**4.8 Report the results**

428 Toxicity and Drug Testing

We now have an interval within which we would expect a large fraction (approximately 95%) of the expected values of the measurand to exist. If we were to assume k=2 to generate an approximate 95% uncertainty interval we would obtain: 0.0827 2 0.00067 0.0827 0.0013 0.0814 0.0840 / *to g dL* . We see that this interval is slightly narrower than that employing the effective degrees of freedom estimate. Choosing the appropriate coverage factor will be a decision made within each forensic laboratory. A 99% interval (k=3) will provide a higher degree of confidence that may be important in forensic applications. This is particularly true where results are near prohibited legal limits. Whatever decision is made, the value for k should be clearly identified in the program policy or SOP manuals and strictly adhered to in practice. In this example we have assumed our expanded interval to be an "uncertainty interval" rather than a "confidence interval". The *GUM* document prefers the term "uncertainty interval" or "level of confidence". (ISO/GUM, 2008) Others, however, interpret U as representing a confidence interval which

Table 2 illustrates one form of an uncertainty budget for our example. The uncertainty budget lists the components contributing to the combined uncertainty along with the percent of their contribution to the total. The percent contributions were determined from the terms under the radical sign in equation 18. This is very useful for identifying which components are the major contributors and which may be reasonably ignored. The *GUM* document states that any contributions less than one-third of the largest contributor can be safely ignored. (ISO/GUM, 2008) Based on this we see that the analytical and dilutor components could be safely ignored in this example. However, from a forensic perspective it may be better to include all components considered, providing full disclosure. We see that the total method contributes the largest component at 59%. This is expected because of all of the contributing sub-components involved: analysts, calibrations, time, dilutions, etc. This analysis does not include, however, the venous blood sampling performed by the phlebotomist who typically performs only one venipuncture. Moreover, many laboratories do not even consider sampling as a component of their combined uncertainty. They simply consider their uncertainty estimates corresponding to the sample "as received in the laboratory". Jones, for example, has considered sampling as a source of uncertainty in some

**Source Type Distribution Standard Uncertainty Percent1** Traceability B Normal 0.0004 g/dL 24% Analyical A Normal 0.0008 g/dL 13% Dilutor B Normal 0.050 ml 4% Total Method A Normal 0.00072 g/dL 59%

0.0019 g/dL

Combined Uncertainty 0.00067 g/dL

95% confidence interval 0.0808 to 0. 0846 g/dL

has a specific definition in the classical statistical sense.

**4.7 Produce the uncertainty budget** 

of his published work. (Jones, 1989)

1Percent of contribution to total combined uncertainty

Table 2. Uncertainty budget for the illustrated example

Expanded Uncertainty

(k=2.776)

One of the most important, yet often overlooked, elements of determining measurement uncertainty is reporting the results. A great deal of thought should be given to this aspect of measurement. The end-user should be consulted to determine exactly what is needed for their application. There should be sufficient information so the results and their associated uncertainty are fully interpretable and unequivocal for a specific application without reference to additional documentation. This will necessitate some textual explanation in addition to the numerical results. One possibility for our blood alcohol example above is:

*The duplicate whole blood alcohol results were 0.082 and 0.081 g/dL with a corrected mean result of 0.0827 g/dL. An expanded combined uncertainty of 0.0019 g/dL assuming a coverage factor of k=2.776 with an effective degrees-of-freedom of 4 and a normal distribution was generated from four principle components contributing to the uncertainty. An approximate 95% confidence interval for the true mean blood alcohol concentration is 0.0808 to 0.0846 g/dL.* 

In addition to the statement, a figure similar to that of figure 3 could be provided which might assist the court in placing the results in some geometric perspective. The format for reporting the results should be considered flexible. As time goes on there will no doubt be the need for revision to ensure clarity in communication and interpretation.
