Annals of Occupational Hygiene Advance Access originally published online on May 22, 2007
Annals of Occupational Hygiene 2007 51(4):397-406; doi:10.1093/annhyg/mem013
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Upper Limits for Exceedance Probabilities Under the One-Way Random Effects Model
1 Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504, USA
2 Department of Mathematics and Statistics, University of Maryland, 1000 Hilltop Circle, Baltimore, MD 21250, USA
3 Division of Environmental Health Sciences, School of Public health, University of Minnesota, MMC 807, 420 Delaware Street SE, Minneapolis, MN 55455, USA
* Author to whom correspondence should be addressed. e-mail: krishna{at}louisiana.edu
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ABSTRACT |
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TOP
ABSTRACT
INTRODUCTIONIn this article, we propose statistical methods for setting upper limits on (i) the probability that the mean exposure of an individual worker exceeds the occupational exposure limit (OEL) and (ii) the probability that the exposure of a worker exceeds the OEL. The proposed method for (i) is obtained using the generalized variable approach, and the one for (ii) is based on an approximate method for constructing one-sided tolerance limits in the one-way random effects model. Even though tolerance limits can be used to assess the proportion of exposure measurements exceeding the OEL, the upper limits on these probabilities are more informative than tolerance limits. The methods are conceptually as well as computationally simple. Two data sets involving industrial exposure data are used to illustrate the methods.
Keywords: between- and within-worker variability • generalized confidence interval • generalized P-value • tolerance interval
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INTRODUCTION |
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Applications of the one-way random effects model for assessing personal exposure level for a job group have been well demonstrated in the industrial hygiene literature for the past two decades. Many authors have postulated this model for analysing exposure data; among others, see Kromhout et al. (1993), Rappaport et al. (1995) and Lyles et al. (1997a,b). If multiple measurements exist for each worker, and both between- and within-worker variability are significant and need to be accounted for, then one should use the random effects model. After postulating the model, personal exposure levels can be assessed based on (i) the probability that the mean exposure of an individual worker exceeds the occupational exposure limit (OEL) and (ii) the probability that an individual exposure measurement exceeds the OEL. At the outset, we need to acknowledge that there is disagreement in the occupational hygiene field about whether one should interpret the OEL as a limit that the mean exposure should not exceed, or a limit that some upper percentile of the exposures should not exceed (Lyles et al., 1997b; Letters to the Editor, 1998). Our goal here is to present a methodology that will work in both cases.
Let 
denote the probability that the mean exposure of a worker exceeds the OEL. Consider testing the hypotheses
 | (1) |
< 0.1 is equivalent to testing if the 90th percentile of the mean exposure distribution is less than the OEL.In this context, this testing problem is addressed in Lyles et al. (1997a) for the case of balanced data,and in Lyles et al. (1997b) for unbalanced data. The problems that come up in this context are somewhat complex as the parameter in the above hypotheses is a function of all the parameters in the one-way random effects model: the mean, as well as the two variance components. Lyles et al. (1997a,b) proposed large sample tests (Wald, likelihood ratio and score-type tests). These authors also suggested suitable modifications to control the type I error probability of the tests. Krishnamoorthy and Mathew (2002) proposed a test procedure based on the generalized variable approach for the case of balanced data, and Krishnamoorthy and Guo (2005) extended the results for unbalanced data. The generalized variable test is simple to use and is applicable for small samples.
Exposure level in a work environment can also be assessed by testing the probability 
that an individual measurement exceeds the OEL. That is,
 | (2) |
Note that is the probability of an exposure measurement exceeding the OEL. Again, as in the preceding testing problem, testing equation (2) is equivalent to testing about the (1 – A)th quantile of the distribution of the exposure measurements, and the quantile testing can be carried out by comparing an appropriate upper tolerance limit with the OEL; see the next section for an explanation of this. The problem of constructing one-sided tolerance limits in a one-way random effects model is also complex, and only approximate methods are available, e.g. Mee and Owen (1983); Bhaumik and Kulkarni (1991); Vangel (1992); and Krishnamoorthy and Mathew (2004). Among these approximate methods, Krishnamoorthy and Mathew's (2004) approach seems to be simple as it does not require any special table values or interpolation. These authors showed, by extensive simulation studies, that their approach is accurate as long as the intraclass correlation coefficient is not very small. In other words, their approach produces accurate results as long as the assumed one-way random model is well-fitted for the exposure data.
Hypothesis testing about an exceedance probability is useful to decide if the personal exposures for a job group are acceptable; however, an upper bound on the probability is more informative than the results based on a significance test. For example, if a significance test indicates that < 0.10 at the level of 5%, then we can conclude that the data provide evidence to indicate that the exceedance probability is <10%; but this is not the least upper bound, and the actual 95% upper limit on could be much <10% (see the example section).
Hewett and Ganser (1997) have provided a method for estimating 90% confidence intervals around the point estimate of the exceedance probability when the exposure sample is assumed to be a simple random sample from a lognormal distribution. This is based on an established procedure for estimating confidence intervals around an estimate of the proportion of observations that fall in one tail of a normal distribution. The Z-value corresponding to the OEL is calculated first, followed by a look-up table or graph that allows the calculation of the 90% confidence intervals for a given sample size. It should be noted that Hewett and Ganser's approach is applicable if the exposure measurements form a simple random sample from a lognormal distribution. This means that Hewett and Ganser's approach is valid only when the between worker variance component is zero, a strict condition which may not be met in many instances.
The purpose of this paper is to bring to the attention of industrial hygienists some simple methods that can be used for constructing tolerance limits in the one-way random effects model, and constructing upper confidence limits for the exceedance probabilities mentioned in the preceding paragraphs. Although the method of Hewett and Ganser (1997) does not consider between- and within-worker variability, the methods we consider do take into account such variances, and our methods are also applicable to small samples with balanced or unbalanced data. As the concept of the generalized variable method is relatively new, we first explain this approach with applications to normal and lognormal parameters in the following section. Then, we describe the one-way random effects model in the context of this problem, and then identify the probabilities of interest in terms of the model parameters. We apply the generalized variable approach given in Krishnamoorthy and Mathew (2004) and Krishnamoorthy and Guo (2005) for constructing one-sided confidence limits for . A confidence interval procedure for is obtained by transforming the approximate tolerance interval procedures given in Krishnamoorthy and Mathew (2004). As the details of constructing generalized variables for various parameters in the one-way random effects model, and that of constructing tolerance limits, are given in the aforementioned papers, we merely outline the basic methodology in the following section. Finally, we illustrate the interval estimation procedures for the exceedance probabilities using two data sets given in Lyles et al. (1997b).
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THE GENERALIZED PIVOTAL QUANTITY (GPQ) AND THE GENERALIZED TEST VARIABLE |
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The generalized P-value approach for hypothesis testing has been introduced by Tsui and Weerahandi (1989) and the generalized confidence interval by Weerahandi (1993). The concepts of generalized P-values and generalized confidence intervals have turned out to be extremely fruitful for obtaining tests and confidence intervals for complex problems where standard procedures are difficult to apply. In particular, the generalized variable approach is useful to develop a generalized pivotal quantity (GPQ) based on which inferential procedures for a parameter of interest can be easily obtained. The classical pivotal quantity is a function of a random sample and the parameter of interest, and its distribution does not depend on any unknown parameters, whereas the GPQ is a function of a random variable, its observed value (the known measurements after a sample has been drawn), the parameter of interest and other parameters. The GPQ is constructed so that, for a given sample, its distribution is free of any unknown parameters. However, the frequentist properties of generalized P-values and generalized confidence intervals could depend on unknown parameters. For example, the actual coverage probability of a generalized confidence interval could be different from the assumed confidence level. However based on numerical results, it has been noted that for a wide variety of problems, generalized P-values and generalized confidence intervals do meet the usual requirements in terms of type I error probability and coverage probability; see Krishnamoorthy and Mathew (2003) for results concerning the lognormal distribution, and see also the book by Weerahandi (1995) for a variety of other applications. An attractive feature of many solutions based on generalized P-values and generalized confidence intervals is that the procedures are applicable to small samples, whereas conventional approaches based on, for example, the likelihood method, do require large samples. This will be clear, for example, from the generalized variable solution for the lognormal mean discussed later.
We shall first explain the generalized variable inferential procedure as given in Krishnamoorthy et al. (2006). Let X be a random variable whose distribution depends on a scalar parameter of interest and a nuisance parameter (parameter that is not of direct inferential interest) 
. Let x denote the observed value of X. That is, x represents the data that has been collected. To obtain a generalized confidence interval for , we need a GPQ, denoted by T1(X;x,,), that is a function of the random variable X, the observed data x, and the parameters and , and satisfying the following two conditions:
- Given x, the distribution of T1(X;x,,) is free of the unknown parameters and ;
- The value of T1(X;x,,) at X = x, namely, T1(x;x,,) is equal to . (3)
The percentiles of T1(X;x,,) can then be used to obtain confidence intervals for . Such confidence intervals are referred to as the generalized confidence intervals. For example, if T1–
denotes the 100(1–)th percentile of T1(X;x,,), then T1– is a generalized upper confidence limit for . A lower confidence limit, or two-sided confidence limits can be similarly defined.
Now suppose we are interested in testing the hypotheses
 | (4) |
0 is a specified quantity. Suppose we can define a generalized test variable T2(X;x,,) satisfying the following conditions:- i. For a given x, the distribution of T2(X;x,,) is free of the nuisance parameter ;
- ii. The value of T2(X;x,
,) at X = x, namely, T2(x;x,,) is free of any unknown parameters;
- iii. For a given x and
, the distribution of T2(X;x,,) is stochastically monotone in i.e. stochastically increasing or decreasing in ). (5)
- ii. The value of T2(X;x,
In general, for a given x and , we can take T2(X;x,,) = T1(X;x,,)-, which is stochastically decreasing in . In this case, the generalized P-value for testing equation (4) is given by
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Because the distribution of T1(X;x,,) is free of any unknown parameters, the generalized P-value at 0 can be obtained using a numerical method or estimated using Monte Carlo simulation.
It should be noted that the generalized confidence intervals are not guaranteed to have exact frequentist coverage properties with respect to the distribution of X, and the distribution of the generalized P-values under a null hypothesis of interest may not be uniform (0, 1). However, a number of simulation and numerical studies suggest that the coverage probabilities are very close to the nominal level, and the Type I error rates are close to the nominal level of significance. Hanning et al. (2006) showed that the generalized variable procedures are asymptotically exact in many situations.
In general, constructing a GPQ is a non-trivial task. Knowledge about the distribution of some basic statistics involved in the problem is necessary to construct a bona fide GPQ. Assuming that the readers are familiar with the distributional results for sample statistics from a normal distribution, we shall illustrate the procedures for finding a GPQ for a normal mean.
A GPQ for a normal mean
Let X1, ..., Xn be a random sample from a N(
,
2) distribution. Define
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and let 
be an observed value of 
. As 
and S2 are sufficient statistics for the normal distribution, we shall construct a GPQ based on them. A GPQ for the mean is obtained, using a sequence of operations and inverse operations, as given below.
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where 
independently of 
. Using the result that 
follows a tm distribution, we can write 
or 
because tm is distributed symmetrically about zero. We shall now show that G in equation (6) satisfies the requirements in equation (3). Using step 1 of equation (6), we see that the value of G at 
is . Also, from step 3 of equation (6), we see that, for a given 
and s, the distribution of G does not depend on any unknown parameters. Therefore, G is a GPQ for the mean . It is easy to see that the lower /2 quantile and the upper /2 quantile of G form the usual t interval for the mean. In other words, inferences on based on the GPQ in equation (6) and the one based on the classical pivotal quantity 
are the same.
An appealing feature of the generalized variable approach is that one can readily develop a GPQ for a function of several parameters provided a GPQ for each parameter is available. Specifically, if the pair (G1,G2) forms GPQs for (1,2), then the GPQ for a function f(1,2)is given by f(G1,G2). As will be seen in the sequel, this particular feature of the generalized variable approach enables us to construct GPQs for various functions of the variance components and the overall mean in a one-way random effects model. As an example, we shall now construct a GPQ for a log normal mean.
A GPQ for a lognormal mean
Let Y1, ..., Yn be a sample from a lognormal distribution with parameters and so that the mean is exp(
), where 
. As the mean is a one–one function of , it is enough to find a GPQ for . Let Xi=ln(Yi),\ i=1, ..., n. Then, the log-transformed data X1, ..., Xn can be regarded as a sample from a normal distribution with mean and standard deviation . Let 
denote respectively the mean and variance of the sample. As we already developed a GPQ for , we shall now develop a GPQ, denoted by G2, for 2 using the fact that 
has the chi-square distribution with df = n – 1. Let s2 be an observed value of S2. Then, a GPQ is given by
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We see from the above expression that the value of G2 at S2 = s2 is 2 and the distribution of G2 does not depend on any parameter when s2 is fixed. Thus, G2 is a bona fide GPQ for 2. The GPQ for 
is given by 
, where G is given in equation (6). Finally, a GPQ for the lognormal mean is given by 
. For a given 
of 
, the distribution of the GPQ does not depend on any unknown parameters, and so its percentiles can be estimated using Monte Carlo simulation or by numerical integration. Specifically, the /2 and 1 – /2 quantiles of the GPQ form a 1 – confidence interval for the lognormal mean. Krishnamoorthy and Mathew (2003) showed that the results based on the GPQ are comparable to those based on the exact method due to Land (1973).
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EXCEEDANCE PROBABILITIES, QUANTILES AND TOLERANCE LIMITS |
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To explain the relationship between exceedance probabilities and quantiles, and to describe the role of tolerance intervals, let us consider a random variable X that is distributed normally with mean µ and variance
2. For a specified limit k, consider the exceedance probability P(X > k). Note that P(X > k) 
A is equivalent to P(X 
k) < 1–A. The latter inequality holds if and only if k is less than or equal to the (1 – A)th quantile of X, namely, µ + z1–A, where z1–A is the (1 – A)th quantile of a standard normal distribution. In other words, statements concerning an exceedance probability translate into statements concerning the quantiles. Thus, the null hypothesis H0 :P(X > k) A is equivalent to H0:µ+z1–A k.
To see the role of tolerance intervals, let us first recall the definition of an upper tolerance limit. Let 
and S2 denote the sample mean and sample variance based on a sample of n observations from the normal distribution. An upper tolerance limit for the normally distributed random variable is a function of 
and S2, denoted by 
,S2), and is constructed so that at least a proportion p of the normal distribution is below 
, with confidence, say, 1 – . The proportion p is also referred to as the content of the tolerance interval. As we know that a proportion p of the normal distribution is less than µ + zp, the upper tolerance limit 
is nothing but a 1 – upper confidence limit for µ + zp. This is because proportion p of the normal distribution is less than µ + zp which is less than or equal to 
with probability 1 – . Therefore, at least a proportion p of the normal distribution is below 
with confidence 1 – . Thus, testing
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Notice that the null hypothesis will be rejected if a 1 – upper confidence limit for µ+z1–A (or 1 – A content— 1 – coverage upper tolerance limit) is less than k. We refer to Guttman (1970) for more details on tolerance intervals, along with the expression for g(
,S2) given above.
For simplicity, we have presented the above results for the normal distribution, where the observations in the sample are all independent. However, the above results also hold more generally. In fact, the present article deals with observations that follow a one-way random effects model, and are not independent. The case of independent observations is already addressed in Krishnamoorthy et al. (2006).
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THE ONE-WAY RANDOM EFFECTS MODEL |
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To describe the problem, let us consider the framework as given in Rappaport et al. (1995) and Lyles et al. (1997a). Let Xij denote the jth shift-long exposure measurement for the ith worker, j=1, ..., ni and i=1, ..., k. Let us assume that Xij follows a lognormal distribution so that Yij = ln(Xij) follows a normal distribution. Under the assumed one-way random effects model, we can write
 | (7) |
where µ is the overall mean, 
i represents the random effect due to the ith worker, and 
ij is the random deviation of the ith worker's exposure around that worker's mean. Furthermore, i and ij are mutually independent with
 | (8) |
i, Yij is normally distributed with mean µ + i and variance e2. Therefore, the mean exposure µxi for the ith worker is given by
 | (9) |
 | (10) |
Thus, the parameter mentioned in equation (1) is a probability that the random variable µxi (or ln(µxi)) exceeds the OEL (or ln(OEL)), and the parameter in equation (2) is a probability that the random variable Xij (or Yij) exceeds the OEL (or ln(OEL)).
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GPQs FOR THE OVERALL MEAN AND VARIANCE COMPONENTS IN THE ONE-WAY RANDOM MODEL |
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As the exceedance probabilities are functions of the overall mean and the variance components, we shall first provide GPQs for these individual parameters. The methods of constructing generalized variables for the mean and variance components, their validity, and the statistical properties of the inferential procedures based on them are well addressed in Krishnamoorthy and Mathew (2002, 2004) and Krishnamoorthy and Guo (2005). As we have already shown that the present problem of exceedance probabilities has one–one relation with that of constructing one-sided tolerance limits, the frequentist coverage probabilities should be similar to those for tolerance limits considered in the aforementioned papers. Specifically, the coverage probabilities of confidence limits are close to the nominal confidence level as long as the one-way random effects model is well fitted for a given sample.
In the following, we shall provide necessary summary statistics and their distributional results that are required to construct GPQs for µ, 2 and e2. Let
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Note that SSe is the usual error sums of squares (or within sums of squares). It is known that
 | (11) |
and these three random variables are independent. The former two distributions are exact, and the approximate chi-square distribution associated with 
is due to Thomas and Hultquist (1978). Because of this approximate chi-square distribution, some of the properties required for GPQ's will hold only approximately.
Let (
) be an observed value of 
. That is, (
) is the computed value of (
) based on a given data set. The generalized variable for the overall mean µ can be constructed as
 | (12) |
N(0,1) independently of 
k–12. To get the second step, we have used the distributional properties in equation (11). In particular, we have used the approximate chi-square distribution associated with 
. We note that the second step in equation (12) holds only as an approximation. Let us verify that Gµ satisfies the conditions (i) and (ii) in equation (3). From step 1 in equation (12), we see that Gµ is equal to µ when 
and 
. Also, we observe from step 2 of equation (12) that, for a given 
and 
, the distribution of Gµ does not depend on any unknown parameters because the joint distribution of 
(Z,k–12) does not depend on any unknown parameters.
We can construct the generalized variables for the variance components similarly. For , we develop the generalized variable (noting that 
) as
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where for any number x, (x)+=max{0,x}, and 
. To get the second step, we used the result that 
independently of 
. It is easy to verify that G satisfies the conditions in equation (3). For 2e, we have 
Again, it is easy to see that 
satisfies the conditions in equation (3).
Finally, the generalized variable for any function 
can be obtained by replacing (µ, 2 and e2 by their generalized variables.
Note that since the chi-square distribution associated with 
is only approximate, conditions (i) and (iii) in equation (5) will hold only approximately. However, the extensive numerical results in Krishnamoorthy and Mathew (2004), and Krishnamoorthy and Guo (2005) show that the generalized variable approach is quite satisfactory in the unbalanced case. Though not pointed out in this article, GPQ's are not unique in the unbalanced case. What we have provided is one solution that performs satisfactorily.
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AN UPPER LIMIT FOR
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Using the distributional result in equation (10), we can express
as
 | (13) |
where 
(.) denotes the standard normal distribution function. As already noted, testing hypotheses about is equivalent to testing about the (1 –A)th quantile of the mean exposure distribution in equation (10), and the quantile is given by 
, where zp is the pth quantile of the standard normal distribution. In particular, the hypotheses in equation (1) are equivalent to
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is the 90th percentile of the distribution of ln(µxi). If a 95% upper confidence limit for 
is less than ln(OEL), then we can conclude that < 0.10 at the level of significance 0.05.Large sample solutions to the above testing problem are given in Lyles et al. (1997a,b). Tests based on the generalized variable approach are given in Krishnamoorthy and Mathew (2002) and Krishnamoorthy and Guo (2005).
Notice that in equation (13) is a function of (µ,e2,), and so we can write as (µ,e2,). If Gµ, G and Ge2 are the generalized variables for µ, and e2, respectively, then a generalized variable for is given by
 | (14) |
 | (15) |
where 
and . Notice that the generalized variable G could be zero; however, this does not cause any problem in computing equation (14) because the argument of the normal distribution function when G=0 is either –
or depending on the sign of the numerator of the argument. In either case, the distribution function is defined and (–)=0 and ()=1.
We see from equation (15) that, for a given 
, the distributions of do not depend on any unknown parameters; so the distribution of 
is also free of unknown parameters. Even then, it is not easy to find the joint sampling distribution of all the independent random variables involved in ). However, one can use Monte Carlo simulation as given in the following algorithm to estimate the percentiles of .
Algorithm 1
For a given data set, compute 
For i = 1, m
Generate ZN(0,1),k–12 and N–k2
compute Gµ,G and Ge2 using equation (11)
compute Q=ln(OEL)-G-Ge2/2
if G=0 and Q < 0 then
set Ti=1
else if G=0 and Q > 0 then
set Ti = 0
else set 
[end do loop]
The 100(1–)th percentile of Ti's is a 1– upper limit for . Based on our experience, we recommend simulation consisting of at least 100 000 (i.e. the value of m in Algorithm 1) runs to get consistent results regardless of the initial seed used for random number generation. The above algorithm can be easily programmed in any programming language. A SAS program for computing one-sided limits for is posted at http://www.ucs.louisiana.edu/~kxk4695. Interested readers can download these files from this website.
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AN UPPER LIMIT FOR
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Notice that the probability that an individual exposure measurement exceeds the OEL is given by
= P(Xij > OEL). Since Yij= ln(Xij)N(µ,2+e2), we have
 | (16) |
, its (1 – A)th quantile is given by 
. Therefore, testing H0 : A versus Ha : < A is equivlent to testing
 | (17) |
The null hypothesis in equation (2) will be rejected at the level if a 1 – upper confidence limit for µ+z1–A2+e2 is less than ln(OEL).
As noted earlier, the above hypotheses can be tested by comparing an upper tolerance limit for N(
) with ln(OEL), where the content of the tolerance interval is to be 1 – A, and the confidence level is to be 1 – . Towards this, we note that Krishnamoorthy and Mathew (2004) provided such an approximate upper tolerance limit, and is given by
 | (18) |
) is the pth quantile of a noncentral t distribution with df = m and noncentrality parameter , and Fk–1,N–k, denotes the th quantile of an F distribution with degrees of freedoms k–1 and N–k.
A 1 – upper confidence bound for can be obtained by identifying the set of values of A for which the null hypothesis in equation (17) will be accepted. Specifically, the maximum value of A for which the null hypothesis in equation (17) is accepted, or equivalently U(A) satisfying U(A) ln(OEL), is a 1 – upper bound for . Notice that in equation (18) is a decreasing function of A while the other quantities are fixed, because z1–A is decreasing with increasing A. Furthermore, for a given m and p, it is known that tm,p() is an increasing function of . As a result, U(A) in equation (18) is a decreasing function of A, and the maximum value of A for which U(A)ln(OEL) is the solution of the equation U(A)=ln(OEL). Thus, a 1 – upper limit for is the solution (with respect to A) of the equation
 | (19) |
 | (20) |
 | (21) |
It should be noted that many other approaches are available for computing an upper tolerance limit in the one-way random model; see for example, Mee and Owen (1983), Bhaumik and Kulkarni (1996), and Liao et al. (2005). The first two references deal with the balanced data situation only, whereas Liao et al. (2005) also consider the unbalanced situation. We have chosen to use the results in Krishnamoorthy and Mathew (2004) since their approximate upper tolerance limit has an explicit expression, and hence is easy to compute. Overall, their approximation is quite accurate, as noted in Krishnamoorthy and Mathew (2004).
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EXAMPLES |
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We shall now illustrate the methods of the preceding sections using two sets of shift-long exposure data reported in Tables D2 and D3 of Lyles et al. (1997b). The data in Tables D2 and D3 represent nickel dust exposure measurements on a sample of maintenance mechanics from a smelter, and on a sample of maintenance mechanics from a mill, respectively. The data were collected from samples of workers from a nickel producing facility. For these data sets, we computed the values of
and sse as given in Tables 1 and 2. An upper limit can be computed by plugging these values in the generalized variables in equation (14) and then using Algorithm 1. To compute the 95% upper limits for , we used Algorithm 1 with m = 100 000.
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The results for the group of maintenance mechanics from a smelter are given in Table 1. Here, we see that the 95% upper limit for
is 0.0004. That is, <0.04% of mean exposures exceed the OEL.
To compute the 95% upper limit for , we need to compute the value of c in equation (20). Using the statistics in Table 1, and F22,11.05 = 0.4428, we computed
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 | (22) |
Using the online calculator (http://calculators.stat.ucla.edu/cdf/), we computed z1–A(4.9112)=15.1994 or z1–A = 3.0948. The last equation implies that 1 – A = (3.0948) or A = 0.0010. Thus, a 95% upper limit for is 0.001. This means that <0.1% of exposure measurements exceed the OEL. To compute a 99% upper limit, we have to use F22,11.01 = 0.3141. Using this value, we computed c as 4.8323. Then solving the equation t22.99(z1–A(4.8323)) = 20.6595, we get z1–A = 13.1995/4.8323 = 2.7315. This yields a 99% upper limit for as 0.0032.
Note: The online calculator mentioned above or the StatCalc 2.0 by Krishnamoorthy (2006) posted at http://www.ucs.louisiana.edu/~kxk4695 computes the missing value satisfying the equation P(tm() x) = q when the other three values are given. In our case, is the missing value. To solve equation (22), we use m = 22, x = 20.6595 and q = 0.95. Using these values, we get = z1–A(4.9112) = 15.1994.
In Table 2, we present the results for the exposure data collected from a group of maintenance mechanics from a mill. The 95% (99%) upper limit for is 0.0002 (0.0045). We also computed 95% (99%) upper limit for as 0.0028 (0.0084).
Remark: We would like to point out that the above data were used by Lyles et al. (1997b) and Krishnamoorthy and Guo (2005) for testing H0 : A versus H0 : < A. Lyles et al. used a large-sample method, and Krishnamoorthy and Guo used a generalized variable approach. If A = 0.1, then both approaches showed that < A, at significance level 0.05, for both sets of data. However, if A = 0.05, then to conclude whether or not < A, one has to carry out the test procedures again at significance level 0.05. On the other hand, in our present setup, the computed 95% upper limits for enable us to conclude < 0.05 for both problems at level 0.05. This is certainly an advantage of setting one-sided limits for .
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CONCLUDING REMARKS |
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The use of a model that includes a random effect is a convenient and practically useful approach to capture the heterogeneity among the exposed group. In this article, we concentrated on a situation where the one-way random effects model is appropriate for the log-transformed exposure data. Problems of interest for the purpose of exposure monitoring now reduce to inference problems concerning the unknown parameters of the model: the overall mean and the two variance components. As opposed to standard applications of the one-way random effects model, where the problems of interest deal with the individual parameters, exposure monitoring applications require inference on parametric functions that involve all the unknown parameters. Novel approaches are required to deal with such problems, especially since small sample procedures are desired. Here, we have investigated the generalized inference idea to come up with confidence intervals and tests for two parametric functions of interest: the probability that the mean exposure of an individual worker exceeds the OEL, and the probability that the exposure of a worker exceeds the OEL. The latter parametric function also comes up in connection with the computation of tolerance intervals. We have also illustrated our methodology by applying them for the analysis of actual exposure data.
For the problems mentioned in this article, large sample confidence bounds could be easily obtained using standard methods; see Lyles et al. (1997a) for details. However, the numerical results in Krishnamoorthy and Mathew (2002) show that the generalized variable approach has a definite edge in terms of maintaining the type I error probability of the tests, and coverage probability of the confidence intervals. As should be clear from the computational algorithm mentioned in this article, the generalized variable approach is quite easy to implement. Furthermore, the fact that they are also applicable to small samples make them attractive options for analysing exposure data.
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ACKNOWLEDGEMENTS |
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This research was supported by grant R01-0H03628-01A1 from the National Institute of Occupational Safety and Health (NIOSH). The authors are thankful to Ms. Yanping Xia for providing computational help and SAS code; they are also grateful to two reviewers and assistant editor David Bartley for providing valuable comments and suggestions.
Received August 2, 2006; in final form February 19, 2007
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