**5. Breath alcohol measurement example**

Our next example illustrates the uncertainty estimation for a breath alcohol measurement. We will assume the following measurement function which was presented earlier as equation 12:

$$
\overline{Y}\_{\text{Corr}} = \frac{\overline{Y}\_0 \cdot \text{GC}\_{\text{Sol}} \cdot R}{\overline{X} \cdot K \cdot \text{GC}\_{\text{Cont}}} \tag{22}
$$

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

0

*Y*

*Y*

*Y*

*Y*

*u*

0

0

0.0031 2

0.0824 0.0290 0.00239 / 210 *Yu g L*

0.0824 0.0824

rewrite equation 24 as follows:

Measurement Uncertainty in Forensic Toxicology: Its Estimation, Reporting and Interpretation 431

2 2 2 2 2 2

*Sol Cont*

2 22 2 22 0.0007 0.0003 0.0012 0.012 0.0006 15 1 1 10 28 0.0985 0.100 0.0795 1.23 0.1015

> 

(24)

(25)

(23)

 

*Y Y*

*(1- / 2)*

The approximate 95% uncertainty interval estimated for this example shows that the lower limit falls below the critical legal driving level of 0.080 g/210L. We may be interested in knowing the probability that the true population mean BrAC is above 0.080 g/210L. This

Since we are interested in determining the probability that µ exceeds the lower limit we

We set the lower limit expressed in equation 25 equal to 0.080 g/210L and solve for Z(1-α/2):

*(1- / 2)* 24 0 00239 0 *(1- / 2) (1- / 2)* 1.0 *<sup>Y</sup> Y - S = 0.080 0.08 - . = .080 = Z ZZ*

Fig. 6. An uncertainty function plotting pooled standard deviation estimates against their

**Breath Alcohol Concentration (g/210L) 0.00 0.05 0.10 0.15 0.20 0.25 0.30**

**Breath Alcohol SD = 0.0260BrAC + 0.00095 n = 27,995 duplicates**

concentration determined from a large number of duplicate breath alcohol results

 

can be estimated by first considering our confidence interval in the following form:

 *P Y - S Y + S = Z Z (1- / 2)* 

*(1- / 2) <sup>Y</sup> P Y - S = <sup>Z</sup>*

**Standard Deviation (g/210L)**

**0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014**

2 2 22 22

*<sup>u</sup> CV CV CV CV CV CV*

*Y X GC R K GC*

*Sol Cont*

*u u u u u u u n n nnn n Y Y GC R X K GC*

*Corr Sol Cont*

*Y GC R K X GC*

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

*K* 1.23 the ratio of partition coefficients

*GCCont* the mean results from measuring the traceable controls on the gas chromatograph

For this example we assume that simulator solutions are prepared and tested by gas chromatography within the toxicology laboratory. Commercially purchased standards (CRM) are used as calibrators and controls on the gas chromatograph. Certificates of analysis are used as Type B uncertainties to establish the traceability. For this example we will assume the following data are available for the six components of equation 22: Duplicate BrAC results: 0.081 and 0.085 g/210L,*Y*<sup>0</sup> 0.0830 g/210L, : *GCSol* mean = 0.0985 g/dL u = 0.0007 g/dL n=15, *R* 0.100 g/dL u = 0.0003 g/dL, *X*: mean = 0.0795 g/210L u = 0.0012 g/210L n=10, *K* 1.23 u = 0.012 and : *GCCont* mean = 0.1015 g/dL u = 0.0006 g/dL n=28. We begin by computing the corrected mean BrAC results according to:

$$\overline{Y}\_{\text{Corr}} = \frac{\left(0.0830 \,\text{g} \,/\, 210 \,\text{L}\right) \left(0.0985 \,\text{g} \,/\,\text{dL}\right) \left(0.100 \,\text{g} \,/\,\text{dL}\right)}{\left(0.0795 \,\text{g} \,/\, 210 \,\text{L}\right) \left(1.23\right) \left(0.1015 \,\text{g} \,/\,\text{dL}\right)} = 0.0824 \,\text{g} \,/\, 210 \,\text{L}$$

The estimate for the uncertainty in*Y*0 will come from an uncertainty function seen in figure 6 and developed from a large number of duplicate breath alcohol tests using equation 15. The total method uncertainty for our example determined from the linear model in figure 6 and using the corrected mean BrAC of 0.0824 g/210L is 0.0031 g/210L. Since our model in equation 22 is multiplicative we employ the RSS for the CV values and assume independence amongst all components. The combined uncertainty estimate is seen in equation 23. Next we estimate the 95% uncertainty interval and obtain:

$$\begin{array}{rcl} \overline{Y} \pm k\mu\_{\mathbb{C}} & \Rightarrow & \overline{Y} \pm \mathcal{U} \Rightarrow & 0.0824 \pm 2(0.00239) \Rightarrow & 0.0824 \pm 0.0048 \\ & & 0.0776 \text{ to } 0.0872 \text{ g } / \, 210 \text{L} \end{array}$$

Since the n for estimating the uncertainty function in figure 6 was very large, we assume an infinite degrees of freedom and use k=2 for estimating an approximate 95% confidence interval. Table 3 shows the uncertainty budget for this analysis. From the uncertainty budget we see that the total method accounted for the majority of the combined uncertainty (84%). This is not surprising since the breath sampling component, contained within the total method uncertainty function of figure 6, has significant variation. The budget also shows that the reference traceability, the GC measurement of the controls and the GC measurement of the simulator solution all provide 1% or less to the combined uncertainty. They could reasonably be ignored in this example. We now report our results as follows:

*The duplicate breath alcohol results were 0.081 and 0.085 g/210L with a corrected mean result of 0.0824 g/210L. An expanded combined uncertainty of 0.0048g/210L assuming a coverage factor of k=2 with an infinite number of degrees-of-freedom and a normal distribution was generated from six principle components contributing to the uncertainty. An approximate 95% confidence interval for the true mean breath alcohol concentration is 0.0776 to 0.0872 g/210L.* 

$$\begin{aligned} \frac{u\_{\overline{Y}}}{Y} &= \sqrt{\text{CV}\_{\overline{Y}}^2 + \text{CV}\_{\text{C}\_{\text{Col}}}^2} + \text{CV}\_{\text{R}}^2 + \text{CV}\_{\text{X}}^2 + \text{CV}\_{\text{K}}^2 + \text{CV}\_{\text{CC}\_{\text{Cont}}}^2} \\ \frac{u\_{\overline{Y}}}{Y} &= \sqrt{\begin{bmatrix} \frac{u\_{\overline{Y}\_0}}{\sqrt{n}} \\ \frac{u\_{\overline{Y}}}{\sqrt{n}} \end{bmatrix}^2 + \left[ \frac{\overline{u\_{\text{C}\_{\text{Col}}}}}{\sqrt{n}} \right]^2 + \left[ \frac{\overline{u\_{\overline{R}}}}{\overline{R}} \right]^2 + \left[ \frac{\overline{u\_{\overline{X}}}}{\overline{X}} \right]^2 + \left[ \frac{\overline{u\_{\overline{R}}}}{\overline{R}} \right]^2 + \left[ \frac{\overline{u\_{\overline{X}}}}{\overline{Q}} \right]^2 \end{aligned} \tag{23}$$

$$\frac{u\_{\overline{Y}}}{0.0824} = \sqrt{\begin{bmatrix} 0.0031\\ \sqrt{2} \\ 0.0824 \end{bmatrix}^2 + \left[ \frac{0.0007}{0.0985} \right]^2 + \left[ \frac{0.0003}{0.100} \right]^2 + \left[ \frac{0.0012}{0.0795} \right]^2 + \left[ \frac{0.012}{1.23} \right]^2 + \left[ \frac{0.012}{0.1015} \right]^2} + \left[ \frac{0.0006}{0.1015} \right]^2} \tag{24}$$

$$\mu\_{\overline{Y}} = 0.0824 \, (0.0290) = 0.00239 \, \_{\mathcal{S}} / \, 210L$$

430 Toxicity and Drug Testing

*X* 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

For this example we assume that simulator solutions are prepared and tested by gas chromatography within the toxicology laboratory. Commercially purchased standards (CRM) are used as calibrators and controls on the gas chromatograph. Certificates of analysis are used as Type B uncertainties to establish the traceability. For this example we will assume the following data are available for the six components of equation 22: Duplicate BrAC results: 0.081 and 0.085 g/210L,*Y*<sup>0</sup> 0.0830 g/210L, : *GCSol* mean = 0.0985 g/dL u = 0.0007 g/dL n=15, *R* 0.100 g/dL u = 0.0003 g/dL, *X*: mean = 0.0795 g/210L u = 0.0012 g/210L n=10, *K* 1.23 u = 0.012 and : *GCCont* mean = 0.1015 g/dL u = 0.0006 g/dL

n=28. We begin by computing the corrected mean BrAC results according to:

equation 23. Next we estimate the 95% uncertainty interval and obtain:

*the true mean breath alcohol concentration is 0.0776 to 0.0872 g/210L.* 

*Y ku Y U <sup>C</sup>*

 

*g L g dL*

0.0776 0.0872 / 210

*to g L* 

Since the n for estimating the uncertainty function in figure 6 was very large, we assume an infinite degrees of freedom and use k=2 for estimating an approximate 95% confidence interval. Table 3 shows the uncertainty budget for this analysis. From the uncertainty budget we see that the total method accounted for the majority of the combined uncertainty (84%). This is not surprising since the breath sampling component, contained within the total method uncertainty function of figure 6, has significant variation. The budget also shows that the reference traceability, the GC measurement of the controls and the GC measurement of the simulator solution all provide 1% or less to the combined uncertainty. They could reasonably be ignored in this example. We now report our results

*The duplicate breath alcohol results were 0.081 and 0.085 g/210L with a corrected mean result of 0.0824 g/210L. An expanded combined uncertainty of 0.0048g/210L assuming a coverage factor of k=2 with an infinite number of degrees-of-freedom and a normal distribution was generated from six principle components contributing to the uncertainty. An approximate 95% confidence interval for* 

0.0830 / 210 0.0985 / 0.100 / 0.0824 / 210 0.0795 / 210 1.23 0.1015 / *Corr g L g dL g dL <sup>Y</sup> <sup>g</sup> <sup>L</sup>*

The estimate for the uncertainty in*Y*0 will come from an uncertainty function seen in figure 6 and developed from a large number of duplicate breath alcohol tests using equation 15. The total method uncertainty for our example determined from the linear model in figure 6 and using the corrected mean BrAC of 0.0824 g/210L is 0.0031 g/210L. Since our model in equation 22 is multiplicative we employ the RSS for the CV values and assume independence amongst all components. The combined uncertainty estimate is seen in

0.0824 2 0.00239 0.0824 0.0048

*K* 1.23 the ratio of partition coefficients

chromatograph

as follows:

The approximate 95% uncertainty interval estimated for this example shows that the lower limit falls below the critical legal driving level of 0.080 g/210L. We may be interested in knowing the probability that the true population mean BrAC is above 0.080 g/210L. This can be estimated by first considering our confidence interval in the following form:

$$P\left[\overline{Y} \cdot \mathbf{Z}\_{\text{(l-a/2)}} \mathbf{S}\_{\overline{Y}} \le \mu \le \overline{Y} \star \mathbf{Z}\_{\text{(l-a/2)}} \mathbf{S}\_{\overline{Y}}\right] = \pi \tag{24}$$

Since we are interested in determining the probability that µ exceeds the lower limit we rewrite equation 24 as follows:

$$P\left[\overline{Y} \text{ - } \mathbf{Z}\_{\left(1 \cdot a/2\right)} \, \mathrm{S}\_{\overline{Y}} \le \, \mu \le \infty\right] = \pi \tag{25}$$

We set the lower limit expressed in equation 25 equal to 0.080 g/210L and solve for Z(1-α/2):

*(1- / 2)* 24 0 00239 0 *(1- / 2) (1- / 2)* 1.0 *<sup>Y</sup> Y - S = 0.080 0.08 - . = .080 = Z ZZ* 

Fig. 6. An uncertainty function plotting pooled standard deviation estimates against their concentration determined from a large number of duplicate breath alcohol results

Measurement Uncertainty in Forensic Toxicology: Its Estimation, Reporting and Interpretation 433

and insignificant and not correct for it. There are ways to handle uncorrected bias as well by adding an additional component to the combined uncertainty. We will consider some examples here. Estimations for bias can come from internal quality control, proficiency test

We will assume in this example that we desire to prepare an ethanol in water solution to be used as a control standard. We want to prepare this solution to have a concentration of approximately 0.10 g/dL. Our measurement function will be as follows: (Philipp et.al., 2010)

> *Etoh Solution m PD <sup>C</sup>*

Preparing a control standard gravimetrically has advantages. (Gates, et.al., 2009) There is better traceability for the mass measurements and no concern regarding the uncertainty in volume measurements. We will assume the purity (P) to be 0.995 with a Type B standard uncertainty of 0.002 determined from the certificate of analysis. We further assume that the density (D) of the solution is 0.997 g/ml (OIML, King and Lawn, 1999) with a Type B standard uncertainty of 0.00054 g/ml (King and Lawn, 1999), determined from the certificate of analysis from the manufacturer of a density meter. For both the purity and the density we will assume the uniform distribution in order to estimate their standard uncertainties. The values for the density are obtained from published tables for ethanol/water solutions. The density of the solution will be a function of the mass fraction of ethanol. The higher the mass fraction of ethanol the closer the density will be to 0.789 g/ml - the density of pure ethanol. The lower the mass fraction of ethanol the closer the density will be to 1.00 g/ml - the density of water. Since the density of the solution depends on the mass fraction of ethanol and we have selected a density of 0.997 g/ml (corresponding to a mass fraction of approximately 0.101%) and we desire a total solution mass of 1800 g, we need to have the mass of ethanol equal to 1.82 g. We will need to weigh 1.82 g of ethanol and place it into solution with water and add water until we have a total mass of 1800g. We will assume that the total solution mass is weighed on a scale that has had replicate measurements (n=30) of a 2 Kg traceable check weight (Type B uncertainty of 0.016 Kg) with a mean result of 1,940 g and a standard uncertainty of 30 g. This will be used to estimate the standard uncertainty in the measurement of *mSolution* . We now recognize that there is a bias in the weighing of the total solution. The measured mass of the solution is low by 3.0%. This will affect the mass of the ethanol necessary to maintain the density of 0.997 and mass fraction of 0.101%. As a result the mass of the ethanol will need to be 1.87 g. The mass of ethanol was weighed on a different scale that also has a set of replicate measurements (n=23) of a 2.0 g traceable check weight (Type B standard uncertainty of 0.014g) with a mean result of 2.08 g and a standard uncertainty of 0.02g. This scale has a bias of +4.0%. We now

mSolution = the mass measurement of the combined solution of ethanol and water

incorporate our assumed measurement information into equation 26:

*<sup>m</sup>* (26)

data, collaborative studies or method validation data. (Kane, 1997)

**6.1 Preparing an alcohol in water control solution** 

where: C = the concentration of ethanol in water mEtoh = the mass measurement of ethanol

P = the purity of the ethanol D = the density of the ethanol


1 Percent of contribution to total combined uncertainty

Table 3. Uncertainty budget for the illustrated breath alcohol example

Next, we rearrange our probability statement, introduce the value for Z(1-α/2), and refer to the standard normal tables:

$$\mathbb{P}\left[\overline{Y} \text{ - } \mathbf{Z}\_{1 \cdot \alpha/2} \, \mathbf{S}\_{\overline{Y}} \le \mu\right] = \mathbb{P}\left[\frac{\overline{Y} \text{ - } \mu}{\mathbf{S}\_{\overline{Y}}} \le \mathbf{Z}\_{1 - \alpha/2}\right] = \mathbb{P}\left[\mathbf{Z} \le \mathbf{Z}\_{1 \cdot \alpha/2}\right] = \mathbb{P}\left[\mathbf{Z} \le \mathbf{1}.0\right] = 0.8413$$

There is a probability of 0.8413 that the individual's true mean BrAC exceeds 0.080 g/100ml. This may or may not rise to the level of proof beyond a reasonable doubt, depending on the opinion of the court. This example illustrated the use of simulator control standards produced within a local toxicology laboratory including their associated uncertainties. Some jurisdictions, however, choose to purchase simulator control standards rather than prepare their own. If that were the case in this example, we could have eliminated the GC solutions and GC controls from our uncertainty estimates. The simulator partition coefficient would have remained while the reference value would have been obtained from the certificate of analysis from the manufacturer and considered a Type B uncertainty. Therefore, rather than having to include the GC solution and GC control components separately in the combined uncertainty estimate, they should already be included within the manufacturer's estimate of combined uncertainty, depending, of course, on how the solution standards were prepared and tested.

#### **6. Dealing with measurement bias**

Our principle objective here will be to illustrate several ways for treating uncorrected bias. Bias or systematic error is common in all measurements. Some consider different types of bias such as: (1) method bias, (2) laboratory bias and (3) run bias. (O'Donnell and Hibbert, 2005) Not all, however, would agree with the need for classifications of bias. (Kadis, 2007, O'Donnell and Hibbert, 2007) Regardless of its classification or source, all forms of bias should ideally be determined and corrected for employing traceable control standards. As this is done, the uncertainty of that correction must be included as one of the components in the combined uncertainty. Occasionally, the analyst may determine that the bias is small and insignificant and not correct for it. There are ways to handle uncorrected bias as well by adding an additional component to the combined uncertainty. We will consider some examples here. Estimations for bias can come from internal quality control, proficiency test data, collaborative studies or method validation data. (Kane, 1997)

#### **6.1 Preparing an alcohol in water control solution**

We will assume in this example that we desire to prepare an ethanol in water solution to be used as a control standard. We want to prepare this solution to have a concentration of approximately 0.10 g/dL. Our measurement function will be as follows: (Philipp et.al., 2010)

$$C = \frac{m\_{Etab} \, PD}{m\_{Solution}} \tag{26}$$

where: C = the concentration of ethanol in water mEtoh = the mass measurement of ethanol P = the purity of the ethanol

D = the density of the ethanol

432 Toxicity and Drug Testing

**Source Type Distribution Standard Uncertainty Percent1** Total Method A Normal 0.0031 g/210L 84% GC Solution A Normal 0.0007 g/dL 0.5% Reference B Normal 0.0003 g/dL 1% Breath Instrument A Normal 0.0012 g/210L 3% Simulator Part. Coef B Normal 0.012 11% GC Controls A Normal 0.0006 g/210L 0.5%

Combined Uncertainty 0.00239 g/210L

(k=2) 0.0048 g/210L

Table 3. Uncertainty budget for the illustrated breath alcohol example

*Y*

1 Percent of contribution to total combined uncertainty

95% confidence interval 0.0776 to 0. 0872 g/210L

Next, we rearrange our probability statement, introduce the value for Z(1-α/2), and refer to

*Y - P Y - S = P Z = P Z = P Z = 0. Z Z <sup>S</sup>*

There is a probability of 0.8413 that the individual's true mean BrAC exceeds 0.080 g/100ml. This may or may not rise to the level of proof beyond a reasonable doubt, depending on the opinion of the court. This example illustrated the use of simulator control standards produced within a local toxicology laboratory including their associated uncertainties. Some jurisdictions, however, choose to purchase simulator control standards rather than prepare their own. If that were the case in this example, we could have eliminated the GC solutions and GC controls from our uncertainty estimates. The simulator partition coefficient would have remained while the reference value would have been obtained from the certificate of analysis from the manufacturer and considered a Type B uncertainty. Therefore, rather than having to include the GC solution and GC control components separately in the combined uncertainty estimate, they should already be included within the manufacturer's estimate of combined uncertainty, depending, of course, on how the solution standards were prepared

Our principle objective here will be to illustrate several ways for treating uncorrected bias. Bias or systematic error is common in all measurements. Some consider different types of bias such as: (1) method bias, (2) laboratory bias and (3) run bias. (O'Donnell and Hibbert, 2005) Not all, however, would agree with the need for classifications of bias. (Kadis, 2007, O'Donnell and Hibbert, 2007) Regardless of its classification or source, all forms of bias should ideally be determined and corrected for employing traceable control standards. As this is done, the uncertainty of that correction must be included as one of the components in the combined uncertainty. Occasionally, the analyst may determine that the bias is small

*1- / 2* 1 /2 *1- / 2* 1.0 8413 *<sup>Y</sup>*

 

Expanded Uncertainty

the standard normal tables:

and tested.

**6. Dealing with measurement bias** 

mSolution = the mass measurement of the combined solution of ethanol and water

Preparing a control standard gravimetrically has advantages. (Gates, et.al., 2009) There is better traceability for the mass measurements and no concern regarding the uncertainty in volume measurements. We will assume the purity (P) to be 0.995 with a Type B standard uncertainty of 0.002 determined from the certificate of analysis. We further assume that the density (D) of the solution is 0.997 g/ml (OIML, King and Lawn, 1999) with a Type B standard uncertainty of 0.00054 g/ml (King and Lawn, 1999), determined from the certificate of analysis from the manufacturer of a density meter. For both the purity and the density we will assume the uniform distribution in order to estimate their standard uncertainties. The values for the density are obtained from published tables for ethanol/water solutions. The density of the solution will be a function of the mass fraction of ethanol. The higher the mass fraction of ethanol the closer the density will be to 0.789 g/ml - the density of pure ethanol. The lower the mass fraction of ethanol the closer the density will be to 1.00 g/ml - the density of water. Since the density of the solution depends on the mass fraction of ethanol and we have selected a density of 0.997 g/ml (corresponding to a mass fraction of approximately 0.101%) and we desire a total solution mass of 1800 g, we need to have the mass of ethanol equal to 1.82 g. We will need to weigh 1.82 g of ethanol and place it into solution with water and add water until we have a total mass of 1800g. We will assume that the total solution mass is weighed on a scale that has had replicate measurements (n=30) of a 2 Kg traceable check weight (Type B uncertainty of 0.016 Kg) with a mean result of 1,940 g and a standard uncertainty of 30 g. This will be used to estimate the standard uncertainty in the measurement of *mSolution* . We now recognize that there is a bias in the weighing of the total solution. The measured mass of the solution is low by 3.0%. This will affect the mass of the ethanol necessary to maintain the density of 0.997 and mass fraction of 0.101%. As a result the mass of the ethanol will need to be 1.87 g. The mass of ethanol was weighed on a different scale that also has a set of replicate measurements (n=23) of a 2.0 g traceable check weight (Type B standard uncertainty of 0.014g) with a mean result of 2.08 g and a standard uncertainty of 0.02g. This scale has a bias of +4.0%. We now incorporate our assumed measurement information into equation 26:

Measurement Uncertainty in Forensic Toxicology: Its Estimation, Reporting and Interpretation 435

*uC* 0.0986 0.0113 0.00111 / *g dL* . The uncertainty budget is shown in table 4 both when ignoring the bias and when including the bias correction. From table 4 we see that including the additional balance bias, the combined uncertainty increased by 25% and contributed 38% to the combined uncertainty. The bias, in this example, is clearly significant and as a result should be corrected for. Before illustrating our next approach to handling uncorrected bias, we will evaluate the bias in our example to determine its significance. To do so we

and, when including the corrected concentration, we obtain:

2.08 2.00 10.9

0.05 and effective degrees of freedom of 51

**Percent1**

**Ignoring Bias Correcting Bias**

(29)

0.02 0.014 23 1 2.08 2.00

**Uncertainty**

Combined Uncertainty 0.00089 g/dL 0.00111 g/dL

Mass of Ethanol A 0.02 g 5% 3% Purity of Ethanol B 0.002 1% 1% Density of Solution B 0.00054 g/ml 1% 1% Mass of Solution A 30 g 12% 7% 2.0 Kg Reference B 16 g 81% 50% 2.0 g Reference B 0.014 g 38%

Table 4. Uncertainty budget for the preparation of the control ethanol solution

from the t-distribution is 0.975,51 *t* 2.01 . The results from equation 29 show the bias to be largely significant and should be corrected for. There are times when measurement bias is known to exist but is not corrected for. The analyst may believe the bias to be small and insignificant or it may be too complex to correct for. There are several methods that have been proposed for including the uncertainty due to uncorrected bias. (Maroto,et.al., 2002, Petersen, et.al., 2001) All of these effectively increase the expanded uncertainty by some amount to account for the uncorrected bias. Moreover, including an uncertainty component

2

*R <sup>g</sup>*

*u*

2

*R*

*g*

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

1 1

0.014

2.00

employ the following t-test:

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

*C R <sup>t</sup> u u*

The critical value for a two-tailed test with

**Source Type Standard** 

1 Percent of contribution to total combined uncertainty

*C R*

$$C = \frac{m\_{\text{Eich}} \, PD}{m\_{\text{Solar}} \cdot \frac{R\_{2Kg}}{\overline{X}}} = \frac{(1.87 \, \text{g})(0.995)(0.997 \, \text{g} \, / \, \text{ml})}{(1800 \, \text{g})\left(\frac{2000 \, \text{g}}{1940 \, \text{g}}\right)} = 0.00100 \, \text{g} \, / \, \text{ml} = 0.1000 \, \text{g} \, / \, \text{dL} \tag{27}$$

where: *R*2*Kg <sup>X</sup>* the correction factor for the bias in the scale used to weigh the total solution

Notice that we only correct for the bias in the scale used to weigh the total solution but not for the scale used to weigh the ethanol. The question now is how to deal with the +4.0% bias in the one scale. We begin by estimating the combined uncertainty ignoring the bias (assuming it is zero) and assuming independence of all variables. Since equation 27 is a multiplicative model we employ the RSS of the CV's squared as in equation 28. Notice that the standard uncertainty in the solution mass measurement comes from the repeatability measurements of the 2.0 Kg traceable check standards. There is no separate uncertainty estimate for the single measurement of the total solution of 1800 g. Employing the Welch-Sattherwaite equation to compute the effective degrees of freedom for our example we obtain:

The 95% confidence interval for our estimated concentration would be:

$$1. Y \pm t\_{0.975,63} \mu\_c \Rightarrow 0.1000 \pm 2.00 \text{(0.00089)} \Rightarrow 0.1000 \pm 0.00178 \Rightarrow 0.0982 \text{ to } 0.1018$$

The next option for dealing with the bias in the mass measurement of the ethanol is to correct for it. This is always the recommended practice and consistent with the *GUM* document. Correcting the ethanol mass for the +4.0% bias yields a result of 1.80 g. Placing this corrected value into equation 27 yields a corrected concentration of 0.000962 g/ml or 0.0962 g/dL. Now we must account for the uncertainty in the 2.0g reference check weight by including its Type B uncertainty in equation 28 where we add the additional term:

1940

*<sup>C</sup> <sup>g</sup> ml <sup>g</sup> dL <sup>R</sup> <sup>g</sup> <sup>m</sup> <sup>g</sup> X g*

Notice that we only correct for the bias in the scale used to weigh the total solution but not for the scale used to weigh the ethanol. The question now is how to deal with the +4.0% bias in the one scale. We begin by estimating the combined uncertainty ignoring the bias (assuming it is zero) and assuming independence of all variables. Since equation 27 is a multiplicative model we employ the RSS of the CV's squared as in equation 28. Notice that the standard uncertainty in the solution mass measurement comes from the repeatability measurements of the 2.0 Kg traceable check standards. There is no separate uncertainty estimate for the single measurement of the total solution of 1800 g. Employing the Welch-Sattherwaite equation to

4

2

0.1000 63.8 63

2

2

2 22 <sup>2</sup> 0.00054 30 16 3 30 1 0.997 1800 2000

4 4 444

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

*Yt u* 0.975,63 *<sup>c</sup>* 0.1000 2.00 0.00089 0.1000 0.00178 0.0982 0.1018 *to*

The next option for dealing with the bias in the mass measurement of the ethanol is to correct for it. This is always the recommended practice and consistent with the *GUM* document. Correcting the ethanol mass for the +4.0% bias yields a result of 1.80 g. Placing this corrected value into equation 27 yields a corrected concentration of 0.000962 g/ml or 0.0962 g/dL. Now we must account for the uncertainty in the 2.0g reference check weight by including its Type B uncertainty in equation 28 where we add the additional term:

*Etoh Sol Kg*

*Etoh Sol Kg*

*m m <sup>R</sup> P D*

*u u u u u u n nn n n Cm P Dm R*

0.00210

 

.02 .002 0.00054 30 16 2.08 0.995 0.997 1800 2000

 

22 29

*<sup>C</sup> m P Dm R*

*<sup>u</sup> CV CV CV CV CV*

<sup>2</sup> 0.02 0.002 23 3

0.1000 0.0089 0.00089 / *Yu g dL*

The 95% confidence interval for our estimated concentration would be:

 

0.1000 2.08 0.995

2 222 2

*Etoh Sol Kg*

1.87 0.995 0.997 / 0.00100 / 0.1000 / <sup>2000</sup> <sup>1800</sup>

*<sup>X</sup>* the correction factor for the bias in the scale used to weigh the total solution

(27)

(28)

 <sup>2</sup>

*m PD g g ml*

compute the effective degrees of freedom for our example we obtain:

*Kg*

*Etoh*

*Solution*

*R*2*Kg*

*eff*

*C*

*C*

*Y*

*u*

where:

2 <sup>2</sup> <sup>2</sup> 2 0.014 1 1 2.00 *R <sup>g</sup> g u R* and, when including the corrected concentration, we obtain:

*uC* 0.0986 0.0113 0.00111 / *g dL* . The uncertainty budget is shown in table 4 both when ignoring the bias and when including the bias correction. From table 4 we see that including the additional balance bias, the combined uncertainty increased by 25% and contributed 38% to the combined uncertainty. The bias, in this example, is clearly significant and as a result should be corrected for. Before illustrating our next approach to handling uncorrected bias, we will evaluate the bias in our example to determine its significance. To do so we employ the following t-test:

$$t = \frac{C - R}{\sqrt{u\_C^2 + u\_R^2}} = \frac{2.08 - 2.00}{\sqrt{\frac{0.02}{\sqrt{23}}}} \left[ \frac{0.014}{+ \left[ \frac{0.014}{2.00} \right]^2} \right] \tag{29}$$

The critical value for a two-tailed test with 0.05 and effective degrees of freedom of 51 from the t-distribution is 0.975,51 *t* 2.01 . The results from equation 29 show the bias to be largely significant and should be corrected for. There are times when measurement bias is known to exist but is not corrected for. The analyst may believe the bias to be small and insignificant or it may be too complex to correct for. There are several methods that have been proposed for including the uncertainty due to uncorrected bias. (Maroto,et.al., 2002, Petersen, et.al., 2001) All of these effectively increase the expanded uncertainty by some amount to account for the uncorrected bias. Moreover, including an uncertainty component


1 Percent of contribution to total combined uncertainty

Table 4. Uncertainty budget for the preparation of the control ethanol solution

Measurement Uncertainty in Forensic Toxicology: Its Estimation, Reporting and Interpretation 437

0.0939 0.1061 /

*Y U bias Y Y U bias Y*

:

Using this approach for our example would yield:

could occur at low concentrations. (Phillips, et.al., 1997)

reference or true measurand value described by:

**6.2 Estimating bias by recovery** 

where: %R = percent recovery C0 = the measured value

CRef = the true value of the measurand

*YU Y YU*

uncertainty (U) estimated as follows:

0.1000 0.00608 0.1000 0.00608

*Y g dL*

0

(33)

0

(34)

0 0

*c*

*if ku bias*

*c*

*if ku bias*

*c c*

*c c*

*ku bias if ku bias and U*

0.1000 2 0.00089 0.0043 0.1000 0 0.0939 0.1000 *Y Y* .

The asymmetry with this method has accounted for the positive bias and yields the same lower limit as the two preceding methods above. This results from the fact that our estimate is biased high by +0.0043 g/dL and was not corrected for. This last approach has more desirable statistical properties compared to the previous methods and has the advantage of avoiding negative expanded uncertainty limits (where the lower limit is below zero) which

Another approach to estimating and handling bias is with recovery analysis. (Thompson, et.al., 1999) Recovery is the ratio, expressed as a percent, of the measurement result to the

> 0 Re % 100 *f*

Percent recovery is a metric more commonly applied in analytical contexts involving complex matrices with several steps of extraction, sample preparation and analysis of a specified sub-sample. The requirements of this complex procedure for extraction and analysis often results in a loss of the analyte prior to its actual quantitative determination. Hence, we have the concept of %Recovery. The accuracy of the analytical method is determined by its ability to quantify (recover) the full amount of the analyte in the original

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

0 0

This clearly would yield the largest uncertainty interval compared to the preceding methods and is probably larger than necessary. The final method we will consider yields an expanded uncertainty interval that is also asymmetric about the measurement result. (Phillips, et.al., 1997) This method computes the confidence interval based on the expanded

*ku bias if ku bias where U*

resulting from a corrected bias is always less than the uncertainty component resulting from uncorrected bias. (Synek, 2005, Linsinger, 2008) One approach is to include the bias within the radical sign and estimate the expanded uncertainty (U) as follows:

$$\mathcal{U} = k \mathbb{C} \sqrt{\mathbb{C} V\_{m\_{\text{Esh}}}^2 + \mathbb{C} V\_P^2 + \mathbb{C} V\_D^2 + \mathbb{C} V\_{m\_{\text{Sol}}}^2 + \mathbb{C} V\_{R\_{2\text{S}\_\text{S}}}^2 + bias^2} \tag{30}$$

Since all of the other terms within the radical sign are dimensionless relative variances, we must transform the bias into dimensionless relative units. Doing this with our example and assuming k=2 we obtain:

$$U = 2(0.1000)\sqrt{\left[\frac{0.02}{\sqrt{23}}\right]^2 + \left[\frac{0.002}{\sqrt{3}}\right]^2 + \left[\frac{0.00054}{\sqrt{3}}\right]^2 + \left[\frac{30}{\sqrt{30}}\right]^2 + \left[\frac{16}{1800}\right]^2 + \left[\frac{16}{2000}\right]^2 + \left[\frac{0.08}{2.00}\right]^2$$

$$\mathcal{U} = 2(0.1000)(0.0122) = 0.0024 \,\text{g }/\,\text{dL}$$

The combined uncertainty with this approach is 0.00122 g/dL compared to 0.00111 g/dL when correcting for the bias and 0.00089 g/dL when ignoring the bias. Another approach is to incorporate the coverage factor k into the radical sign but without effecting the bias term as follows:

$$\mathcal{U} = \mathcal{C}\sqrt{k^2 \left[ \mathcal{C}V\_{m\_{\text{Esh}}}^2 + \mathcal{C}V\_P^2 + \mathcal{C}V\_D^2 + \mathcal{C}V\_{m\_{\text{Sd}}}^2 + \mathcal{C}V\_{R\_{\text{Ng}}}^2 \right] + bias}^2\tag{31}$$

With this approach the combined uncertainty remains the same but the expanded uncertainty becomes 0.00196 g/dL. As expected, this is slightly less than the expanded uncertainty determined from equation 30 which was 0.0024 g/dL. A third approach is basically the same as correcting for the bias and is expressed as:

$$Y \pm \mathcal{U} + bias \quad \Rightarrow \quad \overline{y} - \left(\mathcal{U} + bias\right) \le Y \le \overline{y} + \left(\mathcal{U} - bias\right) \tag{32}$$

For our example, the bias in the mass of the ethanol was +0.08g. The corrected mass of the ethanol should be 1.79 g rather than the 1.87 g value measured. Using the correct value of 1.79 g, the corrected concentration of the ethanol should be 0.0957 g/dL. This indicates that we have a bias in the estimated concentration of +0.0043 g/dL. Using this value for our bias and assuming an approximate 95% confidence interval, equation 32 becomes:

$$\begin{aligned} \text{(0.1000 - \left(2\left(0.00089\right) + 0.0043\right) \le Y \le 0.1000 + \left(2\left(0.00089\right) - 0.0043\right)\right)}\\ \text{0.0939 \le Y \le 0.0975 \text{g } /\text{dL} \end{aligned}$$

Notice that this interval is not symmetric around our estimated, yet biased, concentration of 0.1000 g/dL. Instead, it has accounted for the +0.0043 g/dL bias and adjusted for this. The next proposal for handling uncorrected bias is to simply add the absolute value of the bias to the expanded uncertainty as:*Y U bias* . For our example this would result in:

$$\overline{Y} - \left(\mathcal{U} + \left| bias\right|\right) \le Y \le \overline{Y} + \left(\mathcal{U} + \left| bias\right|\right) \implies \begin{aligned} 0.1000 - 0.00608 &\le Y \le 0.1000 + 0.00608 \\ 0.0939 &\le Y \le 0.1061 \text{ g } /\text{ } \text{dL} \end{aligned}$$

This clearly would yield the largest uncertainty interval compared to the preceding methods and is probably larger than necessary. The final method we will consider yields an expanded uncertainty interval that is also asymmetric about the measurement result. (Phillips, et.al., 1997) This method computes the confidence interval based on the expanded uncertainty (U) estimated as follows:

$$
\overline{Y} - \mathcal{U}\_- \le Y \le \overline{Y} + \mathcal{U}\_+.
$$

$$\text{where}: \qquad \text{UI}\_{+} = \begin{cases} k\mu\_{c} - bias & \text{if } k\mu\_{c} - bias > 0\\ 0 & \text{if } k\mu\_{c} - bias \le 0 \end{cases} \tag{33}$$

0 0 0 *c c c ku bias if ku bias and U if ku bias* 

Using this approach for our example would yield:

$$0.1000 - \left\lceil \Im(0.00089) + 0.0043 \right\rceil \le Y \le 0.1000 + 0 \implies 0.0939 \le Y \le 0.1000\dots$$

The asymmetry with this method has accounted for the positive bias and yields the same lower limit as the two preceding methods above. This results from the fact that our estimate is biased high by +0.0043 g/dL and was not corrected for. This last approach has more desirable statistical properties compared to the previous methods and has the advantage of avoiding negative expanded uncertainty limits (where the lower limit is below zero) which could occur at low concentrations. (Phillips, et.al., 1997)

#### **6.2 Estimating bias by recovery**

436 Toxicity and Drug Testing

resulting from a corrected bias is always less than the uncertainty component resulting from uncorrected bias. (Synek, 2005, Linsinger, 2008) One approach is to include the bias within

Since all of the other terms within the radical sign are dimensionless relative variances, we must transform the bias into dimensionless relative units. Doing this with our example and

23 3 3 30 <sup>1</sup> <sup>23</sup> 2 0.1000

The combined uncertainty with this approach is 0.00122 g/dL compared to 0.00111 g/dL when correcting for the bias and 0.00089 g/dL when ignoring the bias. Another approach is to incorporate the coverage factor k into the radical sign but without effecting the bias term

With this approach the combined uncertainty remains the same but the expanded uncertainty becomes 0.00196 g/dL. As expected, this is slightly less than the expanded uncertainty determined from equation 30 which was 0.0024 g/dL. A third approach is

*Y U bias y U bias Y y U bias* (32)

For our example, the bias in the mass of the ethanol was +0.08g. The corrected mass of the ethanol should be 1.79 g rather than the 1.87 g value measured. Using the correct value of 1.79 g, the corrected concentration of the ethanol should be 0.0957 g/dL. This indicates that we have a bias in the estimated concentration of +0.0043 g/dL. Using this value for our bias

> 0.1000 2 0.00089 0.0043 0.1000 2 0.00089 0.0043 0.0939 0.0975 / *Y*

Notice that this interval is not symmetric around our estimated, yet biased, concentration of 0.1000 g/dL. Instead, it has accounted for the +0.0043 g/dL bias and adjusted for this.

The next proposal for handling uncorrected bias is to simply add the absolute value of the bias to the expanded uncertainty as:*Y U bias* . For our example this would result in:

*Y g dL*

and assuming an approximate 95% confidence interval, equation 32 becomes:

22 2 2 2 2 2 *m P Dm R Etoh Sol Kg U C k CV CV CV CV CV bias*

(31)

2 222 2 2 *m P Dm R Etoh Sol Kg*

*U kC CV CV CV CV CV bias* (30)

2 2 22 <sup>2</sup> <sup>2</sup> 0.02 0.002 0.00054 30 16 0.08

2.08 0.995 0.997 1800 2000 2.00

the radical sign and estimate the expanded uncertainty (U) as follows:

2

assuming k=2 we obtain:

*U*

as follows:

2 0.1000 0.0122 0.0024 /

2

basically the same as correcting for the bias and is expressed as:

*U g dL*

Another approach to estimating and handling bias is with recovery analysis. (Thompson, et.al., 1999) Recovery is the ratio, expressed as a percent, of the measurement result to the reference or true measurand value described by:

$$\%R = \left[\frac{\mathcal{C}\_0}{\mathcal{C}\_{\text{Ref}}}\right] 100\tag{34}$$

where: %R = percent recovery C0 = the measured value CRef = the true value of the measurand

Percent recovery is a metric more commonly applied in analytical contexts involving complex matrices with several steps of extraction, sample preparation and analysis of a specified sub-sample. The requirements of this complex procedure for extraction and analysis often results in a loss of the analyte prior to its actual quantitative determination. Hence, we have the concept of %Recovery. The accuracy of the analytical method is determined by its ability to quantify (recover) the full amount of the analyte in the original

Measurement Uncertainty in Forensic Toxicology: Its Estimation, Reporting and Interpretation 439

The p-value for t = 17.9 with df=44 is <0.00001. We conclude that the mean recovery is very significantly different from 1.0. The recovery estimate should be used to correct the analytical results. Using our mean recovery to correct our analytical results yields:

<sup>0</sup>

*u u CV CV <sup>u</sup> mg dL <sup>C</sup>*

This results in a relative combined uncertainty of approximately 1.3%. Moreover, the analytical component contributed 45% while the recovery component contributed 65% to the combined uncertainty. The same analysis can be done when spiking blank specimens with a known concentration of the analyte. If we added the same 0.1ml of 20mg/dL concentration to 1.0ml of blank specimen, and quantified the specimen with our analytical method and obtained 1.65 mg/dL, this would become the numerator in equation 36 and we

1.65 / %Recovery <sup>100</sup> 100 90.7%

*Measured Concentration mg dL Concentration Added mg dL* 

Both methods of spiking blank samples or spiking samples already containing the analyte are used in recovery studies. Moreover, it is important to remember with recovery studies the assumption that no other bias exists. We have briefly considered several ways that have been proposed to handle uncorrected bias. Ideally, bias should always be corrected for even when statistically insignificant. When the bias is not corrected for, the combined uncertainty statement should include some additional component, thus increasing its magnitude, accounting for the uncorrected bias. Moreover, the customer should be made aware, either in the uncertainty statement or otherwise, when uncorrected bias exists and

This example summarizes work recently published where methadone was measured in post-mortem cases. (Linnet, et.al., 2008) One sample of blood was taken from each femoral vein in 27 post-mortem autopsies. LC-MS/MS was the analytical method used to quantify both methadone and its main metabolite, 2-ethyl-1,5-dimethyl-3,3-diphenylpyrrolinium (EDDP). For our present example we will focus only on the quantitative measurement of methadone. While the study did not explicitly present a measurement function, the

*Meth*

(38)

0

*Corr*

*Cal*

*m P <sup>C</sup> C C <sup>V</sup> <sup>C</sup> C C*

*A A*

0.92 0.06

2 2

multiplicative model and we assume independence according to:

*C R C*

*<sup>R</sup>* . The combined uncertainty in our corrected estimate can now be determined from the RSS method using the CV's squared since we have a

56 45 18.3 0.0126 0.231 / 18.3 18.3 0.84

1.82 /

.

<sup>0</sup> 15.4 18.3 / 0.84 *Corr <sup>C</sup> C mg dL*

2 2

would obtain a recovery estimate of:

how it has been accounted for.

**7. Uncertainty in post-mortem drug analysis** 

following would be a reasonable approximation:

<sup>0</sup>

*C C*

*Corr*

matrix. Simply spiking alcohol in a blood sample and measuring it is not a typical application of percent recovery. The recovery is often determined during the method validation phase where a known blank matrix is spiked with a known mass of the relevant analyte. This is often referred to as a "reference recovery" or a "method recovery". (Barwick and Ellison, 1999) When recovery estimates are applied to correct subsequent samples, it is very important that the concentrations and matrix are appropriately similar and that the same full analytical protocol is followed. Measurements of recovery from several spiked samples may be performed with the mean and standard deviation of the percent estimates determined, providing uncertainty estimates for the percent recovery in future measurements. The fractional recovery can be employed as a correction factor in the measurement equation as follows:

$$\mathbf{C}\_{Corr} = \frac{\mathbf{C}\_0}{\overline{R}} \tag{35}$$

where: CCorr = the corrected analytical result

C0 = the original measurement

*R* the mean fractional recovery

Assume that we are interested in determining the percent recovery of a specific drug for a particular analytical method. Assume that we have two vials of a subject's blood, each containing 1.0 ml and each containing some unknown concentration of the drug of interest. To one tube we add 0.1ml of a known analyte standard having a concentration of 20mg/dL.

We have now added a concentration of: 20 0.1 1.82 0.1 1.0 *mg ml mg dL ml ml dL* . To the other tube

we simply add 0.1 ml of water. We now measure the concentration of the analyte in each tube in replicate (at least twice) and determine the means to be: Tube with added analyte: 10.8 mg/dL Tube with added water: 9.3 mg/dL. We now compute the percent recovery according to:

$$\% \text{Re} \text{cov} \text{very} = \left[ \frac{\text{Measured Difference}}{\text{Concentration} \text{Added}} \right] \cdot 100 = \left[ \frac{10.8 \text{mg} / \text{dL} - 9.3 \text{mg} / \text{dL}}{1.82 \text{mg} / \text{dL}} \right] \cdot 100 = 82.4\% \quad \text{(36)}$$

Assume that we have done this recovery experiment during method validation using blood specimens spiked with the analyte and obtained a mean % recovery of *R* 84% with a standard uncertainty of 6% determined from 45 spiked samples. Assume further that we now have a suspect's blood sample and we wish to provide an unbiased estimate of the analyte's concentration using this recovery data. We determine the suspect's sample results to be C0 = 15.4mg/dL with a standard uncertainty of 0.92mg/dL determined from n=56 measurements of past quality control data. We further assume there are no other significant sources of bias, other than that estimated by the %Recovery. First we could determine whether the mean recovery of 84% was significantly different from 1.0 or not with the following t-test:

$$t = \frac{\left|\overline{R} - 1\right|}{\mu\_{\overline{R}}} = \frac{\left|0.84 - 1\right|}{0.06 \,/\sqrt{45}} = -17.9\tag{37}$$

matrix. Simply spiking alcohol in a blood sample and measuring it is not a typical application of percent recovery. The recovery is often determined during the method validation phase where a known blank matrix is spiked with a known mass of the relevant analyte. This is often referred to as a "reference recovery" or a "method recovery". (Barwick and Ellison, 1999) When recovery estimates are applied to correct subsequent samples, it is very important that the concentrations and matrix are appropriately similar and that the same full analytical protocol is followed. Measurements of recovery from several spiked samples may be performed with the mean and standard deviation of the percent estimates determined, providing uncertainty estimates for the percent recovery in future measurements. The fractional recovery can be employed as a correction factor in the

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

Assume that we are interested in determining the percent recovery of a specific drug for a particular analytical method. Assume that we have two vials of a subject's blood, each containing 1.0 ml and each containing some unknown concentration of the drug of interest. To one tube we add 0.1ml of a known analyte standard having a concentration of 20mg/dL.

we simply add 0.1 ml of water. We now measure the concentration of the analyte in each tube in replicate (at least twice) and determine the means to be: Tube with added analyte: 10.8 mg/dL Tube with added water: 9.3 mg/dL. We now compute the percent recovery

10.8 / 9.3 / %Recovery <sup>100</sup> 100 82.4%

*Measured Difference mg dL mg dL Concentration Added mg dL*

Assume that we have done this recovery experiment during method validation using blood specimens spiked with the analyte and obtained a mean % recovery of *R* 84% with a standard uncertainty of 6% determined from 45 spiked samples. Assume further that we now have a suspect's blood sample and we wish to provide an unbiased estimate of the analyte's concentration using this recovery data. We determine the suspect's sample results to be C0 = 15.4mg/dL with a standard uncertainty of 0.92mg/dL determined from n=56 measurements of past quality control data. We further assume there are no other significant sources of bias, other than that estimated by the %Recovery. First we could determine whether the mean recovery of 84% was significantly different from 1.0 or not with the

0.06 / 45 *<sup>R</sup>*

 

0.1 1.0

*mg ml mg dL ml ml dL* 

1.82 /

17.9

(37)

*<sup>R</sup>* (35)

. To the other tube

(36)

measurement equation as follows:

C0 = the original measurement *R* the mean fractional recovery

according to:

following t-test:

<sup>0</sup>

<sup>1</sup> 0.84 1

*t*

*R*

*u*

We have now added a concentration of: 20 0.1 1.82

where: CCorr = the corrected analytical result

The p-value for t = 17.9 with df=44 is <0.00001. We conclude that the mean recovery is very significantly different from 1.0. The recovery estimate should be used to correct the analytical results. Using our mean recovery to correct our analytical results yields: <sup>0</sup> 15.4 18.3 / 0.84 *Corr <sup>C</sup> C mg dL <sup>R</sup>* . The combined uncertainty in our corrected estimate can now be determined from the RSS method using the CV's squared since we have a multiplicative model and we assume independence according to:

$$\frac{u\_{\text{C}}}{C\_{\text{Corr}}} = \sqrt{\text{CV}\_{\text{C}\_{0}}^{2} + \text{CV}\_{R}^{2}} \implies \frac{u\_{\text{C}}}{18.3} = \sqrt{\left[\frac{\frac{0.92}{\sqrt{56}}}{18.3}\right]^{2} + \left[\frac{\frac{0.06}{\sqrt{45}}}{0.84}\right]^{2}} \implies u\_{\text{C}} = (18.3)(0.0126) = 0.231 \,\text{mg} \,/\text{dL} \cdot \text{m}$$

This results in a relative combined uncertainty of approximately 1.3%. Moreover, the analytical component contributed 45% while the recovery component contributed 65% to the combined uncertainty. The same analysis can be done when spiking blank specimens with a known concentration of the analyte. If we added the same 0.1ml of 20mg/dL concentration to 1.0ml of blank specimen, and quantified the specimen with our analytical method and obtained 1.65 mg/dL, this would become the numerator in equation 36 and we would obtain a recovery estimate of:

$$\% \text{Re} \text{cov} \text{very} = \left[ \frac{\text{Measured Concentration}}{\text{Concentration} \text{ Added}} \right] \cdot 100 = \left[ \frac{1.65 \text{mg} / \text{dL}}{1.82 \text{mg} / \text{dL}} \right] \cdot 100 = 90.7\%$$

Both methods of spiking blank samples or spiking samples already containing the analyte are used in recovery studies. Moreover, it is important to remember with recovery studies the assumption that no other bias exists. We have briefly considered several ways that have been proposed to handle uncorrected bias. Ideally, bias should always be corrected for even when statistically insignificant. When the bias is not corrected for, the combined uncertainty statement should include some additional component, thus increasing its magnitude, accounting for the uncorrected bias. Moreover, the customer should be made aware, either in the uncertainty statement or otherwise, when uncorrected bias exists and how it has been accounted for.

#### **7. Uncertainty in post-mortem drug analysis**

This example summarizes work recently published where methadone was measured in post-mortem cases. (Linnet, et.al., 2008) One sample of blood was taken from each femoral vein in 27 post-mortem autopsies. LC-MS/MS was the analytical method used to quantify both methadone and its main metabolite, 2-ethyl-1,5-dimethyl-3,3-diphenylpyrrolinium (EDDP). For our present example we will focus only on the quantitative measurement of methadone. While the study did not explicitly present a measurement function, the following would be a reasonable approximation:

$$\mathbf{C}\_{Corr} = \frac{\mathbf{C}\_0 \cdot \mathbf{C}\_{Col}}{\overline{\mathbf{C}}\_A} = \frac{\mathbf{C}\_0 \cdot \frac{\mathbf{m}\_{Mch} \, P}{V}}{\overline{\mathbf{C}}\_A} \tag{38}$$

.

Measurement Uncertainty in Forensic Toxicology: Its Estimation, Reporting and Interpretation 441

this study is to compute estimates of r and s as in equation 42. (Altman, et.al., 2000) For our sample size of n=27 and rounding the estimates to the nearest integer we obtain the results seen in equation 43. This would indicate that the 8th and 20th ordered observations would provide an approximate 95% confidence interval for the population median. The exact level of confidence for this example based on the binomial distribution would be 98.1%. (Altman,

> **Source Type %CV Percent1** Pre-Analytical A 18.95% 96% Analytical A 3.65% 3.9% Mass of Methadone A 0.53% 0.08% Purity B 0.29% 0.02% Volume B 0.05% 0% Combined Uncertainty 19.3% 100%

Table 5. Uncertainty budget for the post-mortem measurement of methadone in femoral

22 22 *nn nn r Z*

27 27 27 27 1.96 8.4 8 1 1.96 19.6 20

The unique aspect of this example will be the addition of the uncertainty due to calibration. We will assume that duplicate blood alcohol results of 0.104 and 0.107 g/dL were obtained from the same headspace gas chromatograph. The following is our assumed measurement

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

We have added an additional correction factor *fCalib* in equation 44 which we also set equal to one and also include its uncertainty component. We will assume that the instrument was calibrated with a linear five point calibration curve generated by the use of

 

 

*C R C ff <sup>X</sup>* (44)

(42)

(43)

*s Z*

et.al., 2000)

blood

function:

1Percent of contribution to total combined uncertainty

1 /2 1 /2 1

2 2 2 2

 

*r s*

**8. Uncertainty in a blood alcohol analysis** 

where: Ccorr = the corrected BAC results

R = the traceable reference control value

fdilutor = the correction factor for the dilutor fCalib = the correction factor for the calibration

C0 = the mean of the original measurement results

*XCont* the mean results from measuring the controls

where: CCorr = the corrected measurement of methadone

C0 = the original quantitative measurement result of the methadone by LC-MS/MS CCal = the reference calibration and/or control value

*C <sup>A</sup>* the mean quantitative measurement of the reference value

mMeth = mass of the reference methadone added to the calibration/control solution

P = the purity of the methadone

V = the volume of the calibration/control methadone solution

The study also presented the following uncertainty estimates, expressed as %CV's, for each of the components in equation 38: 3.65% 0.29% 0.53% 0.05% *<sup>C</sup> <sup>A</sup> Pm V Meth u uu u* .

The uncertainty in the purity was determined from employing the uniform distribution and the manufacturer's certificate of analysis stating the purity was 99.99% ± 0.5%. The uncertainty in the original measurements (C0) was determined from the duplicate sampling, one from each femoral vein. The standard uncertainty for a single determination was determined from each of these results according to:

$$\mu\_M = \sqrt{\frac{\sum\_{i=1}^{N} \left(rd\right)\_i^2}{2N}} = \sqrt{\frac{\sum\_{i=1}^{N} d\_i^2}{2N}}\tag{39}$$

Equation 39, expressing the computation in two equivalent forms, was designed to estimate the total method (uM) component of uncertainty. A major part of this was due to the sampling technique from each of the femoral veins. This component was termed preanalytical (PA). Once the computations were determined from equation 39, the preanalytical component was determined according to:

$$\text{CV}\_{M}^{2} = \text{CV}\_{PA}^{2} + \text{CV}\_{A}^{2} \tag{40}$$

Finally, the combined uncertainty was determined according to:

$$\text{CV}\_{T}^{2} = \text{CV}\_{\text{PA}}^{2} + \text{CV}\_{A}^{2} + \text{CV}\_{\text{Cd}}^{2} = \text{CV}\_{\text{PA}}^{2} + \text{CV}\_{A}^{2} + \text{CV}\_{\text{m}\_{\text{Mch}}}^{2} + \text{CV}\_{P}^{2} + \text{CV}\_{V}^{2} \tag{41}$$

Incorporating the uncertainty estimates outlined in Table 1 of the study we obtain:

<sup>22222</sup> *CVT* 18.95% 3.65% 0.53% 0.29% 0.05% 19.3% . With this estimate we, and the authors of the study, have assumed independence of the components and a multiplicative measurement model. The uncertainty budget for this example is shown in Table 5, from which we see that the pre-analytical or sampling component contributes by far the most to the combined uncertainty. This is not unexpected since it represents the sampling component. Sampling, when included as a component in the combined uncertainty estimate, is typically the largest contributor. The study reported that amongst the 27 cases, the concentration of methadone ranged from 0.005 to 2.29 mg/kg with a median value of 0.472 mg/kg. The median was appropriately reported, rather than the mean, because the distribution of results was positively skewed. Therefore, we would be interested in this case in computing a 95% confidence interval for the median. The most common approaches to estimating confidence intervals for a median do not involve uncertainty estimates. This results from the fact that the median is a quantile, specifically, the 50th percentile. One method for estimating the approximate 95% confidence interval for the median presented in

this study is to compute estimates of r and s as in equation 42. (Altman, et.al., 2000) For our sample size of n=27 and rounding the estimates to the nearest integer we obtain the results seen in equation 43. This would indicate that the 8th and 20th ordered observations would provide an approximate 95% confidence interval for the population median. The exact level of confidence for this example based on the binomial distribution would be 98.1%. (Altman, et.al., 2000)


1Percent of contribution to total combined uncertainty

440 Toxicity and Drug Testing

The study also presented the following uncertainty estimates, expressed as %CV's, for each of the components in equation 38: 3.65% 0.29% 0.53% 0.05% *<sup>C</sup> <sup>A</sup> Pm V Meth u uu u* . The uncertainty in the purity was determined from employing the uniform distribution and the manufacturer's certificate of analysis stating the purity was 99.99% ± 0.5%. The uncertainty in the original measurements (C0) was determined from the duplicate sampling, one from each femoral vein. The standard uncertainty for a single determination was

> <sup>2</sup> <sup>2</sup> 1 1 2 2

*rd d*

*N N*

*i i*

(39)

2 22 *CV CV CV M PA A* (40)

*N N*

*i i*

Equation 39, expressing the computation in two equivalent forms, was designed to estimate the total method (uM) component of uncertainty. A major part of this was due to the sampling technique from each of the femoral veins. This component was termed preanalytical (PA). Once the computations were determined from equation 39, the pre-

2 2 22 2 22 22

<sup>22222</sup> *CVT* 18.95% 3.65% 0.53% 0.29% 0.05% 19.3% . With this estimate we, and the authors of the study, have assumed independence of the components and a multiplicative measurement model. The uncertainty budget for this example is shown in Table 5, from which we see that the pre-analytical or sampling component contributes by far the most to the combined uncertainty. This is not unexpected since it represents the sampling component. Sampling, when included as a component in the combined uncertainty estimate, is typically the largest contributor. The study reported that amongst the 27 cases, the concentration of methadone ranged from 0.005 to 2.29 mg/kg with a median value of 0.472 mg/kg. The median was appropriately reported, rather than the mean, because the distribution of results was positively skewed. Therefore, we would be interested in this case in computing a 95% confidence interval for the median. The most common approaches to estimating confidence intervals for a median do not involve uncertainty estimates. This results from the fact that the median is a quantile, specifically, the 50th percentile. One method for estimating the approximate 95% confidence interval for the median presented in

Incorporating the uncertainty estimates outlined in Table 1 of the study we obtain:

*T PA A Cal PA A m P V Meth CV CV CV CV CV CV CV CV CV* (41)

 

C0 = the original quantitative measurement result of the methadone by LC-MS/MS

mMeth = mass of the reference methadone added to the calibration/control solution

where: CCorr = the corrected measurement of methadone

*C <sup>A</sup>* the mean quantitative measurement of the reference value

V = the volume of the calibration/control methadone solution

*M*

*u*

Finally, the combined uncertainty was determined according to:

CCal = the reference calibration and/or control value

determined from each of these results according to:

analytical component was determined according to:

P = the purity of the methadone

Table 5. Uncertainty budget for the post-mortem measurement of methadone in femoral blood

$$r = \frac{n}{2} - \left[ Z\_{1-a/2} \cdot \frac{\sqrt{n}}{2} \right] \qquad \qquad s = 1 + \frac{n}{2} + \left[ Z\_{1-a/2} \cdot \frac{\sqrt{n}}{2} \right] \tag{42}$$

$$r = \frac{27}{2} - \left[1.96 \cdot \frac{\sqrt{27}}{2}\right] = 8.4 \approx 8 \qquad \qquad s = 1 + \frac{27}{2} + \left[1.96 \cdot \frac{\sqrt{27}}{2}\right] = 19.6 \approx 20\tag{43}$$

#### **8. Uncertainty in a blood alcohol analysis**

The unique aspect of this example will be the addition of the uncertainty due to calibration. We will assume that duplicate blood alcohol results of 0.104 and 0.107 g/dL were obtained from the same headspace gas chromatograph. The following is our assumed measurement function:

$$\mathbf{C}\_{corr} = \frac{\mathbf{C}\_0 \,\mathrm{R}}{\mathbf{X}\_{\mathrm{Cont}}} \cdot f\_{dilutor} \cdot f\_{\mathrm{Calib}} \tag{44}$$

where: Ccorr = the corrected BAC results

C0 = the mean of the original measurement results

R = the traceable reference control value

*XCont* the mean results from measuring the controls

fdilutor = the correction factor for the dilutor

fCalib = the correction factor for the calibration

We have added an additional correction factor *fCalib* in equation 44 which we also set equal to one and also include its uncertainty component. We will assume that the instrument was calibrated with a linear five point calibration curve generated by the use of

Measurement Uncertainty in Forensic Toxicology: Its Estimation, Reporting and Interpretation 443

We now combine the standard uncertainty components to determine the combined uncertainty according to equation 49. Estimating an approximate 95% uncertainty interval

0.1029 2 0.0020 0.1029 0.0040 0.0989 0.1069 / *to g dL* .

The percent contribution from each component to the combined uncertainty in this example is: C0 10%, R 2%, *XCont* 1% , *Dilutor f* 1% and *Calib f* 86%. From this we see that the calibration uncertainty contributed by far the most to the combined uncertainty. This may have resulted from the values assumed for this example and may not reflect most forensic programs. Each laboratory would need to determine this for their particular context. It should also be noted that equation 48 includes the uncertainty only of the least squares estimates and not that of the reference standards used as calibrants. These could be added as separate components. There are other methods to account for the uncertainty in calibration as well. For example, the maximum vertical deviation between the line of identify and the least squares regression line can be divided by the square root of three, assuming the

**Variable Estimate Uncertainty n**

C0 0.1055 0.0009 2 R 0.100 0.0003 1 *XCont* 0.1025 0.0008 16 *Dilutor f* 10.65 0.05 10

*Calib f* 0.1058 0.0042 5

*C R X f f*

*u u u u u*

2 2 0.0009 0.0003 0.0008 2 1 16

*Corr dilutor Calib*

*u nnn n n C X CR f f*

0.1029 0.1029 0.100 0.1

0.1029 0.0192 0.0020 / *Yu g dL*

 

> 

Table 6. The values of specific variables assumed for our blood alcohol measurement model

2 2 2 2 2

*dilutor Calib*

22 2 0.050 0.0042 10 5

(49)

025 10.65 0.1058

 

would yield:

uniform distribution, and

0

0

*Corr*

*C*

*u*

*C*

 <sup>0</sup> 0.1055 0.100 1 1 0.1029 / 0.1025 *corr dilutr Calib C R C ff g dL <sup>X</sup>*

five traceable control standards. The calibration curve was generated by linear least squares yielding the following function:

$$Y = a + b \, X \tag{45}$$

where: Y = instrument response, X = known control concentration values and a and b are model parameters. The objective in developing a calibration curve is to estimate the true value of a future unknown concentration (X) given some instrument response (Y). Therefore, we find the inverse of equation 45:

$$X = \frac{Y - a}{b} \,. \tag{46}$$

For our purposes, we are interested in determining the uncertainty in X found in equation 46. The parameters a and b, however, are correlated. We can eliminate the parameter a by solving for a according to *a Y bX* and then substituting this into equation 46 according to:

$$X\_0 = \frac{Y\_0 - \left(\overline{Y} - b\,\overline{X}\right)}{b} \quad \Rightarrow \qquad X\_0 = \frac{Y\_0 - \overline{Y}}{b} + \overline{X} \tag{47}$$

where: X0 = a future single estimate of concentration

Y0 = a future single instrument response

*Y* the mean of the instrument responses during calibration

*X* the mean of the control samples used during calibration

From equation 47 we see that X0 is a function of only three random variables: Y0, *Y* , and b. Solving for the uncertainty in X0 by the method of error propagation we obtain:

$$
\mu\_{X\_0} = \frac{S\_{Y|X}}{b} \sqrt{\frac{1}{m} + \frac{1}{n} + \frac{\left(Y\_0 - \overline{Y}\right)^2}{b^2 \sum\_{i=1}^n \left(X\_i - \overline{X}\right)^2}}\tag{48}
$$

where: SY|X = standard error from regression of Y on X in developing the calibration curve b = the slope of the calibration curve

m = the number of measurements used to estimate X0

n = the number of measurements used to generate the calibration curve

We will assume specific values for the terms in equation 48 and solve for the uncertainty according to:

$$\mu\_{X\_0} = \frac{\left(0.005\right)}{\left(1.02\right)} \sqrt{\frac{1}{2} + \frac{1}{5} + \frac{\left(0.1055 - 0.1516\right)^2}{\left(1.02\right)^2 \left(0.046\right)}} = 0.0042$$

Now, for our example we will assume the variables for equation 44 found in Table 6. For purposes of determining the uncertainties in each of the correction factors we assume *Dilutor f* to be 10.65 and *Calib f* to be 0.1058 g/dL. However, for estimating the corrected blood alcohol concentration in equation 44 we assume each to be 1.0. Next, we can estimate our corrected blood alcohol concentration according to:

five traceable control standards. The calibration curve was generated by linear least squares

*Y a bX* (45)

where: Y = instrument response, X = known control concentration values and a and b are model parameters. The objective in developing a calibration curve is to estimate the true value of a future unknown concentration (X) given some instrument response (Y).

> *Y a <sup>X</sup> b*

For our purposes, we are interested in determining the uncertainty in X found in equation 46. The parameters a and b, however, are correlated. We can eliminate the parameter a by solving for a according to *a Y bX* and then substituting this into equation 46 according to:

> <sup>0</sup> <sup>0</sup> 0 0

*Y Y bX Y Y <sup>X</sup> X X b b*

From equation 47 we see that X0 is a function of only three random variables: Y0, *Y* , and b.

<sup>0</sup>

where: SY|X = standard error from regression of Y on X in developing the calibration curve

We will assume specific values for the terms in equation 48 and solve for the uncertainty

2

Now, for our example we will assume the variables for equation 44 found in Table 6. For purposes of determining the uncertainties in each of the correction factors we assume *Dilutor f* to be 10.65 and *Calib f* to be 0.1058 g/dL. However, for estimating the corrected blood alcohol concentration in equation 44 we assume each to be 1.0. Next, we can estimate our

*b mn b XX*

*S Y Y*

*i*

2

0.0042

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

1

*i*

2

Solving for the uncertainty in X0 by the method of error propagation we obtain:

*Y X* 1 1 *X n*

n = the number of measurements used to generate the calibration curve

<sup>0</sup>

0.005 1 1 0.1055 0.1516

1.02 2 5 1.02 0.046

. (46)

(48)

(47)

yielding the following function:

Therefore, we find the inverse of equation 45:

where: X0 = a future single estimate of concentration

*Y* the mean of the instrument responses during calibration *X* the mean of the control samples used during calibration

*u*

m = the number of measurements used to estimate X0

 

corrected blood alcohol concentration according to:

Y0 = a future single instrument response

b = the slope of the calibration curve

*uX*

according to:

$$C\_{corr} = \frac{C\_0 \, R}{\overline{X}} \cdot f\_{dilutr} \cdot f\_{Calib} = \frac{(0.1055)(0.100)}{(0.1025)} \cdot 1 \cdot 1 \, = 0.1029 \, g/\, d\Omega$$

We now combine the standard uncertainty components to determine the combined uncertainty according to equation 49. Estimating an approximate 95% uncertainty interval would yield:

$$0.1029 \pm 2(0.0020) \Rightarrow 0.1029 \pm 0.0040 \implies 0.0989 \text{ to } 0.1069 \text{ g/dL.}$$

The percent contribution from each component to the combined uncertainty in this example is: C0 10%, R 2%, *XCont* 1% , *Dilutor f* 1% and *Calib f* 86%. From this we see that the calibration uncertainty contributed by far the most to the combined uncertainty. This may have resulted from the values assumed for this example and may not reflect most forensic programs. Each laboratory would need to determine this for their particular context. It should also be noted that equation 48 includes the uncertainty only of the least squares estimates and not that of the reference standards used as calibrants. These could be added as separate components. There are other methods to account for the uncertainty in calibration as well. For example, the maximum vertical deviation between the line of identify and the least squares regression line can be divided by the square root of three, assuming the uniform distribution, and


Table 6. The values of specific variables assumed for our blood alcohol measurement model

$$\begin{aligned} \frac{u\_{\overline{C}\_{corr}}}{\overline{C}\_{Corr}} &= \sqrt{\left[\frac{u\_{\overline{C}\_{0}}}{\overline{C}\_{0}}\right]^{2} + \left[\frac{\overline{u}\_{R}}{\overline{R}}\right]^{2} + \left[\frac{\overline{u\_{\overline{X}}}}{\overline{X}}\right]^{2} + \left[\frac{\overline{u\_{fabx}}}{\overline{f\_{dulator}}}\right]^{2} + \left[\frac{\overline{u\_{fabx}}}{\overline{f\_{cubb}}}\right]^{2} \\\\ \frac{u\_{\overline{C}}}{0.1029} &= \sqrt{\left[\frac{0.0009}{0.1029}\right]^{2} + \left[\frac{0.0003}{0.100}\right]^{2} + \left[\frac{0.0008}{0.1025}\right]^{2} + \left[\frac{\overline{0.050}}{\overline{10}}\right]^{2} + \left[\frac{0.0042}{0.165}\right]^{2} + \left[\frac{0.0042}{0.1058}\right]^{2} \\\\ u\_{\overline{Y}} &= 0.1029(0.0192) = 0.0020 \text{ g/dL} \end{aligned} \tag{49}$$

Measurement Uncertainty in Forensic Toxicology: Its Estimation, Reporting and Interpretation 445

controls were *X g dL* 0.1024 / with n=34 measurements and a standard uncertainty of 0.0009 g/dL. Computing our corrected estimate from equation 56 we obtain 0.1143 g/dL. Using this value to estimate our method uncertainty from the equation found in figure 8 we obtain: *u g <sup>M</sup>* 0.0369 0.1143 0.00129 0.0055 / *dL* . Assuming independence and the multiplicative model of equation 50, we now estimate our combined uncertainty as seen in equation 51. The approximate 95% confidence interval (k=2) for the true mean blood alcohol

2 0.1143 2 0.0039 0.1065 0.1221 / *<sup>Y</sup> Y u to g dL* .

The risk in using proficiency data in this manner is that the actual uncertainty associated with a particular laboratory may be overestimated. Another limitation to keep in mind is that the proficiency data may not have been generated with the same analytical protocol employed within a particular laboratory. Proficiency data, however, does have a large source of variation, which may be acceptable within the forensic context. The uncertainty budget for these results is found in Table 7. The method uncertainty determined from the proficiency test data in this example, contributed by far the most to the combined uncertainty while the reference and analytical components could effectively be ignored.

**Data from 2004 to 2008**

**n = 63**

**Blood Alcohol Concentration (g/dL) 0.00 0.05 0.10 0.15 0.20 0.25 0.30**

Fig. 7. Plot of the standard deviation against concentration and determination of an

uncertainty function from CTS proficiency test blood alcohol data

**Uncertainty Function SD = 0.0369 BAC + 0.00129**

concentration in this example would be:

**Standard Deviation (g/dL)**

**0.000**

**0.005**

**0.010**

**0.015**

**0.020**

divided by the concentration value of X at that point. This is often termed a "lack of linearity" component.

The preceding examples presented here have been illustrative only. There was no intention that the uncertainty estimates assumed were the only ones to be considered or even represented any specific laboratory program. They were presented simply to illustrate the computations involved. Indeed, there are surely other components to be considered. (Sklerov and Couper, 2011) These must be identified by the forensic toxicologist considering their particular laboratory, protocol, instruments, customers and the required fitness-forpurpose.

#### **9. Different methods for estimating uncertainty**

We have illustrated above several examples for estimating the combined uncertainty in contexts relevant to forensic toxicology. These examples have presented the standard bottom-up approach recommended largely by the *GUM* document. There are, however, several other approaches to dealing with uncertainty that have been proposed in the forensic toxicology and metrological literature. Wallace, for example, has proposed a number of different methods for estimating measurement uncertainty. (Wallace, 2010)

#### **9.1 Use of proficiency test data**

One method advocated by Wallace is the use of proficiency test data. (Wallace, 2010) Proficiency testing basically consists of an organizing laboratory which, employing well established and traceable methods, prepares and tests the concentrations of several samples. These samples are then sent blindly to participating laboratories with instructions on how the measurements are to be performed, recorded and then returned to the organizing laboratory. The samples are to be treated by the participating laboratories as routine case samples and tested according to their routine protocols. The organizing laboratory summarizes the data reporting means, standard deviations and various plots, including, for example, Z-scores. The standard deviations at various mean concentrations can be used to generate uncertainty functions. Clearly, these estimates will exhibit rather large variation due to the different laboratories, instruments, protocols, analysts, time, etc. These estimates, conditioned on the appropriate concentration, can be used as the total method component in the combined uncertainty estimate. Consider an example where we have duplicate blood alcohol results obtained in the toxicology laboratory of 0.118 and 0.116 g/dL. The laboratory participated in a proficiency study which yielded the uncertainty function observed in figure 7. This figure was actually generated from data available from Collaborative Testing Services [CTS]. For this example we will assume the following measurement function:

$$\mathbf{C}\_{Corr} = \frac{\mathbf{C}\_0 \, R}{\overline{X}} \tag{50}$$

where: CCorr = the corrected measurement result

C0 = the mean of the original duplicate measurements

R = the reference value for the controls

*X* the mean result for measuring the reference controls

The mean of our assumed duplicate results is 0.1170 g/dL. The reference value is R=0.100 g/dL with a Type B standard uncertainty of 0.0003 g/dL. The mean measurement of the

divided by the concentration value of X at that point. This is often termed a "lack of

The preceding examples presented here have been illustrative only. There was no intention that the uncertainty estimates assumed were the only ones to be considered or even represented any specific laboratory program. They were presented simply to illustrate the computations involved. Indeed, there are surely other components to be considered. (Sklerov and Couper, 2011) These must be identified by the forensic toxicologist considering their particular laboratory, protocol, instruments, customers and the required fitness-for-

We have illustrated above several examples for estimating the combined uncertainty in contexts relevant to forensic toxicology. These examples have presented the standard bottom-up approach recommended largely by the *GUM* document. There are, however, several other approaches to dealing with uncertainty that have been proposed in the forensic toxicology and metrological literature. Wallace, for example, has proposed a number of different methods for estimating measurement uncertainty. (Wallace, 2010)

One method advocated by Wallace is the use of proficiency test data. (Wallace, 2010) Proficiency testing basically consists of an organizing laboratory which, employing well established and traceable methods, prepares and tests the concentrations of several samples. These samples are then sent blindly to participating laboratories with instructions on how the measurements are to be performed, recorded and then returned to the organizing laboratory. The samples are to be treated by the participating laboratories as routine case samples and tested according to their routine protocols. The organizing laboratory summarizes the data reporting means, standard deviations and various plots, including, for example, Z-scores. The standard deviations at various mean concentrations can be used to generate uncertainty functions. Clearly, these estimates will exhibit rather large variation due to the different laboratories, instruments, protocols, analysts, time, etc. These estimates, conditioned on the appropriate concentration, can be used as the total method component in the combined uncertainty estimate. Consider an example where we have duplicate blood alcohol results obtained in the toxicology laboratory of 0.118 and 0.116 g/dL. The laboratory participated in a proficiency study which yielded the uncertainty function observed in figure 7. This figure was actually generated from data available from Collaborative Testing Services [CTS]. For this example we will assume the following measurement function:

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

The mean of our assumed duplicate results is 0.1170 g/dL. The reference value is R=0.100 g/dL with a Type B standard uncertainty of 0.0003 g/dL. The mean measurement of the

*<sup>X</sup>* (50)

linearity" component.

**9.1 Use of proficiency test data** 

**9. Different methods for estimating uncertainty** 

<sup>0</sup>

where: CCorr = the corrected measurement result C0 = the mean of the original duplicate measurements

*X* the mean result for measuring the reference controls

R = the reference value for the controls

purpose.

controls were *X g dL* 0.1024 / with n=34 measurements and a standard uncertainty of 0.0009 g/dL. Computing our corrected estimate from equation 56 we obtain 0.1143 g/dL. Using this value to estimate our method uncertainty from the equation found in figure 8 we obtain: *u g <sup>M</sup>* 0.0369 0.1143 0.00129 0.0055 / *dL* . Assuming independence and the multiplicative model of equation 50, we now estimate our combined uncertainty as seen in equation 51. The approximate 95% confidence interval (k=2) for the true mean blood alcohol concentration in this example would be:

$$
\overline{Y} \pm 2\,\mu\_{\widetilde{Y}} \Rightarrow 0.1143 \pm 2(0.0039) \Rightarrow 0.1065 \text{ to } 0.1221 \,\text{g/} \,\text{dL} \dots
$$

The risk in using proficiency data in this manner is that the actual uncertainty associated with a particular laboratory may be overestimated. Another limitation to keep in mind is that the proficiency data may not have been generated with the same analytical protocol employed within a particular laboratory. Proficiency data, however, does have a large source of variation, which may be acceptable within the forensic context. The uncertainty budget for these results is found in Table 7. The method uncertainty determined from the proficiency test data in this example, contributed by far the most to the combined uncertainty while the reference and analytical components could effectively be ignored.

Fig. 7. Plot of the standard deviation against concentration and determination of an uncertainty function from CTS proficiency test blood alcohol data

Measurement Uncertainty in Forensic Toxicology: Its Estimation, Reporting and Interpretation 447

Fig. 8. Illustrating the construction and use of a "guard band" to determine compliance with

The corrected mean breath alcohol results in our example is found from equation 52 to be:

0.0950 0.0824 0.0914 / 210 0.0856 *C g Corr <sup>L</sup>* . We will assume that our method uncertainty is

determined from the uncertainty function found in figure 6 which results in: *u* 0.0260 0.0914 0.00095 0.0033 / 210 *g L* . Given that our measurement function in equation 52 is multiplicative, we now find our combined uncertainty, assuming

2 1 <sup>46</sup> 0.0914 0.0274 0.0025 / 210 0.0914 0.0914 0.0824 0.0856

We now find the relative combined uncertainty by removing the values of n according to:

<sup>222</sup> 0.0033 0.0008 0.0010 0.0914 0.0391 0.0036 / 210 0.0914 0.0914 0.0824 0.0856

Now we determine our effective degrees of freedom using the Welch-Satterthwaite equation

4

0.0036

.0033 .0008 0.0010 0.0914 0.0824 0.0856

444

Notice that in the Welch-Satterthwaite equation we have changed our degrees of freedom for the total method component to infinity. This is because the standard uncertainty estimate

*Y*

*Y*

0.0914 5814.8

45

*<sup>u</sup> g L* .

*u g L*

.

*X* the mean of replicate (n=18) measurements of the reference control standard

where: CCorr = the corrected breath alcohol concentration

R = the traceable control reference value

a specified limit

*Y*

*Y*

as follows:

*u*

*u*

 

independence, as in equation 51:

*eff*

2 2 <sup>2</sup> 0.0033 0.0008 0.0010

$$\begin{aligned} \frac{u\_C}{C} &= \sqrt{C}V\_{C\_0}^2 + CV\_R^2 + CV\_X^2} \\\\ \frac{u\_C}{C} &= \sqrt{\left[\frac{\frac{u\_C}{\sqrt{n}}}{C\_0}\right]^2 + \left[\frac{\frac{u\_R}{\sqrt{n}}}{R}\right]^2 + \left[\frac{\frac{u\_R}{\sqrt{n}}}{R}\right]^2} \\\\ \frac{u\_\overline{\gamma}}{0.1143} &= \sqrt{\left[\frac{0.0055}{0.1143}\right]^2 + \left[\frac{0.0003}{0.100}\right]^2 + \left[\frac{0.0009}{0.1024}\right]^2} \\\\ \end{aligned} \tag{51}$$


1 Percent of contribution to total combined uncertainty

Table 7. Uncertainty budget resulting from the use of proficiency test data as the estimate for method uncertainty

#### **9.2 Using the guard band approach**

Employing a guard band is another approach to accounting for measurement uncertainty. (EURACHEM/CITAC, 2000) Use of the guard band is a tool for determining compliance within specified limits. It establishes a decision rule, particularly relevant where there are critical or prohibited analytical limits which may define, for example, binary outcomes such as pass/fail, guilty/not guilty, etc. These can be important in drunk-driving prosecution where alcohol results (either blood or breath) are introduced to establish whether the subject exceeded the legal limit. Consider the example where an individual provided duplicate breath alcohol results of 0.092 and 0.098 g/210L. A traceable commercially purchased simulator control standard having a reference value of 0.0824 g/210L and a Type B combined uncertainty of 0.0008 g/210L was measured by the breath test instrument. The mean of n=46 measurements with this control was 0.0856 g/210L with a standard uncertainty of 0.0010 g/d10L. We wish to determine an upper limit to the guard band, above which we will be 99% confident that the individual's true mean breath alcohol concentration exceeds 0.080 g/210L. This can be visualized in figure 8 where we see that the upper limit of the guard band is the value 0.080 + kuC. We must first find the combined uncertainty (uC) and then the appropriate value of k. The value of k will actually be from the t-distribution in this example and will need to correspond to a 98% confidence interval. The degrees of freedom will be determined from the Welch-Satterthwaite equation. We begin by identifying our measurement function as follows:

$$C\_{Corr} = \frac{C\_0 \ R}{\overline{X}} \tag{52}$$

where: CCorr = the corrected breath alcohol concentration

R = the traceable control reference value

446 Toxicity and Drug Testing

2 2 2

*Y*

*C R X*

*u u u*

0

0

2 1 <sup>34</sup> 0.1143 0.0342 0.0039 / 0.1143 0.1143 0.100 0.1024

**Source Type %CV Percent1** Method (Proficiency) A 5% 99% Reference B 0.3% 0.8% Analytical A 0.9% 0.2% Total 100%

Table 7. Uncertainty budget resulting from the use of proficiency test data as the estimate for

Employing a guard band is another approach to accounting for measurement uncertainty. (EURACHEM/CITAC, 2000) Use of the guard band is a tool for determining compliance within specified limits. It establishes a decision rule, particularly relevant where there are critical or prohibited analytical limits which may define, for example, binary outcomes such as pass/fail, guilty/not guilty, etc. These can be important in drunk-driving prosecution where alcohol results (either blood or breath) are introduced to establish whether the subject exceeded the legal limit. Consider the example where an individual provided duplicate breath alcohol results of 0.092 and 0.098 g/210L. A traceable commercially purchased simulator control standard having a reference value of 0.0824 g/210L and a Type B combined uncertainty of 0.0008 g/210L was measured by the breath test instrument. The mean of n=46 measurements with this control was 0.0856 g/210L with a standard uncertainty of 0.0010 g/d10L. We wish to determine an upper limit to the guard band, above which we will be 99% confident that the individual's true mean breath alcohol concentration exceeds 0.080 g/210L. This can be visualized in figure 8 where we see that the upper limit of the guard band is the value 0.080 + kuC. We must first find the combined uncertainty (uC) and then the appropriate value of k. The value of k will actually be from the t-distribution in this example and will need to correspond to a 98% confidence interval. The degrees of freedom will be determined from the Welch-Satterthwaite equation. We begin by

0

*<sup>X</sup>* (52)

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

 

0

*Y*

method uncertainty

*u*

2 22

*C C C R X*

2 2 <sup>2</sup> 0.0055 0.0003 0.0009

1 Percent of contribution to total combined uncertainty

identifying our measurement function as follows:

**9.2 Using the guard band approach** 

*u u nnn CV CV CV C CC R X*

*u g dL*

(51)

*X* the mean of replicate (n=18) measurements of the reference control standard

Fig. 8. Illustrating the construction and use of a "guard band" to determine compliance with a specified limit

The corrected mean breath alcohol results in our example is found from equation 52 to be:

 0.0950 0.0824 0.0914 / 210 0.0856 *C g Corr <sup>L</sup>* . We will assume that our method uncertainty is

determined from the uncertainty function found in figure 6 which results in: *u* 0.0260 0.0914 0.00095 0.0033 / 210 *g L* . Given that our measurement function in equation 52 is multiplicative, we now find our combined uncertainty, assuming independence, as in equation 51:

$$\frac{\mu\_{\overline{Y}}}{0.0914} = \sqrt{\frac{0.0033}{0.0914}}^2 \left[ + \left[ \frac{0.0008}{\sqrt{1}} \right]^2 + \left[ \frac{0.0010}{0.0856} \right]^2 \right] \quad \Rightarrow \quad \mu\_{\overline{Y}} = 0.0914 \{ 0.0274 \} = 0.0025 \,\mathrm{g} \,/\, 210L \dots$$

We now find the relative combined uncertainty by removing the values of n according to:

$$\frac{\mu\_{\overline{Y}}}{0.0914} = \sqrt{\frac{0.0033}{0.0914}}^2 + \left[\frac{0.0008}{0.0824}\right]^2 + \left[\frac{0.0010}{0.0856}\right]^2 \quad \Rightarrow \quad \mu\_{\overline{Y}} = 0.0914 \left(0.0391\right) = 0.0036 \,\mathrm{g} \,/\, 210L \,\mathrm{L}$$

Now we determine our effective degrees of freedom using the Welch-Satterthwaite equation as follows:

$$\nu\_{eff} = \frac{\left[\frac{0.0036}{0.0914}\right]^4}{\left[\frac{.0033}{0.0914}\right]^4 + \left[\frac{.0008}{0.0824}\right]^4 + \left[\frac{0.0010}{0.0856}\right]^4} = 5814.8 \approx \infty$$

Notice that in the Welch-Satterthwaite equation we have changed our degrees of freedom for the total method component to infinity. This is because the standard uncertainty estimate

Measurement Uncertainty in Forensic Toxicology: Its Estimation, Reporting and Interpretation 449

selected a coverage factor of k=2, we would obtain: *TEa* 4% 2 2% 8% . This would provide an upper limit estimate for the customer who could be assured, with a high degree of probability, that the total error would not exceed this limit. One might report the final results in this context as: *The whole blood alcohol results were 0.094 and 0.096 g/dL having a mean of 0.0950 g/dL which did not have an associated total allowable error of more than 8% with approximately 95% probability.* One context appropriate for the application of the total error method is where a single control is measured as part of an analytical run. If the control exceeded the total allowable error, one would not know whether it was due to bias or random sources. However, the result would be caught and the system corrected before resuming routine measurements. One of the criticisms of the method of total allowable error method is that it allows bias to exist without correcting for it. (Dybkaer, 1999, Dybkaer, 1999) Admittedly, the total error method provides a very conservative estimate, a maximum

Monte Carlo methods are simulation techniques that are more computationally intensive. With faster computers available, these methods are becoming more popular. Monte Carlo methods require assumptions regarding the measurement function along with the distributional form and parameters for each of the input components, being themselves random variables. Random data are then simulated from each of the component distributions, placed into the measurement function, followed by the computation of the measurand. This is done a large number of times, generating a distribution of response values. From these results, the distribution, the expected value and the standard uncertainty of the response variable can be determined. As a result we do not need to assume some distributional form for the response variable and we have a direct, empirically determined estimate of uncertainty. Monte Carlo methods also avoid two limitations of the GUM method – the required linear relationship between the response variable and the components and the justified application of the central limit theorem. (Fernandez, et.al., 2009) Consider the following example of a breath alcohol measurement function where we

*Cont*

distributions for each of the six input variables: <sup>2</sup> *Y N* <sup>0</sup> ~ 0.1250,0.0047 the mean of the

original n measurements, <sup>2</sup> ~ 0.1025,0.0008 *GC N Sol* the mean of the simulator solution

measurements by gas chromatography, <sup>2</sup> *R N*~ 0.100,0.0003 the traceable reference value,

<sup>2</sup> *X N*~ 0.0825,0.0012 the mean of the breath test instrument measuring the simulator

solution, *K Unif* ~ 1.21,1.25 the ratio of partition coefficients in the simulator heated to 340C and <sup>2</sup> ~ 0.0980,0.0008 *GC N Cont* the results from measuring the traceable controls on the gas chromatograph. We employ a routine written in R that simulates random results from each of these distributions and computes the response variable *YCorr* . This is done 10,000 times. The resulting distribution for the response variable is seen in figure 9. The

and where we assume the following

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

actually, for interpreting measurement uncertainty.

have six input variables: <sup>0</sup> *Sol Corr*

**9.4 Monte Carlo methods** 

(0.0033 g/210L) from figure 6 is based on much more than one degree of freedom (n>27,000). The degrees of freedom for the reference standard in the Welch-Satterthwaite equation is set to infinity because it is a Type B uncertainty without information on the degrees of freedom provided. Since we have essentially an infinite number of effective degrees of freedom we select our k (or t) value of 1.96. We can now compute the upper limit for our guard band: 0.080 + 2.33(0.0025) = 0.0858 g/210L. Since the subject's corrected mean breath alcohol concentration exceeds the upper guard band limit of 0.0858 g/210L we conclude there is 99% confidence that the individual's true mean breath alcohol concentration exceeds 0.080 g/210L. Values exceeding 0.080 + kuc could be considered within the "rejection zone". For a measurement in this region, the probability of a "false rejection" is less than α, the probability of the false-positive error. (Desimoni and Brunetti, 2007) One must also keep in mind that for guard band estimates at different concentrations, the combined uncertainty estimates need to incorporate the method uncertainty appropriate to that concentration. The guard band approach could also be generated based on a large set of historical data and then employed for a period of time. The estimates could be updated annually, for example, to ensure the system remains in statistical control. The assumptions with this approach is that the individuals continue to be tested on the same instrumentation and protocols used to generate the guard band limits and that the system remains in statistical control. The United Kingdom is one jurisdiction that employs a guard band approach. (Walls and Brownlee, 1985) A value of 6mg/dL is subtracted from the mean of duplicate blood alcohol results below 100 mg/dL and 6% is deducted from results over 100 mg/dL. The results of this deduction must exceed their legal limit of 80 mg/dL for prosecution. Denmark employs a similar approach where they deduct 0.1 g/Kg to compute their level for prosecution. (Kristiansen and Petersen, 2004) Similarly, Sweden employs the guard band approach to uncertainty estimation by requiring that the lower 99.9% confidence interval limit for mean results must exceed their legal limit. (Jones and Schuberth, 1989) Guard band calculations could also be incorporated into computerized breath test instruments for immediate determination of critical limits for purposes of prosecution.

#### **9.3 Uncertainty estimation from total allowable error**

There is considerable debate regarding the best method for estimating measurement uncertainty and whether it is even necessary. Many argue that measurement uncertainty is unnecessary because it may be misunderstood by the customer or confuse the interpretation. Since bias is only determined with regard to a reference standard, many analytes do not have standards available while others have several. As a result, it is argued that bias may not be validly determined in the first place. Some that argue against the use of measurement uncertainty would advocate the use of total allowable error (TEa). (Westgard, 2010) Total allowable error is determined from the following linear model:

$$TE\_a = \left| bias \right| + k \, u\_{\odot} \, \, . \tag{53}$$

The total allowable error combines both bias and random components and estimates the upper limit. In some cases this may over estimate the actual capability of the analytical method or laboratory performance. Moreover, the method of total allowable error does not correct for bias - it simply includes the maximum level allowable. If we were to allow a maximum bias of 4% and the relative combined uncertainty for the method was 2% and we selected a coverage factor of k=2, we would obtain: *TEa* 4% 2 2% 8% . This would

provide an upper limit estimate for the customer who could be assured, with a high degree of probability, that the total error would not exceed this limit. One might report the final results in this context as: *The whole blood alcohol results were 0.094 and 0.096 g/dL having a mean of 0.0950 g/dL which did not have an associated total allowable error of more than 8% with approximately 95% probability.* One context appropriate for the application of the total error method is where a single control is measured as part of an analytical run. If the control exceeded the total allowable error, one would not know whether it was due to bias or random sources. However, the result would be caught and the system corrected before resuming routine measurements. One of the criticisms of the method of total allowable error method is that it allows bias to exist without correcting for it. (Dybkaer, 1999, Dybkaer, 1999) Admittedly, the total error method provides a very conservative estimate, a maximum actually, for interpreting measurement uncertainty.

#### **9.4 Monte Carlo methods**

448 Toxicity and Drug Testing

(0.0033 g/210L) from figure 6 is based on much more than one degree of freedom (n>27,000). The degrees of freedom for the reference standard in the Welch-Satterthwaite equation is set to infinity because it is a Type B uncertainty without information on the degrees of freedom provided. Since we have essentially an infinite number of effective degrees of freedom we select our k (or t) value of 1.96. We can now compute the upper limit for our guard band: 0.080 + 2.33(0.0025) = 0.0858 g/210L. Since the subject's corrected mean breath alcohol concentration exceeds the upper guard band limit of 0.0858 g/210L we conclude there is 99% confidence that the individual's true mean breath alcohol concentration exceeds 0.080 g/210L. Values exceeding 0.080 + kuc could be considered within the "rejection zone". For a measurement in this region, the probability of a "false rejection" is less than α, the probability of the false-positive error. (Desimoni and Brunetti, 2007) One must also keep in mind that for guard band estimates at different concentrations, the combined uncertainty estimates need to incorporate the method uncertainty appropriate to that concentration. The guard band approach could also be generated based on a large set of historical data and then employed for a period of time. The estimates could be updated annually, for example, to ensure the system remains in statistical control. The assumptions with this approach is that the individuals continue to be tested on the same instrumentation and protocols used to generate the guard band limits and that the system remains in statistical control. The United Kingdom is one jurisdiction that employs a guard band approach. (Walls and Brownlee, 1985) A value of 6mg/dL is subtracted from the mean of duplicate blood alcohol results below 100 mg/dL and 6% is deducted from results over 100 mg/dL. The results of this deduction must exceed their legal limit of 80 mg/dL for prosecution. Denmark employs a similar approach where they deduct 0.1 g/Kg to compute their level for prosecution. (Kristiansen and Petersen, 2004) Similarly, Sweden employs the guard band approach to uncertainty estimation by requiring that the lower 99.9% confidence interval limit for mean results must exceed their legal limit. (Jones and Schuberth, 1989) Guard band calculations could also be incorporated into computerized breath test instruments for immediate determination of critical limits for purposes of

There is considerable debate regarding the best method for estimating measurement uncertainty and whether it is even necessary. Many argue that measurement uncertainty is unnecessary because it may be misunderstood by the customer or confuse the interpretation. Since bias is only determined with regard to a reference standard, many analytes do not have standards available while others have several. As a result, it is argued that bias may not be validly determined in the first place. Some that argue against the use of measurement uncertainty would advocate the use of total allowable error (TEa). (Westgard,

The total allowable error combines both bias and random components and estimates the upper limit. In some cases this may over estimate the actual capability of the analytical method or laboratory performance. Moreover, the method of total allowable error does not correct for bias - it simply includes the maximum level allowable. If we were to allow a maximum bias of 4% and the relative combined uncertainty for the method was 2% and we

*TE bias k u a C* . (53)

2010) Total allowable error is determined from the following linear model:

prosecution.

**9.3 Uncertainty estimation from total allowable error** 

Monte Carlo methods are simulation techniques that are more computationally intensive. With faster computers available, these methods are becoming more popular. Monte Carlo methods require assumptions regarding the measurement function along with the distributional form and parameters for each of the input components, being themselves random variables. Random data are then simulated from each of the component distributions, placed into the measurement function, followed by the computation of the measurand. This is done a large number of times, generating a distribution of response values. From these results, the distribution, the expected value and the standard uncertainty of the response variable can be determined. As a result we do not need to assume some distributional form for the response variable and we have a direct, empirically determined estimate of uncertainty. Monte Carlo methods also avoid two limitations of the GUM method – the required linear relationship between the response variable and the components and the justified application of the central limit theorem. (Fernandez, et.al., 2009) Consider the following example of a breath alcohol measurement function where we

have six input variables: <sup>0</sup> *Sol Corr Cont Y GC R <sup>Y</sup> X K GC* and where we assume the following

distributions for each of the six input variables: <sup>2</sup> *Y N* <sup>0</sup> ~ 0.1250,0.0047 the mean of the original n measurements, <sup>2</sup> ~ 0.1025,0.0008 *GC N Sol* the mean of the simulator solution measurements by gas chromatography, <sup>2</sup> *R N*~ 0.100,0.0003 the traceable reference value, <sup>2</sup> *X N*~ 0.0825,0.0012 the mean of the breath test instrument measuring the simulator solution, *K Unif* ~ 1.21,1.25 the ratio of partition coefficients in the simulator heated to 340C and <sup>2</sup> ~ 0.0980,0.0008 *GC N Cont* the results from measuring the traceable controls on the gas chromatograph. We employ a routine written in R that simulates random results from each of these distributions and computes the response variable *YCorr* . This is done 10,000 times. The resulting distribution for the response variable is seen in figure 9. The

Measurement Uncertainty in Forensic Toxicology: Its Estimation, Reporting and Interpretation 451

uncertainty. (Gleser, 1998, Weise and Woger, 1992, Kacker and Jones, 2003, Phillips, et.al., 1998). Space does not permit further discussion of these important and useful methods.

Several examples have been presented here for estimating measurement uncertainty in the context of forensic toxicology. By no means do these examples imply that all possible uncertainty components have been considered. These examples were intended primarily to illustrate the general approach and computations involved. Moreover, while an example may have assumed a blood alcohol context, it could just as well have been applied in the context of breath or drug analysis. While the general approach will be relevant to most methods in forensic toxicology, each laboratory will need to identify and quantify its uncertainty components unique to its protocols and instrumentation. The examples and discussion presented here have also assumed independence among the input or predictor variables. This is certainly not always a valid assumption. In some measurement contexts there will be significant correlation between input variables which must be accounted for. (GUM, EURACHEM/CITAC, Ellison, 2005) While these concepts may be new to some practicing toxicologists, the concept of measurement uncertainty should not raise concerns for the forensic sciences. The emphasis should be on their ability to quantify confidence of measurement results. They should be presented in a manner that emphasizes and demonstrates their fitness-for-purpose. Modern technology should enhance and simplify these computations as well. Spreadsheet programs can be developed which require only the entry of specific values followed by the generation of all uncertainty results. Moreover, such computations can even be incorporated into the software of aalytical instruments. Such

Several factors are responsible for the emphasis today on reporting measurement results along with their uncertainty. These include legal, economic, liability, accrediting and technological considerations. As professional toxicologists concerned with providing measurement results of the highest possible quality, we must be prepared to make this extra effort of providing the relevant uncertainty. Since there is no consensus regarding the best approach for computing uncertainty at this time, toxicologists should be familiar with the several approaches suggested here and then select and validate the one which best suits their analytical, procedural and legal context. The literature is rich with material regarding measurement uncertainty and should be carefully reviewed by toxicologists. (Drosg, 2007, Williams, 2008, Fernandez, 2011 Ekberg,et.al., 2011) This effort wll enhance the quality and interpretability of our measurement results and help establish a foundation of "evidence based forensics". The unavoidable fact of measurement uncertainty results in the risk of making incorrect decisions. While ignoring the uncertainty increases this risk, providing the uncertainty reduces and quantifies the risk for the decision maker. This fact alone should motivate the legal community to request and forensic toxicologists to rigorously estimate

Altman, D.G., Machin, D., Bryant, T.N. and Gardner, M.J. (2000). *Statistics with confidence*,

(2nd edition), British Medical Journal, London, ISBN 0-7279-1375-1

technology, when validated, should greatly simplify the process.

**11. Discussion** 

and provide such estimates.

**12. References** 

expected value for the response variable is 0.1287 g/210L with an empirical 95% confidence interval of 0.1215 to 0.1360g/210L, determined from the distribution of results in figure 9. The sampling/method component was also correctly identified as having the largest contribution to total uncertainty of 85%.

Fig. 9. A distribution of 10,000 Monte Carlo simulated measurement results
