Environ. Sci. Technol. 2000, 34, 2826-2829
Detection of Statistically Significant Trends in Atmospheric Concentrations of Semivolatile Compounds DONALD R. CORTES AND RONALD A. HITES* School of Public and Environmental Affairs and Department of Chemistry, Indiana University, Bloomington, Indiana 47405
This paper examines the sampling and analytical requirements of observing statistically significant temporal trends in atmospheric concentrations of semivolatile organic pollutants. Data precision and study duration are found to be the most important factors that are subject to experimental control. We define a detectable half-life as the longest half-life for which statistical significance can be observed for a given length of study and show how this can be calculated. Finally, we show how to predict the length of a study that will find a statistically significant half-life. These concepts are applied to a recently updated temporal trends study from the Integrated Atmospheric Deposition Network. Updated half-lives for gas-phase pesticides at three U.S. sampling sites are given. There are now enough data to observe statistically significant temporal trends for most of the gas-phase pesticides except for R-chlordane and trans-nonachlor at Eagle Harbor. Using the concept of detectable half-life and based on observations of halflives at other IADN sites, at least 2 more years of study are required to observe statistically significant trends for R-chlordane and trans-nonachlor. The concepts presented in this paper can also be applied to the design of sampling strategies for measuring long-term temporal trends in media other than air.
Introduction The determination of the environmental half-lives of pollutants is often important. Experimental half-lives are needed to validate models that predict future pollutant concentrations. Such half-lives can also validate models that predict the environmental fate of contaminants. For example, according to the global fractionation theory (1), the levels of persistent organic pollutants should decay more slowly in the higher latitudes than in the lower latitudes. Estimating half-lives in environmental data requires the identification of a meaningful trend in data that are often quite scattered. Classical statistical methods have been modified for trend detection in environmental data and are in the literature (2-9). For example, the Kendall test is a powerful, nonparametric test for detecting monotonic trends, but it cannot be used when the data contains cyclic variations (8). Hirsch et al. developed the Seasonal Kendall test, which is a modification of the Mann-Kendall test, to detect trends in seasonal data (7). Later, Hirsch and Slack updated his method to handle serial dependence in the data (5). * Corresponding author e-mail:
[email protected] 2826
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Although nonparametric tests usually place fewer restrictions on environmental data, there are also useful parametric tests in the literature. Hillary and Hites used a multiple linear regression technique to estimate half-lives for atmospheric PCBs near the Great Lakes (10). Using this same technique, we previously determined that there were statistically significant trends in atmospheric pesticides near the Great Lakes and reported these trends in terms of half-lives ranging from 1 to 12 years (11). However, even with nearly 5 years of data, statistically significant temporal trends were not observed for some other gas-phase pesticides. This led to the question of whether the concentrations of these pesticides were not declining or whether the study was not yet long enough to observe significant trends. This paper examines the requirements for observing statistically significant half-lives for the atmospheric concentrations of semivolatile compounds. The concept of a detectable half-life is defined as the longest half-life that can be statistically detected based on the length of study and the variability in the data. Also, a simple method is presented to predict the length of a study that will detect a given half-life. Recently updated atmospheric pesticide trends from the Great Lakes Atmospheric Deposition Network (IADN) (10) are used to illustrate these concepts. These findings will be helpful in designing sampling strategies for determining trends and for estimating whether the continuation of current studies is likely to be fruitful.
Experimental Section The sampling and analytical methodology for the IADN gasphase pesticides has been given previously (11). In summary, the gas-phase samples are collected on XAD-2 (Sigma, Amberlite 20-60 mesh) resin. The organic compounds are extracted with 50% hexane in acetone for 24 h, and the pesticides are isolated by fractionation on 3.5% w/w waterdeactivated silica gel. The pesticide concentrations are then determined by using a gas chromatograph (Hewlett-Packard 5890) with an electron capture detector. Only the three U.S. IADN sites were included in this study. These sites are located at Eagle Harbor, MI, on the shore of Lake Superior; Sleeping Bear Dunes State Park, MI, on the shore of Lake Michigan; and Sturgeon Point, NY, on the shore of Lake Erie. Sampling began in November 1990 at Eagle Harbor and in December 1991 at Sleeping Bear Dunes and Sturgeon Point. However, analysis of chlordane and t-nonachlor did not begin until July 1992 at any of the sites. This study includes samples acquired through December 1997.
Results and Discussion Theory. If one wanted to determine a rate of decline in the concentration of some pollutant in the environment, an experiment could be designed whereby samples were taken at some specified frequency and analyzed for the specified pollutant. Typically, first-order kinetics are used to model decreasing environmental concentrations; that is, the rate at which the concentration decreases is proportional to the concentration at any time. With the assumption of first-order kinetics, the data could be analyzed by plotting the natural logarithm of the concentration of the pollutant versus the time when the sample was collected. Linear regression of data plotted in this way would give a straight line with a slope that is the negative of the first-order rate constant
y ) β0 + βix +
(1)
where y is the natural logarithm of concentration of the 10.1021/es990466l CCC: $19.00
2000 American Chemical Society Published on Web 06/02/2000
FIGURE 1. Semilogarithmic plots of atmospheric partial pressures (in atmospheres) of γ-chlordane at Eagle Harbor before (A) and after (B) temperature correction to a reference temperature of 288 K. The lines define the range of the variance about the regression line (solid line, syx2) and of the variance about the mean of ln P (dashed line, sy2); n ) 129. pollutant, β0 is the intercept of the regression line, β1 is the negative of the first-order rate constant, x is time, and is the error in the data. The rate constant is often reported as a half-life by dividing it into the natural logarithm of 2. Depending on the scatter of the data, the rate constant (-β1) describing the temporal trend may or may not be statistically significant at a predetermined level (in this paper, we use the 95% confidence level). The requirements of observing a significant temporal trend translate into two concepts over which the researcher has some control: variability in the data and the duration of the study. A relatively small trend can be discerned in relatively few data if they have been measured with high precision, but a relatively large trend may be invisible in a relatively large number of data if they have been measured with poor precision. The quantitative relationship between variability and significance is given by (12)
r2 ) 1 -
(n - 2)syx2 (n - 1)sy2
(2)
where r2 is the square of the correlation coefficient (which can be used as a measure of statistical significance at a predetermined level), n is the number of data, syx2 is the variance of the measurements (in this case, the natural logarithms of concentrations) about the regression line plotted as a function of the independent variable (in this
case, time), and sy2 is the total variance of the measurements about the mean of all measurements. It is clear from this equation that for n larger than 2, a decrease in syx2 relative to sy2 is needed to produce the higher r-values needed to show statistical significance. For the researcher, reductions in syx2 translate into improving the precision in sampling and analytical techniques and removing variations due to other known causes. Given the heterogeneity of environmental sampling and the imprecision of analytical measurements, a reduction in variability about the regression line is not always possible. Even when it is possible, reductions in syx2 are usually accompanied by reductions in the total variance, sy2. Since it is the reduction in syx2 relative to sy2 that is important in maximizing r2, we must consider how to increase sy2 independently of syx2. For the researcher, a maximum sy2 translates to a relatively large trend and a relatively long sampling duration. Unfortunately, the researcher has no control over environmental degradation rates (the trend); thus, the continuation of a sampling program is the only option if significance has not yet been observed and the precision in the data cannot be improved. In establishing the study, the researcher might ask “What is the optimum sampling frequency that is needed to observe significant trends?”. Many factors go into determining an optimum sampling frequency. For the method described here, it is critical that enough samples be taken to adequately capture the random variability of the data. Other important considerations include dependence in the error terms, the duration of high concentration pulses, and nonrandom variability (for example, a representative sampling of workweek days) (8, 13). Strictly in terms of obtaining a high r-value, however, eq 2 indicates that frequency of sampling is not an important factor, since (n-2)/(n-1) is close to one given a reasonable sampling campaign. It is true that as n increases, the critical value of r decreases, thus making statistical significance more likely for a given set of data. However, additional samples are more effective when they increase the overall length of the study than when they increase the sampling frequency. As long as a reasonable number of samples are taken to adequately characterize data variance, sampling frequency is not nearly as important to the detection of statistically significant temporal trends as is the overall length of the study. The next issue is the duration of the study. The researcher might ask “How much longer should I sample in order to observe a significant trend?”. To answer this question we must define a detectable rate constant, which is given by the following relation (12)
kdet ) rcrit
sy sx
(3)
where rcrit is the positive critical value for significance at, say, the 95% confidence interval, sy is the square root of the variance of the natural logarithms of the concentrations, and sx is the square root of the variance of the time values. Values of rcrit are available in tables and depend on the number of degrees of freedom in the data, which is n - 2. As the duration of a study increases, the values of both sx and sy will increase but in different ways. We can eliminate sy as a variable by combining eqs 2 and 3
kdet )
rcrit sx
x
zsyx2
(4)
(1 - rcrit2)
where z ) (n - 2)/(n - 1). The detectable rate is now given in terms of variables related to the length of the study (sx) and the unexplained variance about the detectable trend VOL. 34, NO. 13, 2000 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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TABLE 1. Atmospheric Half-Lives of Nine Pesticides Measured near the Great Lakes (Top) and Corresponding Average Atmospheric Concentrations (Bottom)a Eagle Harbor
DDT DDE R-HCH γ-HCH HCB R-chlordane γ-chlordane trans-nonachlor dieldrin
Sleeping Bear Dunes
Sturgeon Point
t1/2 (yrs)
rel SE (%)
P > |T|
t1/2 (yrs)
rel SE (%)
P > | T|
t1/2 (yrs)
rel SE (%)
P > |T|
4.1 3.9 3.0 4.4 14 23 6.9 33 3.7
25 16 6.2 14 23 110 42 160 18
0.0001 0.0001 0.0001 0.0001 0.0001 0.3663 0.0199 0.5460 0.0001
3.5 3.8 3.2 3.4 7.7 9.7 5.2 6.0 2.4
24 17 11 19 23 50 33 35 17
0.0001 0.0001 0.0001 0.0001 0.0001 0.0488 0.0003 0.0045 0.0001
6.2 6.0 3.0 4.4 7.4 5.9 3.2 5.0 2.9
44 27 10 21 29 25 18 30 21
0.0248 0.0004 0.0001 0.0001 0.0001 0.0001 0.0001 0.0012 0.0001
Eagle Harbor
Sleeping Bear Dunes
Sturgeon Point
av concn (pg/m3)
rel SE (%)
av concn (pg/m3)
rel SE %
av concn (pg/m3)
rel SE (%)
3.5 2.4 140 24 74 3.8 3.9 2.7 13
12 8.3 3.1 8.9 2.6 7.3 12 8.9 9.7
5.6 9.4 110 62 82 7.4 7.4 5.4 28
14 8.4 5.9 24 3.0 8.3 8.9 9.9 11
11 20 120 37 84 13 12 8.4 30
9.8 7.3 5.9 9.5 2.5 6.7 6.5 7.5 9.3
DDT DDE R-HCH γ-HCH HCB R-chlordane γ-chlordane trans-nonachlor dieldrin
a Half-lives calculated from parameters which are significant with greater than a 95% confidence level are in normal font, and those that are not significant are in italics.
(syx2). Let us assume that we have sufficient data to get a good estimate of syx2, which is largely due to sampling and measurement errors. Pending the introduction of new sampling or measurement technology, we can assume that this value will not change much as the duration of the study is increased. On the other hand, the value of sx does increase as the duration of the study increases. In fact, the length of this study is implied by sx, which is simply the square root of the variance in x (time) about the mean of x
x
n
∑(x - xj)
sx )
2
i
i)1
n-1
(5)
Note that x1 is the first sample time (typically given in relative Julian Days) and xn is the last sample time. Thus, the overall study length is given by xn - x1. To determine the length of a study, one could guess at how much longer the study should be, calculate sx from eq 5, and calculate the detectable rate constant from eq 4. Given a knowledge of the environmental behavior of the compounds in question, it may then be possible to decide if this duration is sufficient to observe the expected trend. Examples from the Integrated Atmospheric Deposition Network. Let us look at some specific examples to make these concepts clear. Figure 1A shows the natural logarithms of the partial pressures of γ-chlordane measured in the atmosphere at Eagle Harbor, MI every 12 days from 1992 to 1997. Although there may appear to be a slight decreasing trend here, it is not significant at the 95% confidence level (r ) -0.07; rcrit ) (0.15). The first step in observing a significant trend is to remove known causes of variation in partial pressure. In the case of the atmospheric measurements of semivolatile compounds, such as organochlorine pesticides, the largest source of variability is related to atmospheric temperature (10, 11, 1416). Gas-phase partial pressures increase as temperature increases, and the functional relationship between these two 2828
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parameters is given by the Clausius-Clapeyron equation
ln P ) -
(∆HR)(T1) + const
(6)
where P is the partial pressure (in atm) of the atmospheric pollutant (P is calculated from the measured concentration using the ideal gas law), ∆H is an energy of phase change (in kJ/mol), R is the gas constant, and T (in K) is the temperature of the atmosphere during sampling. Thus, a plot of the natural logarithm of the partial pressures of a pesticide in the individual samples versus the reciprocal of the atmospheric temperature during sampling gives a straight line with a slope of -∆H/R. The resulting ∆H value can then be used to correct all of the partial pressures to a standard temperature (we use 288, the average temperature of the troposphere) using eq 6 solved at two pressures and temperatures
P288 ) Pe(∆H/R)[1/T - 1/288]
(7)
This correction was applied to the data in Figure 1A to produce the data shown in Figure 1B. Note that the variance about the regression line (syx2) and the total variance (sy2) decrease substantially from 1.082 to 0.521 and from 1.079 to 0.542, respectively. The reduction in syx2 relative to sy2 contributes to obtaining a significant trend (r ) -0.21; rcrit ) (0.15) in Figure 1B. Clearly, correcting for the temperature effect reduces the bias caused by year-to-year differences in temperature, and this correction is necessary to observe a significant trend. Incidentally, the errors ( in eq 1) of this regression (and all others) are normally distributed as judged by χ2-tests. This regression of the temperature corrected partial pressures vs time was repeated for all nine of the pesticides at all three of the U.S. IADN sampling sites. The temperature corrected half-lives (and the average concentrations) are given in Table 1. With two exceptions, all of these half-lives are statistically significant and range from 3 to 14 years, similar to the range observed before (11) with 2 years less data. Let
TABLE 2. Variables Used for Estimating Required Sampling Duration r-chlordane Eagle Harbor
trans-nonachlor Eagle Harbor
study duration
1984 days ∼ 5.4 yrs
2560 days ∼ 7.0 yrs
1984 days ∼ 5.4 yrs
2560 days ∼ 7.0 yrs
n sx rcrit syx t1/2
137 588.8 0.141 0.5608 14.0 yrs
185 744.5 0.121 0.5608 20.7 yrs
124 600.3 0.148 0.5601 13.6 yrs
172 765.8 0.126 0.5601 20.5 yrs
be about 14 years, which is just at our current detectable half-life (14.0 years in Table 2). If we extended this study to a full 7 years, the detectable half-life is 20.7 years which well exceeds our estimate of 14 years. Thus, if our assumption about the ratio of γ- to R-chlordane degradation rates is correct, we should see a statistically significant trend for this compound by the end of a 7-year study. The detectable halflife of trans-nonachlor after 7 years is about the same as R-chlordane. Because the half-life of trans-nonachlor falls between the two chlordane isomers at the other sites, we would expect to see statistical significance after 7 years for this compound as well.
Acknowledgments us focus on the exceptions that do not show statistically significant trends: R-chlordane and trans-nonachlor at Eagle Harbor. Let us consider adding one and a half more years to the current study of 1984 days to give a total study time of 2560 days (about 7 years total). The number of samples for the 1984-day study should be about 165 if samples were taken and analyzed every 12 days; in fact, the actual number of analyses is 137 for R-chlordane and 124 for trans-nonachlor because of missing samples, values below the detection limits, or analytical difficulties. We have assumed that all future samples will be analyzed, and therefore, by extending the study from 1984 days to 2560 days, we have added 48 samples to give 185 samples for R-chlordane and 172 samples for trans-nonachlor (see Table 2). Knowing the sampling dates for the 1984- and 2560-day studies for both compounds, we can calculate values of sx for the two study durations using eq 5, and we can look up rcrit values at the 95% confidence limit for all values of n. These results are given in Table 2. We assume the variances about the regression line (syx2) will remain constant with increased sampling duration. To check this assumption, consider that at the end of the 1995 study, syx2 for R-chlordane and trans-nonachlor were 0.3226 and 0.3195 compared to 0.3145 and 0.3137, respectively, at the end of the 1997 study. This is a difference of only 2.6% and 1.8%, even though the study length increased by 60%. Using eq 4 we can calculate kdet for the two cases. Each of these rate constants is expressed as a half-life in Table 2. How are we to interpret these calculated half-lives? Let us look at the R-chlordane at Eagle Harbor case first. We know from other studies that R-chlordane (the cis isomer) degrades more slowly than γ-chlordane (the trans isomer) in vivo (17, 18), and there is some evidence to suggest the same is true in the environment (19). At both Sleeping Bear Dunes and Sturgeon Point, the half-life of R-chlordane is about twice that of γ-chlordane (see Table 1). Thus, based on the half-life of γ-chlordane at Eagle Harbor, a reasonable guess is that the half-life of R-chlordane at this location would
The authors thank all the members of the Integrated Atmospheric Deposition Network team and the U.S. Environmental Protection Agency’s Great Lakes National Program Office for funding (Grant GL995656).
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Received for review April 26, 1999. Revised manuscript received April 2, 2000. Accepted April 18, 2000. ES990466L
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