**Measurement Uncertainty in Forensic Toxicology: Its Estimation, Reporting and Interpretation**

 Rod G. Gullberg *Clearview Statistical Consulting, Snohomish, WA, USA* 

#### **1. Introduction**

414 Toxicity and Drug Testing

Yu, Y., Huang, T., Yang, B., Liu, X. & Duan, G. (2007). Development of gas chromatography–

*Pharmaceutical and Biomedical Analysis*, Vol. 43, pp. 24-31.

mass spectrometry with microwave distillation and simultaneous solid-phase microextraction for rapid determination of volatile constituents in ginger. *Journal of* 

> All measurements, regardless of their purpose, context or quality, possess uncertainty. No measurement is performed with absolute perfection since all are approximations. Uncertainty, however, does not mean there is anything wrong or inappropriate with the results. Uncertainty is simply a measure of the confidence we have in our best estimate and results from limitations in our technology, our methods, our standards and our limited understanding of the property being measured. [Drosg] Uncertainty is a fundamental property of the natural world in which we live and work. Moreover, no measurement is fully interpretable within a given context until the full process generating the result is understood. The general additive measurement function observed in equation 1 illustrates this basic limitation of all measurements:

$$Y = \mu + \beta + \varepsilon \tag{1}$$

where: Y = the measurement result

µ = the true value of the measurand

β = measurement error due to bias

ε = random measurement error

Our measurement is an imperfect representation of the measurand due to bias and random error components. Bias may be corrected for when reliably determined with traceable controls. Random error, on the other hand, cannot be corrected for but can be minimized to an acceptable level. Figure 1 illustrates how these two contributors to uncertainty influence measurement results - where we have assumed a normal distribution. Bias is simply the difference between the mean and the reference value while random error, determined by the variance or standard deviation, defines the width of the distribution. Figure 1 also illustrates another important property of measurement - all results are random variables that arise from a specified distribution. As a result they have a fixed mean and variance from which confidence intervals can be determined – an useful metric for defining uncertainty. The fact that uncertainty exists in our measurements, however, should not alarm us. We simply need to understand it, acknowledge it, estimate it in a statistically valid way, report it and ensure that it is fit-for-purpose.

Measurement Uncertainty in Forensic Toxicology: Its Estimation, Reporting and Interpretation 417

Moreover, providing the uncertainty along with measurement results is one important step in ensuring evidence-based inference. (Mnookin, et.al., 2011) We intend to illustrate and

Very basically, measurement uncertainty is best described by an interval, symmetric about the measurement result and within which we claim that the true value (the measurand) exists with some level of probability. The end points of this interval are called uncertainty or confidence limits. This interval quantifies the precision of the measurement result. Figure 2 illustrates this concept of uncertainty. The classical statistical view would state that the measurand (µ) is a fixed quantity and the measurement result along with the interval limits are random variables. The probability, therefore, relates to the random interval actually encompassing the fixed true value (µ). This involves some subtle distinctions between classical and Bayesian statistics which will not be discussed further here. Suffice it to say, our general approach regarding the estimation of measurement uncertainty will be classical

Fig. 2. Measurement uncertainty is best viewed as an interval symmetric about the mean and within which we claim the measureand lies with some stated level of probability

approach that can be justified to both the legal and accrediting communities.

Not all measurement processes are capable of providing a rigorous and statistically valid estimate of uncertainty. This fact is acknowledged by metrologists and by the ISO 17025 document in particular. (IEC/ISO 17025, 2000) For these situations, ISO 17025 requires that the analyst or laboratory at least identify the uncertainty components and make a reasonable effort to express the uncertainty. All of the published guides on measurement uncertainty recognize that every measurement context is different and there are multiple ways for estimation. Accordingly, forensic toxicologists should develop a well reasoned documented

Consider the following two separate blood alcohol concentrations measured on samples from two different individuals: **0.086 g/dL, 0.104 g/dL.** Which result presents the stronger inference that the subject's true blood alcohol concentration exceeds 0.080 g/dL? Very simply, we do not know. We have no information regarding the measurement process or the uncertainty for each. Now consider the same two results along with their two standard deviation uncertainty estimates: **0.086 ± 0.005 g/dL, 0.104 ± 0.027 g/dL.** From this we now see that the first results (0.086 ± 0.005 g/dL) provide the stronger evidence that the individual's true blood alcohol concentration exceeds 0.080 g/dL. Figure 3 illustrates this as well. The

explain here several practical ways this can be accomplished.

in nature.

Fig. 1. Measurement results, Y, are random representations from a distribution having a fixed mean and variance. The variance defines the random error while the mean relative to a reference defines their bias

Forensic toxicologists have a conceptual understanding of measurement uncertainty. However, most would probably find it difficult to actually compute a statistically valid estimate of the uncertainty, accounting for all relevant factors, and report it in an intuitive and comprehendible fashion for a jury to understand. For most analytical measurements performed by forensic toxicologists, both quantitative and qualitative, the formalization of measurement uncertainty is not generally considered or provided. This is due, in large part, to the lack of customer demand. The primary customers of forensic toxicologists are the courts and members of the legal community. They do not understand measurement uncertainty and are not aware of its relevance or importance. This, however, is changing. The legal community is becoming more aware of the concept and is now demanding it in several jurisdictions. The uncertainty allows the user to judge the quality and validity of the measurement results for a given application. Several factors have contributed to this renewed interest in measurement uncertainty. One is a recent report from the National Academy of Sciences in 2009. The NAS report states, "All results for every forensic science method should indicate the uncertainty in the measurements that are made,...". (NAS, 2009) The report was largely critical of the forensic sciences arguing the lack of a strong scientific foundation for their claims and practices. Another influencing factor has been the US Supreme Court decision in 1993 of Daubert vs. Merrell Dow Pharmaceuticals. The court required one of four criteria for admissibility to be "...the technique's known or potential rate of error...". (Daubert vs. Merrell Dow, 1993) The ruling requires that uncertainty be considered and accompany the introduction of measurement results in court. Finally, accrediting agencies are now requiring that forensic laboratories perform and report measurement uncertainty as part of their analytical protocol. The ASCLD/LAB-International accreditation program, for example, has adopted the ISO/IEC 17025 program and requires in part that, "...the laboratory estimate the measurement uncertainty for any area of testing or calibration where the customer makes the request or the jurisdiction or statute requires such". (ASCLD/LAB, 2011) These and other factors have now brought attention on this issue to measurement uncertainty. Forensic toxicologists need to address the issue and be prepared to compute, report and explain measurement uncertainty.

Fig. 1. Measurement results, Y, are random representations from a distribution having a fixed mean and variance. The variance defines the random error while the mean relative to a

Forensic toxicologists have a conceptual understanding of measurement uncertainty. However, most would probably find it difficult to actually compute a statistically valid estimate of the uncertainty, accounting for all relevant factors, and report it in an intuitive and comprehendible fashion for a jury to understand. For most analytical measurements performed by forensic toxicologists, both quantitative and qualitative, the formalization of measurement uncertainty is not generally considered or provided. This is due, in large part, to the lack of customer demand. The primary customers of forensic toxicologists are the courts and members of the legal community. They do not understand measurement uncertainty and are not aware of its relevance or importance. This, however, is changing. The legal community is becoming more aware of the concept and is now demanding it in several jurisdictions. The uncertainty allows the user to judge the quality and validity of the measurement results for a given application. Several factors have contributed to this renewed interest in measurement uncertainty. One is a recent report from the National Academy of Sciences in 2009. The NAS report states, "All results for every forensic science method should indicate the uncertainty in the measurements that are made,...". (NAS, 2009) The report was largely critical of the forensic sciences arguing the lack of a strong scientific foundation for their claims and practices. Another influencing factor has been the US Supreme Court decision in 1993 of Daubert vs. Merrell Dow Pharmaceuticals. The court required one of four criteria for admissibility to be "...the technique's known or potential rate of error...". (Daubert vs. Merrell Dow, 1993) The ruling requires that uncertainty be considered and accompany the introduction of measurement results in court. Finally, accrediting agencies are now requiring that forensic laboratories perform and report measurement uncertainty as part of their analytical protocol. The ASCLD/LAB-International accreditation program, for example, has adopted the ISO/IEC 17025 program and requires in part that, "...the laboratory estimate the measurement uncertainty for any area of testing or calibration where the customer makes the request or the jurisdiction or statute requires such". (ASCLD/LAB, 2011) These and other factors have now brought attention on this issue to measurement uncertainty. Forensic toxicologists need to address the issue and be prepared to compute, report and explain measurement uncertainty.

reference defines their bias

Moreover, providing the uncertainty along with measurement results is one important step in ensuring evidence-based inference. (Mnookin, et.al., 2011) We intend to illustrate and explain here several practical ways this can be accomplished.

Very basically, measurement uncertainty is best described by an interval, symmetric about the measurement result and within which we claim that the true value (the measurand) exists with some level of probability. The end points of this interval are called uncertainty or confidence limits. This interval quantifies the precision of the measurement result. Figure 2 illustrates this concept of uncertainty. The classical statistical view would state that the measurand (µ) is a fixed quantity and the measurement result along with the interval limits are random variables. The probability, therefore, relates to the random interval actually encompassing the fixed true value (µ). This involves some subtle distinctions between classical and Bayesian statistics which will not be discussed further here. Suffice it to say, our general approach regarding the estimation of measurement uncertainty will be classical in nature.

Fig. 2. Measurement uncertainty is best viewed as an interval symmetric about the mean and within which we claim the measureand lies with some stated level of probability

Not all measurement processes are capable of providing a rigorous and statistically valid estimate of uncertainty. This fact is acknowledged by metrologists and by the ISO 17025 document in particular. (IEC/ISO 17025, 2000) For these situations, ISO 17025 requires that the analyst or laboratory at least identify the uncertainty components and make a reasonable effort to express the uncertainty. All of the published guides on measurement uncertainty recognize that every measurement context is different and there are multiple ways for estimation. Accordingly, forensic toxicologists should develop a well reasoned documented approach that can be justified to both the legal and accrediting communities.

Consider the following two separate blood alcohol concentrations measured on samples from two different individuals: **0.086 g/dL, 0.104 g/dL.** Which result presents the stronger inference that the subject's true blood alcohol concentration exceeds 0.080 g/dL? Very simply, we do not know. We have no information regarding the measurement process or the uncertainty for each. Now consider the same two results along with their two standard deviation uncertainty estimates: **0.086 ± 0.005 g/dL, 0.104 ± 0.027 g/dL.** From this we now see that the first results (0.086 ± 0.005 g/dL) provide the stronger evidence that the individual's true blood alcohol concentration exceeds 0.080 g/dL. Figure 3 illustrates this as well. The

Measurement Uncertainty in Forensic Toxicology: Its Estimation, Reporting and Interpretation 419

2. EURACHEM/CITAC Guide, *Quantifying Uncertainty in Analytical Measurement:* (EURACHEM/CITAC, 2000) This document is similar to the *GUM* and provides all of the basic terminology and computations. The illustrated examples are more relevant to

3. NIST Technical Note 1297, *Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results:* (NIST, 1994) This document is brief but includes the key

All of these documents are available on the internet and can be downloaded free of charge. There are also a large number of other documents and guidelines regarding measurement uncertainty available on the internet. As one begins to read this large body of literature it soon becomes apparent that there is no consensus in the analytical sciences on the best

The measurement model is a mathematical function where the measurement result (the response variable) is expressed explicitly as a function of several input (predictor) variables.

The values of X in equation 2 may represent quality control results, bias estimates, traceability components, a total measurement method component, calibrant materials, etc. Moreover, the values of X may themselves be functions of other input variables. The

For additive models with independent input variables, the uncertainty is found from the root sum square (RSS) of the variance terms for each component as illustrated in equation 4:

1 2

For the multiplicative model with independent variables the uncertainty is found by

1 2

22 2 ... *<sup>n</sup>*

*XX X*

*CV CV CV*

*Y fX X X* 1 2 , ,..., *<sup>n</sup>* (2)

1 2 ... *YXX X <sup>n</sup>* (3)

22 2 ... *Y XX Xn u uu u* (4)

1 2 ... *Y XX X <sup>n</sup>* (5)

(6)

chemistry and may be more helpful to toxicologists.

approach to estimating measurement uncertainty.

function f may be additive as illustrated in equation 3:

*u* the variance estimate for the ith variable

The function f may, on the other hand, be multiplicative as in equation 5:

employing the RSS of the coefficients of variation squared as in equation 6:

*Y*

*Y*

*u*

**2. The measurement model** 

Equation 2 shows the general form:

where: Y = the measurement result Xi = the predictor or input variables

where: <sup>2</sup>

*Xi*

concepts and definitions. There are very few illustrated examples.

propagation.

"bottom-up" approach to uncertainty estimation. They generally begin with an assumed measurement model and then proceed to employ the general method of error

result of 0.104 g/dL actually has a significant probability that the true value is below 0.080 g/dL. This illustrates the additional value provided by measurement uncertainty, particularly in the cases near critical prohibited limits. Such information would be important for a court to consider.

Fig. 3. Including measurement uncertainty adds considerable information when interpreting measurement results near critical concentrations

#### **1.1 The meaning of Fit-for-purpose**

Fitness-for-purpose (FFP) is a very important concept in analytical measurements designed to be used in important decision making contexts. FFP is the assurance that a measurement result will be suitable or appropriate for its intended applications. FFP is closely associated with uncertainty and the confidence that is necessary for a measurement result in a particular application. Measurement results in forensic toxicology have significant implications for the rights and property of individuals. Major consequences result from their interpretation in a legal context. For this reason, measurement results generated by forensic toxicologists must have a high level of confidence with minimum uncertainty to ensure their FFP. Determining the FFP in forensic toxicology can be challenging. (Thompson and Fearn, 1996) Toxicologists and customers should both contribute to establishing the appropriate FFP in a forensic context. Forensic toxicologists should continually strive to optimize their process and enhance the quality.

#### **1.2 Published resources**

There are a few important resource documents regarding measurement uncertainty that should be read and kept as references by the forensic toxicologist. These represent standards in the field of metrology. They are rigorous and well grounded theoretically. However, this does not mean there is uniform acceptance of these documents. There is a great deal of literature debating their application and interpretation. (Bich and Harris, 2006, Deldossi and Zappa 2009, Kacker,et.al. 2007, Kacker,et.al. 2010, Krouwer, 2003, Kristiansen, 2003) Three references of significant importance are:

1. *Guide to the Expression of Uncertainty in Measurement (GUM):* (ISO, 2008) This is commonly referred to as the *GUM* document and is published by ISO along with several other international standards organizations. The *GUM* provides primarily a "bottom-up" approach to uncertainty estimation. They generally begin with an assumed measurement model and then proceed to employ the general method of error propagation.


All of these documents are available on the internet and can be downloaded free of charge. There are also a large number of other documents and guidelines regarding measurement uncertainty available on the internet. As one begins to read this large body of literature it soon becomes apparent that there is no consensus in the analytical sciences on the best approach to estimating measurement uncertainty.

#### **2. The measurement model**

418 Toxicity and Drug Testing

result of 0.104 g/dL actually has a significant probability that the true value is below 0.080 g/dL. This illustrates the additional value provided by measurement uncertainty, particularly in the cases near critical prohibited limits. Such information would be important

Fig. 3. Including measurement uncertainty adds considerable information when interpreting

Fitness-for-purpose (FFP) is a very important concept in analytical measurements designed to be used in important decision making contexts. FFP is the assurance that a measurement result will be suitable or appropriate for its intended applications. FFP is closely associated with uncertainty and the confidence that is necessary for a measurement result in a particular application. Measurement results in forensic toxicology have significant implications for the rights and property of individuals. Major consequences result from their interpretation in a legal context. For this reason, measurement results generated by forensic toxicologists must have a high level of confidence with minimum uncertainty to ensure their FFP. Determining the FFP in forensic toxicology can be challenging. (Thompson and Fearn, 1996) Toxicologists and customers should both contribute to establishing the appropriate FFP in a forensic context. Forensic toxicologists should continually strive to optimize their

There are a few important resource documents regarding measurement uncertainty that should be read and kept as references by the forensic toxicologist. These represent standards in the field of metrology. They are rigorous and well grounded theoretically. However, this does not mean there is uniform acceptance of these documents. There is a great deal of literature debating their application and interpretation. (Bich and Harris, 2006, Deldossi and Zappa 2009, Kacker,et.al. 2007, Kacker,et.al. 2010, Krouwer, 2003, Kristiansen, 2003) Three

1. *Guide to the Expression of Uncertainty in Measurement (GUM):* (ISO, 2008) This is commonly referred to as the *GUM* document and is published by ISO along with several other international standards organizations. The *GUM* provides primarily a

for a court to consider.

measurement results near critical concentrations

**1.1 The meaning of Fit-for-purpose** 

process and enhance the quality.

references of significant importance are:

**1.2 Published resources** 

The measurement model is a mathematical function where the measurement result (the response variable) is expressed explicitly as a function of several input (predictor) variables. Equation 2 shows the general form:

$$Y = f\left(X\_1, X\_2, \dots, X\_n\right) \tag{2}$$

where: Y = the measurement result

Xi = the predictor or input variables

The values of X in equation 2 may represent quality control results, bias estimates, traceability components, a total measurement method component, calibrant materials, etc. Moreover, the values of X may themselves be functions of other input variables. The function f may be additive as illustrated in equation 3:

$$Y = X\_1 + X\_2 + \dots + X\_n \tag{3}$$

For additive models with independent input variables, the uncertainty is found from the root sum square (RSS) of the variance terms for each component as illustrated in equation 4:

$$
\mu\_Y = \sqrt{\mu\_{X\_1}^2 + \mu\_{X\_2}^2 + \dots + \mu\_{X\_n}^2} \tag{4}
$$

where: <sup>2</sup> *Xi u* the variance estimate for the ith variable

The function f may, on the other hand, be multiplicative as in equation 5:

$$Y = X\_1 \cdot X\_2 \cdot \dots \cdot X\_n \tag{5}$$

For the multiplicative model with independent variables the uncertainty is found by employing the RSS of the coefficients of variation squared as in equation 6:

$$\frac{\mu\_{\overline{Y}}}{\overline{Y}} = \sqrt{\text{CV}\_{X\_1}^2 + \text{CV}\_{X\_2}^2 + ... + \text{CV}\_{X\_n}^2} \tag{6}$$

commercial vendor

**3. Traceability** 

Measurement Uncertainty in Forensic Toxicology: Its Estimation, Reporting and Interpretation 421

*C C R X*

<sup>0</sup> *Sol Corr*

*Y GC R <sup>Y</sup> X K GC*

*R* the traceable reference value of alcohol in water solutions purchased from a

*X* the mean of the breath test instrument measuring the simulator solution heated to 340C

*GCCont* the mean results from measuring the traceable controls on the gas chromatograph Notice also that equation 12 is simply a set of correction factors that adjust for bias in the gas

> 0 0 *Sol Sol Corr Inst GC Cont Cont Y GC R GC <sup>R</sup> <sup>Y</sup> <sup>Y</sup> Yf f X K GC X K GC*

The uncertainty estimates for R and K will generally be Type B estimates available from certificates of analysis or other documentation. The other four factors will be Type A estimates since they are based on actual experimental results. The uncertainty computation for equation 13 can be determined from employing either the RSS method of equation 6 (since the function is multiplicative) or the error propagation method of equation 8. Both will yield the same estimate. We have illustrated only a few of the many measurement functions that may be relevant for forensic toxicologists. More examples are found in the *EURACHEM/CITAC Guide* as well as other literature sources*.* (Kristiansen and Peterson, 2004) The important point is to try and develop a model best describing the measurement process which will facilitate selecting the most appropriate uncertainty computation to perform. Where the measurement model is unknown it is common to assume a multiplicative form. The justification for this is the fact that variation generally increases

Traceability is defined within the *VIM* document as a "...property of a measurement result whereby the result can be related to a reference through a documented unbroken chain of

with concentration, a property of a multiplicative model. (Kristiansen, 2001)

*u n u n*

*C C R X* 

*GCSol* the mean of the simulator solution measurements by gas chromatography

*K* 1.23 the ratio of partition coefficients relating to the simulator heated to 340C

0

where: *Y*<sup>0</sup> the mean of the original n measurements

chromatograph as well as in the breath test instrument:

0

*GC f* correction factor for the gas chromatograph

where: *Inst f* correction factor for the breath test instrument

*Corr*

*Corr*

2 22

*C R X*

*CV CV CV*

0 0

0

*Cont*

<sup>2</sup> <sup>2</sup> 2

(12)

(11)

(13)

*<sup>C</sup> <sup>X</sup>*

*u u*

Notice also that equation 6 incorporates the mean *Y* and yields the standard deviation of the mean. This will result when we incorporate the appropriate sample sizes (values of n) for each term within the radical sign of equation 6. The function f may even be a combination of additive and multiplicative terms as in equation 7:

$$Y = \frac{X\_1 \cdot X\_2}{X\_3 + X\_4} - X\_5 \tag{7}$$

In this case the uncertainty must be estimated by employing the general method of error propagation. The equation for this estimation is derived from the first-order (linear term) of the Taylor series expansion: (Ku, 1966)

$$\frac{\mu\_{\overline{Y}}}{\overline{Y}} = \sqrt{\left[\frac{\partial Y}{\partial X\_1}\right]^2 \mu\_{X\_1}^2 + \left[\frac{\partial Y}{\partial X\_2}\right]^2 \mu\_{X\_2}^2 + \dots + \left[\frac{\partial Y}{\partial X\_n}\right]^2 \mu\_{X\_n}^2} \tag{8}$$

Equation 8 also assumes that all of the input variables are independent. When this is not the case, a covariance term must be added as seen in equation 9:

$$\frac{\mu\_{\overline{Y}}}{\overline{Y}} = \sqrt{\sum\_{i=1}^{n} \left[\frac{\partial Y}{\partial X\_{i}}\right]^{2}} \mu\_{X\_{i}}^{2} + 2 \left[\frac{\partial Y}{\partial X\_{i}}\right] \left[\frac{\partial Y}{\partial X\_{j}}\right] \text{Cov}\left(X\_{i}, X\_{j}\right) \tag{9}$$

where:

$$\operatorname{Cov} \left( \mathbf{X}\_{i'} \mathbf{X}\_j \right) = r\_{\left( \mathbf{X}\_{i'} \mathbf{X}\_j \right)} \mathbf{S}\_{X\_i} \mathbf{S}\_{X\_j}$$

The value of r in equation 9 is the correlation coefficient between the two input variables. For each pair of input variables that are correlated an additional covariance term would need to be added. A simple example of a concentration measurement function that could apply to either blood or breath alcohol measurement is shown in equation 10:

$$C\_{Corr} = \frac{C\_0 \, R}{\overline{X}} \tag{10}$$

where: CCorr = the corrected measurement concentration result C0 = the raw measurement results (either a mean or a single observation) R = the traceable reference control value

*X* the mean results from measuring the control reference standard (R)

Since equation 10 is multiplicative and we assume all three variables are independent we could employ the RSS for the CV's squared according to equation 11. Notice that we have incorporated the values of n, which may vary for each term, where this information is known. This will result in *CCorr u* representing the standard deviation (or standard error) of the mean. Equation 12 illustrates a more complicated model that may represent the measurement of breath alcohol concentration. Bias in the breath test instrument is adjusted for by measuring controls which have been measured by gas chromatography and which in turn has had its bias accounted for by measuring other traceable controls.

$$\frac{\mu\_{\text{C}\_{\text{Corr}}}}{\text{C}\_{\text{Corr}}} = \sqrt{\text{CV}\_{\text{C}\_{0}}^{2} + \text{CV}\_{R}^{2} + \text{CV}\_{X}^{2}} = \sqrt{\left[\frac{\frac{\mu\_{\text{C}\_{0}}}{\sqrt{n\_{\text{C}\_{0}}}}}{\text{C}\_{0}}\right]^{2} + \left[\frac{\mu\_{R}}{R}\right]^{2} + \left[\frac{\frac{\mu\_{X}}{\sqrt{n\_{X}}}}{\overline{X}}\right]^{2}} \tag{11}$$

$$\overline{Y}\_{\text{Corr}} = \frac{Y\_0 \cdot \text{GC}\_{\text{Sol}} \cdot R}{\overline{X} \cdot K \cdot \text{GC}\_{\text{Cont}}} \tag{12}$$

where: *Y*<sup>0</sup> the mean of the original n measurements

*GCSol* the mean of the simulator solution measurements by gas chromatography

*R* the traceable reference value of alcohol in water solutions purchased from a commercial vendor

*X* the mean of the breath test instrument measuring the simulator solution heated to 340C *K* 1.23 the ratio of partition coefficients relating to the simulator heated to 340C

*GCCont* the mean results from measuring the traceable controls on the gas chromatograph Notice also that equation 12 is simply a set of correction factors that adjust for bias in the gas chromatograph as well as in the breath test instrument:

$$\overline{Y}\_{\text{Corr}} = \frac{\overline{Y}\_0 \cdot \text{GC}\_{\text{Sol}} \cdot R}{\overline{X} \cdot K \cdot \text{GC}\_{\text{Cont}}} = \overline{Y}\_0 \cdot \left[\frac{\text{GC}\_{\text{Sol}}}{\overline{X} \cdot K}\right] \cdot \left[\frac{R}{\text{GC}\_{\text{Cont}}}\right] = \overline{Y}\_0 \cdot f\_{\text{Inst}} \cdot f\_{\text{GC}} \tag{13}$$

where: *Inst f* correction factor for the breath test instrument

*GC f* correction factor for the gas chromatograph

The uncertainty estimates for R and K will generally be Type B estimates available from certificates of analysis or other documentation. The other four factors will be Type A estimates since they are based on actual experimental results. The uncertainty computation for equation 13 can be determined from employing either the RSS method of equation 6 (since the function is multiplicative) or the error propagation method of equation 8. Both will yield the same estimate. We have illustrated only a few of the many measurement functions that may be relevant for forensic toxicologists. More examples are found in the *EURACHEM/CITAC Guide* as well as other literature sources*.* (Kristiansen and Peterson, 2004) The important point is to try and develop a model best describing the measurement process which will facilitate selecting the most appropriate uncertainty computation to perform. Where the measurement model is unknown it is common to assume a multiplicative form. The justification for this is the fact that variation generally increases with concentration, a property of a multiplicative model. (Kristiansen, 2001)

#### **3. Traceability**

420 Toxicity and Drug Testing

Notice also that equation 6 incorporates the mean *Y* and yields the standard deviation of the mean. This will result when we incorporate the appropriate sample sizes (values of n) for each term within the radical sign of equation 6. The function f may even be a combination of

1 2

3 4 *X X Y X X X* 

In this case the uncertainty must be estimated by employing the general method of error propagation. The equation for this estimation is derived from the first-order (linear term) of

1 2

 

Equation 8 also assumes that all of the input variables are independent. When this is not the

 , , *<sup>i</sup> <sup>j</sup> i j Cov X X r S S <sup>i</sup> <sup>j</sup> X X X X*

0

Since equation 10 is multiplicative and we assume all three variables are independent we could employ the RSS for the CV's squared according to equation 11. Notice that we have incorporated the values of n, which may vary for each term, where this information is known. This will result in *CCorr u* representing the standard deviation (or standard error) of the mean. Equation 12 illustrates a more complicated model that may represent the measurement of breath alcohol concentration. Bias in the breath test instrument is adjusted for by measuring controls which have been measured by gas chromatography and which in

The value of r in equation 9 is the correlation coefficient between the two input variables. For each pair of input variables that are correlated an additional covariance term would need to be added. A simple example of a concentration measurement function that could

1 2

2 2

*u Y YY*

*Y X XX* 

apply to either blood or breath alcohol measurement is shown in equation 10:

C0 = the raw measurement results (either a mean or a single observation)

*X* the mean results from measuring the control reference standard (R)

turn has had its bias accounted for by measuring other traceable controls.

*Corr C R <sup>C</sup>*

*u YY Y*

*Y XX X*

*i i i j*

2 2 2 22 2

*XX X*

2 , *<sup>i</sup>*

*u Cov X X*

*X i j*

*uu u*

... *<sup>n</sup>*

(9)

*n*

*<sup>X</sup>* (10)

5

(7)

(8)

additive and multiplicative terms as in equation 7:

the Taylor series expansion: (Ku, 1966)

where:

*Y*

case, a covariance term must be added as seen in equation 9:

1

where: CCorr = the corrected measurement concentration result

R = the traceable reference control value

*<sup>n</sup> <sup>Y</sup>*

Traceability is defined within the *VIM* document as a "...property of a measurement result whereby the result can be related to a reference through a documented unbroken chain of

Measurement Uncertainty in Forensic Toxicology: Its Estimation, Reporting and Interpretation 423

following eight basic steps for estimating measurement uncertainty that should generally

5. Combine the standard uncertainties for each component and compute the combined

Next, we present these steps in some detail. In addition we will present an example of blood alcohol measurement by gas chromatography and illustrate how each of the steps can be applied. We will assume duplicate blood alcohol results of 0.081 and 0.082 g/dL for this

It is very important that the customer and the toxicologist have a clear understanding of exactly the property being measured. Interpretation will then be applied to a specific measurand in a specific context where FFP can be appropriately determined. For our example we will assume that the measurand is the venous whole blood alcohol

We will assume the following basic model for our measurement of blood alcohol

<sup>0</sup> *corr dilutor*

Equation 14 is a basic multiplicative model that includes four components of uncertainty

From equation 14 we see four components that contribute to the combined uncertainty in the corrected BAC. These include: (1) the original duplicate measurement results of the blood alcohol concentration, (2) the reference value (R) representing a traceable unbiased control standard purchased from a commercial laboratory having a certificate of analysis, (3) the mean of the replicate measurements *X* of the traceable control and (4) the correction factor *fdilutor* for the dilutor used in preparing both the controls and blood samples before

*C R C f <sup>X</sup>* (14)

concentration collected from a specific individual at a specific time and location.

apply for most quantitative measurements in forensic toxicology: 1. Clearly define the property to be measured (the measurand)

6. Compute the expanded uncertainty and the confidence interval

4. Quantify the standard uncertainty for each component

3. Identify the components contributing to the measurement uncertainty

2. Identify the measurement function

7. Produce the uncertainty budget

**4.1 Clearly define the measurand** 

**4.2 Identify the measurement function** 

where: Ccorr = the corrected BAC results

R = the traceable reference control value

fdilutor = the correction factor for the dilutor

**4.3 Identify the components of uncertainty** 

and corrects for analytical bias.

analysis. We will assume 1 *dilutor f* .

C0 = the mean of the original measurement results

*X* the mean results from measuring the controls

concentration (BAC) by headspace gas chromatography:

uncertainty

8. Report the results

example.

calibrations, each contributing to the measurement uncertainty". (ISO/VIM, 2008) Figure 3 illustrates this concept of traceability which links a measurement result (breath alcohol) to a national metrological authority with each link propagating its own uncertainty. The magnitude of uncertainty will increase with each additional level of the metrological chain. Since standards are imperfect there is the associated uncertainty that must be included as part of the final combined measurement uncertainty. The ultimate reference is usually a property maintained and defined by some metrological authority such as a National Metrological Institute (NMI). Chemical analytes are generally considered traceable to a method or standard reference material (SRM) such as NIST 1828b. There are other intermediate standards often used between the measurement result and the NMI. These are referred to as Certified Reference Materials (CRM) or simply Reference Materials (RM). (Thompson, 1997) Traceability is important for establishing the property of comparability and to determine and correct for bias. Uncertainty information regarding traceable standards are found on the certificates of analysis (COA).

Fig. 4. Illustrating traceability where a measurement result is linked through an unbroken chain of comparisons to the national metrological authority

#### **4. Practical steps for estimating measurement uncertainty**

There are several valid approaches to estimating and quantifying measurement uncertainty. For our present purposes, we will present a very general "bottom-up" corresponding to the *GUM* document. Later, we will discuss other approaches as well. We will assume the following eight basic steps for estimating measurement uncertainty that should generally apply for most quantitative measurements in forensic toxicology:


422 Toxicity and Drug Testing

calibrations, each contributing to the measurement uncertainty". (ISO/VIM, 2008) Figure 3 illustrates this concept of traceability which links a measurement result (breath alcohol) to a national metrological authority with each link propagating its own uncertainty. The magnitude of uncertainty will increase with each additional level of the metrological chain. Since standards are imperfect there is the associated uncertainty that must be included as part of the final combined measurement uncertainty. The ultimate reference is usually a property maintained and defined by some metrological authority such as a National Metrological Institute (NMI). Chemical analytes are generally considered traceable to a method or standard reference material (SRM) such as NIST 1828b. There are other intermediate standards often used between the measurement result and the NMI. These are referred to as Certified Reference Materials (CRM) or simply Reference Materials (RM). (Thompson, 1997) Traceability is important for establishing the property of comparability and to determine and correct for bias. Uncertainty information regarding traceable

u1

u2

u3

u4

Fig. 4. Illustrating traceability where a measurement result is linked through an unbroken

There are several valid approaches to estimating and quantifying measurement uncertainty. For our present purposes, we will present a very general "bottom-up" corresponding to the *GUM* document. Later, we will discuss other approaches as well. We will assume the

chain of comparisons to the national metrological authority

**4. Practical steps for estimating measurement uncertainty** 

standards are found on the certificates of analysis (COA).

Next, we present these steps in some detail. In addition we will present an example of blood alcohol measurement by gas chromatography and illustrate how each of the steps can be applied. We will assume duplicate blood alcohol results of 0.081 and 0.082 g/dL for this example.
