## Mandel and Westgard

Readers may know that I been known to critique Westgard’s total error model.

But let’s step it back to 1964 with Mandel’s representation of total error (1), where:

Total Error (TE) = x-R = (x-mu) + (mu-R) with

x= the sample measurement
R=the reference value and
mu=the population mean of the sample

Thus, mu-R is the bias and x-mu the imprecision – the same as the Westgard model. There is an implicit assumption that the replicates of x which estimate mu are only affected by random error. For example, if the observations of the replicates contain drift, the Mandel model would be incorrect. For replicates sampled close in time, this is a reasonable assumption, although it is rarely if ever tested.

Interferences are not a problem because even if they exist, there is only one sample. Thus, interference bias is mixed in with any other biases in the sample.

Total error is often expressed for 95% of the results. I have argued that 5% of results are unspecified but if the assumption of random error is true for the repeated measurements, this is not a problem because these results come from a Normal distribution. Thus, the probability is extremely remote that high multiples of the standard deviation will occur.

But outliers are a problem. Typically for these studies, outliers (if found) are deleted because they will perturb the estimates – the problem is the outliers are usually not dealt with and now the 5% unspecified results becomes a problem.

If no outliers are observed, this is a good thing but here are some 95% confidence levels for the maximum outlier rate given the number of sample replicates indicated where 0 outliers have been found.

N                             Maximum outlier rate (95% CI)

10                           25.9%
100                         3.0%
1,000                      0.3%

So if one is measuring TE for a control or patient pool and keeping the time between replicates short, then the Westgard model estimate of total error is reasonable, although one still has to worry about outliers.

But when one applies the Westgard model to patient samples, it is no longer correct since each patient sample can have a different amount of interference bias. And while large interferences are rare, interferences can come in small amounts and affect every sample – inflating the total error. Moreover, other sources of bias can be expected with patient samples, such as user error in sample preparation. And with patient samples, outliers while still rare, can occur.

This raises the question as to the interpretation of results from a study that uses the Westgard model (such as a Six Sigma study). These studies typically use controls but the implication is that they inform about the quality of the assay – meaning of course for patient samples. This is a problem for the reasons stated above. So one can say that if an assay has a bad six sigma value, the assay has a problem, but if the assay has a good six sigma value, one cannot say the assay is without problems.

Reference

1. Mandel J. The statistical analysis of experimental data Dover, NY 1964, p 105.