## To the Editor:

The critical systematic error (ΔSE_{C}) and the critical random error (ΔRE_{C}) are indicators used in quality-control planning (1). ΔSE_{C} is calculated with the following formula: where TE_{A} 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. ΔRE_{C} is calculated with the formula: where *z*_{2} is a factor specifying the two-tailed significance.

ΔSE_{C} and ΔRE_{C} allow the selection of appropriate quality-control procedures through power function graphs (2).

In practice, critical errors are only estimators of the true ΔSE_{C} and ΔRE_{C} 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, TE_{A}, *z*, and *z*_{2} 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 ΔSE_{C} and ΔRE_{C} 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 C_{pk}, a process capability index that has been thoroughly studied and is well known in industrial quality management (4). In medical laboratory settings, C_{pk} may be calculated with the following formula:

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

Several methods for determining CIs for C_{pk} have been proposed (5). In 1990, Bissell(6) described an approximate two-sided CI for C_{pk} by assuming that the distribution of C_{pk} is gaussian. In Bissell’s approach, this CI is given by: 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 ΔSE_{C}, ΔRE_{C}, and C_{pk}, it is possible to find approximate two-sided CIs for ΔSE_{C} and ΔRE_{C}. By rearranging Eqs. 4 and 5, we can derive the confidence interval for ΔSE_{C}: and the confidence interval for ΔRE_{C}:

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.

- © 2006 The American Association for Clinical Chemistry