PII: S0003-4878(99)00022-8
Ann. occup. Hyg., Vol. 43, No. 3, pp. 201±213, 1999
# 1999 British Occupational Hygiene Society
Published by Elsevier Science Ltd. All rights reserved
Printed in Great Britain
0003±4878/99/$20.00 + 0.00
Computational Method for Ranking Task-speci®c
Exposures Using Multi-task Time-weighted Average
Samples
MARGARET L. PHILLIPS* and NURTAN A. ESMEN
Department of Occupational and Environmental Health, College of Public Health, University of
Oklahoma Health Sciences Center, 801 N.E. 13th Street, Oklahoma City, OK 73104, USA
A method is presented for ranking task-speci®c exposures using time-weighted average samples
collected during the performance of multiple tasks. The task ranking can be used for purposes
such as prioritizing further assessment or control. No a priori estimates of the individual task
concentration distributions are required. Sample concentrations and task-speci®c concentrations
are assumed to be log-normally distributed, and each known sample concentration is modeled
as the geometric time-weighted average of the unknown task concentrations. Since the task
durations are usually not known, the task time-weights are estimated as a crude fraction of
sampling period. Log transformed sample concentrations are aggregated based on the nonoccurrence of each task during some samples, resulting in a set of linear equations which are
solved to yield estimates of the log-transformed task median concentrations. The performance
of the method was tested under a variety of conditions using simulated sample data. The
method was found to yield remarkably reliable ranking of task median concentrations,
especially for the high exposure tasks, provided that the number of samples was adequate and
the task concentration distributions were not highly overlapped. The performance of the method
can be easily modeled using simulated data over the range of plausible task concentration
distributions for any number of samples and any job scenario. Even under the conservative
assumption that some task concentration distributions are highly overlapped, the assigned
ranking can be usefully interpreted in the light of the modeling to determine whether a task is
relatively high exposure or low exposure. # 1999 British Occupational Hygiene Society.
Published by Elsevier Science Ltd. All rights reserved.
Keywords: task speci®c exposures; sampling strategy; ranking exposures; prioritising control
INTRODUCTION
and, in some cases, the prudent reduction of exposures that are already below current OELs.
Indeed, it has been suggested that occupational hygiene standards may increasingly shift from emphasis on compliance-driven exposure measurement to
emphasis on e€ective controls (Vincent, 1998). If a
job consists of a number of di€erent tasks, possibly
performed at di€erent workstations or with di€erent
materials and equipment, it would be sensible to
direct control e€orts toward the tasks that contribute most signi®cantly to the exposure in a given job
category. In this case, task-speci®c exposure estimates might be needed for the purpose of prioritizing activities for control. In a di€erent context, taskspeci®c exposure estimates could also be useful for
predicting how reapportionment of job tasks arising
from process redesign or stang changes would
a€ect workers' exposure.
Many occupational hygiene sampling strategies
revolve around the collection of full-shift or shorter
duration samples for the purpose of assessing exposures in job categories or homogeneous exposure
groups (HEGs) relative to long term or short term
occupational exposure limits (OELS) (Leidel and
Busch, 1985; Rappaport et al., 1988; Esmen, 1992).
Such job-speci®c exposure assessments are useful for
making decisions as to whether the exposure is
acceptable or not acceptable. However, the mission
of the occupational hygienist goes beyond the mere
determination of compliance or non-compliance to
address the abatement of unacceptable exposures
Received 1 September 1998; in ®nal form 15 January 1999.
*Author to whom correspondence should be addressed.
Tel.: ++1-405-2712070; Fax: ++1-405-2711971.
201
202
M. L. Phillips and N. A. Esmin
Single-task sampling would obviously be the most
straightforward approach to estimating task-speci®c
exposure. In fact, Nicas and Spear (1993a) have
shown that a sampling strategy based on simple random sampling or strati®ed random sampling of
short term (15 minute) single-task work periods can
be an ecient measurement strategy for the estimation of the means and variances of both shortterm exposure and full-shift exposure. On the other
hand, in many situations the collection of singletask samples could be quite labor intensive for the
occupational hygienist, as well as potentially disruptive to the work activity. It may be more practical
to collect pre-planned two-hour, four-hour or fullshift samples. Moreover, much historical sampling
data is likely to consist of full-shift monitoring. Use
of real-time personal monitors with datalogging
capability would be the ideal solution, but this technology is currently limited to a narrow range of
physical and chemical agents. Given the historical
and continuing prevalence of full-shift time-weighted
average (TWA) exposure sampling in occupational
hygiene practice, it would therefore be useful for the
occupational hygienist to be able to extract estimates of task-speci®c exposures from traditional
full-shift or partial-shift time-weighted average
sampling data. In several studies involving empirical
modeling of exposure (Kromhout et al., 1994;
Preller et al., 1995; Burstyn et al., 1997), taskspeci®c exposures have been estimated from TWA
samples using the results of multiple linear regression modeling in which individual tasks emerged
as signi®cant factors. However, as noted by
Kromhout et al. (1994), for a meaningful regression
analysis a large number of samples may be necessary
to ensure adequate variation in the values of the
task durations. Additionally, since the task durations are constrained to add up to the length of the
sample duration, it is unclear how the resulting collinearities in the durations of the various tasks might
a€ect the validity of the multiple regression model.
This paper presents an alternative method for estimating and ranking task-speci®c median concentrations of hazardous agents using a modest number
of time-weighted average samples collected during
the performance of multiple tasks.
THEORY
Model of time-weighted average concentrations
Consider a job consisting of I di€erent tasks a1,
a2,..., aI, each task ai having a characteristic distribution of contaminant concentrations ci. If a worker
in this job performs more than one task during
sample period j, the time-weighted average concentration over that sampling period, c‡
j , can be
expressed as a time-weighted average of the individual task concentrations:
c‡
j ˆ
I
X
cij yij ,
…1†
iˆ1
where cij is the average concentration associated
with task ai during sampling period j and yij is the
fraction of the sampling period j spent doing task ai.
The time-weights yij will be considered random variables which are constrained to meet the following
conditions:
I
X
yij ˆ 1; 0Ryij R1:
…2†
iˆ1
Since the log normal central tendency of exposure
measurements is well established (Esmen and
Hammad, 1977), we will assume that the concentrations associated with each task are log-normally
distributed:
log cij ˆ log cmed
‡ uij log sg,i ,
i
…3†
where cmed
is the median concentration associated
i
with task ai, uij is a unit normal variate and sg,i is
the geometric standard deviation (GSD) of the concentration distribution for task ai. We will further
assume that the uij for di€erent tasks are mutually
independent and that the time series of instantaneous or short-term average concentration values
for each task is stationary, i.e. that the means and
variances of the underlying instantaneous task concentration distributions are not changing over the
time period spanned by the available sample data.
This assumption allows us to consider each task
concentration distribution to have a parameter of
central tendency which is more or less stable and so
can be meaningfully ranked relative to the others.
For simplicity during the development of the
model, it will be assumed that the median log cmed
i
and the variance log2 sg,i of the log-transformed
task concentration is independent of the averaging
time, i.e. the task duration. This would be the case if
the concentrations averaged over successive short
intervals during performance of a task were completely autocorrelated over the time scale of the sample
(Nicas and Spear, 1993b). If a lower degree of autocorrelation were assumed, with the autocorrelation
being a decreasing function of the lag between short
intervals, then the variance of task concentration
and the variance of log-transformed task concentration both would decrease with increasing task
duration (Spear et al., 1986). After developing the
model for the total autocorrelation case, we will
examine the e€ect on the model of the opposite
extreme, i.e. zero autocorrelation.
The sample concentrations c‡
j may tend to be
right-skewed like the underlying task distributions.
If so, we can stabilize the variability of the sample
results by performing a log transformation of c‡
j .
We then proceed to model the log-transformed
sample concentrations, log c‡
j , as a time-weighted
Computational method for ranking task-speci®c exposures
average of the log-transformed task concentrations,
log ci:
log c‡
j ˆ
I
X
iˆ1
log cmed
yij ‡
i
I
X
uij log sg,i yij :
…4†
iˆ1
This is equivalent to estimating that the sample
concentration is the time-weighted geometric mean
of the task concentrations. (If the c‡
j are not skewed,
the log transformations can be omitted from the
model).
This model preserves the appropriate rank order
of the relative contributions of the di€erent tasks to
the overall measured concentration; that is, if two
tasks a1 and a2 with median concentrations cmed
>
1
cmed
were equally time-weighted, then task a1 would
2
contribute more to the sample concentration c‡
j .
Analysis of time-weighted average sample data sets
Consider a data set for a particular job classi®cation, consisting of J time-weighted average
samples collected during J multi-task work periods.
Using the model de®ned in Eq. (4), the overall average concentration for this job is estimated as:
hlog c‡ i ˆ
J X
I
1X
log cmed
yij
i
J jˆ1 iˆ1
J X
I
1X
‡
uij log sg,i yij :
J jˆ1 iˆ1
…5†
The second term on the right-hand side of Eq. (5)
contains all the inherent variability of the task concentrations and tends toward zero as the number of
samples J becomes large. This type of term will be
referred to for brevity as an `error' term.
For an intuitive understanding of the next step in
the analysis, consider two possible subsets of the
sample data: Subset `not-1' consists of samples
during which all tasks except task a1 were performed, and Subset `not-2' consists of samples
during which all tasks except task a2 were performed. If the average concentration of Subset `not2' samples were greater than the average concentration of Subset `not-1' samples, we might infer
med
that cmed
> cmed
and cmed
are the median
1
2 , where c1
2
concentrations associated with tasks a1 and a2, respectively. However, this inference would be based
on the assumption that for all other tasks, the task
time-weights averaged over the `not-1' samples are
the same as the task time-weights averaged over the
`not-2' samples. Unless this assumption can be justi®ed, a more sophisticated method such as matrix
inversion must be used to control for the variability
in task time weighting between samples.
For each task, we construct an average over all
the samples in which that task did not occur:
203
P1 ˆ
1 X
log c‡
j for all j such that y1j ˆ 0:
J1 j
P2 ˆ
1 X
log c‡
j for all j such that y2j ˆ 0:
J2 j
..
.
PI ˆ
1 X
log c‡
j for all j such that yIj ˆ 0:
JI j
…6†
In other words, in the sum P1, we have screened
the data to use only those samples in which task 1
does not occur. If more than one task was omitted
during a particular sample, then that sample would
be used in the P-sum for each of the omitted tasks.
Using Eq. (5), each P-sum can then be expressed as
a time-weighted average over all other tasks:
"
#
X
X
1
Pi 1
log cmed
y
…7†
kj ‡ ei ,
k
Ji j such that y ˆ0
k6ˆi
ij
where ei is the `error' term which, as noted above,
tends toward zero as Ji becomes large. The expression in brackets is the time-weighting for the
task ak, averaged over the samples in which task ai
did not occur. For brevity, this averaged timeweighting will be referred to henceforth as the Pscreened time-weight or Pyik.
If the sample concentrations c‡
j and the task durations yij are known for a suitable set of samples,
then the I simultaneous equations generated using
Eq. (7) can be solved to obtain estimates of the I
unknown task median concentrations cmed
. Due to
i
the log transforms, the model does not ensure
unbiased estimation of the magnitudes of the task
median concentrations. (It should be noted that task
exposure estimates derived from multiple linear regression of log-transformed TWA exposures on task
durations might likewise be subject to bias).
However, given a sucient number of samples for
each P-sum, the rank ordering of the task median
concentrations should be correct. The procedure is
therefore most appropriately regarded as a method
for ranking task concentrations and will be evaluated on that basis.
Dependence of task distribution parameters on averaging time
As noted above, for stationary processes if the
autocorrelation between concentrations at successive
short time increments is low, then the variance of
time-weighted average concentrations will decrease
with increasing averaging time. Then a di€erent task
concentration distribution exists for each averaging
time. Due to the assumption of stationarity, the
arithmetic mean of the concentration distribution
for a given task is a constant irrespective of aver-
204
M. L. Phillips and N. A. Esmin
aging time. If the autocorrelation function is zero
and all distributions are assumed to be log-normal,
then it can be shown using the relationships between
the arithmetic mean, the arithmetic variance, the
geometric mean (i.e. the median of the log-normal
distribution) and the GSD (Leidel et al., 1977) that
the GSD of the TWA task concentration distribution will decrease with increasing averaging time.
The median of the TWA task concentration distribution will increase with increasing averaging time,
asymptotically approaching the arithmetic mean of
the distribution for long averaging times. Thus longtime averages would be less skewed than short-time
averages.
Two potential diculties with our model might be
anticipated if we allow the cmed
and the sg,i to be
i
dependent on the averaging time:
1. the method of estimating the cmed
by solving the
i
simultaneous equations de®ned by Eq. (7) would
appear to be invalid, because the cmed
would not
i
be independent of the magnitudes of their coecients, the Pyik; and
2. the time-dependence of the GSD could adversely
a€ect the convergence to zero of the ei.
Regarding the second issue, it should be noted
that the task GSD, which is largest at short averaging times, is weighted in the `error' term by the
task time-weight, which is proportional to the averaging time. This weighting should restrain the size
of the `error' term for short task durations.
Likewise, the nature of the time-dependence of the
cmed
should serve to mitigate the ®rst issue to some
i
extent. For relatively long time averaging, the cmed
i
are bounded from above by the means of their respective task concentration distribution. Thus the
relative contributions of the di€erent task medians
to the sample concentrations would tend to follow
the ordering of the task mean concentrations. For
short time averaging, a cmed
may be much smaller
i
than the distribution mean, but its time-weighting yij
would also be small, reducing the importance of
these low-end values of cmed
in the sample conceni
trations. If the variances are suciently similar
among the tasks, the rank ordering of the central
tendencies of the short-time task concentration distributions based on the task medians will be much
the same as the rank ordering based on the task
means. The method would then still provide meaningful ranking of the task concentration distributions. The only constraint on application of the
method would be that the data used must have
roughly equal sample durations.
The conclusion of these theoretical arguments in
support of the method under zero autocorrelation
conditions was con®rmed empirically using simulated data, as described below in Section 4,
``Results''.
COMPUTATIONAL PROCEDURE
The relationship between the P-sums, the Pscreened time-weights and the task median concentrations can be expressed in matrix form:
0 1 0
1
P
P1
y12 P y13 P y1I
0
B C B P
C
P
B P2 C B y21 0
y23 P y2I C
B C B
C
B P C B Py
P
y32 0
P y3I C
B 3 C ˆB
C
31
B C B
C
B. C B.
C
.
.
.
.
. . ..
..
..
B .. C B ..
C
@ A @
A
P
P
P
PI
yI1
yI2
yI3 0
1 0 1
0
log cmed
e1
1
C B C
B
B
C
C
B log cmed
2
C B e2 C
B
B log cmed C B e C
C‡B 3C
B
3
…8†
C B C
B
C B. C
B.
C B .. C
B ..
A @ A
@
eI
log cmed
I
The ei are assumed to be negligible. Given the
sample concentrations and task time-weights for
each sample, the P-sum vector and the Py matrix are
easily calculated.
Typically, the task time-weights yij for each
sample are not known. In the absence of detailed information on task duration for each sample, more
or less crude estimates of these time-weights can be
used instead in the construction of the P-screened
time-weight matrix. In the crudest method of estimation, if it is known that m di€erent tasks were
performed for a non-trivial fraction (e.g., >5%) of
the sample period j, we assume an equal division of
the sample period among those m tasks; therefore
the estimated time-weights y^ ij for those tasks are set
equal to 1/m. All other task time-weights for the
sample are set equal to zero. If it is further known
that a given task took a large, moderate, or small
fraction of the sampling period, the estimated timeweights can be adjusted accordingly for that task.
To guarantee that the matrix equation will have a
solution, the sample data set must meet the following conditions:
1. for each task, there is at least one sampling
period in which the task did not occur;
2. there are no tasks that always occur together;
and
3. the P y^ matrix has a nonzero determinant.
It is permissible for tasks to be mutually exclusive.
If conditions 1±3 are met, a suitable P y^ matrix and
P-sum vector can be constructed and Eq. (8) can be
solved by matrix inversion to yield the vector of logtransformed task median concentrations. The construction of the P y^ matrix and P-sum vector from a
data set and the solution of the simultaneous
equations can be readily performed using a commer-
Computational method for ranking task-speci®c exposures
205
Table 1. Example of procedure using crude estimate of task durations
Sample Data
c‡
j
Sample No. (j)
1
2
3
4
5
6
7
8
9
10
11
12
y^ matrix (crude estimate)
P-screen matrix
Major tasks performed log c‡
j
6.9
29.0
3.3
3.8
17.8
12.7
6.2
9.7
4.3
24.7
3.5
13.8
a2, a3, a4
a2, a3, a4
a1, a2, a3
a1, a2, a3
a1, a3, a4
a1, a3, a4
a1, a2, a4
a1, a2, a4
a1, a2
a1, a3
a1, a2, a4
a2, a4
1.93
3.37
1.20
1.32
2.88
2.55
1.82
2.27
1.47
3.21
1.26
2.62
a1
a2
a3
a4
1
0
0
0
1
0
0
0
0
0
0
1
0
0
0
1
0
1
0
0
0
1
0
0
0
0
1
0
0
0
1
0
0
0
1
1
0
1
0
1
0
0
1
0
1
0
1
0
J1=3 J2=3 J3=5 J4=4
a1
a2
a3
a4
0
0
1/3
1/3
1/3
1/3
1/3
1/3
1/2
1/2
1/3
0
1/3
1/3
1/3
1/3
0
0
1/3
1/3
1/2
0
1/3
1/2
1/3
1/3
1/3
1/3
1/3
1/3
0
0
0
1/2
0
0
1/3
1/3
0
0
1/3
1/3
1/3
1/3
0
0
1/3
1/2
P-sum vector
Raw
Results:
P^
Normalized
7.9
8.6
9.4
7.2
y matrix
2.64
2.88
1.89
1.80
0
0.389
0.300
0.417
0.389
0
0.400
0.292
0.222
0.389
0
0.292
cially available software package such as Microsoft
Excel.
An example of the procedure is provided in
Tables 1 and 2. Simulated sample data were created
for a job consisting of four tasks: a1, a2, a3 and a4,
with task median concentrations 2, 4, 8 and 16, respectively, in arbitrary units. The vector of sample
concentrations c‡
j for 12 samples is presented in
Table 1, together with a notation of the tasks performed during that sample. Tasks of duration less
than 5% of the sample period were omitted. The
sample concentrations were log transformed. The
Inverted
0.389
0.222
0.300
0
P^
Task log cmed
Rank
i
y matrix
ÿ1.97 1.01 1.81 0.15
1.12 ÿ2.03 0.06 1.86
1.70 0.59 ÿ2.64 1.35
0.48 1.70 1.45 ÿ2.63
a1
a2
a3
a4
1.40
0.55
3.62
4.17
2
1
3
4
selective summations in Eqs (6) and (7) are conveniently performed by matrix multiplication using
the P-screen matrix, which is constructed simply by
entering a zero (0) for each task that occurred
during the sample and a one (1) for each task that
did not occur (or occurred only brie¯y) during the
sampling period. The transpose of the P-screen
matrix is multiplied by the log c‡
j vector to yield the
`raw' P-sum vector. The P-sum vector is normalized
by dividing each element Pi by the number of
samples Ji that contributed to that sum.
Table 2. Time-weights and results of procedure using re®ned estimates of task durations
y^ matrix (re®ned estimate)
Sample No. (j)
1
2
3
4
5
6
7
8
9
10
11
12
a1
0
0
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0
a2
0.4
0.4
0.4
0.4
0
0
0.6
0.6
0.8
0
0.6
0.8
a3
0.4
0.4
0.4
0.4
0.6
0.6
0
0
0
0.8
0
0
a4
0.2
0.2
0
0
0.2
0.2
0.2
0.2
0
0
0.2
0.2
Results:
P^
y matrix (re®ned)
0
0.200
0.160
0.200
0.533
0
0.680
0.400
0.267
0.667
0
0.400
0.200
0.133
0.160
0
Task
log cmed
i
Rank
a1
a2
a3
a4
0.58
1.25
2.96
5.92
1
2
3
4
206
M. L. Phillips and N. A. Esmin
The estimated time-weight matrix y^ in Table 1
was created using the crude estimate that the sample
period was equally divided among the two or three
tasks performed. The P-screened time-weight matrix
P^
y was then calculated by ®rst multiplying the
transpose of the P-screen matrix by the y^ matrix
and then dividing each row i of the resulting 4 4
matrix product by the corresponding scalar Ji. The
P^
y matrix was then inverted and the inverted matrix
was multiplied by the P-sum vector to yield the vector of estimated log transformed task median concentrations log cmed
. The ranks of the estimated task
i
median concentrations were found to be correct for
tasks a3 and a4, but the ranking of tasks a1 and a2
was reversed. The magnitudes of the estimated task
median concentrations cmed
were 4, 2, 37 and 65 for
i
tasks a1, a2, a3 and a4, respectively; thus the estimates were found to be within an order of magnitude of the true medians.
In the job exposure scenario used to generate the
data in this example, the actual durations of tasks a1
and a4 were consistently shorter than the actual durations of tasks a2 and a3 during any sampling
period. If this fact were known, then we could re®ne
the task time-weight estimates by assigning lower
time-weights to tasks a1 and a4. In the re®ned y^
matrix shown in Table 2, if tasks a1 and a4 occurred,
they were assigned time-weights of 0.2 and the
remaining fraction of sample time was apportioned
equally between the other tasks that occurred during
the sample. The P-screen matrix and P-sum vector
are not a€ected by this re®nement. The re®ned P y
matrix and the results obtained by using this matrix
in the matrix inversion solution are presented in
Table 2. All four tasks were ranked correctly when
the re®ned task time weight matrix y^ was used in
the procedure. However, the magnitude of the task
median concentration for task a4 was overestimated
by more than one order of magnitude.
RESULTS USING SIMULATED DATA
The performance of the method was evaluated
under a variety of underlying conditions using synthetically generated sample data sets. Conditions
varied included the spread and overlap of individual
task concentration distributions, the actual task
time-weights in a sample set, worker propensity to
elevated or reduced concentration relative to the median, number of samples available and time-dependence of distribution parameters.
Unless otherwise noted below, to simulate a lack
of knowledge about actual task durations, crude
estimates of task time-weights were used in the construction of the P-screened time-weight matrices in
all of the data analyses. Each task ai accounting for
more than 5% of the sampling period j was assigned
an equal estimated time-weight y^ ij , while tasks
which were not performed or which accounted for
less than 5% of the sampling period were assigned a
zero estimated time weight.
It was not practical to perform a large number of
simulations using commercially available software.
Therefore nearly all calculations, including the generation of the simulated data sets and the solution
of the simultaneous equations, were performed
using original programs written by the authors in
Microsoft QuickBASIC.
Generating the simulated data
In general, the I task time-weights yij for each
sample j were created by assigning to one randomly
chosen task a time-weight of 0, assigning to each of
Iÿ2 tasks a di€erent time-weight which was randomly selected from the discrete uniform distribution {0, 1/96, 2/96,..., 95/96}, and assigning to the
remaining task a time-weight such that the yij
summed to unity. For a full-shift, 480 minute
sample, the task durations would thus be multiples
of 5 minutes and would sum to a total sample duration of 480 minutes. A ®nished set of task timeweights for J samples would be handled as a I J
array.
When the task time-weights were generated in this
way, the expectation of the time-weight was Iÿ1 for
each task. This would correspond to a work pattern
in which each task would take an equal share of the
work time over the long run, i.e. over many samples.
Unless otherwise noted, this was the pattern used in
the simulations. Alternative work pattern scenarios,
in which the tasks had unequal long-run time
weights, were created for some simulations, as
described below in Section 4.5, ``Number of
Samples''.
The task concentrations cij during each sample j
were drawn randomly from a family of log-normal
distributions having medians cmed
and GSDs sg,i.
i
Unless otherwise noted, the parameters cmed
and sg,i
i
were not dependent on the task time-weights, representing the case of total autocorrelation of the instantaneous concentrations for a given task during a
given sample. Concentrations during di€erent
samples or di€erent tasks were uncorrelated. Where
noted below, in some simulations the sg,i used were
allowed to di€er from a central GSD value by as
much as 20.5, 21, or 22.
When comparisons of model performance were
made between the total autocorrelation case and the
zero autocorrelation case, 8-hour sampling was
assumed. The job scenario involved six tasks with
equal long-run time-weights. The underlying shortterm (5 minute) task distribution parameters in the
zero autocorrelation simulations were contrived so
that the nominal values of cmed
and sg,i for tasks
i
with 80-minute averaging times (480 minutes 6 6
tasks) were equal to the medians and nominal GSDs
of the task concentration distributions used in the
total autocorrelation simulations.
Computational method for ranking task-speci®c exposures
The simulated sample data c‡
j were calculated as
time-weighted averages over the tasks, in accordance
with Eq. (1). The simulated sample data thus existed
as a J-dimensional vector. The c‡
j data tended to be
right skewed, justifying the use of the log transform
in the data analysis, in accordance with the model in
Eq. (4).
The analysis of each set of simulated sample data
was, in essence, a single Monte Carlo trial of the
method. Multiple trials were created by generating
at least 100 data sets for each set of time-weights,
using independent random selection of cij values for
each set. Because the performance of the method
was found to depend idiosyncratically on the underlying set of task time-weights, multiple time-weight
sets are also used in some collections of trials.
E€ect of task concentration distributions
A number of simulations were run to determine
how the ability of the method to rank tasks by their
median concentrations is a€ected by the spread
within and between task distributions.
Two families of task concentration distributions
were used to generate the synthetic data; each family
consisted of six task distributions with equallyspaced log-transformed task median concentrations
log cmed
. In the more closely spaced family of distrii
butions, hereafter referred to as Type 1, the medians
of adjacent task distributions di€ered by a factor of
2. The task median concentrations for tasks a1
through a6 were 1, 2, 4, 8, 16 and 32, in arbitrary
units. The less closely spaced family of distributions,
hereafter referred to as Type 2, had task median
concentrations of 0.5, 1.4, 4, 11.3, 32 and 90.5 in
arbitrary units, based on a factor of 21.5 ratio
207
between the medians of adjacent distributions. To
assess the e€ect of the absolute magnitude of the
task median concentrations, two additional families
of distributions with Type 2 spacing were tested:
Type 2', with task median concentrations of 0.18,
0.5, 1.4, 4, 11.3 and 32, and Type 20, with task median concentrations 1, 2.8, 8, 22.6, 64 and 181.
Simulations are run using GSD values ranging from
1.5 to 4.5. Within a given simulation run, the GSD
was the same for all task distributions. The task
ranking method was evaluated in trials of 500 di€erent data sets for each combination of distribution
family and GSD. Each data set consisted of 100 synthetic samples. The same task time-weight matrix
was used in all trials. The results of these trials are
presented in Fig. 1. Not surprisingly, the fraction of
trials in which all tasks were correctly ranked
decreases as the GSD of the underlying task concentration distributions increases. Correct ranking
occurred less frequently for the Type 1 task distributions than for Type 2 spaced distributions; this is
expected because the Type 1 distributions would be
more overlapped for a given GSD. However, when
the performance of the method is evaluated as a
function of the overlap between adjacent task distributions, as shown in Fig. 2, the Type 1 distributions
yielded better results than the Type 2 spaced distributions. This suggests that the ability of the method
to rank tasks correctly is in¯uenced by both the
degree of overlap between distributions and the
within-distribution spread as measured by sg,i. No
clear trend was apparent in the di€erences in performance among Type 2, Type 2' and Type 20 distributions.
Fig. 1. E€ect of task concentration variability (as the geometric standard deviation of the task concentration distribution) on the performance of the method in ranking all six tasks correctly. Each data point is based on 500 trials.
208
M. L. Phillips and N. A. Esmin
Fig. 2. E€ect of overlap between adjacent task concentration distributions on the performance of the method in ranking
all six tasks correctly. Each data point is based on 500 trials.
Table 3. Fraction of tasks assigned to a given ranking in 500 trials using 100-sample data sets for best case family of
task concentration distributions*
Assigned Rank
True Rank
1
2
3
4
5
6
1
2
3
4
5
6
.856
.144
0
0
0
0
.144
.856
0
0
0
0
0
0
.974
.026
0
0
0
0
.026
.974
0
0
0
0
0
0
1
0
0
0
0
0
0
1
* underlying Type 2 family of distributions had an overlap of 20.9% between adjacent task distributions and task geometric standard deviation of 1.5.
Even for relatively low degrees of overlap between
distributions (020±40% overlap), the method
showed only a moderate reliability in ranking all
tasks correctly (060±85% correct). However, the
ranking of the higher task median concentrations
was considerably more reliable than the ranking of
the lower task median concentrations. Table 3 summarizes the assigned task ranks versus true task
ranks for the best case investigated: the Type 2 dis-
tribution with sg,i=1.5 (20.9% overlap between
adjacent distributions). A comparison of true and
assigned task ranks for the worst case investigatedÐthe Type 1 distribution with sg,i=4.5
(80.5% overlap between adjacent distributions)Ðis
presented in Table 4. The worst-case Type 1 and
best-case Type 2 distributions used are illustrated in
Fig. 3. If the goal of the analysis were to identify the
two most highly exposed tasks, not necessarily in
Table 4. Fraction of tasks assigned to a given ranking in 500 trials using 100-sample data sets for worst case family of
task concentration distributions*
Assigned Rank
True Rank
1
2
3
4
5
6
1
2
3
4
5
.598
.318
.066
.016
.002
0
.286
.392
.236
.080
.006
0
.098
.210
.452
.214
.024
.002
.016
.070
.220
.502
.184
.008
.002
.010
.026
.178
.620
.164
6
0
0
0
.010
.164
.826
* Underlying Type 1 family of distributions with an overlap of 80.5% between adjacent task distributions and task geometric standard deviation of 4.5.
Computational method for ranking task-speci®c exposures
209
Fig. 3. (a) The Type 1 `worst case' family of distributions has a ratio of 2 between adjacent task medians and a withintask GSD of 4.5, resulting in an 80.5% overlap between adjacent distributions. (b) The Type 2 `best case' family of distributions has a ratio of 21.5 between adjacent task medians and a within-task geometric standard deviation of 1.5, resulting in a 20.9% overlap between adjacent distributions.
order, the method succeeded in more than 78% of
trials in the worst case and in 100% of trials in the
best case. If the occupational hygienist needed only
to rule out certain tasks as being the source of highest concentration, then even in the worst case it is
evident that the tasks assigned the lowest three
rankings by the method had less than a 0.2% probability of actually being the highest task with the
highest median concentration.
Task time-weight distribution e€ects
Apart from the e€ects of task concentration variability, the task rankings obtained by the use of the
computational method were found to be subject to
biases characteristic of the speci®c matrix of task
time-weights used to generate the synthetic sample
data. The con®guration of task time-weights in a
sample data set can give rise to two di€erent sources
of potential bias:
1. the use of estimated time-weights instead of exact
time-weights in the creation of the P-screened
time-weight matrix and
2. the modeling of the sample concentrations as
geometric time-weighted averages, which do not
necessarily follow the same rank order as the corresponding arithmetic time-weighted average
sample concentrations.
The ®rst source of bias can be mitigated by using
more re®ned time-weight estimates in the procedure,
provided that some information were available
about the relative duration of the tasks occurring
during each sample. In large (100 sample) synthetic
data sets without consistently high or low timeweighting of any one task, the second source of bias
was found to a€ect only the task ranking in the
lower tier of task median concentrations.
Worker e€ect
The e€ect of inter-worker di€erences in task median concentrations on the overall performance of
the method was evaluated under the assumption of
a multiplicative model which is consistent with the
log-normality of multiple worker concentration dis-
210
M. L. Phillips and N. A. Esmin
tributions. In this simulation, a pool of ®ve `virtual'
workers were sampled randomly; each worker had a
tendency to be exposed in all tasks at a consistently
higher or lower level than the other workers.
Accordingly, the characteristic task median concentrations were decreased or increased relative to their
null values by a constant factor for each worker
(nÿ1, nÿ0.5, 1, n0.5 and n for the ®ve workers, respectively), where n is a measure of the `worker e€ect'.
The overlap between task concentration distributions for each individual worker was not a€ected
by the shift in task median concentrations, but
obviously the overall multiple-worker task distributions would become more overlapped with
increasing worker e€ect n.
The performance of the method under these conditions for a job consisting of six tasks was evaluated in 3000 trials for values of n ranging from 1
(for no worker e€ect) to 8; the trials consisted of
100 di€erent data sets for each of 30 di€erent task
time-weight matrices. Each data set consisted of 100
simulated samples, with random selection of one out
of the ®ve workers for each sample. The results of
these trials are presented in Fig. 4 for Type 1 distributions (ratio between adjacent task medians=2,
sg,i in range 3.5 to 4.5) and Type 2 distributions
(ratio between adjacent task medians=21.5, sg,i in
range 1.5 to 2.5). The performance of the method,
expressed as the probability of correctly ranking the
two most highly exposed tasks, declined moderately
with increasing size of the worker e€ect n. Note that
the size of the worker e€ect used in the modeling, n
r2, was comparable to the size of the task e€ect as
measured by the ratio between task median concentrations. The performance of the method is reduced
only minimally when the worker e€ect is smaller
than the task e€ect.
Number of samples
As noted in Sections 2 and 3, ``Theory'' and
``Computational Procedure'', above, the method
depends on having a sucient number of samples Ji
contributing to each P-sum so that the ei in Eq. (7)
is negligible. If this condition is not met, the solution obtained through the method, i.e. the vector
of estimated log cmed
values, may contain errors
i
large enough to change the ranking of the task median concentrations.
The performance of the method when applied to
small data sets was evaluated for several job scenarios. In Scenario 1, the job was assumed to consist
of four tasks occurring randomly in a sample period
and with equal long-run time weighting. Scenario 2
was designed to represent a job with more consistent
work patterns: task a1 (`set up') could take between
15 and 90 minutes, task a4 (`clean-up') could take
20±120 minutes and the rest of the shift was devoted
to task a2 and/or task a3 (two types of `production').
Since samples in which all four tasks occur cannot
be used in the computation, simulated samples were
generated with task time-weights corresponding to
full shift samples in which only one type of `production' took place (task a2 or task a3) or to partial
shift samples in which most or all of the `set-up'
and/or `clean-up' work was not sampled. (The
example worked in Section 3, ``Computational
Procedure'', was based on Scenario 2). Scenario 3
was similar to Scenario 2 except that task a2 represented `clean-up' with duration 20±120 minutes
and task a4 represented a `production' activity. In
all scenarios, the concentration distributions for
tasks a1, a2, a3 and a4 were assumed to have medians of 2, 4, 8 and 16 respectively. All task distributions had GSDs of 2, resulting in a 60% overlap
between adjacent task distributions.
The performance of the method for data sets of
12 samples was evaluated using 2600 simulated data
sets for each scenario, based on 26 di€erent sets of
time-weights. Crude time-weight estimates were used
in the method for Scenario 1, while re®ned timeweight estimates were used in the application of the
method for Scenarios 2 and 3. Comparisons of
assigned rank to true rank for the three scenarios
are presented in Tables 5±7. Not surprisingly, the
Fig. 4. E€ect of between-worker di€erences in task median concentrations on the performance of the method in correctly
ranking the two most highly exposed tasks in a job consisting of six tasks. Each data point represents 3000 trials.
Computational method for ranking task-speci®c exposures
211
Table 5. Fraction of tasks assigned to a given ranking in 2600 trials using data sets of 12 samples for Scenario 1*
Assigned Rank
Actual Rank
1
2
3
4
1
2
3
4
0.554
0.296
0.138
0.012
0.282
0.399
0.287
0.032
0.121
0.248
0.423
0.208
0.043
0.057
0.152
0.748
* Four tasks occur randomly in shift with equal long-term probability.
Table 6. Fraction of tasks assigned to a given ranking in 2600 trials using data sets of 12 samples for Scenario 2*
Assigned Rank
Actual Rank
1
2
3
4
1
2
3
4
0.579
0.336
0.045
0.041
0.169
0.486
0.255
0.091
0.180
0.162
0.543
0.116
0.073
0.017
0.158
0.752
* Highest and lowest exposure tasks have relatively shorter duration than intermediate exposure tasks.
Table 7. Fraction of tasks assigned to a given ranking in 2600 trials using data sets of 12 samples for Scenario 3*
Assigned Rank
Actual Rank
1
2
3
4
1
2
3
4
0.391
0.375
0.207
0.026
0.243
0.240
0.380
0.138
0.189
0.180
0.308
0.323
0.203
0.220
0.097
0.480
* Two lowest exposure tasks have relatively shorter duration than higher exposure tasks.
method did not reliably rank any of the task median
concentrations when the number of samples is low.
However, at least one potentially useful inference
may still be drawn from the rankings obtained: if a
task was assigned the lowest rank by the method,
there was less than a 5% probability that the task
actually had the highest median concentration. Thus
in the scenarios tested, the method could be used
with 95% con®dence to rule out the lowest-ranked
task as being in fact the task with the highest
median concentration.
E€ect of distribution parameters' dependence on averaging time
When the log-normal task concentration distributions used to generate the simulated data were
made dependent upon averaging time (task duration) under the zero autocorrelation assumption,
the method proved to be able to identify the highest
concentration task at least as reliably as under the
total autocorrelation assumption. The zero vs. total
autocorrelation comparisons were made for a job
consisting of six tasks with equal long-run timeweights. The task distributions used were Type 1
and Type 2 with nominal task GSDs of 2, 220.5, 4,
421 and 422. Data sets of 20 samples and 100
samples were analyzed. Each combination of conditions was assessed in 3000 trials based on 30 sets
of time-weights.
The worst performance was found in 20-sample
data sets with Type 1 distributions and nominal sg,i
in the range 422 (average overlap approximately
80%). The highest concentration task was correctly
ranked in about 50% of the total autocorrelation
trials and 52% of the zero autocorrelation trials.
The highest concentration task was incorrectly
ranked as the lowest in 2% of the total autocorrelation trials and 2% of the zero autocorrelation trials.
The lowest concentration task was incorrectly
ranked as the highest in 5% of the total autocorrelation trials and 4% of the zero autocorrelation trials.
DISCUSSION
We have described a computational method for
estimating and ranking the median concentrations
associated with job tasks based on time-weighted
average sample data, and tested the performance of
the method under various conditions using simulated sample data. The method was found to yield
remarkably reliable ranking of task median concentrations, especially for the high exposure tasks, pro-
212
M. L. Phillips and N. A. Esmin
vided that the number of samples was adequate and
the task concentration distributions were not highly
overlapped. These are reasonable limitations. If the
task concentration distributions were highly overlapped, then knowing the exact ranking of the tasks
would be of little practical value, since the central
range of concentrations during performance of such
tasks would be substantially similar. More relevant
would be a general categorization of tasks as low,
intermediate, or high concentration. Inspection of
the assigned ranks vs. actual task ranks in Table 4
indicates that the method provides a credible basis
for such categorization even for jobs in which some
task concentration distributions are highly overlapped.
The number of samples which is adequate to
ensure reliable performance of the method is de®ned
by the limit in which the `error' term ei in Eq. (7)
becomes negligible:
X
X
1
ei ˆ
ukj log sg,k ykj
Ji j such that y ˆ0 k6ˆi
ij
…9†
ÿ
ÿ40
ÿ
for large Ji :
The expected magnitude of the ei is scaled by the
standard error of sampling which decreases as J ÿ0:5
,
i
but is further attenuated by a factor which may be
as small as (Iÿ1)ÿ0.5 due to the averaging over tasks
in each sample. The results from simulated small
data sets with a moderate degree of overlap between
adjacent distributions (60%) indicate that crude
judgments concerning the ranking of task-speci®c
concentrations could be made when the Ji are as
small as 3; that is, when as few as three samples contributed to each P-sum. Larger sample sizes increase
the reliability of the ranking.
In the development of the model, it was assumed
that the task concentrations and the sample concentrations were log-normally distributed. A log transformation was applied to stabilize the variability of
the concentrations. The log transformed sample
concentration was then modeled as a time-weighted
average of the log transformed task concentration
[Eq. (4)]. In actuality, of course, the sample concentration is the time-weighted average of the task concentrations [Eq. (1)]. The substitution of log
transformed concentrations for actual concentrations, albeit necessary for the handling of linear
equations when the underlying task distributions are
assumed to be right-skewed, led to sizable errors in
the estimation of task median concentrations: the
absolute magnitudes and the ratios of the estimated
task median concentrations often di€ered from the
true values by as much as an order of magnitude,
even when the estimates showed the correct rank
order.
If there were reason to believe that the task concentrations and the sample concentrations are nearly
normally distributed, then the log transformations
would be unnecessary. Log normal samples with sg
< 1.8 would ®t this criterion. The P-sums could
then be calculated using the untransformed sample
concentrations and the solution vector would consist
of estimates of the untransformed task median concentrations. Preliminary results from simulated data
indicate that reasonably accurate estimates of task
median concentrations can be obtained under these
circumstances.
CONCLUSIONS
A practical strength of the computational procedure presented here is its ability to extract taskspeci®c concentration information from multipletask samples. While the procedure would not be a
good substitute for direct single-task sampling in the
estimation of parameters of task-speci®c log-normal
concentration distributions, it may be useful to occupational hygienists as a method for ranking task
concentration distributions using a modest number
of full-shift or partial-shift samples which might be
collected more easily than single-task samples.
The reliability of the ranks assigned through this
method depends upon the spread within and
between task concentration distributions. Of course,
if such information about the task concentration
distributions were available, the tasks could readily
be ranked without resorting to the method described
in this paper. Therefore, under circumstances in
which the method would be useful, the user will not
be in a position to determine the precision of the
method. A conservative response to this inherent
uncertainty would be to interpret the assigned rank
of a task in less precise terms, such as `high exposure', `intermediate exposure', or `low exposure'.
Since a conservative interpretation of the rankings
assigned by the method restricts its use to the modest goal of sorting tasks into two or three general exposure categories, it might seem as if the method
o€ers no advantage over professional judgment in
identifying `high exposure' tasks. However, there is
considerable evidence that professional judgment is
quite unreliable in `data free' exposure assessment
(Kromhout et al., 1987; Post et al., 1991); also, the
reliability of an individual's professional judgment
cannot be modeled, but can only be calibrated relative to measured concentrations, if measurements
are available. In contrast, the reliability of the
method presented here can be easily modeled over
the range of plausible task concentration distributions using simulated sample data for any number
of samples and any job scenario.
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