The Random-Effects Model Applied to Refractory Ceramic Fiber Data

Regulatory Toxicology and Pharmacology 32, 190 –199 (2000) doi:10.1006/rtph.2000.1420, available online at http://www.idealibrary.com on

The Random-Effects Model Applied to Refractory Ceramic Fiber Data L. D. Maxim, 1,* J. N. Allshouse,* and D. E. Venturin† *Everest Consulting Associates, Cranbury, New Jersey 08512; and †Unifrax Corporation, Amherst, New York 14228 Received May 10, 2000

Refractory ceramic fiber (RCF) is a valuable, hightemperature, insulating material with a variety of industrial uses. Because some fibers are respirable by humans and RCF is relatively durable in simulated lung fluids, RCF may pose a health hazard in the workplace. The RCF industry has established a comprehensive product stewardship program (PSP) to identify, quantify, and manage risks. One key element of this PSP is a workplace monitoring program. This paper analyzes monitoring data collected as part of a Consent Agreement with the U.S. Environmental Protection Agency over the period from 1993 to 1998. More specifically, this paper applies the random-effects model (REM) to data collected at several Unifrax plants and applicable to several groups of workers. The REM fits the RCF data well. Depending upon the plant and the functional job category values of the variance of the log-transformed time-weighted average workplace concentrations range from slightly less than 0.5 to 1.0. The estimated intraclass correlations (ratio of the between-worker variance to the total variance) were less than 0.4, and most were less than 0.2. Implications of these findings are examined. Use of the REM in the development of a workplace respiratory policy is described. Finally, two possible criteria for measuring compliance with an occupational exposure limit are reviewed: an “overexposure” criterion developed by Rappaport and co-workers and a conventional “no exceedance” criterion reportedly used by regulatory agencies. The overexposure criterion is logically correct for potential toxicants with chronic effects. For representative values of statistical parameters for RCF from the plants considered, the overexposure criterion is less stringent. © 2000 Academic Press

INTRODUCTION

Refractory ceramic fiber (RCF) is a valuable hightemperature synthetic vitreous fiber (SVF) used as an insulating refractory for diverse industrial applica1

To whom correspondence should be addressed.

0273-2300/00 $35.00 Copyright © 2000 by Academic Press All rights of reproduction in any form reserved.

tions (see ECA, 1992, 1999 for background). Because RCF is relatively durable in biological fluids and some fibers are in the respirable size range, there is concern that RCF may present a hazard by inhalation. Chronic inhalation bioassays on laboratory animals have indicated that fibrosis and tumors result when animals are exposed to high [ca. 200 fibers per milliliter (f/ml)] airborne concentrations of RCF. Specifically, there were excess lung tumors in rats and mesothelioma in hamsters. However, epidemiological studies on present and former workers in RCF production plants have not indicated any fibrosis or mesothelioma or an elevated incidence of lung cancer (see Maxim et al., 1994, 1997, 1998, and references therein). Under the auspices of the Refractory Ceramic Fibers Coalition (RCFC) RCF producers in the United States have developed a product stewardship program (PSP) to identify and manage workplace risks (Barrows et al., 1993; Walters, 1995). A key component of this PSP is a workplace monitoring program, one portion of which was conducted pursuant to a Consent Agreement with the U.S. Environmental Protection Agency (EPA). Details of this program are presented elsewhere (e.g., RCFC, 1993; Maxim et al., 1994, 1997, 1999a,b). Key points pertinent to this paper include: ● Workers sampled were classified into one of eight functional job categories (FJCs). FJCs were the basis for stratification in the sampling plan. These include fiber production, mixing/forming, assembly, auxiliary, finishing, installation, removal, and other (not elsewhere classified). The sampling plan was also stratified on the basis of the type of sample. Samples were gathered from plants operated by RCFC producers (termed internal samples) and those operated by customers/end users (termed external samples). ● For internal samples, replicates on individual workers were quite common over the years. For external samples, replicates were less common because the number of customer facilities is large. In the case of internal samples, the time interval between replicated samples varied from a few days to years. No specific minimum time interval was established to avoid the

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RANDOM-EFFECTS MODEL APPLIED TO RCF DATA

effects of possible autocorrelation. Most studies (e.g., Rappaport, 1991; Francis et al., 1989; George et al., 1995), however, conclude that autocorrelation is not a problem in practical terms. ● Workplace samples were gathered for individual workers. Cassettes were changed as necessary to attempt to capture differences in workplace concentration if a worker changed FJCs and to minimize the likelihood that the cassette would be overloaded. ● Fiber concentrations were measured using phase contrast optical microscopy (PCOM) and counted using the National Institutes for Occupational Safety and Health (NIOSH) 7400 B method. ● Task length average (TLA) and time-weighted average (TWA) workplace concentrations were measured. ● Data analyzed in this paper were collected and analyzed as part of the Consent Agreement over the period from 1993 to 1998. Numerous papers have been written to analyze various aspects of these data (Maxim et al., 1994, 1997, 1998, 1999a,b; Rice et al., 1994, 1996, 1997). This paper applies the random-effects model (REM) to these data and details the resulting implications in terms of the degree of control necessary to attain a recommended exposure guideline. This paper also indicates how one RCF producer has used this model to establish a respirator policy. WORKPLACE RCF CONCENTRATIONS ARE LOGNORMALLY DISTRIBUTED

The lognormal model has been found to provide an adequate representation of the distribution of environmental contaminants (Gilbert, 1987), workplace concentrations for many airborne substances (Esmen and Hammad, 1977; Leidel et al., 1977; Lyles et al., 1997; Kromhout et al., 1993; Rappaport, 1991; Rappaport et al., 1993, 1995), geological data (Koch and Link, 1980), and many other applications (Johnson and Kotz, 1970; Aitchison and Brown, 1969). More specifically, the lognormal model has been shown to be applicable to RCF workplace concentration data. For example, Maxim et al. (1994, 1997) evaluated a series of possible data transformations suggested by Box and Cox (1964) and found that the log-transformation was optimal among a broad range of alternatives for RCF workplace concentration data. The suitability of the lognormal model is easily demonstrated with histograms and fitted density functions. Figure 1, for example, shows a histogram of TWA concentration data (natural logarithm units) for the “mixing/forming” FJC (one of eight used to classify the population of workers) and a fitted normal distribution. The fit is excellent (as are those for other FJCs) and supports the use of the lognormal model to describe RCF workplace concentrations.

191

FIG. 1. Histogram and density function for mixing/forming functional category.

SIGNIFICANT DIFFERENCES AMONG FJCs

As noted, the worker population was stratified on the basis of FJC and type of sample (internal versus external). Sampling strata were initially regarded as “observational groups” as suggested by Rappaport et al. (1995). One-way analysis of variance (ANOVA) on logtransformed data indicated that there are significant differences in mean exposure (1) among FJCs, (2) among certain jobs within FJCs, (3) among industry sectors within FJCs (customer data), and (4) among individual RCFC plants within FJCs (Maxim et al., 1999b). In this analysis, observational groups were narrowed down to individual FJC–plant combinations for internal samples. Following the exposure assessment protocol suggested by Rappaport et al. (1995), the observational group was defined as an FJC–plant combination. A random-effects ANOVA model was fit to many observational groups in plants belonging to one of the major RCFC producers to see if these observational groups can be termed “monomorphic groups” (to use the term coined by Rappaport (1991)). THE RANDOM-EFFECTS MODEL

Simply put, the REM can be described as follows. Let X ij represent the jth ( j ⫽ 1, . . . , n i ) time-weighted average measurement on the ith (i ⫽ 1, . . . , k) worker among those randomly sampled from an observational group. In total, there are N observations on the k workers. Let Y ij be the natural logarithm of X ij (Y ij ⫽ ln(X ij )). The REM (Lyles et al., 1997) is Y ij ⫽ ln共X ij 兲 ⫽ ␮ y ⫹ ␤ i ⫹ ⑀ ij ,

(1)

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MAXIM, ALLSHOUSE, AND VENTURIN

where

␤ i ⬃ N共0, ␴ b2兲

and

⑀ ij ⬃ 共0, ␴ w2兲.

(2)

Here ␤ i represents the random deviation of the ith worker’s mean ( ␮ yi ⫽ ␮ y ⫹ ␤ i ) of log-transformed observations from the population mean ( ␮ y ) and ⑀ ij represents the random deviation of the jth exposure measurement on the ith worker from ␮ yi . The parameters ␴ b2 and ␴ w2 represent the between- and withinworker components of the total variance ␴ y2 ⫽ ␴ b2 ⫹ ␴ w2, and Y ij ⬃ N( ␮ y , ␴ y2 ). Note also that the intraclass correlation coefficient, ␳, represents (Tornero-Velez et al., 1997) “the proportion of the total variance that is explained by the between-worker variance component,” ␳ ⫽ ␴ b2/ ␴ y2 . In order to fit the REM it is necessary to have multiple shift-long (TWA) workplace concentration measurements on at least some of the workers in an observational group. The specific equations used to estimate the various quantities defined above depend upon whether the data set is balanced or unbalanced (Searle et al., 1992). For this application, a balanced data set is one in which the same number of repeated TWA measurements are available for each of the sampled workers in the observational group. Rappaport et al. (1995) present the assessment protocol and necessary estimating equations for analysis of a balanced data set; Lyles et al. (1997) present the corresponding equations for the analysis of unbalanced data. For several statistical reasons, balanced data sets are preferable. However, as noted by Lyles et al. (1997), “few existing databases contain completely balanced exposure data, and the practical issues of environmental sampling often make the attainment of balanced data problematic.” Such is the case for RCF; the protocol was not specifically designed to support the REM and historical data are not balanced. Therefore, the estimating equations used in Lyles et al. (1997) for unbalanced designs were used. To avoid having to define terms, unless otherwise noted the same notation used in Lyles et al. (1997) is used in this paper. Even if the REM is directly applicable, negative ANOVA estimates of ␴ b2 occasionally occur (Lyles et al., 1997; Searle et al., 1992, provide equations which enable calculation of the probability of a negative estimate). By definition, a negative value of ␴ b2 is impossible. However, it is possible that the estimate will be negative—particularly if the true value of ␴ b2, the number of classes, or the number of observations per class is small. A commonly used decision rule is to set negative estimates of ␴ b2 equal to zero. An improved procedure is set forth in Lyles et al. (1997). REM ESTIMATES FOR RCF DATA

We have applied the REM to data from several plants operated by one RCF producer (Unifrax). To

FIG. 2. tions.

Summary of results for several plant–FJC combina-

illustrate some of the above concepts, data were analyzed from plant “C” in the assembly FJC. This data set consists of 99 (N ⫽ 99) personal monitoring TWA values taken from 29 (k ⫽ 29) workers in the assembly FJC at this plant. For these data (using the equations presented in Appendix A of Lyles et al., 1997) the following estimates were determined: the withinworker variance ␴ˆ w2 ⫽ 0.64, and the between-worker variance ␴ˆ b2 ⫽ 0.0246. The total variance, ␴ˆ y2, is the sum of the between- and within-worker variances or approximately 0.66 in this case. The intraclass correlation coefficient ␳ˆ ⫽ 0.037. The overall mean of the transformed TWA concentrations ␮ˆ y is estimated as ⫺1.52 (using the equation presented in Appendix A of Lyles et al. (1997)) and that for the untransformed TWA concentration measurements, ␮ˆ x , is approximately 0.303 as determined from the equation

␮ x ⫽ exp共 ␮ y ⫹ 0.5共 ␴ b2 ⫹ ␴ w2兲兲.

(3)

The standard deviation of the untransformed concentration measurements, ␴ x , can be calculated from the equation

␴ x ⫽ 关exp共2 ␮ y ⫹ ␴ y2 兲共exp共 ␴ y2 兲 ⫺ 1兲兴 0.5

(4)

and is equal to 0.2933 in this case. The coefficient of variation (CV x ) of the untransformed data equals 96.79% (⫽0.2933/0.303). Figure 2 summarizes a series of similar calculations for several other FJC–plant location combinations. Kromhout et al. (1993) assembled workplace concentration data from many industries and found that the REM was applicable for most groups studied. Among 165 worker groups adequately described by the REM,

RANDOM-EFFECTS MODEL APPLIED TO RCF DATA

193

the median estimated between- and within-worker geometric standard deviations were 1.43 and 2.0, respectively— equivalent to ␴ b2 ⫽ 0.1279 and ␴ w2 ⫽ 0.4805. These values are consistent in order-of-magnitude terms with the estimates provided in Fig. 2. Specifically, we find (for this manufacturer) that ␴ b2 values are generally small in comparison with ␴ w2—i.e., ␳ is typically close to zero. Several estimated values of ␴ b2 were negative, necessitating use of the alternative method of analysis presented in Lyles et al. (1997). UTILITY OF THE REM

The REM is conceptually relevant because it may provide a more accurate characterization of worker exposure. From an operational perspective, the REM is useful in three important respects. First, the REM offers guidance on the suitability of various control strategies. Second, the REM provides the foundation for an improved measure of compliance with an occupational exposure limit (OEL). Third, the REM enables calculation of a realistic target concentration to ensure compliance with an OEL. These applications are discussed below. Guidance on Selection of Control Strategies As noted by Rappaport et al. (1995): The presence of substantial between-worker variation is a major consideration regarding the control of exposure and has thus far been ignored. Suppose, for example, that personal practices of a few workers produces shift-average exposures well in excess of their co-workers. In such a situation, engineering or administrative controls (which affect everyone more or less equally) are arguably unnecessary for protecting most of the workers and may be of limited effectiveness in reducing the exposures of even those few individuals who are highly exposed. Indeed, it could well be more effective to identify the highly exposed individuals so that their practices can be modified and they might then receive the benefits of low exposure enjoyed by their colleagues.

Rappaport et al. (1995) and Lyles et al. (1997) have proposed the use of empirical Bayes-like estimates of the random variables ( ␮ xi ) to help select an intervention strategy for balanced and unbalanced data sets, respectively. For the unbalanced case, where the estimate of ␴ b2, ␴ˆ b2 is ⬎0,

再

␮ˆ xi ⫽ exp

冎

៮ i • ⫹ ␮ˆ y ⫹ ␥ˆ ␴ˆ w2/2兲 共n i ␥ˆ Y ⫹ ␴ˆ w2/2 , 共n i ␥ˆ ⫹ 1兲

(5)

៮ i䡠 ⫽ (n i ) ⫺1 ⌺Y ij and ␮ˆ y, ␴ˆ b2, and ␴ˆ w2 are where ␥ˆ ⫽ ␴ˆ b2/␴ˆ w2, Y the estimates (the caret denotes an estimate) given in Appendix A of Lyles et al. (1997). In the event that ␴ˆ b2 ⬍ 0, Lyles et al. (1997) suggest ៮ i • ⫹ 关共n i ⫺ 1兲/共2n i 兲兴 ␴ w2其 ␮ˆ xi ⫽ exp兵Y to estimate ␮ˆ xi .

(6)

FIG. 3. Plot of estimated mean exposures ␮៮ xi and approximate 100(1 ⫺ 1/k)% CI for population mean exposure. Data are for Plant C, assembly FJC. N ⫽ 99, k ⫽ 29, ␴ w ⫽ 0.80, ␴ b ⫽ 0.16.

The quantity in Eqs. (5) and (6) is an estimator of worker-specific mean exposure. Rappaport et al. (1995) recommend calculating an approximate 100(1 ⫺ 1/k)% confidence interval (CI) for the overall population mean exposure ␮ x . Equations for calculation of the boundaries of the CI are presented in Lyles et al. (1997) for unbalanced data (Eq. (8) and Appendixes A and C) and are omitted here for the sake of brevity. These equations are tedious (requiring a computer), but straightforward. The necessary calculations can easily be made using common “spreadsheet” software. It is useful to construct a plot that shows the estimated mean exposures for each worker together with the CI. If all the ␮ xi values lie within the boundaries of the CI, then overall engineering or administrative controls aimed at reducing exposure for the entire group are probably required if exposures are excessive. If, alternatively, one or more of the ␮ xi values fall outside the bounds of the CI, then it may be worthwhile to consider control measures designed to reduce the exposures of specific individual workers. In this regard, exposures that fall outside of the CI on the low side may be just as interesting as those which exceed the upper limit. An indication that one or more workers differ from the others in an observational group may suggest that the group be redefined in some way. It certainly provides information relative to the determinants of exposure. Figure 3 shows this plot for the assembly FJC at plant “C.” In this case, the estimated mean exposure levels for all 29 workers in this FJC fall within the limits of the CI. The overall mean fiber concentration is approximately 0.3 f/ml. If this is acceptable (see below) then no further action is required. If this is determined to be excessive, the plot if Fig. 3 suggests that controls should be developed (if feasible) to reduce all exposures in the group. Figure 4 shows a similar plot for the mixing/forming

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MAXIM, ALLSHOUSE, AND VENTURIN

paport, 1984; Tornero-Velez et al., 1997; Leidel et al., 1997; U.S. Department of Labor, 1996) is to gather and analyze a TWA sample from a worker selected at random on a day selected at random. If the TWA exceeds the OEL, noncompliance is alleged. For the lognormal distribution of exposures this probability, denoted ␣ (others have used the symbol ␥, but this is avoided here because ␥ is used in another equation in this paper), can be calculated from the equation

␣ ⫽ P兵X ij ⬎ OEL其 ⫽ 1 ⫺ ⌽

FIG. 4. Plot of estimated mean exposures ␮៮ xi and approximate 100(1 ⫺ 1/k)% CI for population mean exposure. Data are for Plant B, mixing/forming FJC. N ⫽ 26, k ⫽ 10, ␴ w ⫽ 0.86, ␴ b ⫽ 0.41.

FJC at plant “B.” Exposures for all but “worker 2” fall within the CI. The exposure for worker 2 is beneath the lower end of the CI. If this concentration was excessively high the worker’s job might be studied to identify an “assignable cause” and/or the job placed in the “respirator required” category (see below). Even if below average for the group (as it is here) a follow-up study might provide useful ideas for an exposure control strategy. For the plants and FJCs studied in this analysis, most plots resembled that shown in Fig. 3. That is, past engineering efforts tended to focus on individual jobs with relatively high exposures. As a consequence, if additional controls prove necessary, it is likely that a broader strategy might be required. All plants studied in this analysis have relatively low exposures, however. Calculation of “Overexposure” Probabilities A second major use of the REM is to provide estimates necessary to calculate improved measures of compliance with OELs. Numerous papers have been written on the subject of compliance testing and the use of the REM for this purpose (see, e.g., Rappaport, 1984, 1991; Rappaport et al., 1995; Lyles et al., 1997; Spear and Selvin, 1989; Tornero-Velez et al., 1997). For reasons that are not entirely clear, some of this work has generated controversy (see, e.g., Hewett, 1998; Rappaport et al., 1998, and references contained therein). At issue are the definition(s) of compliance with an OEL and the precise meaning of overexposure. We confine the discussion here to potential toxicants (such as RCF) with chronic, rather than acute, effects. The conventional (compliance-based) means to determine if an employer meets an OEL (see, e.g., Rap-

再

冎

ln共OEL兲 ⫺ ␮ y , ␴y

(7)

where ⌽( z) denotes the probability that a standard normal deviate would fall beneath the value z, i.e., ⌽( z) is the cumulative distribution function (CDF) of the standard normal distribution, and ␣ represents the probability of noncompliance. Because the compliance decision can be (and reportedly is) made by a regulatory authority on the basis of only one sample, it has come to be called the “no exceedance” or “one strike, you’re out” criterion. A conscientious employer to be judged by this criterion would seek to minimize ␣. This can be done in principle by installing additional controls (if technically and economically feasible) in order to lower ␮ y or by placing workers in this observational group in respirators (see Maxim et al., 1998, for respirator statistics) so as to lower effective worker exposure. To anyone familiar with the large variability of workplace exposure measurements, the statistical properties (e.g., the operating characteristic curve) of a one-sample test are clearly unattractive. Beyond this practical difficulty, the compliance probability approach is flawed conceptually for toxicants with exclusively chronic effects; the appropriate concern for these materials is not an exceedance of the OEL on any given day, but rather whether or not the average concentration exceeds the OEL on a long-term basis. This latter outcome has been termed “overexposure” by Rappaport and others. A statistically based criterion relative to overexposure is to the probability ␪ that a randomly chosen worker from a monomorphic group has a mean exposure greater than the OEL. This probability can be calculated from estimates in the REM by the equation

␪ ⫽ P兵 ␮ xi ⬎ OEL其 ⫽ 1 ⫺ ⌽

再

冎

ln共OEL兲 ⫺ ␮ y ⫺ 0.5 ␴ w2 . ␴b (8)

Reporting on the analysis of a substantial database of workplace exposures, Tornero-Velez et al. (1997) found that ␣ exceeded ␪ for most (83%) workplace groups evaluated, indicating that the “one strike, you’re out” criterion was often a conservative estimator of overexposure, a finding supported by these data. Statistical tests to assess possible overexposure (as defined above)

RANDOM-EFFECTS MODEL APPLIED TO RCF DATA

are found in Rappaport et al. (1995) and Lyles et al. (1997). Application to RCF At present, there is no permissible exposure limit (PEL) for RCF. RCFC established a recommended exposure guideline (REG), however. (This guideline has been established on the basis of prudence and technical/economic feasibility, not demonstrated risk.) In the mid-1980s an REG of 2 f/ml was established. In 1991 this was lowered to 1 f/ml. In 1997 it was lowered again to 0.5 f/ml, reflecting progress in reducing exposures and RCFC’s goal of continuous improvement. Unifrax employed the REM model to identify which, if any, operations required additional exposure controls and to establish an internal respirator policy. For these purposes, the REG is regarded as an OEL. To illustrate the above equations, consider one of the examples presented above, the data for the mixing/forming FJC at plant “A” shown in Fig. 4. In this example,

␮ˆ y ⫽ ⫺2.375 ␴ˆ w2 ⫽ 0.743 ␴ˆ b2 ⫽ 0.172 ␴ˆ b ⫽ 0.414 ␴ y ⫽ 0.956 ␮ˆ x ⫽ 0.147 k ⫽ 10 N ⫽ 26 and OEL ⫽ REG ⫽ 0.5 f/ml. From Eq. (6) the probability ␣ that one sample selected at random would exceed the REG is approximately 0.039. From Eq. (7) the calculated probability of overexposure, ␪, is approximately 0.0007—substantially smaller than ␣. In this example, the likelihood of overexposure is very small, but the probability of noncompliance, ␣, is higher even though the average workplace concentration (approximately 0.15 f/ml) is well beneath the REG. Lyles et al. (1997) suggested a Waldtype test generalized for the unbalanced case (Eq. (2) and Appendix A of their paper). In this example, the computed value of the test statistic, Zˆ w ⫽ ⫺1.91, is beneath the threshold for proving that overexposure is possible. (Lyles et al. (1997) provide another test staˆ , to use in the event that ␴ˆ b2 ⬍ 0.) tistic, T We found with RCF data that, for most observational

195

groups in plants operated by this manufacturer, values of ␪ were beneath corresponding values for ␣. That is, the simple compliance criterion was conservative compared to the preferred overexposure criterion. Where possible, the ␪ criterion and associated test statistics were used for identifying respirator-required jobs. Practical Ideas for Respirator Decisions Specification of a respirator policy is a practical issue facing RCF producers. The logic suggested here has worked well in practice. It consists of the following broad steps: ● First, a qualitative industrial hygiene (IH) analysis is performed to evaluate potential hazards and assess the need for a respirator. The policy enables IH personnel to mandate respirator use (for entire jobs or for selected tasks within jobs) based on this qualitative analysis alone. ● Second, the available exposure data are assembled and a decision is made whether or not the data are adequate to support the use of the REM. If so, the steps described below are followed. If not, a conventional analysis is undertaken to estimate the mean workplace exposure and an upper-confidence limit (UCL) on this mean (see, e.g., Gilbert, 1987; Rappaport and Selvin, 1987). Despite the fact that the exposure-monitoring program for RCF collected over 300 internal TWA samples annually, there are certain FJC–plant combinations (generally for jobs that are performed only on an intermittent basis) with insufficient replicates to support the use of the REM. ● Third, the available exposure data are analyzed using the procedure described by Lyles et al. (1997). Values of ␴ b2 and ␴ w2 are calculated and a plot similar to those shown in Fig. 3 or Fig. 4 is prepared. In the event that one or more workers fall outside the confidence limits, these workers may be shifted to another group or placed in respirators if their exposure appears high. Next, statistical tests are conducted to determine if ␪ exceeds a prespecified threshold (e.g., 0.1). The specific equations used (see Lyles et al. (1997)) for the test Zˆ w ˆ depend on whether or not ␴ b2 ⬎ 0. Although there or T are normative reasons to use ␪ as the relevant criterion, we recommend calculation of ␣ as well. This is because the “one strike, you’re out” criterion may be used by a regulatory agency and the employer should have some idea of the likelihood of a finding of noncompliance. ● Whether the REM (step 3) or a test on the mean exposure (step 2) is used, there are three possible outcomes of the statistical tests. If the UCL on the mean is beneath the OEL (conventional procedure) or the appropriate test statistic on ␪ supports this same conclusion, respirators are not required for this group of workers. If not, respirators may be required depending upon the estimated mean exposure and the OEL. If the

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MAXIM, ALLSHOUSE, AND VENTURIN

mean is greater than the OEL then respirators are required. If the mean is beneath the OEL, but the UCL on the mean exceeds the OEL, then the data suggest— but do not prove—that respirators are not required. In this event, more data should be collected and the procedure repeated. The above steps have been simplified for clarity. Additional steps may also prove useful or necessary. For example: ● In the second step (data assembly) it may be important to test for trends in the data and/or to note the times when engineering controls have been installed or new workplace practices have been implemented. The objective is to isolate a representative data set. ● If plots of the type shown in Fig. 3 or Fig. 4 indicate that some workers appear to have higher (or lower averages) it may be appropriate to examine the jobs and tasks performed and reallocate workers to groups. Perhaps respirators (or other intervention) are required only for a subset of the workers.

In the initial development of the policy, respirators were mandated for only a few individuals and (based on IH analysis) for a few tasks. Accelerated monitoring was instituted for certain FJC–plant combinations. Implications for Required Level of Control The above discussion shows how the overexposure criterion can be used to set a respirator policy. This section shows how the overexposure criterion can be used to define a target value for the mean exposure to ensure compliance with an OEL. To begin, it is useful to examine the consequences of the no exceedance criterion. As noted, a conscientious employer will attempt to control the mean workplace concentration (if possible) so that the probability of noncompliance ␣ defined by Eq. (7) is small. Assume that the employer seeks a statistical “guarantee,” g ⫽ (1 ⫺ ␣ ), that no one sample chosen at random will exceed the OEL. Let CV x be the coefficient of variation of the untransformed exposure distribution (i.e., CV x ⫽ ␴ x / ␮ x ). Rappaport (1991) provides a simple equation relating the standard deviation of the logged concentrations ( ␴ y ) to CV x ,

␴ y ⫽ 关ln共1 ⫹ CV 兲兴 2 x

0.5

.

(9)

Now if the employer wants to have a probability g that no one sample selected at random will exceed the OEL, it follows from Eq. (7) that

␮ y ⫹ zg ␴ y ⫽ ln共OEL兲,

(10)

where z g is the standard normal variate associated with a cumulative area g. Finally, for any lognormal

FIG. 5. Relation between the CV of TWA workplace concentration measurements and the mean level necessary to provide a guarantee, g, that a single TWA measurement does not exceed the OEL.

distribution (Rappaport, 1991), the mean of the untransformed variable is related to the mean and variance of the transformed variable by means of the equation

␮ x ⫽ exp共 ␮ y ⫹ 0.5 ␴ y2 兲.

(11)

Combining the above and simplifying it follows that, in order to provide a probability g of no exceedance in a single sample, the ratio of the mean workplace concentration to the OEL can be no greater than

␮ x /OEL ⫽ exp共 ␴ y共0.5 ␴ y ⫺ zg 兲兲

(12)

(Rappaport, 1991, derived a similar equation). Figure 5 shows how ␮ x /OEL varies with ␴ y or CV x for various values of the probability, g. Note that— over the range of values plotted—the maximum value of the ratio ␮ x /OEL decreases rapidly as ␴ y or CV x increase. As a practical matter, the CVs for many workplace data sets are relatively large; CVs for RCF data (even for individual FJC–plant combinations) are often 1.0 (100%) or greater. Moreover, although the imposition of controls and altered workplace practices diminishes

197

RANDOM-EFFECTS MODEL APPLIED TO RCF DATA

when ␳ is unity, the between-worker variance accounts for the total variability. If ␳ is less than unity, the relative stringency of each criterion depends upon the probability ( g), the standard deviation of the log-transformed data ( ␴ y ), and ␳. Comparing Eq. (12) with Eq. (13) there is a “breakeven” value of ␴ y , for which the two criteria are equally stringent. This break-even value of ␴ y can be shown to be

␴y ⫽ FIG. 6. Relation between ␳ , ␴ y , and ␮ x to have 95% assurance that mean of worker selected at random does not exceed OEL.

␴ x as well as ␮ x , our experience suggests that the CVs may stay the same. From Eqs. (9) and (12) or Fig. 5, it follows that at a CV of 1.0, the mean workplace concentration must be reduced to no more than 36% of the OEL to have a probability g of 0.95 that no single sample selected at random will exceed the OEL. This result has very important implications in terms of the economic and technical feasibility determinations that underlie the establishment of a PEL. In effect, the “one strike, you’re out” criterion implies that the employer must reduce workplace concentrations to an average level substantially more stringent than the established OEL. Now suppose that the overexposure criterion (Eq. (8)) is used as a starting point rather than Eq. (7). Following the same logic used to derive Eq. (12) an analogous equation can be derived to calculate the maximum workplace concentration to provide a guarantee g of compliance with Rappaport’s overexposure criterion. The equation is ␮ x/OEL ⫽ exp共 ␴ y共0.5 ␳␴ y ⫺ zg冑␳ 兲兲,

2z g 共1 ⫺ 冑␳ 兲 . 共1 ⫺ ␳ 兲

When ␳ is equal to zero, the break even value of ␴ y is 2z g . Equation (14) is indeterminate when ␳ is unity. However, the ratio of the derivatives of the top and the bottom terms in Eq. (14) shows that the limit when ␳ approaches unity is z g . Thus, as ␳ varies from 0 to 1.0, the break-even value for ␴ y varies from 2z g to z g . Figure 6 shows how ␮ x /OEL varies with ␳ and ␴ y for the overexposure model (Eq. (13)). Figure 7 provides contours of constant ␮ x /OEL in terms of ␳ and ␴ y . Figure 8 compares the overexposure criterion (solid lines) with the day-to-day or no exceedance criterion (dashed line) for values of CV between 0.0 and 2.0 and values of ␳ between 0.0 and 0.6. In this region, the overexposure criterion is less stringent than the no exceedance criterion. As noted above, analyses of RCF data for several plants and several FJCs indicate that the CV x values are typically close to 1.0 (i.e., ␴ y close to 0.83) and that ␳ values are typically beneath 0.4. Thus, for these data, the overexposure criterion will be less stringent than the no exceedance criterion. The principal point in favor of the overexposure criterion is that it deals correctly with overexposure for chronic toxicants. That it also is less stringent (in terms of the necessary degree of control) is potentially of economic interest.

(13)

where,

␳ ⫽ ␴ b2 / ␴ y2 . This equation is only slightly more complex than that for the no exceedance criterion. The critical ratio ␮ x / OEL depends not only upon ␴ y (hence CV x ), but also on the intraclass correlation ␳. If ␳ is zero (i.e., the between-worker variance component is zero), then the critical ratio ␮ x /OEL can be as high as 1. As ␳ increases the criterion becomes more stringent in terms of the degree of control necessary to avoid overexposure. When ␳ is unity both criteria are equally stringent. To see this, compare Eq. (12) with Eq. (13) as ␳ approaches 1.0. This makes sense on reflection because,

(14)

FIG. 7.

Contours of ␮ x /OEL.

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MAXIM, ALLSHOUSE, AND VENTURIN

FIG. 8. Relation between CV of TWA workplace concentration measurements and true mean (as ratio to OEL) to provide 95% guarantee of compliance no overexposure.

CONCLUDING COMMENTS

The data analysis reported in this paper illustrates the use of the REM applied to RCF. This model, coupled with use of the overexposure criterion, has proven useful for designing control strategies, evaluating the need for exposure reduction, and developing a rational respirator policy. The REM is applicable to RCF and (at least for the data analyzed to date) it appears that the between-worker variance is small relative to the total variance—a result that has significant implications for the workplace concentrations necessary to meet the REG of 0.5 f/ml. ACKNOWLEDGMENTS The overall monitoring program was sponsored by RCFC. Unifrax Corporation supported this particular analysis. The constructive comments of the referees have improved the clarity of this paper.

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