Journal of Analytical Toxicology 2015;39:411 –414 doi:10.1093/jat/bkv025 Advance Access publication March 31, 2015
Technical Note
How Many Digits Should Be Reported In Forensic Breath Alcohol Measurements?† Rod G. Gullberg* Clearview Statistical Consulting, 20119 61st Avenue SE, Snohomish, WA 98296, USA *Author to whom correspondence should be addressed. Email: [email protected]
Uncertainty is an inherent property of all measurements. The magnitude of this uncertainty will determine the number of meaningful digits that should be reported in a measurement result. Several statistical arguments are considered providing evidence that three digit truncated results are more appropriate than two since the first significant digit of the combined uncertainty (standard deviation) in breath alcohol measurement is found in the third decimal place. Probably, the most compelling reason for reporting three digits is the significant reduction in combined uncertainty compared with the use of two digits. For a breath alcohol concentration of 0.089 g/210 L, the combined uncertainty for two digit results is ∼0.0042 g/210 L, compared with 0.0031 g/210 L for three digit results. The historical practice of reporting two digit truncated results in forensic breath alcohol analysis has been largely based on the use of analog scale instruments with 0.01 g/210 L scale resolution. With today’s modern digital instrumentation, this practice should be reconsidered. While the focus of this paper is on breath alcohol analysis, the general principles will apply to any quantitative analytical measurement.
Introduction The results of forensic breath alcohol measurement in a per se legal context (i.e., 0.080 g/210 L) provide significant and compelling evidence in the prosecution of drunk-driving cases. The numerical results possess significant evidential weight with each reported digit being carefully considered by the court. Unfortunately, the courts often assign unmerited confidence to numerical results without appreciation for their uncertainty. For our present discussion, the combined uncertainty shall refer to the standard deviation estimated from combining all known sources of uncertainty in the manner outlined in the GUM document (1). In addition, when mention is made here of ‘two digit’ or ‘three digit’ results, this shall refer to truncated results in the second or third position beyond the decimal point (i.e., 0.12 or 0.126 g/210 L, respectively). The recommended practice in analytical chemistry is to report all digits that are known exactly along with one additional digit (the last or least significant digit) that is uncertain and results from appropriate rounding (or truncation in our case) (2). The objective is to report all digits in a forensic measurement that possess meaningful signal and avoid extra digits possessing uninformative noise. In addition, the combined uncertainty (standard deviation) should be rounded to no more than two significant digits. The recommended practice espoused by the National Safety Council Committee on Alcohol and Other Drugs, however, has been to truncate the third digit of breath alcohol results and report to only two decimal place precision (3). The principle reason for this is to assist in meeting the prosecutorial burden of ‘proof beyond a reasonable doubt’ as well as reporting to the †
The R code for generating the simulated data is available from the author.
same statutory precision found in most jurisdictions (i.e., 0.08). This standard, however, was recommended at a time (1986) when the Breathalyzer instrument with its analog scale of 0.01 g/ 210 L resolution was widely used. Today’s computerized instruments with digital results offer the advantage of ‘reported certainty’ for all observed or printed digits. Therefore, the practice of routinely truncating the third digit should be reconsidered. Analytical and statistical arguments will be presented that suggest breath alcohol measurements obtained from modern instrumentation should be reported to the third decimal place, helping to ensure their fitness-for-purpose. Indeed, the third decimal place is where we see the first significant digit in the combined uncertainty (standard deviation) of forensic breath alcohol measurement (4, 5). General rules for reporting analytical results Consider two arbitrary yet typical sets of replicate breath alcohol measurements: Set 1
Set 2
0.127 0.124 0.123 0.125 0.127 0.126
0.119 0.118 0.122 0.121 0.120 0.122
Both sets of data have the same standard deviation (SD ¼ 0.0016 g/210 L). From Set 1, we see that the first and second digits to the right of the decimal are certain—they have no variability. Only the third digit shows variation. Therefore, the measurement results should be reported to the third decimal place. If, however, the set of data had a mean near 0.120, as in Set 2 above, variation would be observed in the second digit. This should not be interpreted, however, as requiring the retention of only two digits. The criteria for how many digits to retain should be determined by the position of the most significant digit in the standard deviation—the third decimal position in both sets above. Consider n ¼ 5,000 simulated measurements distributed as Yi N ð0:080; 0:00052 Þ. This will show variation in the second digit and yet four digit data should be retained. This results from observing that the standard deviation has its most significant digit in the fourth decimal place (0.0005). The standard deviation and the position of its most significant digit, therefore, indicate the appropriate decimal position for reported results (6, 7). Instrumental considerations Modern forensic breath alcohol instruments provide digital display and printing of results, eliminating any uncertainty associated with reporting results or estimation from an analog scale. This also eliminates the concern of ‘digit preference’—a problem where individuals transcribe observed measurement results (8).
# The Author 2015. Published by Oxford University Press. All rights reserved. For Permissions, please email: [email protected]
Analytical instruments, however, possess certain constraints due to hardware and software limitations that determine how many digits can be reliably reported in final results. Appropriate computational procedures will ensure that sufficient precision is maintained throughout. Rounding or truncation should only done on the final reported result. Evaluating actual breath alcohol data When n ¼ 5,000 breath alcohol results were selected for BAC Datamaster results in Washington State during 2006, the mean + SD was 0.1409 + 0.054 for three digit data and 0.136 + 0.054 for two digit truncated data. While the mean estimate is understandably lower for two digit truncated data, the standard deviation remains unchanged due to the large range (0.01 – 0.372 g/210 L). Therefore, two decimal places are sufficient to reproduce a valid standard deviation estimate when summarizing between-subject breath test data. Figure 1 shows uncertainty functions for both two and three digit breath alcohol data, similar to that found in other jurisdictions (9). The standard deviation estimates were determined from the following equation: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pk 2 i¼1 di S ¼ 2k
ð1Þ
where di is the difference between duplicate results for the ith subject and k, the number of subjects. Throughout the entire forensically relevant range, the standard deviation remains ,0.01 g/210 L for both the two and three digit data. The estimates, however, are larger for the two digit data. At 0.080 g/210 L, for example, the standard deviation is 0.0025 g/210 L determined from three digit data and 0.0038 g/ 210 L from two digit data—a 52% increase. This implies that within-subject results should be reported to the third decimal place, corresponding to the position of the first significant figure in the standard deviation.
Analyzing simulated breath test results A terminal digit that conforms to a discrete uniform distribution is a useful diagnostic (10). The distribution form, however, depends on the ratio of the span of all data to the magnitude of the digit: ratio ¼ span/digit magnitude. For example, a normal distribution of breath alcohol results having a mean of 0.085 g/ 210 L and a standard deviation of 0.003 g/210 L may yield in 10,000 simulations a span of 0.022 g/210 L. This would yield a ratio for the third digit of: ratio ¼ 0:022=0:001 ¼ 22:0. This ratio needs to be 30 or more to yield a discrete uniform distribution for the digits. Figure 2 illustrates this with n ¼ 10,000 normally distributed ðY N ð0:085; 0:0052 ÞÞ simulated breath test data considering digits one through four. These plots were generated using statistical software R V 3.1.0. The first digit to conform to the discrete uniform distribution should be considered the terminal yet informative digit. A previous study (11) showing a uniform distribution for the third digit employed betweensubject data and should not be used to infer the appropriate number of digits to report for within-subject data. Figure 3 shows the standard deviation estimates determined from 1 to 10,000 simulated duplicates having from one to four digit truncated results. This analysis was done in R V 3.1.0. The simulated data had a mean of 0.085 with a standard deviation of 0.0036 g/210 L, representing a typical withinsubject distribution. Both the three and four digit results are indistinguishable and closely approach the known value of 0.0036 g/210 L. For a population standard deviation of 0.0036 g/210 L, the four digit result does not improve the estimate beyond that of the three digit results. One can simulate data with different combinations of mean and standard deviation to observe the effect. Relevance to measurement uncertainty An important issue in forensic toxicology today is the need to provide an estimate of measurement uncertainty as noted in the recent National Academy of Sciences report and required for accreditation (12, 13). The influence of measurement resolution on the combined uncertainty would be important to consider. Table I summarizes several components contributing to the uncertainty in breath alcohol analysis—one of which is the digit resolution. Assuming that the truncated digit has a discrete uniform distribution, the uncertainty associated with truncating to a specified digit will be determined from half the scale resolution. Table I reveals that when two digit data are employed, both the combined uncertainty and the confidence interval are increased over the three digit results by 37%. In addition, the contribution to total uncertainty when using two digit data is 47%, compared with 1% when using three digit data. Clearly, the scale resolution will be a major contributor to the combined uncertainty (14). Moreover, the duplicate test agreement requirement (typically +0.02 g/210 L) will also be influenced by how many digits are retained and should be evaluated. Discussion
Figure 1. Uncertainty functions for computing the standard deviation determined from two and three digit truncated breath alcohol results obtained from arrested drivers in Washington State using the BAC Datamaster.
412 Gullberg
Courts and juries often have the ‘illusion of certainty’ (15) and yet are asked to interpret numerical data in support of critical decisions. They should be provided with the most informative
Figure 2. Distributions of the first four digits determined from simulated breath alcohol results from a normal distribution with a mean of 0.085 g/210 L and a standard deviation of 0.005 g/210 L.
Table I Estimates Associated With Seven Components Contributing to the Combined Uncertainty in Breath Alcohol Analysis Component
Mean
Sampling GC solution Breath instrument Simulator partition coefficient Reference GC reference Scale two digit Scale three digit
0.0897 0.1020 0.0825 1.23 0.100 0.0980 0.0897 0.0897
Uncertainty
n
(g/210 L or g/dL) 0.0039 0.0008 0.0006 0.0124 0.0003 0.0006 0.00289 0.000289
Percent Two digit
2 15 10 1 1 3 1 1 Total
47 0.2 0.2 5 0.2 0.4 47 1 100
Three digit 88 0.2 0.4 9 0.4 1
100
Combined uncertainty two digit 0.00420 g/210 L Combined uncertainty three digit 0.00306 g/210 L 99% confidence interval two digit 0.0789 –0.1006 g/210 L 99% confidence interval three digit 0.0819 –0.0976 g/210 L The effects of two and three digit results on the combined uncertainty are significant.
Figure 3. Standard deviation estimates determined as a function of n from duplicate simulated breath alcohol results based on one through four digits. The simulating distribution had a mean of 0.085 g/210 L and a standard deviation of 0.0036 g/210 L. The horizontal line is 0.0036 g/210 L.
results as possible—including all relevant and meaningful digits. Hence, retaining one figure too many is better than retaining one too few (16).
The reporting of measurement results for forensic purposes must be consistent with local regulations and administrative rules. A recent Ontario court decision required that duplicate three digit measurements from the Intoxilyzer 5000C not differ by .20 mg/dL (0.02 g/210 L). While analysis showed that a small percentage (3%) of cases had differences between 21 and 29 mg/dL (0.021–0.029 g/210 L) for three digit results, these individuals were nevertheless affected (17). Where administrative Resolution for Reporting Breath Alcohol Measurements 413
rules allow for only two digits to be reported, the third can always be preserved in the database for future analyses. The third digit can always be shown on the printout as the analytical results while the truncated two digit results (or the lower of the two) can be reported for legal purposes. Three digit results from modern digital instruments should always be employed in a research or method validation context. The relevance of this was recently demonstrated in a study (18) where such variables as time-to-peak, concentration increase following the last drink and length of concentration plateau were influenced by the number of digits employed in the data. Indeed, the ability to resolve relevant information in the study was lost due to the truncation to two decimal places. Control standard results (simulator or dry gas) should clearly be reported to three digits where within-run variation is in the third digit and standard deviations are typically 0.0015 g/ 210 L. No variation would be observed if recording only two digit results (19). In addition, the allowed limits for the control results should be specified to three digits, consistent with the measurement results. A control result of 0.096 g/210 L does not comply with an upper limit of 0.090 g/210 L, but would be unclear with an upper limit specified as 0.09 g/210 L.
Conclusion Several statistical arguments have been presented here to show that forensic breath alcohol results should be reported and analyzed employing three digits rather than two. These arguments include: (i) uncertainty functions, (ii) two and three terminal digit distributions and (iii) measurement uncertainty. We must emphasize that these analyses refer to within-subject data (forensically the most relevant) and not between-subject data. The position of the most significant figure in the standard deviation should determine the position of the least significant digit in the reported results. Failing to retain the third digit means it can never be recovered while its retention can always allow for subsequent elimination if desired. Moreover, we have shown that analysis of within-subject results up to 0.28 g/210 L indicates the third digit is informative, relevant and carries an important part of the signal that should be reported and used in subsequent computations. Finally, while the focus of this work was on breath alcohol measurement, the issues presented here apply equally as well to blood alcohol analysis or any other analytical results.
414 Gullberg
References 1. International Organization for Standardization. Guide to the Expression of Uncertainty in Measurement; ISO: Geneva, 2008. http://www.bipm.org/utils/common/documents/jcgm/JCGM_100_ 2008_E.pdf (accessed Mar 17, 2004). 2. Wilrich, P. (2005) Rounding of measurement values or derived values. Measurement, 37, 21– 30. 3. Committee on Alcohol and Other Drugs, National Safety Council: Committee Handbook; National Safety Council: Chicago, IL, 1996, p. 145. 4. Adam, C. (2010) Essential Mathematics and Statistics for Forensic Science; Wiley-Blackwell Publications, New York. p. 17. 5. Groth, T. (1980) Methods and approaches to determine ‘optimal’ quality for clinical chemical data. Scandinavian Journal of Clinical and Labororatory Investigation, 40, 47 –63. 6. Badrick, T., Hickman, P.E. (2008) Significant figures. Clinical Biochemist Review, 29, S89 –S91. 7. Coleman, D., Vanatta, L. (2009) Part 35—Reporting data and significant figures. American Laboratory, 41, 34 –35. 8. Bland, M. (1987) An Introduction to Medical Statistics; Oxford University Press: Oxford, UK, 278– 280. 9. Gullberg, R.G. (2003) Breath alcohol measurement variability associated with different instrumentation and protocols. Forensic Science International, 131, 30– 35. 10. Mohapl, J. (2000) Measurement diagnostics by analysis of last digits. Environmental Monitoring and Assessment, 61, 407– 417. 11. Gullberg, R.G. (1988) Statistical evaluation of truncated breath-alcohol test measurements. Journal of Forensic Sciences, 33, 507–510. 12. National Academy of Science. Strengthening Forensic Science in the United States: A Path Forward; National Research Council, The National Academies Press: Washington, DC, 2009. 13. ASCLD/LAB-International. ASCLD/LAB Policy on Measurement Uncertainty; ASCLD/LAB: Garner, NC, 2011. http://www.ascld-lab.org/ wp-content/uploads/2013/05/AL-PD-3060-Policy_on_Measurement_ Uncertainty_v1.1_unmked.pdf (accessed Mar 17, 2004). 14. Hawkins, R.C.W., Johnson, R.N. (1990) The significance of significant figures. Clinical Chemistry, 36, 824. 15. Gigerenzer, G. (2002) Calculated Risks: How to Know When Numbers Deceive You; Simon and Schuster, New York. 16. Drosq, M. (2007) Dealing with Uncertainties: A Guide to Error Analysis; Springer, New York. 17. Langille, R.M., Patrick, J. (2005) Precision of breath alcohol testing in the field using the Intoxilyzer 5000C and the paradox of truncation. Canadian Society of Forensic Science Journal, 39, 55– 64. 18. McElrea, A., Su, C. (2011) Truncation of breath alcohol values and its effect on peak concentrations: a comparative intra-subject study. Canadian Society of Forensic Science Journal, 44, 13– 21. 19. ASTM International (2004) Standard practice for using significant digits in test data to determine conformance with specifications ASTM E29-13, ASTM International, West Conshohocken, PA 194282959.
Technical Note
How Many Digits Should Be Reported In Forensic Breath Alcohol Measurements?† Rod G. Gullberg* Clearview Statistical Consulting, 20119 61st Avenue SE, Snohomish, WA 98296, USA *Author to whom correspondence should be addressed. Email: [email protected]
Uncertainty is an inherent property of all measurements. The magnitude of this uncertainty will determine the number of meaningful digits that should be reported in a measurement result. Several statistical arguments are considered providing evidence that three digit truncated results are more appropriate than two since the first significant digit of the combined uncertainty (standard deviation) in breath alcohol measurement is found in the third decimal place. Probably, the most compelling reason for reporting three digits is the significant reduction in combined uncertainty compared with the use of two digits. For a breath alcohol concentration of 0.089 g/210 L, the combined uncertainty for two digit results is ∼0.0042 g/210 L, compared with 0.0031 g/210 L for three digit results. The historical practice of reporting two digit truncated results in forensic breath alcohol analysis has been largely based on the use of analog scale instruments with 0.01 g/210 L scale resolution. With today’s modern digital instrumentation, this practice should be reconsidered. While the focus of this paper is on breath alcohol analysis, the general principles will apply to any quantitative analytical measurement.
Introduction The results of forensic breath alcohol measurement in a per se legal context (i.e., 0.080 g/210 L) provide significant and compelling evidence in the prosecution of drunk-driving cases. The numerical results possess significant evidential weight with each reported digit being carefully considered by the court. Unfortunately, the courts often assign unmerited confidence to numerical results without appreciation for their uncertainty. For our present discussion, the combined uncertainty shall refer to the standard deviation estimated from combining all known sources of uncertainty in the manner outlined in the GUM document (1). In addition, when mention is made here of ‘two digit’ or ‘three digit’ results, this shall refer to truncated results in the second or third position beyond the decimal point (i.e., 0.12 or 0.126 g/210 L, respectively). The recommended practice in analytical chemistry is to report all digits that are known exactly along with one additional digit (the last or least significant digit) that is uncertain and results from appropriate rounding (or truncation in our case) (2). The objective is to report all digits in a forensic measurement that possess meaningful signal and avoid extra digits possessing uninformative noise. In addition, the combined uncertainty (standard deviation) should be rounded to no more than two significant digits. The recommended practice espoused by the National Safety Council Committee on Alcohol and Other Drugs, however, has been to truncate the third digit of breath alcohol results and report to only two decimal place precision (3). The principle reason for this is to assist in meeting the prosecutorial burden of ‘proof beyond a reasonable doubt’ as well as reporting to the †
The R code for generating the simulated data is available from the author.
same statutory precision found in most jurisdictions (i.e., 0.08). This standard, however, was recommended at a time (1986) when the Breathalyzer instrument with its analog scale of 0.01 g/ 210 L resolution was widely used. Today’s computerized instruments with digital results offer the advantage of ‘reported certainty’ for all observed or printed digits. Therefore, the practice of routinely truncating the third digit should be reconsidered. Analytical and statistical arguments will be presented that suggest breath alcohol measurements obtained from modern instrumentation should be reported to the third decimal place, helping to ensure their fitness-for-purpose. Indeed, the third decimal place is where we see the first significant digit in the combined uncertainty (standard deviation) of forensic breath alcohol measurement (4, 5). General rules for reporting analytical results Consider two arbitrary yet typical sets of replicate breath alcohol measurements: Set 1
Set 2
0.127 0.124 0.123 0.125 0.127 0.126
0.119 0.118 0.122 0.121 0.120 0.122
Both sets of data have the same standard deviation (SD ¼ 0.0016 g/210 L). From Set 1, we see that the first and second digits to the right of the decimal are certain—they have no variability. Only the third digit shows variation. Therefore, the measurement results should be reported to the third decimal place. If, however, the set of data had a mean near 0.120, as in Set 2 above, variation would be observed in the second digit. This should not be interpreted, however, as requiring the retention of only two digits. The criteria for how many digits to retain should be determined by the position of the most significant digit in the standard deviation—the third decimal position in both sets above. Consider n ¼ 5,000 simulated measurements distributed as Yi N ð0:080; 0:00052 Þ. This will show variation in the second digit and yet four digit data should be retained. This results from observing that the standard deviation has its most significant digit in the fourth decimal place (0.0005). The standard deviation and the position of its most significant digit, therefore, indicate the appropriate decimal position for reported results (6, 7). Instrumental considerations Modern forensic breath alcohol instruments provide digital display and printing of results, eliminating any uncertainty associated with reporting results or estimation from an analog scale. This also eliminates the concern of ‘digit preference’—a problem where individuals transcribe observed measurement results (8).
# The Author 2015. Published by Oxford University Press. All rights reserved. For Permissions, please email: [email protected]
Analytical instruments, however, possess certain constraints due to hardware and software limitations that determine how many digits can be reliably reported in final results. Appropriate computational procedures will ensure that sufficient precision is maintained throughout. Rounding or truncation should only done on the final reported result. Evaluating actual breath alcohol data When n ¼ 5,000 breath alcohol results were selected for BAC Datamaster results in Washington State during 2006, the mean + SD was 0.1409 + 0.054 for three digit data and 0.136 + 0.054 for two digit truncated data. While the mean estimate is understandably lower for two digit truncated data, the standard deviation remains unchanged due to the large range (0.01 – 0.372 g/210 L). Therefore, two decimal places are sufficient to reproduce a valid standard deviation estimate when summarizing between-subject breath test data. Figure 1 shows uncertainty functions for both two and three digit breath alcohol data, similar to that found in other jurisdictions (9). The standard deviation estimates were determined from the following equation: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pk 2 i¼1 di S ¼ 2k
ð1Þ
where di is the difference between duplicate results for the ith subject and k, the number of subjects. Throughout the entire forensically relevant range, the standard deviation remains ,0.01 g/210 L for both the two and three digit data. The estimates, however, are larger for the two digit data. At 0.080 g/210 L, for example, the standard deviation is 0.0025 g/210 L determined from three digit data and 0.0038 g/ 210 L from two digit data—a 52% increase. This implies that within-subject results should be reported to the third decimal place, corresponding to the position of the first significant figure in the standard deviation.
Analyzing simulated breath test results A terminal digit that conforms to a discrete uniform distribution is a useful diagnostic (10). The distribution form, however, depends on the ratio of the span of all data to the magnitude of the digit: ratio ¼ span/digit magnitude. For example, a normal distribution of breath alcohol results having a mean of 0.085 g/ 210 L and a standard deviation of 0.003 g/210 L may yield in 10,000 simulations a span of 0.022 g/210 L. This would yield a ratio for the third digit of: ratio ¼ 0:022=0:001 ¼ 22:0. This ratio needs to be 30 or more to yield a discrete uniform distribution for the digits. Figure 2 illustrates this with n ¼ 10,000 normally distributed ðY N ð0:085; 0:0052 ÞÞ simulated breath test data considering digits one through four. These plots were generated using statistical software R V 3.1.0. The first digit to conform to the discrete uniform distribution should be considered the terminal yet informative digit. A previous study (11) showing a uniform distribution for the third digit employed betweensubject data and should not be used to infer the appropriate number of digits to report for within-subject data. Figure 3 shows the standard deviation estimates determined from 1 to 10,000 simulated duplicates having from one to four digit truncated results. This analysis was done in R V 3.1.0. The simulated data had a mean of 0.085 with a standard deviation of 0.0036 g/210 L, representing a typical withinsubject distribution. Both the three and four digit results are indistinguishable and closely approach the known value of 0.0036 g/210 L. For a population standard deviation of 0.0036 g/210 L, the four digit result does not improve the estimate beyond that of the three digit results. One can simulate data with different combinations of mean and standard deviation to observe the effect. Relevance to measurement uncertainty An important issue in forensic toxicology today is the need to provide an estimate of measurement uncertainty as noted in the recent National Academy of Sciences report and required for accreditation (12, 13). The influence of measurement resolution on the combined uncertainty would be important to consider. Table I summarizes several components contributing to the uncertainty in breath alcohol analysis—one of which is the digit resolution. Assuming that the truncated digit has a discrete uniform distribution, the uncertainty associated with truncating to a specified digit will be determined from half the scale resolution. Table I reveals that when two digit data are employed, both the combined uncertainty and the confidence interval are increased over the three digit results by 37%. In addition, the contribution to total uncertainty when using two digit data is 47%, compared with 1% when using three digit data. Clearly, the scale resolution will be a major contributor to the combined uncertainty (14). Moreover, the duplicate test agreement requirement (typically +0.02 g/210 L) will also be influenced by how many digits are retained and should be evaluated. Discussion
Figure 1. Uncertainty functions for computing the standard deviation determined from two and three digit truncated breath alcohol results obtained from arrested drivers in Washington State using the BAC Datamaster.
412 Gullberg
Courts and juries often have the ‘illusion of certainty’ (15) and yet are asked to interpret numerical data in support of critical decisions. They should be provided with the most informative
Figure 2. Distributions of the first four digits determined from simulated breath alcohol results from a normal distribution with a mean of 0.085 g/210 L and a standard deviation of 0.005 g/210 L.
Table I Estimates Associated With Seven Components Contributing to the Combined Uncertainty in Breath Alcohol Analysis Component
Mean
Sampling GC solution Breath instrument Simulator partition coefficient Reference GC reference Scale two digit Scale three digit
0.0897 0.1020 0.0825 1.23 0.100 0.0980 0.0897 0.0897
Uncertainty
n
(g/210 L or g/dL) 0.0039 0.0008 0.0006 0.0124 0.0003 0.0006 0.00289 0.000289
Percent Two digit
2 15 10 1 1 3 1 1 Total
47 0.2 0.2 5 0.2 0.4 47 1 100
Three digit 88 0.2 0.4 9 0.4 1
100
Combined uncertainty two digit 0.00420 g/210 L Combined uncertainty three digit 0.00306 g/210 L 99% confidence interval two digit 0.0789 –0.1006 g/210 L 99% confidence interval three digit 0.0819 –0.0976 g/210 L The effects of two and three digit results on the combined uncertainty are significant.
Figure 3. Standard deviation estimates determined as a function of n from duplicate simulated breath alcohol results based on one through four digits. The simulating distribution had a mean of 0.085 g/210 L and a standard deviation of 0.0036 g/210 L. The horizontal line is 0.0036 g/210 L.
results as possible—including all relevant and meaningful digits. Hence, retaining one figure too many is better than retaining one too few (16).
The reporting of measurement results for forensic purposes must be consistent with local regulations and administrative rules. A recent Ontario court decision required that duplicate three digit measurements from the Intoxilyzer 5000C not differ by .20 mg/dL (0.02 g/210 L). While analysis showed that a small percentage (3%) of cases had differences between 21 and 29 mg/dL (0.021–0.029 g/210 L) for three digit results, these individuals were nevertheless affected (17). Where administrative Resolution for Reporting Breath Alcohol Measurements 413
rules allow for only two digits to be reported, the third can always be preserved in the database for future analyses. The third digit can always be shown on the printout as the analytical results while the truncated two digit results (or the lower of the two) can be reported for legal purposes. Three digit results from modern digital instruments should always be employed in a research or method validation context. The relevance of this was recently demonstrated in a study (18) where such variables as time-to-peak, concentration increase following the last drink and length of concentration plateau were influenced by the number of digits employed in the data. Indeed, the ability to resolve relevant information in the study was lost due to the truncation to two decimal places. Control standard results (simulator or dry gas) should clearly be reported to three digits where within-run variation is in the third digit and standard deviations are typically 0.0015 g/ 210 L. No variation would be observed if recording only two digit results (19). In addition, the allowed limits for the control results should be specified to three digits, consistent with the measurement results. A control result of 0.096 g/210 L does not comply with an upper limit of 0.090 g/210 L, but would be unclear with an upper limit specified as 0.09 g/210 L.
Conclusion Several statistical arguments have been presented here to show that forensic breath alcohol results should be reported and analyzed employing three digits rather than two. These arguments include: (i) uncertainty functions, (ii) two and three terminal digit distributions and (iii) measurement uncertainty. We must emphasize that these analyses refer to within-subject data (forensically the most relevant) and not between-subject data. The position of the most significant figure in the standard deviation should determine the position of the least significant digit in the reported results. Failing to retain the third digit means it can never be recovered while its retention can always allow for subsequent elimination if desired. Moreover, we have shown that analysis of within-subject results up to 0.28 g/210 L indicates the third digit is informative, relevant and carries an important part of the signal that should be reported and used in subsequent computations. Finally, while the focus of this work was on breath alcohol measurement, the issues presented here apply equally as well to blood alcohol analysis or any other analytical results.
414 Gullberg
References 1. International Organization for Standardization. Guide to the Expression of Uncertainty in Measurement; ISO: Geneva, 2008. http://www.bipm.org/utils/common/documents/jcgm/JCGM_100_ 2008_E.pdf (accessed Mar 17, 2004). 2. Wilrich, P. (2005) Rounding of measurement values or derived values. Measurement, 37, 21– 30. 3. Committee on Alcohol and Other Drugs, National Safety Council: Committee Handbook; National Safety Council: Chicago, IL, 1996, p. 145. 4. Adam, C. (2010) Essential Mathematics and Statistics for Forensic Science; Wiley-Blackwell Publications, New York. p. 17. 5. Groth, T. (1980) Methods and approaches to determine ‘optimal’ quality for clinical chemical data. Scandinavian Journal of Clinical and Labororatory Investigation, 40, 47 –63. 6. Badrick, T., Hickman, P.E. (2008) Significant figures. Clinical Biochemist Review, 29, S89 –S91. 7. Coleman, D., Vanatta, L. (2009) Part 35—Reporting data and significant figures. American Laboratory, 41, 34 –35. 8. Bland, M. (1987) An Introduction to Medical Statistics; Oxford University Press: Oxford, UK, 278– 280. 9. Gullberg, R.G. (2003) Breath alcohol measurement variability associated with different instrumentation and protocols. Forensic Science International, 131, 30– 35. 10. Mohapl, J. (2000) Measurement diagnostics by analysis of last digits. Environmental Monitoring and Assessment, 61, 407– 417. 11. Gullberg, R.G. (1988) Statistical evaluation of truncated breath-alcohol test measurements. Journal of Forensic Sciences, 33, 507–510. 12. National Academy of Science. Strengthening Forensic Science in the United States: A Path Forward; National Research Council, The National Academies Press: Washington, DC, 2009. 13. ASCLD/LAB-International. ASCLD/LAB Policy on Measurement Uncertainty; ASCLD/LAB: Garner, NC, 2011. http://www.ascld-lab.org/ wp-content/uploads/2013/05/AL-PD-3060-Policy_on_Measurement_ Uncertainty_v1.1_unmked.pdf (accessed Mar 17, 2004). 14. Hawkins, R.C.W., Johnson, R.N. (1990) The significance of significant figures. Clinical Chemistry, 36, 824. 15. Gigerenzer, G. (2002) Calculated Risks: How to Know When Numbers Deceive You; Simon and Schuster, New York. 16. Drosq, M. (2007) Dealing with Uncertainties: A Guide to Error Analysis; Springer, New York. 17. Langille, R.M., Patrick, J. (2005) Precision of breath alcohol testing in the field using the Intoxilyzer 5000C and the paradox of truncation. Canadian Society of Forensic Science Journal, 39, 55– 64. 18. McElrea, A., Su, C. (2011) Truncation of breath alcohol values and its effect on peak concentrations: a comparative intra-subject study. Canadian Society of Forensic Science Journal, 44, 13– 21. 19. ASTM International (2004) Standard practice for using significant digits in test data to determine conformance with specifications ASTM E29-13, ASTM International, West Conshohocken, PA 194282959.
