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Clinical Chemistry 43: 1100-1101, 1997;
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(Clinical Chemistry. 1997;43:1100-1101.)
© 1997 American Association for Clinical Chemistry, Inc.

Letters

Equivalence of Critical Error Calculations and Process Capability Index Cpk

Doug Chesher and Leslie Burnetta

Dept. of Clin. Chem., Inst. of Clin. Pathol. and Medical Res., Westmead Hosp., Westmead NSW 2145, Australia
a Author for correspondence.


To the Editor:

The concept of process capability has been used by the manufacturing industry to quantify the relation between product specifications and the measured process performance (1). Various ratios and indices have been developed to describe this relation. We have previously reported the application of the simplest of these, Cp (the capability index or capability ratio), to the selection of quality-control (QC) algorithms appropriate to the specification limits and analytical imprecision (2). Cp is defined as (USL - LSL)/6{sigma}, where USL and LSL are the upper and lower specification limits of an analytical process, and {sigma} is the standard deviation of the process.

In contrast to the approach taken by us, the use of medically important critical systematic error ({Delta}SEC) and critical random error ({Delta}REC) calculations for the selection of QC algorithms has been promulgated (3)(4)(5). Westgard and Burnett (6) have also described the relation between Cp and {Delta}SEC and have shown that, assuming zero bias, Cp can be directly related to {Delta}SEC by equation:

(1)
where z is a factor for a one-tailed test of significance (usually set at 1.65 for 95% confidence, assuming a gaussian distribution).

A limitation of the use of Cp is that this particular capability index does not consider any bias present within the analytical process. Cpk is a related capability index used by the manufacturing industry that considers bias as well as imprecision (1) and is defined as:

where USL, LSL, and {sigma} are as for Cp, and µ is the mean of the process.

We have now derived the mathematical relation between the process capability index, Cpk, and medically important critical errors, {Delta}SEC and {Delta}REC, for cases of nonzero bias.

If the bias is nonzero and the specification limits are symmetrical about the "true value" (0), then:

and

Therefore, Cpk =

or

From ref. 6:

By substitution:

(2)
Similarly:

(3)
where z is as defined above, and z2 is a factor for a two-tailed test of significance (usually set at 1.96 for 95% confidence, assuming a gaussian distribution). The previously derived relation (6) between {Delta}SEC and Cp shown in Eq. 1Up can now be seen to be a limiting value of Eq. 2Up for the special case of zero bias.

We have found that implementation of a QC strategy based on capability indices may be conceptually easier to understand than one based on {Delta}SEC and {Delta}REC (2). By rearranging Eqs. 2Up , and 3Up , we can derive:

(4)
and

(5)
This form of the relation may facilitate the educational and training process for those laboratories that choose this perspective. Conversely, for those laboratories that have been using QC strategies based on process capabilities, a simple pathway is now offered to convert their strategies to the more widely adopted approach of using critical error parameters.

There exists ready availability of computational aids for the selection of QC algorithms based on critical errors, such as the "QC Validator" software program (7) and the use of QC selection grids (8). As Eqs. 2 through 5Up Up Up Up permit interconversion of Cpk with {Delta}SEC and {Delta}REC, there appears to be no need to develop parallel sets of tools based on Cpk.

The demonstration of the mathematical equivalence of {Delta}SEC, {Delta}REC, and Cpk now permits a unification of those approaches based around process capability (2) and those around medically important critical errors (3)(4)(5). Finally, the equivalence between approaches based on critical error and Cp (6), and now also Cpk, means that there is available a common language between clinical chemistry and quality professionals in the manufacturing industry.


References

  1. Gryna FM. Manufacturing planning. In: Juran JM, Gryna FM, eds. Juran's quality control handbook, 4th ed. New York: McGraw-Hill, 1988;16:1–59..
  2. Burnett L, Hegedus G, Chesher D, Burnett J, Costaganna G. Application of process capability indices to quality control in a clinical chemistry laboratory [Tech Brief]. Clin Chem 1996;42:2035-2037. [Free Full Text]
  3. Groth T, Falk H, Westgard JO. An interactive computer simulation program for the design of statistical control procedures in clinical chemistry. Comput Programs Biomed 1981;13:73-86. [ISI][Medline] [Order article via Infotrieve]
  4. Westgard JO, Barry PL. Cost-effective quality control: managing the quality and productivity of analytical processes 1986:47-49 AACC Press Washington, DC. .
  5. Koch DD, Oryall JJ, Quam EF, Feldbruegge DH, Dowd DE, Barry PL, Westgard JO. 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]
  6. Westgard JO, Burnett RW. Precision requirements for cost-effective operation of analytical processes. Clin Chem 1990;36:1629-1632. [Abstract/Free Full Text]
  7. QC Validator® program manual, version 1.1. Ogunquit, ME: WesTgard® Quality Corp., 1993..
  8. Westgard JO, Quam EF, Barry PL. Selection grids for planning quality control procedures. Clin Lab Sci 1990;3:273-280.



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