- Email: [email protected]
Assessment of precision of a passive sampler by duplicate measurements
~
Pergamon
Environment International, Vol. 21, No. 4, pp. 407-412, 1995 Copyright O1995 Elsevier Science Ltd Printed in the USA. All fights reserved
0160-4120/95 $9.50+.00
0160-4120(95)00034-8 ASSESSMENT OF PRECISION OF A PASSIVE SAMPLER BY DUPLICATE M E A S U R E M E N T S Kiyoung Lee, Yukio Yanagisawa, and John D. Spengler Department of Environmental Health, Harvard School of Public Health, Boston, MA 02115, USA Roger Davis Department of Biostatistics, Harvard School of Public Health, Boston, MA 02115 and Division of General Medicine and Primary Care, Beth Israel Hospital, Boston, MA 02215, USA
El 9409-243 M (Received 19 September 1994; accepted 17 March 1995)
Duplicate measurement is a common method to determine precision of a sampling method in an air pollution field study. Several evaluating measures for the precision of the sampling method are explored with duplicate measurementsof two types of passive samplers, such as nitrogen dioxide diffusive badge and carbon monoxide passive sampler. Presentation of duplicates in a graph as one of the evaluating measures can be affected by different assignment of each measurement to vertical and horizontal axes. Although absolute difference or relative error among duplicates can quantitatively represent the precision, interpretation of these values depends on the measured concentration range. Intraclass correlation coeffieient~often used for an indicator of the reliability of measurements, is an appropriate statistical measure to present a relative similarity of duplicate measurements. Intraclass correlation coefficientis an appropriate way to indicate precision of passive samplers which is usually determined by duplicate field measurements in air pollution studies.
INTRODUCTION Potential error sources in manufacturing, sampling, shipping, and analysis processes prevent precision of measurement by the passive sampler. The variation in the amount of pollutant contained in an unexposed passive sampler provides a consistent absolute difference of duplicates, regardless of measured concentrations. Variations in the diffusion barrier, i.e., in length and crosssectional area, can result in variations in the amount of pollutant collected by duplicates. Contamination or leakage during shipping and errors in analysis do not usually produce a typical pattern of difference in duplicates. The determination o f precision using duplicate measurements is based on the assumption that both duplicate samplers are exposed to identical concentrations. The actual concentration difference can occur due to the concentration gradient such as the presence of a pollutant plume.
A passive sampler is an important tool for studying an air pollutant in epidemiological investigations. The passive sampler is generally simple in structure and easily used by people o f all ages with simple instruction. Since the passive sampler can measure air pollution levels without using a pump, personal exposures can be measured without interrupting an individual's daily activities. The passive sampler collects the target air pollutant by molecular diffusion through a diffusion barrier. The sampling rate of the passive sampler is controlled by certain configurations, such as length and cross-sectional area of the diffusion barrier. The sampling rate of passive samplers must be appropriately determined and evaluated in laboratory experiments and field studies (Brown et al. 1984). The performances of passive samplers should be evaluated on the bases of accuracy, precision, and environmental effects. 407
408
The standard deviation of more than or equal to three duplicates may be statistically appropriate to determine the precision of passive samplers (ACS 1972). In addition to the standard deviation, there are several measures for precision with more than three duplicates, such as variance, relative standard deviation, mean deviation, and range. West et al. (1978) determined precision using four duplicates with nine different concentrations in laboratory experiments. In field studies, the precision of the measurement is often determined by two duplicate measurements in one exposure location. Published studies have used different measures to present the precision of duplicate measurements, such that duplicates were analyzed by difference and relative error (Lynch et al. 1974), the relationship between relative error and average concentration was used (Lee et al. 1993), and duplicates were presented in a graph along with means and standard deviations of absolute difference and percentage of difference (Treitman et al. 1990). Although investigators have often used duplicate measurements to gauge the precision of the sampling methods, the quantitative measures to present the precision have rarely been used. This study evaluated several measures for precision, including intraclass correlation coefficient, using the duplicate data of a nitrogen dioxide diffusive badge and a carbon monoxide passive sampler. DUPLICATE DATA
An N O 2 diffusive badge was developed by Yanagisawa et al. (1982) for the measurement of personal NO2 exposures. In the badge, NO2 is absorbed to a cellulose fiber filter containing triethanolamine solution after molecular diffusion through five layers of hydrophobic fiber filters. The NO2 badge had a large ratio of a cross-sectional area to length of diffusion barrier in order to increase sensitivity of the passive sampler. The sampling rate of a passive sampler with higher sensitivity could be affected by a surface wind velocity. The wind velocity effects on sampling rate were determined in laboratory experiments (Lee et al. 1992). The precision of the NO2 badge was evaluated in three different indoor environments: an unoccupied research house, three residential houses, and an office (Lee et al. 1993). Duplicate NO2 badges were exposed for 24 h. The distance between the collecting surfaces of the duplicate NO2 badges was less than 5 cm in order to ensure both badges were exposed to the identical NO2 level. Indoor NO2 concentrations ranged from 5.8 to 79.7 nL/L. A total of 50 duplicates were used in this evaluation. The carbon monoxide passive sampler adsorbs CO by solid adsorbent, Zn-Y-zeolite (Lee et al. 1992). Carbon
K. Lee et al.
monoxide passive samplers with a 2.5 cm long diffusion tube were used to measure CO concentrations inside and outside of ten underground garages in the Metropolitan Boston area. Carbon monoxide levels in four randomly selected garages were measured on two separate days. The CO passive samplers were placed in six or eight locations inside the underground garage. The same number of sampling locations were selected in the first floor of the same building with the underground garages. Duplicate CO passive samplers were placed less than 1 cm apart to ensure that they were exposed to identical CO levels. The sampling times were varied from 6 to 24 h and included rush hour. A total of 176 CO passive samplers were deployed in this investigation. Among these, 10 duplicates samplers were lost or broken during deployment. Seven passive samplers were missed due to analysis error including integrator error and mishandling of samplers. Finally, 79 duplicates were available to determine the precision of the CO passive sampler. PRESENTATION OF DUPLICATES IN A GRAPH
The relationship between two variables can be determined by linear regression using a correlation coefficient and a regression coefficient, when one dependent variable can be expressed as the function of an independent variable. The Pearson correlation coefficient is the indicator of a linear relationship between the two variables. The regression coefficient is the change in the dependent variable for one unit increase of the independent variable. In the duplicate measurements, each observation can be classified as either an independent or a dependent variable. When the classification of each observation is switched, the reference statistical values, i.e., a regression coefficient and a correlation coefficient of linear regression model, can be different, making precision indication confusing, The NO2 badge duplicate measurements were presented in a graph by two arbitrary designations of one measurement to the horizontal axes and the other to the vertical axis, as shown in Fig. la and lb. However, the same data can produce disparate conclusions with the different designations. Figure la shows the designation of the duplicates which have a regression coefficient of 1.005. The NO2 badge duplicates were intentionally designated for having regression coefficient close to 1.0. This designation may lead to the conclusion that there is agreement between duplicates. The same data of Fig. 1a were redesignated to indicate overestimation of one duplicate, as shown in Fig. lb. Linear regression models of Fig. la and lb show different intercepts, regression coefficients, and correlation coefficients.
Precision of a passive sampler
409
9O
0 n~
8070-
D
60-
v
O z
5040-
o. 30-1 C3 20-
Y = 0.002 + 1.005X R2=0.9581 N=50
100
0
lb
2'0
3'0
40
go
7'0
6'o
8'o
90
NO2 (nL/L)
Fig. la. Presentationo f N O 2 badge duplicateswith designationto have regression coefficientof 1,005. 90 807060O z
5040-
-I
a
30-
o
20-
Y=-2.642+ 1.097X R2=0.9707 N=50
100
0
1'0
2'0
3'0
4'0
5'0
6'0
7.0
8'0
90
NOz (nL/L)
Fig. lb. Presentationo f N O 2 badge duplicateswith designationto overestimate. The presentation of the duplicate data in a graphic format may not be a reliable way to present the precision. Also it does not provide any numerical values for precision. However, presentation in a graph may be useful as an auxiliary tool with an appropriate measure of precision. When duplicates are designated to either axes prior to the measurement, it may help to accomplish the random designation for presentation in a graph. Presentation in a graph may help readers understand the dispersion of duplicates. RELATIVE ERROR OR A B S O L U T E DIFFERENCE
The relative error, defined as a ratio of the difference of duplicates to their average, and the absolute difference of the duplicates can be used to present reproducibility. The duplicate measurements at more than one location can be presented by mean and standard deviation of relative
errors and absolute differences of each duplicate (Treitman et al. 1990). However, relative error and absolute difference can imply different precision depending on levels of measured concentration range. The same difference in the duplicates can mean a lower relative error in higher concentration. Absolute difference also does not provide enough information of precision without measured concentration. The same difference can imply different precision with an increase in the measurement value. The uses of the relative error and absolute difference can sometimes provide characteristics of the passive sampler, although it is often difficult to distinguish clearly the source of the difference. Laboratory errors, such as contamination of unexposed sampler by pollutant, generally provide consistent differences of the duplicates against measured pollutant concentrations. As an example, differences in the CO passive sampler duplicates
410
K. Lee et al.
10. .
A
_1
0 t,-
®
7" 6" 5-
"0 -I
4"
"5
[]
[]
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0 []
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[]
[]
_
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[] !
0
2
4
6 ab Average of C O duplicates (uL/L)
[]
1'2
a'4
Fig. 2. Relationship between differences and averages o f duplicates o f CO passive sampler.
1 0.9 0.80.70.6-
0.5" I1)
rr"
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[]
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0.1 0
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0
10
2'0
3'0
40
50
60
7b
80
A v e r a g e N O z (nL/L) Fig. 3. Relationship between relative error and average o f duplicates o f NO 2 badge.
are shown in Fig. 2. Absolute differences of the CO passive samplers were lower than 3.8 ~zL/L, except in two duplicate cases, regardless of the CO concentrations. Measurement errors can be caused by variations in the diffusion barrier. The passive sampler duplicates, which are significantly affected by the sampling rate, show an increase in absolute differences with an increase in measurement levels. The relationship between the
relative error and the average of duplicate N O 2 m e a s u r e ments is shown in Fig. 3. The relative errors of the NO2 measurements were less than 20%, except for two duplicates, regardless of measured concentrations. The potential sources of the differences are as follows: variations in length and cross-sectional area of the diffusion barrier, face wind velocity, and real concentration gradients.
Precision of a passive sampler
411
INTRACLASS CORRELATION COEFFICIENT The relationship between a dependent variable and an independent variable can be determined by a linear regression method. For example, when a dependent variable can be expressed as a linear function of an independent variable, a regression coefficient and a Pearson correlation coefficient are indicators of similarity between the two variables. Since duplicate measurements cannot be elassifted by dependent and independent variables, each observation can be classified as either an independent or a dependent variable. The precision measures can be affected by the classification, as shown in Fig. 1a and 1b. It is desirable to use a unique statistical measure to evaluate the precision. The intraclass correlation coefficient was designed to calculate the correlation coefficient between members of the same class (Koch 1983), a class implying a sampling location. Intraclass correlation coefficient is a good measure to describe the precision as a relative ~imilarity of a duplicate measurements at the same sampling location. In duplicates, observations Xe (i=l ,...,n, j=l,...,d) can be expressed by Xij (i=l ..... n, j=l,2), since n is the number of the measurement location and d is the number of the duplicate, usually equal to 2 in field duplicate measurements. The estimator of intraclass correlation coefficient (r~) of duplicates is calculated by the following equation (Koch 1983): $2 _ 5,2. _
rI
.
,,
(1)
5'2 + $ 2 ,.1
w
where S a2 = ( 2
n
Zi= 1 (X't. - . ~ t ' ) 2 / ( n
-
1)),
The s,2 is the between-classes mean square (BMS) with respect to the within-class sample mean. The s2 is the within-classes mean square (WMS). The Xi. = (Xn+Xi2)/2; and X is the overall sample mean. The estimator of the intraclass correlation coefficient is obtained by two unbiased estimators, BMS and WMS. However, the estimator of the intraclass correlation coefficient itself is slightly biased from the intraclass correlation coefficient (Olkin et al. 1958). The bias of the estimator can be significantly reduced with an increase in number of duplicates. Therefore, it is necessary to provide the number of duplicates with the estimator of the intraclass correlation coefficient.
A two-sided 100(1 - ~)% confidence interval for the estimator of the intraclass correlation coefficient can be calculated by the following equation. S2
$2 ..2_~ -
5' 2
G
F
2
W
82
rz
S2
-&+F $2
N
-- F
2
(2)
s 2
_2_a + F 8
11¢
1
W
2
1
w
where
F I = F±a((n-l),n), F2 = Fl_±,,((n-l),n). 2
The confidence interval can be used to determine the hypothesis of zero intraclass correlation coefficient and to indicate the limit of uncertainty in precision. The intraclass correlation coefficient is usually positive, with values between 0 to +1. The intraclass correlation coefficient of 0 indicates that BMS is equal to W/vlS. In this case, the duplicates do not contain any information of measurement in a sampling location. The intraclass correlation coefficient of 1 indicates an ideal precision of the measurement method, since it occurs when two measurements of duplicates are equal to each other. Therefore, an intraclass correlation coefficient closer to 1 indicates excellent precision. According to the above equations, the estimator of the intraclass correlation coefficient of the N O 2 duplicate measurements was 0.9779 with 50 sampling locations, with a 95% confidence interval of 0.9626 and 0.9956. The CO duplicate measurements had an estimator of intraclass correlation coefficient of 0.8765 with 79 sampling locations, with a 95% confidence interval of 0.8323 and 0.9276. Since the intraclass correlation coefficient provides one value indicating the relative similarity of duplicate measurements, the intraclass correlation coefficient can be used for a criterion of a passive sampler. CONCLUSION Several possible measures to indicate precision of duplicates were evaluated using duplicate data from NO2 badges and CO passive samplers. Duplicates can be represented in a graph with random designation of duplicates, however, it is probably the least appropriate measure. Relative error or absolute difference of duplicates depends on measured concentration: the same difference may reflect a smaller error at higher concentration.
412
The intraclass correlation coefficient is a statistically appropriate measure to present precision o f duplicate measurements. It provides a quantitative value without an effect from the measured concentration. Intraclass correlation coefficients o f NO2 badge and C O passive sampler duplicates were 0.9779 and 0.8765, respectively.
Acknowledgment--Theauthors thank Sarah Spengler for her assistance in the field works and Joan Arnold for editing this paper.
REFERENCES ACS (American Chemical Society) Guide to the use of terms in reporting data in analytical chemistry. Anal. Chem. 44: 2420; 1972. Brown, R.H.; Harvey, R.P.; Purnell, C.J.; Sannders, K.J. A diffusive sampler evaluation protocol. Am. Ind. Hyg. Assoc. J. 45: 67-75; 1984. Koch, G.G. Intraclass correlation coefficient, Encyclopedia of Statistical Sciences, Vol.4. In: Kotz, S.; Johnson, N.L., eds. pp 212217. New York. NY: John Wiley & Sons; 1983. Lee, K.; Yanagisawa, Y.; Spengler, J.D.; Billick, I.H. Sampling rate evaluation for NO2 badge: (I) in indoor environment. Indoor Air 3: 124-130; 1993.
K. Lee et al.
Lee, K.; Yanagisawa, Y.; Spengler, J.D.; Billick, I.H. Wind velocity effects on sampling rate of NO2 badge. J. Exp. Anal. Environ. Epidemiol. 2: 207-219; 1992. Lee, K.; Yanagisawa, Y.; Hishinuma, M.; Spengler, J. D.; Billick, I.H. A passive sampler for measurement of carbon monoxide exposure using a solid adsorbent. Environ. Sci. Technol. 26: 697-702; 1992. Lynch, J.J.; Burgess, W.A. A personal exposure sampler for carbon monoxide. Am. Ind. Hyg. Assoc. J. 35: 354-358; 1974. Olkin, I.; Pratt, J.W. Unbiased estimation of certain correlation coefficients. Ann. Math. Stat. 29:201-211; 1958. Treitman, R.D.; Ryan. P.B.; Harlos, D.P.; Soczek, M.L.; Yanagisawa, Y.; Spengler, J.D.; Billick, I.H. Sampling and analysis of nitrogen dioxide and respirable particles in the indoor environment. In: W.L. Zielinski, Jr.; W.D. Dorko, eds. Monitoring methods for toxics in the atmosphere. ASTM STP 1052. American Society for Testing and Materials, Philadelphia, PA; 1990: 197-212. West, P.W.; Reiszner, K.D. Field tests of a permeation-type personal monitor for vinyl chloride. Am. Ind. Hyg. Assoc. J. 39: 645-650; 1978. Yanagisawa, Y.; Nishimura, H. A badge-type personal sampler for measurements of personal exposure to NO2 and NO in ambient air. Environ. Int. 8: 235-242; 1982.
Pergamon
Environment International, Vol. 21, No. 4, pp. 407-412, 1995 Copyright O1995 Elsevier Science Ltd Printed in the USA. All fights reserved
0160-4120/95 $9.50+.00
0160-4120(95)00034-8 ASSESSMENT OF PRECISION OF A PASSIVE SAMPLER BY DUPLICATE M E A S U R E M E N T S Kiyoung Lee, Yukio Yanagisawa, and John D. Spengler Department of Environmental Health, Harvard School of Public Health, Boston, MA 02115, USA Roger Davis Department of Biostatistics, Harvard School of Public Health, Boston, MA 02115 and Division of General Medicine and Primary Care, Beth Israel Hospital, Boston, MA 02215, USA
El 9409-243 M (Received 19 September 1994; accepted 17 March 1995)
Duplicate measurement is a common method to determine precision of a sampling method in an air pollution field study. Several evaluating measures for the precision of the sampling method are explored with duplicate measurementsof two types of passive samplers, such as nitrogen dioxide diffusive badge and carbon monoxide passive sampler. Presentation of duplicates in a graph as one of the evaluating measures can be affected by different assignment of each measurement to vertical and horizontal axes. Although absolute difference or relative error among duplicates can quantitatively represent the precision, interpretation of these values depends on the measured concentration range. Intraclass correlation coeffieient~often used for an indicator of the reliability of measurements, is an appropriate statistical measure to present a relative similarity of duplicate measurements. Intraclass correlation coefficientis an appropriate way to indicate precision of passive samplers which is usually determined by duplicate field measurements in air pollution studies.
INTRODUCTION Potential error sources in manufacturing, sampling, shipping, and analysis processes prevent precision of measurement by the passive sampler. The variation in the amount of pollutant contained in an unexposed passive sampler provides a consistent absolute difference of duplicates, regardless of measured concentrations. Variations in the diffusion barrier, i.e., in length and crosssectional area, can result in variations in the amount of pollutant collected by duplicates. Contamination or leakage during shipping and errors in analysis do not usually produce a typical pattern of difference in duplicates. The determination o f precision using duplicate measurements is based on the assumption that both duplicate samplers are exposed to identical concentrations. The actual concentration difference can occur due to the concentration gradient such as the presence of a pollutant plume.
A passive sampler is an important tool for studying an air pollutant in epidemiological investigations. The passive sampler is generally simple in structure and easily used by people o f all ages with simple instruction. Since the passive sampler can measure air pollution levels without using a pump, personal exposures can be measured without interrupting an individual's daily activities. The passive sampler collects the target air pollutant by molecular diffusion through a diffusion barrier. The sampling rate of the passive sampler is controlled by certain configurations, such as length and cross-sectional area of the diffusion barrier. The sampling rate of passive samplers must be appropriately determined and evaluated in laboratory experiments and field studies (Brown et al. 1984). The performances of passive samplers should be evaluated on the bases of accuracy, precision, and environmental effects. 407
408
The standard deviation of more than or equal to three duplicates may be statistically appropriate to determine the precision of passive samplers (ACS 1972). In addition to the standard deviation, there are several measures for precision with more than three duplicates, such as variance, relative standard deviation, mean deviation, and range. West et al. (1978) determined precision using four duplicates with nine different concentrations in laboratory experiments. In field studies, the precision of the measurement is often determined by two duplicate measurements in one exposure location. Published studies have used different measures to present the precision of duplicate measurements, such that duplicates were analyzed by difference and relative error (Lynch et al. 1974), the relationship between relative error and average concentration was used (Lee et al. 1993), and duplicates were presented in a graph along with means and standard deviations of absolute difference and percentage of difference (Treitman et al. 1990). Although investigators have often used duplicate measurements to gauge the precision of the sampling methods, the quantitative measures to present the precision have rarely been used. This study evaluated several measures for precision, including intraclass correlation coefficient, using the duplicate data of a nitrogen dioxide diffusive badge and a carbon monoxide passive sampler. DUPLICATE DATA
An N O 2 diffusive badge was developed by Yanagisawa et al. (1982) for the measurement of personal NO2 exposures. In the badge, NO2 is absorbed to a cellulose fiber filter containing triethanolamine solution after molecular diffusion through five layers of hydrophobic fiber filters. The NO2 badge had a large ratio of a cross-sectional area to length of diffusion barrier in order to increase sensitivity of the passive sampler. The sampling rate of a passive sampler with higher sensitivity could be affected by a surface wind velocity. The wind velocity effects on sampling rate were determined in laboratory experiments (Lee et al. 1992). The precision of the NO2 badge was evaluated in three different indoor environments: an unoccupied research house, three residential houses, and an office (Lee et al. 1993). Duplicate NO2 badges were exposed for 24 h. The distance between the collecting surfaces of the duplicate NO2 badges was less than 5 cm in order to ensure both badges were exposed to the identical NO2 level. Indoor NO2 concentrations ranged from 5.8 to 79.7 nL/L. A total of 50 duplicates were used in this evaluation. The carbon monoxide passive sampler adsorbs CO by solid adsorbent, Zn-Y-zeolite (Lee et al. 1992). Carbon
K. Lee et al.
monoxide passive samplers with a 2.5 cm long diffusion tube were used to measure CO concentrations inside and outside of ten underground garages in the Metropolitan Boston area. Carbon monoxide levels in four randomly selected garages were measured on two separate days. The CO passive samplers were placed in six or eight locations inside the underground garage. The same number of sampling locations were selected in the first floor of the same building with the underground garages. Duplicate CO passive samplers were placed less than 1 cm apart to ensure that they were exposed to identical CO levels. The sampling times were varied from 6 to 24 h and included rush hour. A total of 176 CO passive samplers were deployed in this investigation. Among these, 10 duplicates samplers were lost or broken during deployment. Seven passive samplers were missed due to analysis error including integrator error and mishandling of samplers. Finally, 79 duplicates were available to determine the precision of the CO passive sampler. PRESENTATION OF DUPLICATES IN A GRAPH
The relationship between two variables can be determined by linear regression using a correlation coefficient and a regression coefficient, when one dependent variable can be expressed as the function of an independent variable. The Pearson correlation coefficient is the indicator of a linear relationship between the two variables. The regression coefficient is the change in the dependent variable for one unit increase of the independent variable. In the duplicate measurements, each observation can be classified as either an independent or a dependent variable. When the classification of each observation is switched, the reference statistical values, i.e., a regression coefficient and a correlation coefficient of linear regression model, can be different, making precision indication confusing, The NO2 badge duplicate measurements were presented in a graph by two arbitrary designations of one measurement to the horizontal axes and the other to the vertical axis, as shown in Fig. la and lb. However, the same data can produce disparate conclusions with the different designations. Figure la shows the designation of the duplicates which have a regression coefficient of 1.005. The NO2 badge duplicates were intentionally designated for having regression coefficient close to 1.0. This designation may lead to the conclusion that there is agreement between duplicates. The same data of Fig. 1a were redesignated to indicate overestimation of one duplicate, as shown in Fig. lb. Linear regression models of Fig. la and lb show different intercepts, regression coefficients, and correlation coefficients.
Precision of a passive sampler
409
9O
0 n~
8070-
D
60-
v
O z
5040-
o. 30-1 C3 20-
Y = 0.002 + 1.005X R2=0.9581 N=50
100
0
lb
2'0
3'0
40
go
7'0
6'o
8'o
90
NO2 (nL/L)
Fig. la. Presentationo f N O 2 badge duplicateswith designationto have regression coefficientof 1,005. 90 807060O z
5040-
-I
a
30-
o
20-
Y=-2.642+ 1.097X R2=0.9707 N=50
100
0
1'0
2'0
3'0
4'0
5'0
6'0
7.0
8'0
90
NOz (nL/L)
Fig. lb. Presentationo f N O 2 badge duplicateswith designationto overestimate. The presentation of the duplicate data in a graphic format may not be a reliable way to present the precision. Also it does not provide any numerical values for precision. However, presentation in a graph may be useful as an auxiliary tool with an appropriate measure of precision. When duplicates are designated to either axes prior to the measurement, it may help to accomplish the random designation for presentation in a graph. Presentation in a graph may help readers understand the dispersion of duplicates. RELATIVE ERROR OR A B S O L U T E DIFFERENCE
The relative error, defined as a ratio of the difference of duplicates to their average, and the absolute difference of the duplicates can be used to present reproducibility. The duplicate measurements at more than one location can be presented by mean and standard deviation of relative
errors and absolute differences of each duplicate (Treitman et al. 1990). However, relative error and absolute difference can imply different precision depending on levels of measured concentration range. The same difference in the duplicates can mean a lower relative error in higher concentration. Absolute difference also does not provide enough information of precision without measured concentration. The same difference can imply different precision with an increase in the measurement value. The uses of the relative error and absolute difference can sometimes provide characteristics of the passive sampler, although it is often difficult to distinguish clearly the source of the difference. Laboratory errors, such as contamination of unexposed sampler by pollutant, generally provide consistent differences of the duplicates against measured pollutant concentrations. As an example, differences in the CO passive sampler duplicates
410
K. Lee et al.
10. .
A
_1
0 t,-
®
7" 6" 5-
"0 -I
4"
"5
[]
[]
[]
< .
0 []
[]
[]
[]
_
[] []
[] !
0
2
4
6 ab Average of C O duplicates (uL/L)
[]
1'2
a'4
Fig. 2. Relationship between differences and averages o f duplicates o f CO passive sampler.
1 0.9 0.80.70.6-
0.5" I1)
rr"
0.40.30.2-
[]
r-7
[]
[] []
0.1 0
t-
0
10
2'0
3'0
40
50
60
7b
80
A v e r a g e N O z (nL/L) Fig. 3. Relationship between relative error and average o f duplicates o f NO 2 badge.
are shown in Fig. 2. Absolute differences of the CO passive samplers were lower than 3.8 ~zL/L, except in two duplicate cases, regardless of the CO concentrations. Measurement errors can be caused by variations in the diffusion barrier. The passive sampler duplicates, which are significantly affected by the sampling rate, show an increase in absolute differences with an increase in measurement levels. The relationship between the
relative error and the average of duplicate N O 2 m e a s u r e ments is shown in Fig. 3. The relative errors of the NO2 measurements were less than 20%, except for two duplicates, regardless of measured concentrations. The potential sources of the differences are as follows: variations in length and cross-sectional area of the diffusion barrier, face wind velocity, and real concentration gradients.
Precision of a passive sampler
411
INTRACLASS CORRELATION COEFFICIENT The relationship between a dependent variable and an independent variable can be determined by a linear regression method. For example, when a dependent variable can be expressed as a linear function of an independent variable, a regression coefficient and a Pearson correlation coefficient are indicators of similarity between the two variables. Since duplicate measurements cannot be elassifted by dependent and independent variables, each observation can be classified as either an independent or a dependent variable. The precision measures can be affected by the classification, as shown in Fig. 1a and 1b. It is desirable to use a unique statistical measure to evaluate the precision. The intraclass correlation coefficient was designed to calculate the correlation coefficient between members of the same class (Koch 1983), a class implying a sampling location. Intraclass correlation coefficient is a good measure to describe the precision as a relative ~imilarity of a duplicate measurements at the same sampling location. In duplicates, observations Xe (i=l ,...,n, j=l,...,d) can be expressed by Xij (i=l ..... n, j=l,2), since n is the number of the measurement location and d is the number of the duplicate, usually equal to 2 in field duplicate measurements. The estimator of intraclass correlation coefficient (r~) of duplicates is calculated by the following equation (Koch 1983): $2 _ 5,2. _
rI
.
,,
(1)
5'2 + $ 2 ,.1
w
where S a2 = ( 2
n
Zi= 1 (X't. - . ~ t ' ) 2 / ( n
-
1)),
The s,2 is the between-classes mean square (BMS) with respect to the within-class sample mean. The s2 is the within-classes mean square (WMS). The Xi. = (Xn+Xi2)/2; and X is the overall sample mean. The estimator of the intraclass correlation coefficient is obtained by two unbiased estimators, BMS and WMS. However, the estimator of the intraclass correlation coefficient itself is slightly biased from the intraclass correlation coefficient (Olkin et al. 1958). The bias of the estimator can be significantly reduced with an increase in number of duplicates. Therefore, it is necessary to provide the number of duplicates with the estimator of the intraclass correlation coefficient.
A two-sided 100(1 - ~)% confidence interval for the estimator of the intraclass correlation coefficient can be calculated by the following equation. S2
$2 ..2_~ -
5' 2
G
F
2
W
82
rz
S2
-&+F $2
N
-- F
2
(2)
s 2
_2_a + F 8
11¢
1
W
2
1
w
where
F I = F±a((n-l),n), F2 = Fl_±,,((n-l),n). 2
The confidence interval can be used to determine the hypothesis of zero intraclass correlation coefficient and to indicate the limit of uncertainty in precision. The intraclass correlation coefficient is usually positive, with values between 0 to +1. The intraclass correlation coefficient of 0 indicates that BMS is equal to W/vlS. In this case, the duplicates do not contain any information of measurement in a sampling location. The intraclass correlation coefficient of 1 indicates an ideal precision of the measurement method, since it occurs when two measurements of duplicates are equal to each other. Therefore, an intraclass correlation coefficient closer to 1 indicates excellent precision. According to the above equations, the estimator of the intraclass correlation coefficient of the N O 2 duplicate measurements was 0.9779 with 50 sampling locations, with a 95% confidence interval of 0.9626 and 0.9956. The CO duplicate measurements had an estimator of intraclass correlation coefficient of 0.8765 with 79 sampling locations, with a 95% confidence interval of 0.8323 and 0.9276. Since the intraclass correlation coefficient provides one value indicating the relative similarity of duplicate measurements, the intraclass correlation coefficient can be used for a criterion of a passive sampler. CONCLUSION Several possible measures to indicate precision of duplicates were evaluated using duplicate data from NO2 badges and CO passive samplers. Duplicates can be represented in a graph with random designation of duplicates, however, it is probably the least appropriate measure. Relative error or absolute difference of duplicates depends on measured concentration: the same difference may reflect a smaller error at higher concentration.
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The intraclass correlation coefficient is a statistically appropriate measure to present precision o f duplicate measurements. It provides a quantitative value without an effect from the measured concentration. Intraclass correlation coefficients o f NO2 badge and C O passive sampler duplicates were 0.9779 and 0.8765, respectively.
Acknowledgment--Theauthors thank Sarah Spengler for her assistance in the field works and Joan Arnold for editing this paper.
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