The NIOSH Action Level - ACS Symposium Series (ACS Publications)

29 The NIOSH Action Level A Closer Look JAMES C. R O C K

Downloaded by UNIV OF ARIZONA on January 7, 2013 | http://pubs.acs.org Publication Date: April 2, 1981 | doi: 10.1021/bk-1981-0149.ch029

The U S A F Occupational and Environmental Health Laboratory, Brooks A i r Force Base, San Antonio TX 78235

Most practicing industrial hygienists in America today must manage their programs with at least two objectives in mind: to limit the physiological risk to employees from their occupational exposures, and to limit the legal risk to employers from OSHA compliance inspections. This paper compares the probability of noncompliance during an OSHA inspection to three different measures of physiological risk: the average long-term exposure; the NIOSH Action Level as originally proposed (1); and the legal action level which is usually set equal to half of the permissible exposure limit (PEL). The comparison is conducted by identifying those occupational environments which will pass or f a i l each set of decision c r i t e ria. This paper deals only with occupational stresses which pose a chronic health hazard; acute health hazards require different treatment. The most surprising conclusion is that there are some commonly encountered work environments which are very l i k e l y to pass the NIOSH Action Level Test even though there is better than a 50% chance that an OSHA compliance officer collecting six samples during one v i s i t would be able to identify a citable violation of an 8-hour PEL. The Model of the Occupational Environment The model of occupational exposures which forms the basis for the proposed NIOSH Action Level i s introduced in this f i r s t section; the NIOSH decision c r i t e r i a are generalized in the second section; and the decision p r o b a b i l i t i e s are compared in the l a s t section of t h i s paper. A conscious e f f o r t has been made to present important results graphically, but the appendix contains equations so that an interested, mathematically-inclined reader can check or extend the r e s u l t s . Daily Exposures. For the model to be as general as possible, i t must represent a l l l i k e l y occupational exposure situations. It i s , therefore, useful to define a normalized exposure equal to the This chapter not subject to U . S . copyright. Published 1981 American Chemical Society

In Chemical Hazards in the Workplace; Choudhary, G.; ACS Symposium Series; American Chemical Society: Washington, DC, 1981.

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actual 8-hour time weighted average exposure (TWAE) divided by the applicable 8-hour standard (PEL). (1)

χ = TWAE/ΡEL


Since the goal of an occupational health program should be to control the exposures for each and every employee, i t i s conven ient to consider the group of daily exposures received by an employee during one year of employment. A typical 50-week work year would include 250 separate daily exposures. Let Ε represent the number of days during the year that the exposure exceeds the standard. Then the probability of an overexposure on any given day i s given by: (2)

P(x > 1) = e, and P(x < 1) = (1 - e ) , where e = E/250.

Distribution of Daily Exposures. From Equation 1, χ represents normalized daily exposures. Ft i s generally recognized that χ i s a lognormally distributed random variable ( I j . Therefore, the probability density function for the population of exposures which i s experienced by one worker i s completely characterized by two parameters: i t s geometric mean, GM, and i t s geometric stan dard deviation, GSD ( £ ) . Let χ represent the true long-term average of a l l normalized d a i l y exposures and recall from Equa tion 2 that e i s the fraction of tkose daily exposures which exceeds the standard. GM can be calculated from χ and GSD using Equation A-7 or from e and GSD using Equations A-2 and A-13 (3, 4). Thus, the lognormal probability density function, pdf(x), i s completely characterized by any one__pf the following pairs of parameters: (GM,GSD), (e,GSD), or (x,GSD). In each pair of parameters, the f i r s t parameter i s a measure of the "dirtiness" of the occupational environment, while GSD is a measure of the day-to-day v a r i a b i l i t y in the environment. In this paper, e and GSD are used to identify the various possible occupational envi ronments. I n t u i t i v e l y , GSD can be considered to be inversely related to the efficacy of i n s t a l l e d engineering controls. A GSD of 1.0 means that a l l exposures are i d e n t i c a l , a condition nearly r e a l ized in some laminar flow clean rooms. When GSD > 2.5, as may be the case for some kinds of maintenance work, i t i s l i k e l y that there are no functioning engineering controls. The exposure v a r i a b i l i t y for most American workers i s characterized by a GSD lying between 1.2 and 2.5 (3>). It i s d i f f i c u l t to determine how much "dirtiness" is too much. Nearly everyone would agree that i f more than half of the daily exposures exceed the standard (e > h) or i f the long-term average exposure i s greater than the standard (x > 1), then the workplace i s too d i r t y . There is much less agreement on when an environment i s clean enough to be considered acceptable.



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Long-Term Average Daily Exposure. __To shed some l i g h t on that question, i t i s useful to consider how χ behaves as a function of e and GSD. Figure 1 reveals that even in work environments where exposures are highly variable and where the daily exposure exceeds the standard 20% of the time, the average daily exposure is s t i l l less than the standard. Figure 2 i s a contour plot of the surface represented by Equation A-7![ In Figure 2, each pair of values for (e,GSD) specifies the average daily exposure received by one hypothetical worker and this average i s plotted as one point on the surface. For convenience in the rest of this paper I propose to coin a term which I shall c a l l the Average Exposure Limit (AEL). I have selected for the AEL the contour where the average daily exposure equals 95% of the PEL (AEL = 0.95). Just as the PEL is a standard against which one tests the value of each daily exposure, the AEL i s a standard against which one tests the average of many daily exposures. In Figure 2, a l l those workers whose exposure d i s t r i butions l i e to the l e f t of the AEL contour are deemed to work in an acceptable workplace because, for them, χ < AEL; those on the right are deemed to work in an unacceptable workplace because, for them, χ > AEL. By this c r i t e r i o n , six of the workplaces represented in Figure 1 are OK and three of them are NOT OK. This selection of AEL = 0.95 i s a convenient means to enhance the i n t u i t i v e understanding of the sections which follow. Any readers who disagree with my choice of contour are encouraged to select another contour and to make their own comparisons. Con tours not plotted can be easily computed from Equation A-7. Also, please note that the AEL concept does not apply to occupational exposures which have c e i l i n g standards. Further, discussion of the relationship between the AEL, the PEL, and short-term excur sion l i m i t s i s beyond the scope of this paper. Distribution of Daily Exposure Estimates. It might seem that the mathematical model of the industrial hygiene sampling problem i s complete once i t has been shown that two parameters are s u f f i cient to uniquely represent the set of daily exposures experienced by an employee during one year. Unfortunately, this i s not so. Employee exposures are not d i r e c t l y observable. Each exposure measurement i s only an estimate of the true exposure. Let TWAEE be the time weighted average exposure estimate resulting from laboratory analysis of a sample. Then define the normalized exposure estimate as £. (3) * = TWAEE/PEL Note that because each sample i s l i k e l y to be different from the exposure i t estimates, the distribution of exposure estimates, péd{%) i s different from the distribution of exposures, pdf(x). Leidel, Busch, and Crouse showed that i f the uncertainty of the sampling and analytical process i s represented by i t s coefficient


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4.5.

H A Z A R D S IN

VS.

THE WORKPLACE

β AND GSD 4.5

_

X =1 .752

UJ — · - J 00 CD Œ 1.5

en Œ

>

4.5

r

ce *—« ^ II LU ^ 1 1.5

α

> Œ

CO CD 1.5 4.5

" X=0.79&

(Ό

X=1 .038

CD Œ 1 1

l S

1.5

1.5

CO CO Ο 0

0.5

1

e=0.024 CLEAN

1.5

0

0.5

1

ez0.20 ACCEPTABLE

1.5

0

0.5

1.5

ezO.60 DIRTY

Figure 1. Nine charts showing how the probability density function, pdf(x), and the long-term average exposure, x, vary as a function of e, the fraction of daily ex posures that exceed the standard and GSD, the variability of the work environment.


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of variation (CV), and i f CV < 0.3, then % can be modelled as a lognormally distributed random variable as shown in Equations A-14 and A-15. For purposes of i l l u s t r a t i o n in this paper CV = 0.1, a value more or less typical of approved NIOSH sampling and analytical methods for which published values range from 0.01 to 0.25 (5). Comparing the Population of Exposures to that of Exposure Estimates. The relationship between the distribution of exposures, x, and the distribution of exposure estimates, £, is summarized in the Appendix. The distribution of exposures i s characterized completely by two parameters: e and GSD. The d i s tribution of 250 exposure estimates (or samples) which could be observed each year for each employee i s broader than the d i s t r i bution of exposures, and i s characterized by three parameters: e, GSD, and CV. Equations A-16, A-17, and A-18 clearly show that although both distributions have the same mean, the median and mode of samples are respectively smaller than the median and mode of exposures. To put this into practical terms, recall that the only data available to an industrial hygienist are a small fraction of a l l possible samples; no exposure i s d i r e c t l y observable. The average of several industrial hygiene samples i s a good estimate of the long-term average exposure, but the median and mode of sample data underestimate the median and mode of the true exposures. Summary of Key Concepts. The following concepts are so important to understanding the major conclusions of this paper that i t i s worthwhile to summarize them for repeated reference: a. The exposures experienced by any one worker over a long period of time are lognormally distributed. b. The distribution of exposures experienced by each worker can be completely specified by two numbers: e, and GSD. c. The distribution of sampling and analytical errors i s completely specified by the coefficient of variation, CV. d. The distribution of samples which can be collected from each worker i s characterized by three numbers: e, GSD and CV. A Generalization of the Action Level Decision C r i t e r i a The Decision Rules. The NIOSH action level i s a s t a t i s t i c a l decision threshold designed to help employers attain a high degree of confidence that no more than a small fraction of any employee's daily exposures exceed the standard. After developing the action l e v e l , NIOSH provided a set of decision rules to be used by employers who want to determine i f their workplaces meet this objective: an exposure estimate smaller than the action level indicates that the employer has probably achieved the objective; an estimate larger than the standard probably indicates a serious


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problem; and an estimate between the action level and the standard means that a decision cannot be made with s u f f i c i e n t l y high con fidence on the basis of one sample (3). These decision rules can be generalized to permit easy com parison between OSHA decisions and NIOSH decisions by defining several new terms. Let UAL represent the upper action l e v e l , which i s to be used in place of the standard. Let OK represent an acceptable workplace and l e t NOT OK represent an unacceptable workplace. Let Ρ indicate the probability that the indicated decision i s made. Thus, P(?) represents the probability that a decision cannot be made with s u f f i c i e n t l y high confidence on the basis of one breathing zone sample. Under these d e f i n i t i o n s , the decision rules become: (4)

P(0K) = P(* < AL)

(5)

P(N0T OK) = P(* > UAL)

(6)

P(?) = P(AL < S < UAL)

The challenge to the s t a t i s t i c i a n i s to select values for AL and UAL such that when the decisions are made, they are made correctly and with s u f f i c i e n t confidence. It i s beyond the scope of this paper to debate the assumptions underlying the various competing derivations for values to be assigned to the AL and the UAL. Instead, I propose to compare the decision probabilities which result from the various proposed decision thresholds. A l l that i s required for this comparison i s knowledge of pdi(%) and repeated applications of Equations A - l l and A-12, with AL = Xq and UAL = Xg, to compute the decision p r o b a b i l i t i e s . Once the data are computed in this fashion, there are three numbers to associate with each point on the surface of Figure 2: the probability that the environment i s OK; the probability that i t i s NOT OK; and the probability that no decision can be made on the basis of one exposure estimate. Therefore, a complete com parison between Figure 2 and the three decision c r i t e r i a in this report requires careful consideration of ten 3-dimensional sur faces. This i s not p r a c t i c a l , so i t i s necessary to define contours of significance which can be used to provide a rapid, easily understood comparison of the different behavior of the three decision c r i t e r i a . The f i r s t i s the contour of unbiased decisions, and the other three are the contours marking the boundaries of the regions where one decision i s the most l i k e l y outcome of evaluating the meaning of one representative exposure estimate. The Unbiased Decision Contour. A workplace which produces an unbiased decision i s defined for purposes of this paper as one for which P(0K) = P(N0T OK). From Equation 4, P(0K) i s repre sented by the area in the t a i l of pdl(%) below the AL. From Equation 5, Ρ(NOT OK) i s represented by the area in the t a i l of


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pdi(t) above the UAL. From Equation 6, P(?) i s represented by the area under the pd^U) curve lying between the AL and the UAL. There are two important observations to be made concerning the contour of unbiased decisions. The f i r s t i s that for workplaces lying to the l e f t of the unbiased decision contour, P(0K) > P(N0T OK); while for workplaces lying to the right of the unbiased decision contour, P(N0T OK) > P(0K). The second i s that since our decision c r i t e r i a have been derived to provide high confidence in the decisions which are made, i t frequently happens that the most l i k e l y outcome from a t r i a l involving one sample taken from a workplace lying close to the unbiased decision contour i s that a decision cannot be made with s u f f i c i e n t c o n f i dence. Because of t h i s , i t i s important to define additional decision boundaries. The Single Decision Contours. A workplace which i s more l i k e l y to produce a decision than to produce no decision and for which one decision i s c l e a r l y predominant i s defined for purposes of this paper as lying in a single decision region. There are two such regions: (7)

P(0K) > P(?) + P(N0T OK)

(8)

P(N0T OK) > P(?) + P(0K)

The boundaries of these regions may be computed from Equations A-23 and A-24. The region where the predominant decision i s OK l i e s to the l e f t of the contour defined by Equation A-23. The region where the predominant decision i s NOT OK l i e s to the right of the contour defined by Equation A-24. The region between these two contours i s the region where the decision rule makes most of i t s errors. Note that there are three types of error: no decision when one should be made; NOT OK when the decision should be OK; and OK when the decision should be NOT OK. The No Decision Region. A workplace which i s more l i k e l y to produce no decision than to produce either or both of the decisions, OK and NOT OK, i s defined for purposes of this paper as lying in the no decision region. There i s one such region: (9)

P(?) > P(0K) + P(N0T OK)

There i s no closed form expression for the boundary of this region. It can only be calculated by numerical methods. This region i s of interest because i t i d e n t i f i e s those workplaces for which many samples w i l l be required to insure that a decision can be made. If workplaces lying in this region need to be sampled, then i t may be wise to develop alternate decision c r i t e r i a which exhibit higher decision p r o b a b i l i t i e s , rather than to bear the unwarranted expense of excess sampling.


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LONG T E R M A V E R A G E E X P O S U R E

0

.1

.2

.3

.4

.5

1

.6

.7

.8

e = FRACTION OF EXPOSURES EXCEEDING THE STANDARD

Figure 2. Contour plot showing the long-term average exposure, x, as a function of e, and GSD. The heavy line, where χ = 0.95 PEL is the proposed Average Ex posure Limit, A EL. Acceptable workplaces lie in the region to the left of the AEL. Each marks the location of one of the charts from Figure 1.

Tabulated Decision Probabilities For Workplaces Shown in Figure 1. e = O.bU e = 0.024 1 e = 0.20 P(0K) P(?) P(N0K)|P(0K) P(?) P(N0K) P(0K) P(?) P(N0K) Table 2.306 1.519 1.127

I. NIOSH C r i t e r i a . AL = ffGSD. .0943 .8812 .0245 .0073 .7927 .0932 .8803 .0265 .0076 .7892 .0824 .8572 .0605 .0118 .7394

Table 2.306 1.519 1.127

II. Leaal Action Level C r i t e r i a . AL = .8740 .1016 .0245 .5070 .2930 .2000 .6261 .3474 .0265 .2167 .5801 .2032 .0019 .9377 .0605 .0001 .7512 .2488

Table 2.306 1.519 1.127

III. 0SHA C r i t e r i a . .9604 .0238 .0158 .9353 .0537 .0110 .6544 .3399 .0057

2.306 .7847 .1241 .0911 1.519 .6694 .2664 .0642 1.127 .0785 .8877 .0337 P(0K) P(?) P(N0K) GSD e = 0.024

CV) and UAI = 1. .2000 .0002 .4028 .2032 .0002 .4069 .2488 .0010 .4344

0.5 and UAL = 1. .1424 .Sôôé .5570 .0323 .3749 .5928 .0000 .4354 .5646

One SamDle. AL = 0. R35 and UAL = .7350 .1118 .1532 .3230 .1512 .6598 .2222 .1181 .2568 .2908 .3173 .6338 .0489 .0940 .6984 .1577 ?2TTÔ .Ôàl3 .0825 .3879 .5295 .0010 .7392 .2598 P(0K) P(?) Ρ(Ν0Κ) e = 0.20

.5970 .5928 .5646

.0011 .0003 .0000 P(0K) e

1.165. .5258 .4524 .2076

.0102 .9886 .0267 .9730 .2476 7524 P(?) P(N0K) = 0.60


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Decisions Made by Various Decision C r i t e r i a . To gain the maximum amount of insight from the following discussion, i t i s necessary for the reader to make frequent com parisons between the various figures. To make this as easy as possible, Figures 2, 3, 4, 5, and 6 have been reproduced to the same scale and have been annotated by "+" to mark the points which correspond to the nine charts in Figure 1. The three decision p r o b a b i l i t i e s , P(0K), P(?) and P(N0T OK), are computed for each of these nine sample workplaces using each of the deci sion c r i t e r i a . Tables I, II, III, and IV summarize the results of this computation. In the course of the discussion, these sample workplaces are referred to by their coordinates (e,GSD). Thus, the workplace in the lower l e f t corner i s (0.024, 1.13), while the middle one in the right-hand column i s (0.60, 1.52). Decisions Made by the NIOSH Action Level C r i t e r i a . Figure 3 shows the decision contours for the NIOSH Action Level Decision C r i t e r i a , and Table I summarizes the decision probabilities for each of the nine sample workplaces. Recall from Equation A-19 that the AL i s computed from GSD to provide 95% confidence that no more than 5% of the daily exposures exceed the standard i f one randomly collected sample i s less than the AL. In terms of the variables used in this paper, (e > 0.05 with ρ :> 0.05 i f X > AL). Figure 3 helps to demonstrate that the requirement for 95% confidence makes the NIOSH c r i t e r i a rather i n e f f i c i e n t and con servative. The inefficiency i s i l l u s t r a t e d by the size of the no decision region, which includes most of the workplaces where less than half of the daily exposures exceed the standard (6). The conservativeness i s i l l u s t r a t e d by the virtual absence of work places for which P(0K) > In Table I, the NIOSH decision c r i t e r i a i s shown to have poor efficiency by the three dirty workplaces for which e = 0.6. In these cases, a worker would be exposed above the standard three days out of f i v e , and his long-term average exposure would be greater than the 8-hour PEL. Very few people would disagree with the decision to c a l l these workplaces NOT OK. Nevertheless, these workplaces w i l l be declared NOT OK by the NIOSH Action Level decision c r i t e r i a only about 60% of the time. This i n e f f i ciency i s further i l l u s t r a t e d by the fact that only one of the three average workplaces with e = 0.2 has P(?) < 0.75. That one is (0.2, 1.13) and i t also i l l u s t r a t e s the conservât!veness of the NIOSH c r i t e r i a since on those infrequent occasions when a decision i s made, the odds are 21 to 1 to decide NOT OK. However, Table I most clearly i l l u s t r a t e s the conservât!veness of the NIOSH c r i t e r i a by the fact that P(0K) < 0.1 for the three clean workplaces where e = 0.024. Decisions Made with the Legal Action Level C r i t e r i a . Proper application of the NIOSH Action Level (Equation A-19) requires


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DECISION CONTOURS: NIOSH ACTION L E V E L

0

.1

.2

.3

.4

.5

.6

.7

.8


Figure 3. Decision contours resulting from the NIOSH Action Level Criteria: AL = f(GSD), and UAL = 1. Each marks the location of one of the charts from Figure 1. Note that the NO DECISION region includes nearly the whole area between the GSD axis and the NOT OK region.

DECISION CONTOURS: L E G A L ACTION L E V E L

= FRACTION OF EXPOSURES EXCEEDING THE STANDARD

Figure 4. Decision contours resulting from the Legal Action Level Criteria: AL = V2, and UAL = 1. Each marks the location of one of the charts from Figure 1.


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DECISION CONTOURS: OSHA CRITERIA

0

.1

.2

.3

.4

.5

.6

M=l

.7

.8


Figure 5. Decision contours resulting from OSHA Compliance Criteria: AL = 0.835, and UAL = 1.165. Each marks the location of one of the charts from Figure 1. Note that P(NOT OK) = probability of a citation based on one sample.

DECISION CONTOURS: OSHA CRITERIA

0

.1

.2

.3

.4

.5

.6

M=6

.7

.8


Figure 6. Decision contours resulting from OSHA Compliance Criteria: AL = 0.835, and UAL = 1.165, and assuming that the decision is based on six statistically independent samples. In comparison with Figure 5, many more environments are subject to a citation. This even includes some environments within the acceptable region bounded by the A EL contour in Figure 2.



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knowledge of both CV and GSD. It i s impossible to estimate CV and GSD from one sample, and so i t i s clear that in the real world the action level c r i t e r i a have to be modified. A currently common practice i s to set AL = h and UAL = 1. Figure 4 shows the decision contours for this case. Since UAL = 1, as i t did in Figure 3, the single decision contour for P(N0T OK) does not change. For GSD > 1.22, setting AL = h amounts to increasing the AL as compared with i t s value from Equation A-19. The effect of this i s to increase P(0K) at the expense of P(?) as i s c l e a r l y demonstrated by comparing Tables I and II for workplaces (0.024, 2.31), (0.20, 2.31), (0.024, 1.52), and (0.20, 1.52). For GSD < 1.22, setting AL = h amounts to decreasing the AL as compared with i t s value from Equation A-19. This increases P(?) at the expense of P(0K) as clearly demonstrated by comparing Tables I and II for workplaces (0.024, 1.13), (0.20, 1.13), and (0.6, 1.13). There are two features of Figure 4 which deserve comment. F i r s t , the Legal Action Level C r i t e r i a i s l i k e l y to declare as OK only three of the six workplaces declared OK by the Average Expo sure Limit concept of Figure 2. Second, and in contrast with the NIOSH Action Level C r i t e r i a of Figure 3 where nearly a l l common workplaces were in the no decision region, the Legal Action Level C r i t e r i a i s s l i g h t l y more e f f i c i e n t in making decisions; only those r e l a t i v e l y stable workplaces with GSD < 1.7 are in i t s no decision region. Decisions Made with the OSHA Compliance C r i t e r i a . The problem of the OSHA compliance o f f i c e r i s s i g n i f i c a n t l y different than the problem addressed by the NIOSH Action Level. The compli ance o f f i c e r i s not nearly so interested in how many exposures exceed the standard as in whether i t can be shown with 95% c o n f i dence that the standard was exceeded on the basis of the sample which was collected. Thus, the OSHA Compliance C r i t e r i a i s not a function of workplace v a r i a b i l i t y , GSD, but only of the random sampling and analytical errors, CV (5). As explained e a r l i e r , CV = 0.1 for a l l i l l u s t r a t i o n s in this paper. In this case, the OSHA Decision C r i t e r i a i s characterized by AL = 0.8355 and UAL = 1.1645. In comparison with the Legal Action Level C r i t e r i a , both the AL and the UAL for the OSHA C r i t e r i a are shifted to the right. Shifting the UAL reduces Ρ(NOT OK) while increasing P(?). Shifting the AL reduces P(?) while increasing P(0K). Since the AL is shifted proportionately further than the UAL, the net result i s that the compliance o f f i c e r has a higher P(0K) and a lower P(N0T OK) than does an employer who makes his decisions using the Legal Action Level C r i t e r i a . In addition, for most workplaces the compliance o f f i c e r has a much lower P(?) than does the employer. These comments can be reinforced by careful comparison of Figures 4 and 5 with the aid of Tables II and III.


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It i s informative to compare the decision regions of the OSHA Compliance C r i t e r i a as shown in Figure 5, with the Average Exposure Limit Concept of Figure 2. Five of the six workplaces identified as OK in Figure 2 are l i k e l y to be declared OK by an OSHA compliance o f f i c e r . However, only one of the three unaccept able workplaces of Figure 2 i s l i k e l y to be declared NOT OK by the c r i t e r i a of Figure 5. Perhaps the most surprising feature of Figure 5 i s the fact that only those workplaces where more than half of the daily expo sures exceed the PEL give the compliance o f f i c e r better than a 5050 chance of collecting a sample which w i l l support a c i t a t i o n ! How, Then, Does OSHA Issue So Many Citations? To support a c i t a t i o n , a compliance o f f i c e r must demonstrate with a high degree of confidence that at least one employee was exposed above the standard on the day of the inspection. Assuming that day-to day v a r i a b i l i t y in exposures i s approximately equal to the workerto-worker v a r i a b i l i t y on the day of the inspection, then i t would be possible, in p r i n c i p l e , for a compliance o f f i c e r to c o l l e c t η s t a t i s t i c a l l y independent samples. Under these conditions, the probability that the workplace w i l l be found in compliance i s equal to the probability that a l l η samples are less than the AL. Let Pn(OK) represent the probability of compliance after η samples are collected and Ρ(OK) represent the probability that a single sample i s less than AL. Then one has: (10)

Pn(OK) = (P(0K))

n

Likewise, the probability that at least one sample i s greater than the UAL i s given by one minus the probability that none of the samples i s greater than the UAL. (11)

Pn(N0T OK) = ( ! - ( ! - P(N0T 0K)) ) n

Figure 6 shows the decision contours for a compliance o f f i cer who takes six s t a t i s t i c a l l y independent samples during an inspection. Table IV l i s t s the decision probabilities for each of our nine sample workplaces. Clearly, collecting more samples increases the probability of demonstrating noncompliance and decreases the probability of demonstrating compliance. By comparing Figure 6 with Figure 2, i t i s clear that by taking as few as six samples, a compliance o f f i c e r can increase to better than 50-50 the odds of finding evidence to support a c i t a t i o n , even for some workplaces which l i e inside the presumably acceptable region bounded by the AEL = 0.95 contour. The principle i s very easily generalized. By collecting enough samples, a compliance o f f i c e r can move the single decision contour for a c i t a t i o n as far to the l e f t as he chooses. Given enough samples, every workplace could be subjected to a c i t a t i o n ! For example, by collecting 14 samples, a compliance o f f i c e r moves


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the single decision contour for issuance of a c i t a t i o n far enough to the l e f t to include five sample workplaces not included in the contour of Figure 5: (0.2, 1.13), (0.2, 1.52), (0.2, 2.31), (0.6, 1.13) and (0.6, 1.52). That i s , 14 samples guarantee that P ( c i t a tion) = P14(N0T OK) i s greater than 1/2 for these workplaces. The most t e l l i n g comparison i s that between Figures 4 and 6, There are many workplaces with GSD > 1.5 which are l i k e l y to be declared OK by the Legal Action Level C r i t e r i a but which could be cited by a conscientious compliance o f f i c e r who collects more than five or six samples! Conclusion. A l l of the decision c r i t e r i a examined in this paper make a decision on the basis of one sample. It should be clear from the discussion that one sample i s usually not enough. An employer who wants to document conditions in his workplace would be well advised to c o l l e c t enough samples to use a more e f f i c i e n t decision rule such as the one-sided tolerance test suggested recently by Tuggle (_7). To be truly representative, samples must be drawn from a l l operating conditions; i . e . , from each season of the year, from each day of the week, from each process condition. In addition, a compliance o f f i c e r should not expect to be able to sustain a c i t a t i o n based solely on evidence that on the day of his v i s i t one sample exceeded the upper action level defined in the OSHA Industrial Hygiene Field Operations Manual. He should also be required to present corroborative evidence to demonstrate that the sample in question i s a representative sample. If i t i s the only one of a large number of representative samples which exceeds the UAL, a c i t a t i o n i s probably not warranted unless a c e i l i n g standard has been violated. F i n a l l y , any decision strategy can be compared to any other on the basis of the decision contours defined in this paper. It i s recommended that every candidate decision c r i t e r i a , whether i t is based upon one or several samples, be evaluated to determine which workplaces are accepted and which are rejected before i t i s adopted. None of the decision c r i t e r i a examined in this paper are able to f a i r l y determine the quality of a l l workplaces. Acknowledgement. My sincere appreciation to Dr David Cohoon of the USAF School of Aerospace Medicine for his continuing support, his constructive c r i t i c i s m , and the use of his computer graphic software. Also to Dr R.M. Tuggle of the US Army Environmental Hygiene Agency for his timely c r i t i c a l review and suggestions for improving my manuscript.


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LIST OF MATHEMATICAL SYMBOLS AEL

= Average Exposure Limit:

If χ < AEL, Decide OK.

AL

= Action Level:

Ε

= The number of days/year when χ > PEL.

e

= Fraction of work days when χ > PEL.

exp(.)

= Raise e, the base of l n ( . ) , to the indicated power.

GM

= Geometric Mean = Median of x.

GM

= Geometric Mean = Median of X.

GSD

= Geometric Standard Deviation of x.

GSV

= Geometric Standard Deviation of X.

ln(.)

= Take the natural logarithm of the indicated quantity.

If χ > AL, Decide OK.

Note that GM < GM. GSV > GSD.

NOK/NOT OK = The decision one makes whenever X > UAL. OK

= The decision one makes whenever % £ AL.

P(.)

= The probability that the indicated event occurs.

ρ

= The (usually small) probability that e > Θ.

pdf(x)

= The lognormal probability density function of x.

pdi(%)

= The lognormal probability density function of £.

PEL

= 8-hour Permissible Exposure Limit.

Pn(OK)

= The probability that η s t a t i s t i c a l l y independent industrial hygiene samples are a l l less than AL.

Pn(N0T OK)

= The probability that at least one of η s t a t i s t i c a l l y independent industrial hygiene samples i s > UAL.

UAL

= Upper Action Level:

χ

= The random variable representing true daily exposures.

x"

= The arithmetic average of a l l true daily exposures.

If £ > UAL, Decide NOT OK.

£

= The random variable for daily exposure estimates.

Xg

= The gth^ f r a c t i l e of x:

Xq

= The qth_ quantile of x:

y

= I n ( χ ) , a normal random variable.

Za

= The ath_ quantile of y:

μ = ln(GM)

= The arithmetic average of y = l n ( x ) .

P(x > Xg) = g. P(x < Xq) = q. P([(y-y)/a] PEL.


486

CHEMICAL

H A Z A R D S IN T H E W O R K P L A C E

Literature Cited. 1. Leidel, N.A., K.A. Busch, and W.E. Crouse. "Exposure Measurement Action Level and Occupational Environmental Varia b i l i t y , " Department of Health, Education, and Welfare, National Institute for Occupational Safety and Health, 4676 Columbia Parkway, Cincinnati, NIOSH Technical Information, HEW Pub. No. (NIOSH) 76-131, Cincinnati, Ohio 45226, April 1975.


2. Aitchison, J., and J . A . C . Brown. "The Lognormal D i s t r i bution," Cambridge University Press, New York, 1976. 3. Liedel, N.A., K.A. Busch, and J.R. Lynch. "Occupational Exposure Sampling Strategy Manual," Department of Health, Education, and Welfare, National Institute for Occupational Safety and Health, NIOSH Technical Information, DHEW (NIOSH) Pub. No. 77137, 4676 Columbia Parkway, Cincinnati, Ohio 45226, January 1977. 4. Natrella, M.G. "Experimental Statistics," National Bureau of Standards Handbook 91, Superintendent of Documents, US Government Printing Office, Washington DC 20402, October 1966. 5. OSHA Instruction CPL 2-2.20, Industrial Hygiene Field Operation Manual, US Department of Labor, OSHA, Washington DC 20210, 30 April 1979. 6. Rock, J . C . "The NIOSH Sampling Strategy...A Useful Industrial Hygiene Tool or an Economic Disaster?," Paper 131. 1979 American Industrial Hygiene Conference, Chicago, I l l i n o i s , May 31, 1979. 7. Tuggle, R.M. "The NIOSH Decision Scheme--A User's Perspective," Paper 321. 1980 American Industrial Hygiene Conference, 23 May 1980.


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487

Appendix--Key Mathematical Concepts and Equations Normal Random Variable. The probability density function of a normally distributed random variable, y, i s completely charac terized by i t s arithmetic mean, y, and i t s standard deviation, σ. This i s abbreviated as y - Ν ( μ , σ ) and written as: 2

(A-l)

pdf(y) = ( l / ^ 7 o ) e x p -^ (((y - y ) / o ) ) 2

2


The Z-variate can be defined so that there i s a one-to-one mapping, (Za a) expressed by a definite integral. (A-2)

a = (l/ν^π )

Za / exp(-x /2) άτ 2

Because (a) i s the area in the lower t a i l of the normal d i s t r i b u t i o n , Za i s called the atji quantile of the standard normal d i s t r i b u t i o n , (or the (100)(a)th~percentile). A useful identity follows d i r e c t l y from the symmetry of the Gaussian distribution in Equation A-2 (4). (A-3)

Z(l-a) = -Za

Note that the area (a) also has a more general p r o b a b i l i s t i c interpretation. (A-4)

P(y < (oZa + y ) )

=a

Lognormal Random Variable. Every normally distributed random variable, y, i s uniquely associated with a lognormal1y distributed random variable, x, whose probability density function i s com pletely characterized by i t s geometric mean, GM, and geometric standard deviation, GSD (2). y = ln(x) y = ln(GM) σ = ln(GSD)

(A-5)

The distribution of χ i s abbreviated as In χ - Ν [ln(GM), (ln(GSD)) ] and written: 2

(A-6) pdf(x) = [l/(x^?ln(GSD))]exp(-% [(ln(x)-ln(GM))/ln(GSD)] ) 2

Since (x) has a skewed d i s t r i b u t i o n , i t s central measures are d i s t i n c t with Mean χ > Median χ > Mode χ {2). (A-7)

χ = Mean χ = exp[ln(GM) + 0.5 (In(GSD)) ]

(A-8)

Median χ = GM

2


488

(A-9)

CHEMICAL

HAZARDS


Mode χ = exp[ln(GM) - (In(GSD)) ] 2

The cumulative distribution function for the lognormal random variable χ i s derived by combining Equations A-4 and A-5. (A-10)

P(lnx < [Za In(GSD) + ln(GM)]) = a

Separating Equation A-10 into two equations helps to c l a r i f y the procedure needed to quickly calculate the qth quantile, Xq, of pdf(x) when χ i s a lognormally distributed random variable. Since q and Zq are related by Equation A-2, Equation A - l l can be used to solve for Xq given q or for q given Xq. Downloaded by UNIV OF ARIZONA on January 7, 2013 | http://pubs.acs.org Publication Date: April 2, 1981 | doi: 10.1021/bk-1981-0149.ch029

(A-ll)

ln(Xq) = ln(GM) + Zq In(GSD), where P(x < Xq) = q

Equation A-12 can be derived from Equations A-3 and A - l l . i s presented here for easy reference. (A-12)

ln(Xg) = ln(GM) - Zg In(GSD),

It

where Ρ (χ > Xg) = g

Recall from Equation 2 that e = P(x > 1). Then the following useful relationship between GM, GSD, and e, follows d i r e c t l y from Equation A-12. (A-13)

ln(GM) = Ze In(GSD)

Thus, although a lognormal distribution i s usually charac terized by GM and GSD, i t i s equally valid to characterize i t by e and GSD, since Equation A-13 permits GM to be calculated uniquely from e and GSD. The Effect of Imperfect Sampling. Let £ be the random variable which estimates the true value of the lognormally d i s tributed random variable (x). The value of (fc) i s determined by sampling (x) and analyzing with a process which has inherent uncertainty associated with i t . The uncertainty i s described by the coefficient of variation of the analysis, CV. If CV < 0.3, then (£) can be modelled adequately as a lognormally distributed random varible characterized by GM and GSV as defined below {}). (A-14) (A-15)

(In(GSP)) = (In(GSD)) + l n ( l + CV ) 2

2

2

ln(GM) = ln(GM) - h l n ( l + CV ) 2

It i s possible to combine Equations A-7, A-8, A-9, A-14, and A-15 to derive expressions for the mean, median and mode of £. (A-16)

Mean S = exp(ln(GM) + ^(In(GSO)) ) = Mean χ

(A-17)

Median X = GM =

2

Median χ exp[h l n ( l + CV )) Z


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(A-18)

489

NIOSH Action Level

Mode % = exp(ln(GM) - (ln(GSP)) ) = 2

Mode χ exp(1.5 l n ( l + CV*))

Note that since GSV > GSD, the median and Mode of S are respectively smaller than the Median and Mode of x, while t h e i r means are equal. The Action Level. By combining the last equation in Ref 2 with Equation A-14, one can derive an e x p l i c i t equation for the NIOSH Action Level. (A-19)

ln(AL) = (Zp)(ln(GSD)) + (Z6)(ln(GSD)) - h ( l n ( l + CV )) 2

ln(AL) = (Zp)/ln(l + CV ) + (In(GSD)) - h ( l n ( l + CV )) 2

2

+ (Z0)(ln(GSD))


2

In this equation, ρ & Zp and θ & ΖΘ are related through Equation A-2. θ i s the fraction of daily exposures to be allowed to exceed the standard, and ρ i s the (normally small) probability that at least (100)(θ)% of the daily exposures represented by one sample exceeds the standard i f that sample exceeds the action l e v e l . Numerical examples in this paper assume ρ = θ = 0.05. Decision Contours. Let AL = Xq in Equation A - l l and UAL = Xg in Equation A-12. Rearranging these equations: (A-20) Zq = (ln(AL)-ln(GM))/ln(GSP), where P{% < AL) = q (A-21) Zg = (ln(GM)-ln(UAL))/ln(GSP), where P(X > UAL) = g To derive the unbiased decision contour, note that P(£ < AL) = P{% > UAL) along the contour. This i s achieved when Zq = I g , which means setting Equations A-20 and A-21 equal. Combine that expression with Equations A-14 and A-15 to derive Equation A-22, which specifies the unbiased contour in terms of e and GSD. (A-22)

Ze = [ln(AL) + In(UAL) + l n ( l + CV )]/(2 In(GSD)) 2

Note that the single decision contours occur when P(X < AL) = 0.5 or P(£ > UAL) = 0.5. These conditions are met respectively when Zq = 0 or Zg = 0. Substitute zero into Equations A-20 and A-21 and combine with Equations A-14 and A-15 to derive equa tions for the single decision contours: (A-23)

Ze = (2(ln(AL)) + l n ( l + CV ))/(2 In(GSD))

(A-24)

Ze = (2(ln(UAL)) + l n ( l + CV ))/(2 In(GSD))

2

2

Equation A-23 i s the contour for OK. contour for NOT OK.

Equation A-24 i s the

R E C E I V E D October 27, 1980.


The NIOSH Action Level - ACS Symposium Series (ACS Publications)

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