEC and {Delta}REC is not straightforward because these estimators depend jointly on B and s.

To find the solution for this problem, we can refer to the mathematical equivalence of critical errors and Cpk, a process capability index that has been thoroughly studied and is well known in industrial quality management (4). In medical laboratory settings, Cpk may be calculated with the following formula:

Formula 3(3)

By rearranging the above equations, Chesher and Burnett (4) derived:

Formula 4(4)

Several methods for determining CIs for Cpk have been proposed (5). In 1990, Bissell(6) described an approximate two-sided CI for Cpk by assuming that the distribution of Cpk is gaussian. In Bissell’s approach, this CI is given by:

Formula 5(5)
where n is the number of measurements used in calculating s and B. Kushler and Hurley (7) tested Bissell’s method and concluded that it is easily computed and gives reasonably accurate results.

Taking into account the mathematical equivalence of {Delta}SEC, {Delta}REC, and Cpk, it is possible to find approximate two-sided CIs for {Delta}SEC and {Delta}REC. By rearranging Eqs. 4 and 5, we can derive the confidence interval for {Delta}SEC:

Formula 6(6)
and the confidence interval for {Delta}REC:

Formula 7(7)

CIs calculated for critical errors depend on the number of measurement results used in calculating them. Decisions on the adequacy of quality-control algorithms may be highly uncertain when based on a small number of measurement results. Such a situation might occur in a medical laboratory; for example, when a measurement method is newly introduced into routine practice. A minimum of 20 results is typically used to form an initial estimation of the measurement’s performance (8). These 20 results are, however, insufficient for reliable quality-control planning. Further updating of the initial estimation as new results are obtained is obligatory.

Through the calculation of CIs for critical errors with the above formulas, it is possible to quantify the reliability of quality judgments. This analysis highlights the significant variability that may exist in such judgments because of the random nature and limited quantity of quality control data. Caution should be exercised when interpreting such data and making decisions on an appropriate quality-control strategy.


References

  1. Westgard JO, Barry PL. Cost-Effective Quality Control: Managing the Quality and Productivity of Analytical Processes 1986:47-49 AACC Press Washington, DC. .
  2. Koch DD, Oryall JJ, Quam EF, Feldbruegge DH, Dowd DE, Barry PL, et al. Selection of medically useful quality-control procedures for individual tests done in a multitest analytical system. Clin Chem 1990;36:230-233.[Abstract/Free Full Text]
  3. Henderson AR. Chemistry with confidence: should Clinical Chemistry require confidence intervals for analytical and other data?. Clin Chem 1993;39:929-935.[Abstract/Free Full Text]
  4. Chesher D, Burnett L. Equivalence of critical error calculations and process capability index Cpk. Clin Chem 1997;43:1100-1101.[Free Full Text]
  5. Kotz S, Johnson NL. Process capability indices: a review, 1992–2000. J Qual Technol 2002;34:2-19.
  6. Bissell AF. How reliable is your capability index?. Appl Stat 1990;39:331-340.[CrossRef]
  7. Kushler RH, Hurley P. Confidence bounds for capability indices. J Qual Technol 1992;24:188-195.
  8. . Clinical and Laboratory Standards Institute. Statistical quality control for quantitative measurements: s and definitions; approved guideline, 2nd ed. CLSI Guideline C24–A2 1999 CLSI Wayne, PA. .




This Article
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Related Collections
Right arrow Laboratory Management
Right arrow Informatics and Statistics

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