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Journal of Loss Prevention in the Process Industries, 25(3), pp. 467-477 (May 2012) doi:10.1016/j.jlp.2011.11.014 Prediction of occupational accident statistics and work time loss distributions using Bayesian analysis Eftychia C. Marcoulaki*, Ioannis A. Papazoglou, Myrto Konstandinidou System Reliability and Industrial Safety Laboratory, National Centre for Scientific Research
“Demokritos”, PO Box 60228, 15310 Aghia Paraskevi, Greece
ABSTRACT
This paper uses Bayesian analysis tools for the stochastic evaluation of work time losses
due to occupational accidents in a workplace. Models are developed for accident
frequencies, duration of recovery from an accident, and the worker unavailability. The
unavailability statistics are hereby derived considering a two state stochastic model, to
provide estimates for the expected work time losses over a base period of workplace
operation. The above models are applied on real multiyear accident data collected from the
Greek Petrochemical Industry.
Keywords: Bayesian analysis, occupational accidents, work time loss, petrochemical
industry
* Corresponding author. Tel: +302106503743; fax: +302106545496.
E-mail addresses: emarcoulaki@ipta.demokritos.gr (Eftychia C. Marcoulaki),
yannisp@ipta.demokritos.gr (Ioannis A. Papazoglou),
myrto@ipta.demokritos.gr (Myrto Konstandinidou)
Bayesian analysis for the estimation of work time loss distributions
page 2
1. INTRODUCTION
Health and safety at work is one of the most important advanced fields of the European
Union (EU) social policy (European Commission, 2011). Based on EU-15 statistics, there
are 4.3 million non-fatal occupational accidents resulting in more than three days absence
from work every year across the EU-15 (Eurostat, average 1998-2007), or around 146
million lost workdays. These accidents cost around 20 billion euro to the EU, and a
considerable share of the above costs falls upon social security systems and public finances
(Konkolewsky, 2001). Employers face costs linked to sick pay, replacement of absent
workers, training/administration and loss of productivity –only a part of these are covered by
insurance. For European workers, the total annual loss of income due to absence from work
is estimated at one billion euro (European Commission, 2011).
This paper presents a set of Bayesian tools, to model the work time lost on sick leave due
to occupational accidents occurring in an industrial workplace. A lot of research has been
done on the quantitative analysis of occupational accident data. This includes tools for
identifying causal factors related to different consequences of incidents (Bellamy et al.,
2007; Konstandinidou et al., 2011;etc), modeling occupational accident frequency (Meel at
al., 2007; Marcoulaki et al., 2011; etc), assessing occupational risk (Marhavilas et al., 2011)
etc. Data on work time losses are typically considered in analyses for accident severity
(Jacinto and Soares, 2008; Blanch et al., 2009; Carnero and Pedregal, 2010; etc).
Acknowledging the financial implications of occupational accidents, Parejo-Moscoso et al.
(2011) estimated costs related to sick-leave, and Jallon et al. (2011) provided data collection
criteria for the development of indirect accident cost calculation models. Ale et al. (2008)
constructed an exposure-based occupational risk model, to derive improvement measures
and support cost-effective risk reduction strategies.
Some of the above works relay on descriptive statistics and factor analysis, others make
use of advanced prediction methods. Bayesian inference methods have long been used for
the analysis of accident databases (Hora and Iman, 1990) to model uncertainties and enable
future predictions. None of the previous works considered sick leaves due to occupational
accidents, and the associated work time loss for the company, within a Bayesian perspective.
The tools presented herein include Bayesian models for the prediction of (a) the number
of accidents during a given time period, (b) the duration of the recovery from an accident,
and (c) the unavailability of labor force in the workplace. The developed models are updated
Bayesian analysis for the estimation of work time loss distributions
page 3
using available evidence from real accident data, to inform predictions of work time losses
and related probability density functions (pdf’s). The proposed framework can be useful in
investigating future trends in the workplace, and enabling a better management of the labor
force.
The paper is organized as follows. Section 2 presents the method of analysis and the
Bayesian models. Section 3 presents and discusses the numerical results using evidence from
real accident reports. Section 4 concludes this work.
2. BAYESIAN MODELS
This work applies Bayesian inference methods for the analysis of occupational accident
data, predict:
(i) the number of accidents expected over a given time period in a company
workplace
(ii) the duration of the recovery period following an accident
(iii) the amount of time that the workers are recovering from the accidents, and not
being available to perform their work.
The Bayesian approach has the following steps:
(i) the quantity x to be estimated is assumed to be a random variable, generated
according to a specific stochastic model. Let ( | )f x be the associated pdf and
be a vector of parameters for x,
(ii) if is not known, we quantify our prior belief about the true value of as a pdf
( ) ( | )g g , where is a vector of parameters of g,
(iii) we collect evidence E through observation of the stochastic process, e.g. regarding
worker accidents in a workplace. We then quantify the relative likelihood of
observing E as ( )L E (likelihood function).
(iv) we calculate the posterior pdf according to the general formulation of Bayes’
theorem ( ) ( ) ( ) ( )g E g L E L E g d . Since the integral at the
denominator is a constant function of θ, we get ( ) ( )g E g L E .
The posterior pdf provides an expression of the remaining uncertainty about the value of
, and consequently about ( | )f x . The more information we have, the less the uncertainty
about the true value of .
Bayesian analysis for the estimation of work time loss distributions
page 4
2.1. Problem description
Workers in chemical companies are often involved in occupational accidents. Some of
these accidents require several days leave before the worker recovers and can return to work.
Others (like a very minor injury) may be dealt within the company premises with negligible
loss of work time. The company pays wages for the total workdays, including the time lost
for accident recovery. It is therefore important to have a reliable estimation of the amount of
work time loss due to the occupational accidents, in total as well as per accident. The
following models aim to support such estimations, starting from the model of section 2.2 to
predict the expected number of accidents over a given period of company operation. Sections
2.3 and 2.4 present models for work time loss predictions, either per individual accident
regardless of the total work time, or over a given period of company operation, respectively.
Figure 1 gives a simple illustration of a workplace with N workers. In general, the set of
workers may include employees or contractors and the employment time is not necessarily
the same for all workers. The figure shows the timeline for each worker during a period of
observation starting at time TS and terminating at TF. Each worker, n = 1, 2, …, N, works
between TS,n TS and TF,n ≤ TF, and during this time the worker is involved in Kn accidents.
The staring times TS,1, TS,5, TS,6, TS,8 and TS,9 coincide with TS. The finishing times TF,1, TF,2,
TF,6, TF,8 and TF,9 coincide with TF. Only workers 1, 6, 8 and 9 are employed during the entire
period of observation, while the others work only for a fraction of this time. The working
time of worker n before accident kn = 1, 2, …, Kn, occurs is denoted by , .
nn kt The respective
recovery time is denoted by , .nn kr The time range between the recovery from the last
accident, Kn, and TF,n is denoted by sn. Worker 3 has no accidents, so s3 = TF,3 TS,3. The
contract of worker 4 expires when she/he recovers from her/his last accident, thus s4 = 0.
2.2. Number of accident occurrences over a period of time
The occurrence of an occupational accident during a time period is not certain, thus the
number of accidents occurring during the same period in a workplace is a random variable.
The first set of models considers the probability distribution of the number of accidents
occurring over a given time interval, τ. Let the times between successive accidents be
randomly distributed according to an Exponential pdf, with rate of accident occurrence λ.
The stochastic process generating the accidents is then a Poisson process, and the number of
accidents κτ occurring over a given time interval τ is distributed according to the pdf:
Bayesian analysis for the estimation of work time loss distributions
page 5
( | ) exp !f
(1)
where ρτ denotes the amount of time within τ that workers are recovering from accidents
(see section 2.4). In the present application, we expect that , thus the models for
accident occurrences assume , and equation (1) becomes:
( | ) exp !f
(2)
Equation (2) can be solved for given values of the rate parameter λ. If λ is not known with
certainty, the Bayesian approach consists in assuming that λ is a stochastic variable
distributed according to a known pdf ( | )g . It is assumed here that the λ prior is a
Gamma pdf, with shape parameter aλ and rate parameter bλ, i.e. | ,g f a b .
Then, integration of the unconditional pdf of on λ gives the following analytical solution:
11( | , )
aa b
f a bb b
(3)
If |g follows other pdfs, numerical integration can be applied. Note that, equation
(3) is a Negative Binomial pdf with success probability ( )b (Forbes et al., 2011).
Figure 2 presents prior distributions for κτ using the pairs of and a b values reported on
Table 1, where and a b denote the prior values of and a b , respectively. Pair L0 is the
Gamma(0.001, 0.001) pdf, used in practice as a prior for Poisson processes (Meel and
Seider, 2006; Lambert et al., 2005). Each of the other three pairs expresses different degrees
of prior belief for the ranges of possible values of κτ and their associated probabilities, and is
derived assuming that λ lies with probability 95% within a specified range (λ1, λ2). The time
base used here is τ=250,000 workdays, which represents the equivalent of 1000 employees
working 8 hours per day, 5 days per week, 50 weeks per year. The four priors of Figure 2
and Table 1 are used in the data analysis of section 3.
The prior knowledge encapsulated in ( ) | ,g g a b and ( | , )f a b can be
updated using evidence collected from system observation. The evidence, Eλ, consists here
of:
the set of times between successive accidents, , ,nn kt for the N workers, and
the set of times without accident, sn, between the recovery from the last accident of
worker n and TF,n.
Bayesian analysis for the estimation of work time loss distributions
page 6
The likelihood of the evidence is the probability of the joint event ,
1 1 1
n
n
n
KN N
n k n
n k n
t s
, so
,
1 1 1
( | ) ( | ) ( 0 )n
n n
n
KN N
n k s
n k n
L E f t f
. Assuming that all the company workers have
identical behavior in terms of generating accidents, thus , ,( | ) exp( )n nn k n kf t t and
( 0 ) exp( )ns nf s , the likelihood function is:
( | ) exp ( )KL E T R (4)
where T is the total work time during the observation period , ,
1
N
F n S n
n
T T T
, and R is
the total time loss due to occupational accidents ,
1 1
n
n
n
KN
n k
n k
R r
. Since the prior for λ is a
Gamma pdf and the Gamma pdf is the conjugate prior to the Poisson pdf, the posterior
distribution for λ is also in the family of Gamma pdfs. In effect:
1( ) and ex ( ) p
a Kb b bg T R a a K T R
(5)
where K is the total number of accidents 1
N
n
n
K K
.
So, when the accidents occur according to a Poisson process, the sufficient statistics for λ
are the total number of accidents, K, and the time difference T R.
The Negative Binomial pdf of equation (3) is also valid for the updated values of the λ
parameters, to derive posterior statistics for κτ. Assuming Gamma and Negative Binomial
pdfs for λ and κτ, respectively, their expected values and variances are formulated as (Forbes
et al., 2011):
2 and
a aE V
b b
(6)
2
( an
)d
a a bE V
b b
(7)
Using the model of equation (3) for posterior pdf calculations, and real data from reported
accidents in a workplace, section 3 provides posterior results on the predicted number of
occupational accidents. Section 3 also discusses how the choices of prior pdfs affect the
predictions.
Bayesian analysis for the estimation of work time loss distributions
page 7
2.3. Duration of the worker recovery from an accident
The second set of models gives the probability distribution of the time required to recover
from an accident. Let the time loss (in number of days), δ, be randomly distributed according
to an Exponential pdf, with rate of accident recovery μ:
( | ) exp( )f (8)
Since the exact value of parameter μ is not known, we quantify our lack of knowledge by
assuming that μ is randomly distributed following a known prior pdf ( ) ( | )g g .
When the μ prior is a Gamma pdf, i.e | ,g f a b , integration of the
unconditional on μ pdf of δ yields the following analytical solution:
( 1)( ) ( )
a af E b a b
(9)
Otherwise numerical integration can be applied.
Figure 3 presents prior distributions for δ using the pairs of and a b values reported on
Table 2, where and a b denote the prior values of and a b , respectively. Pair M0 is the
Gamma(0.001, 0.001) pdf. Τhe other three pairs express different degrees of prior belief for
the range of possible values of δ and their associated probabilities, and are derived assuming
that μ lies with given probability within a specified range (μ2, μ1). The priors of Figure 3 and
Table 2 are used in the data analysis of section 3.
The evidence Eμ, used here to derive posterior statistics for μ and δ, consists of the set of
recovery times, , ,nn kr for the K accidents, with kn = 1, 2, …, Kn and n = 1, 2, …, N. The
likelihood of Eμ is the probability of observing the joint event ,
1 1
n
n
n
KN
n k
n k
r
. Assuming that all
the company workers and the accidents they are involved in have identical behavior in terms
of generating accident recovery times, and that the recovery times are randomly distributed
according to an Exponential pdf, the likelihood function is:
1
( | ) k
Kr K R
k
f E e e
(10)
With respect to the assumptions behind equation (10), the sufficient statistics for μ are the
total number of accidents, K, and their total recovery time R. Since Gamma is a conjugate
prior to the Exponential distribution ( | )f , the posterior distribution for μ is also
according to a Gamma distribution:
Bayesian analysis for the estimation of work time loss distributions
page 8
1exp (( ) and )
a Kg b bR R ba a K
(11)
The δ model in equation (9) is also valid for the updated values of the μ parameters. It can
be shown that, the expected value and variance formulae for the μ and δ pdfs are:
2 and
a aE V
b b
(12)
2
21 a
( 2) ( 1)nd
b a bE V
a a a
(13)
Section 3 presents case study results for predicted accident durations using the prior pdfs
presented above.
2.4. Worker unavailability and work time loss model
Figure 4 provides an illustration of the dynamic behavior of a company worker as two-
state continuous Markov process (Howard, 1971). A worker at the first state is working
normally (available to work as required). Workers found at the second state are recovering
from an accident (unavailable due to an accident). A process randomly generates accidents
with rate λ, so that an available worker (state 1) becomes unavailable (state 2). Another
stochastic process generates accident recovery times, so that a worker at state 2 falls back to
state 1 with rate μ. The two rates λ and μ are assumed independent, and remain constant
when the transition distributions are Exponential.
Looking ahead, the probability of a worker occupying state 2 at a particular instance of
time, U(t), gives the probability that the worker will be unavailable for work at t. This
probability converges to a steady-state value, U , formulated as (Howard, 1971; Henley and
Kumamoto, 1981):
lim ( )t
U U t
(14)
Assuming that the rates λ and μ are statistically independent and each is distributed
according to a Gamma pdf, the joint pdf is the product of individual pdf’s. It can be shown
(see Appendix A.1) that, the pdf of the steady state unavailability, U , is given by:
111
( , , , )( , ) 1
aaa a
a a
b b U Uh U a b a b
a a b U b U
(15)
The associated expected value and variance are formulated as (see Appendices A.2-3):
0
1( 1, )
( , )
m
m
E U a m aa a
(16)
Bayesian analysis for the estimation of work time loss distributions
page 9
2
2
0
11 ( 2, )
( , )
m
m
V U m a m a E Ua a
(17)
where, 1 b b and ,x y denotes the beta function for *,x y (see notation).
Likewise to the models for κτ and δ, equations (15)(17) provide prior and posterior
statistics of the worker unavailability, according to the quartet of , , ,a b a b values
assumed in the calculations.
The overall work time losses during a given time period, τ, of workplace operation are
calculated as the product of the steady-state unavailability, U and τ. Using equation (14):
2, ,U E E U V V U (18)
Note that, the steady-state assumption becomes valid for 1( ) .
Figure 5 presents prior distributions for ρτ using the quartets of , , and a b a b values
reported on Table 3. As in section 2.2, the time base for ρτ calculations is τ=250,000
workdays. Case U0 assumes Gamma(0.001, 0.001) pdfs. In case U1, the quartets {aλ=aμ=1,
bλ=bμ} yield the uniform distribution for ρτ. As the bλ / bμ ratio increases, the pdf curve for
ρτ shifts towards lower unavailabilities, and case U2 uses bλ / bμ = 103 to get
Pr / 1% 1% . Section 3 presents prediction results for work time losses in a
workplace, using equations (15)(18) and the prior cases of Table 3 and Figure 5.
3. ANALYSIS OF OCCUPATIONAL ACCIDENT DATA
3.1. Accident data
The prior pdfs of the Bayesian models presented in section 2 are updated using available
evidence from real accident data. The derived posterior distributions embed knowledge on
the system, model its future behavior, and allow predictions for accident occurrences and
durations, and work time loss. For an observation period of known total work time, the
sufficient statistics for the models of section 2 are the number of accident occurrences and
their recovery times, as these are recorded in the company databases.
An extended database is used, which comprises data from reported accidents in the Greek
Petrochemical Industry over a multi-year period (Nivolianitou et al., 2006; Konstandinidou
et al., 2006). The collected data were acquired directly from the different sites, and the
Bayesian analysis for the estimation of work time loss distributions
page 10
participating companies gave access to their archives and to the initial reports of the
accidents. The sites included refineries, onshore and offshore facilities, storage locations and
extraction sites. The present analysis is based on 406 reported non-fatal accidents, which
occurred at 5 different sites during the years 1997 and 2003. For reasons of confidentiality,
the sites are hereby randomly referred to as A, B, C, D and E. Table 1 aggregates the
available accident data, and reports only the sufficient statistics for the models of section 2.
3.2. Calculations and results
The following calculations consider the cases chosen for the parameters of the Gamma
priors for λ and μ, as presented in section 2 (see also Figures 2,3,5 and Tables 1-3). The prior
choices affect the predictions, as the quartets , , ,a a b b are involved in the calculations
of κτ (equation (3)), δ (equation (9)) and ρτ (equations (15) and (18)).
Figures 6–8 present the posterior results for the five companies and their total, for each
prior case of section 2. Predictions on the number of accident occurrences (Figures 6) and
the work time losses (Figures 8) assume τ = 2.5105 workdays. With the exception of
company B, the differences between the company posterior curves derived using different
priors are negligible. Company B is more sensitive to the prior parameter values, since it has
the shortest record of accidents and the fewest observed workdays. The prediction
differences for this company are particularly evident in the κτ model, whereas the models for
δ and ρτ appear less sensitive to the prior beliefs considered here. To check the validity of
equation (14), U(t) is within 1% of U when 4.6 ( )t (Howard, 1971; Henley and
Kumamoto, 1981). For the sites A, B, C, D and E, this range is reached after 38, 108, 121,
12 and 76 workdays, respectively.
The results reported at Table 5 include the posterior pdf statistics for the predicted
number of accidents κτ, the predicted accident duration δ, and the predicted work time losses
ρτ. Expected values and variances are estimated using the models of Section 2.
Even though the five sites host similar processes and their workers are involved in similar
activities, the analysis yields very different models for the prediction of the number of
accidents, their recovery times and the system unavailabilities. Looking at the expected
values of κτ, δ and ρ reported at Table 5, the number of expected accidents at site C is almost
an order of magnitude higher than the other four sites. The expected recovery times are also
increased, though site B gives slightly higher recovery times (close to a month). The κτ
expected values for sites A and E are quite close. Expected value results for site D are
Bayesian analysis for the estimation of work time loss distributions
page 11
located between the predictions for the other sites. Looking at the variances, in δ predictions
appear relatively higher compared to ρτ and especially κτ.
Figure 9 shows the 50% and 95% probability intervals, and the median for each site and
their total, using the κτ model for the L0 prior case. Figure 10 provides similar results using
the δ model, assuming the M0 prior case. The derived intervals verify the results of Table 5,
also illustrating the increased uncertainties in the results for accident duration, compared to
the number of accidents during the base time used here. Figures 11 show the same
probability intervals for the work time losses in the 5 sites and their total, using the models
of section 2.4 for prior case U0. The work time losses in sites C and B (Figure 11 top) are
predicted an order of magnitude higher than the losses in other sites and the total (Figure 11
bottom). The results for site B seem to have the highest uncertainty, which can be attributed
to the scarcity of the relevant data.
3.3. Discussion
The following analysis reveals that the number of accident records, K, is the most
important statistic to control the uncertainty in the predicted estimates. Uncertainty is hereby
quantified using the posterior coefficients of variation, formulated as:
CV V E (19)
The variances, and the resulting coefficients of variation, are high for κτ and even higher
for δ, while the uncertainty in the work time loss predictions is notably lower. Consider the
κτ model in Section 2.2, and sufficient evidence to ensure that the prior values ,a b have
negligible impact on the final predictions. This assumption is valid for sites A, C, D and E,
which have sufficiently large data pools. Then, using equations (7), the posterior coefficient
of variation for κτ takes the form:
CV T R K (20)
Equation (20) indicates that, for given evidence, the value for CV decreases as the
considered time base τ increases, and reaches its minimal value, K–0.5
, at infinite τ. As the
observation time (T) increases and since T R , the uncertainties in the prediction of
accident occurrences become smaller in “riskier” workplaces, i.e. sites with higher K T
ratios. In effect, site C with 410K T has the lowest CV , while sites A and E with
51.5 10K T exhibit the highest CV .
For the analysis of prediction uncertainties in the accident durations, δ, we use the model
Bayesian analysis for the estimation of work time loss distributions
page 12
of section 2.3. Assuming again that the prior and a b values are negligible, the posterior
coefficient of variation for δ becomes:
2CV K K (21)
Therefore, CV depends solely on the number of reported accidents, and tends to
unity at increasing K. Equation (21) explains the particularly high variances calculated using
the δ model, which is derived under the assumption that accident recovery times follow
Exponential pdfs (equation (8)). Consideration of pdfs other than the Exponential may
hinder the development of analytical formulae for δ, thus call for Monte Carlo simulations.
The models for worker unavailability and work time losses are derived through the λ and
μ pdfs, thus the models for U and ρτ (section 2.4) do not inherit the high variances observed
in the κτ and δ models. Using equations (16)(17) and assuming negligible prior parameter
values, the posterior coefficients of variation for U and ρτ are both formulated as:
2
0 0
( , ) 1 ( 2, ) ( 1, ) 1m m
m m
CV CV U
K K m K m K K m K
(22)
For negligible and b b , equation (36) yields 1 R T , thus the uncertainties in the
ρτ and U predictions depend solely on K and the R T ratio. Figure 12 presents a set of
CV results from numerical experiments using different values of K and R T . The figure
also shows the limiting curve for 0
lim 1R
T
and the points for the five sites. The limiting
curve gives the maximal variation at each value of K, and is bounded by 2K
, for K > 2.
Again, the number of recorded accidents is proven to be the most important variable to
control the prediction uncertainty for work time loss, among the sufficient statistics of the
present analysis. Note that, these trends are valid for the likelihood and prior pdfs assumed
during the model development in section 2, whereas different assumptions may yield
different models and reveal other trends.
4. CONCLUSIONS
This paper presents a Bayesian approach to the statistical analysis of occupational
accident data, including accident recovery times. Recovery times are considered per accident
and over a period of operation and relevant models are derived. A dynamic two state
Bayesian analysis for the estimation of work time loss distributions
page 13
stochastic model is used to derive the worker unavailability statistics, and predict the amount
of time that workers will be recovering from accidents and therefore won’t be available to
perform the job they are paid for. The models developed here can be informed using
available databases of occupational accidents observed in the process industry. Sufficient
statistics to use the models include only the total number of workdays, the work days lost
due to recovering from occupational accidents, and the number of occupational accidents
over the period of observation.
The work discusses various prior pdfs, which are updated using evidence from a database
of over 400 accidents reported in the Greek Petrochemical industry over a multi-year period.
The statistical analysis provides future predictions for the number of accidents, accident
durations and work time loss in each one of the considered sites. The present analysis shows
that the most important statistic to reduce the prediction uncertainties is the number of
observed accidents. Uncertainties in the expected number of accidents are inversely affected
by the totally observed workdays. In sites where there is scarcity of accident data and the
observed workdays are few, the predictions are poor and sensitive to the choice of prior pdf
parameter values. The results also indicate that, the evidence collected at one site, or group
of sites, cannot be used directly to inform predictions at another similar site. Current work
considers the development of Bayesian models to enable integration of evidence collected at
different sites.
APPENDIX: STEADY STATE WORKER UNAVAILABILITY PDF
A.1. Worker unavailability pdf
Assuming that λ and μ are statistically independent and each is distributed according to
Gamma pdfs, the joint pdf is the product of individual pdf’s:
( , | , , , ) ( | , ) ( | , )f a b a b f a b f a b (23)
2
2
11 exp( )exp( )( , )
( ) ( )
aa
a
bbf
b a b a
(24)
Consider the transformation:
( , ) ( , ) ,s v g (25)
Note that, v is the steady state unavailability (see equation (14)).
Let h(s,v) denote the pdf of the random variables s and v, then:
Bayesian analysis for the estimation of work time loss distributions
page 14
1( , ) ( , ) det ( , )h s v f g J s w (26)
According to equation (25) λ = vs and μ = (1v)s, and the Jacobean determinant term in
(26) becomes:
( , ) det ( , )1
v ss vJ s w J s w vs s vs s
v ss v
(27)
Using equations (25) to (27), equation (24) is transformed to:
11 ( ) exp( ( ))( ) exp( )( , )
( ) ( )
s v s b s v sv s b v sh s w s
b a b a
1 11(1 ) exp (1 )
( ) ( )
aa
a a aab b
s v v b v b v sa a
(28)
The pdf of the unavailability variable v U is derived by integrating out the variable s:
1 11
0
(1 ) exp (1 )( ) ( )
aa
a a aab b
h v v v s b v b v s dsa a
(29)
Using the Gamma function definition (Forbes et al., 2011), the integration on (29) gives:
11( )
(1 )( ) ( ) (1 )
aa
aa
a a
b b a ah v v v
a a b v b v
(30)
Since ( ) ( )
( , )( )
a aa a
a a
(Forbes et al., 2011), equation (30) finally becomes:
11(1 )
( ; , , , )( , ) (1 )
aa aa
a a
b b v vh v a b a b
a a b v b v
(31)
A.2. Expected value of the worker unavailability pdf
The expected value of U according to equation (15) is given as follows:
1 1
0 0
(1 )1( )
( , ) 1(1 )
aa
a a
b v b v dvE U v h v dv
a a vb v b v
(32)
The E U is derived with the aid of variable z, defined as:
(1 )
b vz
b v b v
(33)
and equation (15) is transformed to give h U as a function of z:
Bayesian analysis for the estimation of work time loss distributions
page 15
1 (1 )
( , ) (1 )
aaz z
h Ua a v v
(34)
Based on the definition of variable z in equation (33):
2
1 1(1 ) , 1 and
1 1 1
z zv v dv dz
z z z
(35)
where β is defined as:
1b
b
(36)
Using equations (35)(36), the integral (32) becomes:
11 1
0
1(1 ) (1 )
( , )
aaE U z z z dz
a a
(37)
Note that, when v is distributed according to equation (15), the variable z is distributed
according to a Beta pdf. Therefore, function ( )h U is a generalized-inverted-Beta pdf.
For the calculation of E U , the integral of equation (37) is hereby represented as a
series. Provided that 2 1 , the term 1(1 )z can be replaced by the Maclaurin series:
1
0
(1 )m
m
z z
(38)
The assumption that 2 1 holds for b b . Since bμ is the scale parameter for the
“repair” rate while bλ is the scale parameter for the “failure” rate it is safe to assume that
b b . If for any reason b b then the analysis can be performed using 1 b b .
With the aid of equation (38), the integral in of equation (37) becomes:
1 1
00
1(1 )
( , )
aa mm
m
E U z z dza a
(39)
According to the definition of the Beta function, so the integral terms in equation (39)
give:
0
1( 1, )
( , )
m
m
E U a m aa a
(40)
A.3 Variance of the worker unavailability pdf
Similarly, the unavailability variance V U is formulated as:
1 1
2 22
0 0
( ) ( )V U v E U h v dv v h v dv E U (41)
Bayesian analysis for the estimation of work time loss distributions
page 16
With the aid of (33)(35), the integral term in equation (41) yields:
21 11 212
0 0
1( ) 1 1
( , )
aav h v dv z z z dz
a a
(42)
Assuming 2 1 the term 2(1 )z can be replaced by the Maclaurin series:
2
0
(1 ) 1m
m
z m z
(43)
Using equation (43) the integral (42) becomes:
21 1112
00 0
1( ) 1
( , )
aa mm
m
v h v dv z z dza a
(44)
According to the definition of the Beta function, (44) gives:
21
2
00
1( ) 1 ( 2, )
( , )
m
m
v h v dv m a m aa a
(45)
By substituting equation (45) into (41) the variance V U is finally calculated as:
2
2
0
11 ( 2, )
( , )
m
m
V U m a m a E Ua a
(46)
REFERENCES
Ale BJM, Baksteen H, Bellamy LJ, Bloemhof A, Goossens L, Hale A, Mud ML, Oh JIH,
Papazoglou IA, Post J, Whiston JY. Quantifying occupational risk: The development of an
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NOTATION
ax = gamma pdf shape parameter for stochastic variable x
bx = gamma pdf rate parameter for stochastic variable x
11 1
0( , ) (1 ) for , x yx y z z x ydz
= Beta function (Forbes et al., 2011)
CV x V x E x = coefficient of variation for stochastic variable x
E = evidence
, ,nn k nE t s = evidence for λ
, nn kE r = evidence for μ
E x = expected value of stochastic variable x
( | )f x = conditional pdf of stochastic variable vector x given parameter vector
1 1, ( ) for x x xa
x
a
x
b
x x
xe bf x a b x xa
= Gamma pdf (Forbes et al., 2011)
( )g x = prior pdf of variable vector x
g x E = posterior pdf of variable x given evidence E (can be simplified to g x )
1
N
n
n
K K
= total number of accidents of the N workers over time T
Kn = number of accidents occurring to worker n between TS,n and TF,n
Bayesian analysis for the estimation of work time loss distributions
page 19
kn = index for the accidents of worker n, with kn [1, Kn]
( )L E = likelihood of evidence E given parameter vector θ
N = number of workers (employees and contractors) working for the company
n = index for company worker, with n [1, N]
,
1 1
n
n
n
KN
n k
n k
R r
= total time loss due to all the occupational accidents, in workdays
, nn kr = duration of recovery of worker n from accident kn, in workdays
sn = time interval of worker n between the time of recovery from accident Kn and TF,n, in
workdays
, ,
1
N
F n S n
n
T T T
= total work time, for all the N workers, , in workdays
TF = finishing time for the period of observation, in workdays
TF,n = time that worker n stops working at the company, in workdays
, , , ,
1
n
n n
n
K
n F n S n n k n k n
k
T T T t r s
= total work time of worker n, , in workdays
, nn kt = time interval between either TS,n (if kn=1), or the end of recovery from accident kn1
(if kn>1), and the occurrence of accident kn of worker n, in workdays
TS = starting time for the period of observation, in workdays
TS,n = time that worker n starts working at the company, in workdays
U = average value of the worker unavailability over a base time period τ
U = steady state value of worker unavailability reached at infinite time
U(t) = probability that the worker will be unavailable for work at time instance t
V x = variance of stochastic variable x
x = prior instance of entity x
x = posterior instance of entity x
1
0( ) d for x tx t e xt
(Gamma function, Forbes et al., 2011)
δ = predicted duration of recovery from an accident, in workdays
, = vectors of pdf parameters
κτ = predicted number of accidents over a base time period τ
λ = rate of accident occurrence parameter, in accidents workdays -1
μ = rate of accident recovery parameter, in workdays -1
Bayesian analysis for the estimation of work time loss distributions
page 20
ρτ = overall work time losses during a given time period, τ, in workdays
τ = base time period for the prediction of number of accidents, worker unavailability and
work time loss, in workdays
TABLE CAPTIONS
Table 1: Prior pdf parameters for the κτ model (section 2.2)
Table 2: Prior pdf parameters for the δ model (section 2.3)
Table 3: Prior pdf parameters for the ρτ model (section 2.4)
Table 4: Occupational accident data
Table 5: Posterior pdf statistics for κτ, δ and ρτ
FIGURE CAPTIONS
Figure 1: Timelines for a workplace with N workers
Figure 2: Prior pdfs for the number of accidents during 250,000 workdays
Figure 3: Prior pdfs for the duration of accident recovery δ
Figure 4: Stochastic model for worker unavailability
Figure 5: Prior pdfs for the work time losses during 250,000 workdays
Figure 6: Posterior pdfs for the number of accidents during 250,000 workdays
Figure 7: Posterior pdfs for the duration of accident recovery δ
Figure 8: Posterior pdfs for the work time losses during 250,000 workdays
Figure 9: Probability intervals for the predicted number of accidents during 250,000
workdays
Figure 10: Probability intervals for the predicted duration of accident recovery
Figure 11: Probability intervals for the predicted work time losses during 250,000
workdays
Figure 12: Uncertainty in the work time loss predictions
Table 1: Prior pdf parameters for the κτ model (section 2.2)
L0 L1 L2 L3
aλ 10-3 3.358 1.117 3.358
bλ (days) 10-3 7787 393 3894
λ1⋅τ (accidents) – 104 104 5×103
Pr (λ ≥ λ1) – 2.50% 2.50% 2.50%
λ2⋅τ (accidents) – 103 102 5×102
Pr (λ ≤ λ2) – 2.50% 2.50% 2.50%
Table 2: Prior pdf parameters for the δ model (section 2.3)
M0 M1 M2 M3
aµ 10-3 1.509 0.5405 2.212
bµ (days) 10-3 3.577 0.7880 8.872
11µ− (days) – 20 250 20
Pr (µ ≤ µ1) – 5% 5% 5%
12µ− (days) – 3 3 3
Pr (µ ≥ µ2) – 5% 5% 75%
Table 3: Prior pdf parameters for the ρτ model (section 2.4)
U0 U1 U2
aλ 10-3 1 1
bλ 10-3 1 1
aµ 10-3 1 1
bµ 10-3 1 10-4
Table 4: Occupational accident data
Site A Site B Site C Site D Site E
number of accidents 110 5 130 66 95
number of workdays lost 909 117 3424 165 1559
total workdays 7,527,422 161,079 1,238,019 2,271,740 6,452,076
Tables
Table 5: Posterior pdf statistics for κτ, δ and ρτ
model κτ, number of accidents* δ, accident duration (workdays) ρτ, work time loss (workdays)*
prior case L0 L1 L2 L3 M0 M1 M2 M3 U0 U1 U2
Predicted average E τκ′′ , eq. (9) E δ′′ , eq. (16) E τρ′′ , eq. (21), (23)
company A 3.654 3.761 3.691 3.763 8.339 8.258 8.305 8.253 30.47 30.50 30.46
company B 7.767 12.38 9.477 12.68 29.24 21.89 25.94 20.26 226.8 219.6 217.8
company C 26.32 26.84 26.54 26.92 26.54 26.26 26.44 26.16 696.7 696.9 696.7
company D 7.264 7.607 7.385 7.620 2.538 2.535 2.530 2.587 18.44 18.54 18.43
company E 3.682 3.807 3.725 3.810 16.60 16.37 16.51 16.31 61.09 61.12 61.08
Total 5.753 5.798 5.768 5.799 15.25 15.20 15.23 15.19 87.68 87.69 87.68
Predicted variance V τκ′′ , eq. (9) V δ′′ , eq. (16) V U∞′′ , eq. (22), (23)
company A 3.775 3.886 3.813 3.888 70.83 69.44 70.25 69.35 17.11 16.98 16.95
company B 19.83 30.73 24.16 31.90 1425 691.4 1053 568.1 3.068×103 2.202×103 2.165×103
company C 31.66 32.24 31.92 32.35 715.5 700.4 709.8 695.0 7513. 7458. 7454.
company D 8.063 8.442 8.198 8.457 6.645 6.620 6.597 6.894 10.54 10.50 10.37
company E 3.825 3.955 3.869 3.957 281.3 273.7 278.4 271.5 79.79 79.03 78.93
Total 5.834 5.880 5.850 5.881 233.6 232.2 233.1 231.7 37.98 37.90 37.89
* τ = 250,000workdays
Figure 1: Timelines for a workplace with N workers
TS,2 t2,1 r2,1 s2 TF,2
TS,1 t1,1 r1,1 t1,2 r1,2 s1 TF,1
TS,3 s3 TF,3
TS,4 t4,1 r2,1 TS,4
TS,5 t5,1 r5,1=0 s5 TF,5
TS,6 t6,1 r6,1 s6 TF,6
TS,7 s7 TF,7
tS,8 t8,1 r8,1 s8 TF,8
TS,9 t9,1 r9,1 s9 TF,9
1
2
3
4
5
6
7
8
9
start (TS,n) accident/incident back to work finish (TF,n) work time (tn,k; sn) recovery time (rn,k)
TS TF
Figure(s)
Figure 2: Prior pdfs for the number of accidents during 250,000 days
0.E+00
2.E-03
4.E-03
6.E-03
8.E-03
0 200 400 600 800 1000number of accidents per 250,000 workdays, prior pdfs
pro
bab
ilit
y d
ensi
ty
L0
L1
L2
L3
Figure 3: Prior pdfs for duration of accident recovery δ
1.E-06
1.E-04
1.E-02
1.E+00
0 20 40 60 80 100accident duration (workdays), prior pdfs
pro
bab
ilit
y d
ensi
ty
M0
M1
M2
M3
Figure 4: Stochastic model for worker unavailability
random
process 1
random
process 2
failure rate λ
repair rate μ
Figure 5: Prior pdfs for the work time losses during 250,000 workdays
0.01
0.1
1
10
100
1000
10000
0 1000 2000 3000 4000 5000
work time loss (workdays), prior pdfs
pro
bab
ilit
y d
ensi
ty
U0
U1
U2
Figure 6: Posterior pdfs for the number of accidents during 250,000 days
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0 5 10 15 20 25
number of accidents per 250,000 workdays, site B
pro
bab
ilit
y d
ensi
ty
L0
L1
L2
L3
0.00
0.03
0.06
0.09
0.12
0.15
0 4 8 12 16
number of accidents per 250,000 workdays, site D
pro
bab
ilit
y d
ensi
ty
0.00
0.05
0.10
0.15
0.20
0 4 8 12 16number of accidents per 250,000 workdays, total
pro
bab
ilit
y d
ensi
ty0.00
0.05
0.10
0.15
0.20
0.25
0 3 6 9 12
number of accidents per 250,000 workdays, site A
pro
bab
ilit
y d
ensi
ty
0
0.03
0.06
0.09
5 15 25 35 45
number of accidents per 250,000 workdays, site C
pro
bab
ilit
y d
ensi
ty
0.00
0.05
0.10
0.15
0.20
0.25
0 3 6 9 12number of accidents per 250,000 workdays, site E
pro
bab
ilit
y d
ensi
ty
Figure 7: Posterior pdfs for the duration of accident recovery δ
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0 20 40 60 80 100 120
accident duration (workdays), company B
pro
bab
ilit
y d
ensi
ty
M0
M1
M2
M3
0.00
0.10
0.20
0.30
0.40
0.50
0 2 4 6 8 10 12
accident duration (workdays), company D
pro
bab
ilit
y d
ensi
ty
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0 20 40 60 80
accident duration (workdays), total
pro
bab
ilit
y d
ensi
ty0.00
0.02
0.04
0.06
0.08
0.10
0.12
0 10 20 30 40 50
accident duration (workdays), company A
pro
bab
ilit
y d
ensi
ty
0.00
0.01
0.02
0.03
0.04
0 20 40 60 80 100
accident duration (workdays), company C
pro
bab
ilit
y d
ensi
ty
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0 20 40 60 80
accident duration (workdays), company E
pro
bab
ilit
y d
ensi
ty
Figure 8: Posterior pdfs for the work time losses during 250,000 workdays
0
200
400
600
800
1000
1200
0 100 200 300 400 500 600
worktime loss (workdays), company B
0
5000
10000
15000
20000
25000
30000
35000
10 15 20 25 30worktime loss (workdays), company D
pro
bab
ilit
y d
ensi
ty
0
2500
5000
7500
10000
12500
15000
17500
70 80 90 100 110
worktime loss (workdays), total
pro
bab
ilit
y d
ensi
ty0
5000
10000
15000
20000
25000
20 25 30 35 40 45 50
worktime loss (workdays), company A
pro
bab
ilit
y d
ensi
ty
U0
U1
U2
0
250
500
750
1000
1250
400 500 600 700 800 900 1000
worktime loss (workdays), company C
pro
bab
ilit
y d
ensi
ty
0
2000
4000
6000
8000
10000
12000
14000
30 40 50 60 70 80 90worktime loss (workdays), company E
pro
bab
ilit
y d
ensi
ty
Figure 9: Probability intervals for the predicted number of accidents during 250,000
workdays
0 5 10 15 20 25 30
A
B
C
D
E
Total
number of accidents during 250,000 workdays
50% range
95% range
Median
Figure 10: Probability intervals for the predicted accident duration
0 20 40 60 80 100 120 140
E
B
C
D
E
Total
accident duration (workdays)
50% range
95% range
Median
Figure 11: Probability intervals for the predicted work time losses during 250,000
workdays
0 100 200 300 400 500 600 700 800 900
A
B
C
D
E
Total
50% range
95% range
10 20 30 40 50 60 70 80 90 100
A
B
C
D
E
Total
work time loss (workdays) during 250,000 workdays
Median
Figure 12: Uncertainty in the work time loss predictions
0.0
0.2
0.4
0.6
0.8
1.0
0 20 40 60 80 100 120 140
number of reported accidents (K )
po
ster
ior
CV
fo
r w
ork
tim
e lo
ss,
0
max at lim 1R
T
CV
0.52 K
β = 0.990
β = 0.900
site dataCV
B
DE A C
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