Modeling mixed exposures: an application to welding fumes in the construction trades

Abstract

Workers are often exposed to mixtures of airborne contaminants which may complicate health studies, risk assessments, and epidemiological investigations. In situations where these mixed exposures are correlated, multivariate probability distributions may serve as models suitable for description and subsequent inference. This paper presents a methodology for modeling correlated mixed exposures using the Johnson system of multivariate probability distributions. This system involves transformations to normality and includes the multivariate normal and lognormal distributions, among others. The technique is illustrated with manganese exposures for pipefitters and boilermakers conditioned upon a measurement of total particulate exposure. Applications involving compliance and risk assessment for mixed exposure problems are presented using the fitted distributions.

Appendix: The Johnson system of probability distributions

Norman L. Johnson developed a comprehensive system of probability distribution functions based on transformations (translations) to normality. The system is very general and encompasses four different probability distributions that are identified as S _N, S _L, S _B, and S _U for Normal, Lognormal, Bounded and Unbounded respectively. These four distributions are specified as follows:

The y transform is the first step and is defined by:

$$ y = {\frac{x - \varepsilon }{\lambda }} $$

(25)

The interpretation of epsilon and lambda depends upon the distribution of concern.

The subsequent f transforms for each of the four univariate distributions are:

For the Normal distribution:

$$ f = y $$

(26)

For the Lognormal:

$$ f = { \ln }(y) $$

(27)

For the S _B distribution:

$$ f = { \ln }(y/(1 - y)) $$

(28)

And for the S _U distribution:

$$ f = { \ln }\left( {y + \sqrt {1 + y^{2} } } \right) $$

(29)

In all cases the unit normal z is defined by:

$$ z = \gamma + \delta f = {\frac{{f - \mu_{f} }}{{\sigma_{f} }}} $$

(30)

The normal and lognormal distributions are well-known and the unbounded distribution is not often of interest for exposure data (at least in this author’s experience so far). However the S _B distribution is less well-known and more difficult mathematically than the others, for this distribution the mean of y is (Johnson 1949):

$$ \mu (y) = {\frac{\phi }{\psi }};{\text{where}}\,\phi = A - B\,{\text{and}}\,\psi = CD $$

(31)

and:

$$ A = {\frac{1}{2\delta }} + {\frac{1}{\delta }}\sum\limits_{n = 1}^{\infty } {\exp \left( {{{ - n^{2} } \mathord{\left/ {\vphantom {{ - n^{2} } {\left( {2\delta^{2} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {2\delta^{2} } \right)}}} \right)} \cos\;h \left( {{\frac{n(1 - 2\delta \gamma )}{{2\delta^{2} }}}} \right)\sec h\left( {{\frac{n}{{2\delta^{2} }}}} \right) $$

(32)

$$ B = 2\pi \delta \sum\limits_{n = 1}^{\infty } {\exp \left( { - {\frac{1}{2}}(2n - 1)^{2} \pi^{2} \delta^{2} } \right)} \sin ((2n - 1)\pi \delta \gamma ){\text{cosec}\;h}(2n - 1\pi^{2} \delta^{2} ) $$

(33)

$$ C = 1 + 2\sum\limits_{n = 1}^{\infty } {{ \exp }( - 2n^{2} \pi^{2} \delta^{2} )} ){ \cos }(2n\pi \gamma \delta ) $$

(34)

$$ D = \sqrt {2\pi } \,{ \exp }\left( {{\frac{{\gamma^{2} }}{2}}} \right) $$

(35)

The variance of y is:

$$ \sigma^{2} = \mu^{\prime}_{2} - \mu^{2} $$

(36)

where the second moment is:

$$ \mu^{\prime}_{2} = \mu^{\prime}_{1} (1 - \delta \gamma ) + {\frac{\delta }{\psi }}\left[ {{\frac{\partial A}{\partial \gamma }} - {\frac{\partial B}{\partial \gamma }} - \mu {\frac{\partial C}{\partial \gamma }}D} \right] $$

(37)

and,

$$ {\frac{\partial A}{\partial \gamma }} = - {\frac{1}{{\delta^{2} }}}\sum\limits_{n = 1}^{\infty } {n\,{\text{exp(}} - n^{2} /(2\delta^{2} ) )} \sin\;h \left( {{\frac{n(1 - 2\delta \gamma )}{{2\delta^{2} }}}} \right)\sec h\left( {{\frac{n}{{2\delta^{2} }}}} \right) $$

(38)

$$ {\frac{\partial B}{\partial \gamma }} = 2(\pi \delta )^{2} \sum\limits_{n = 1}^{\infty } {(2n - 1)e^{{\left( { - {\frac{1}{2}}(2n - 1)^{2} \pi^{2} \delta^{2} } \right)}} } \, \cos ((2n - 1)\pi \delta \gamma )\text{cosec}\;h{\text{((}}2n - 1 )\pi^{2} \delta^{2} ) $$

(39)

$$ {\frac{\partial C}{\partial \gamma }} = - 4\pi \delta \sum\limits_{n = 1}^{\infty } {n\,{\text{exp(}} - 2n^{2} \pi^{2} \delta^{2} )} { \sin }(2n\pi \gamma \delta ) $$

(40)

Similar infinite sum expressions for the standardized third and fourth moments are given in (Flynn 2005a, b).

Subsequently Johnson extended the univariate models to bivariate systems as: S _IJ where the I, J subscripts denote either normal (N), lognormal (L), bounded (B), or unbounded (U) marginal distributions in accordance with the univariate designations. Thus an S _LB would denote a bivariate normal distribution where the x ₁ variable is lognormal (S _L) and the x ₂ variable is distributed as an S _B variate. This leads to a total of 16 possible bivariate normal distributions.

In the general case of bivariate Johnson distributions (S _IJ) we have:

$$ z_{1} = \gamma_{1} + \delta_{1} f_{I} (y_{1} ) {\text{ and }}z_{2} = \gamma_{2} + \delta_{2} f_{J} (y_{2} ) $$

(41)

For the S _BB this is equivalent to:

$$ z_{1} = \gamma_{1} + \delta_{1} \,{ \ln }\left( {{\frac{{y_{1} }}{{1 - y_{1} }}}} \right){\text{and }}z_{2} = \gamma_{2} + \delta_{2} \,{ \ln }\left( {{\frac{{y_{2} }}{{1 - y_{2} }}}} \right) $$

(42)

For an S _BL this would be

$$ \begin{gathered} z_{1} = \gamma_{1} + \delta_{1} \,{ \ln }\left( {{\frac{{y_{1} }}{{1 - y_{1} }}}} \right){\text{ and }}z_{2} = \gamma_{2} + \delta_{2} {\text{ln\;(}}y_{2} )\hfill \end{gathered} $$

(43)

and note in the 2 parameter lognormal case y ₂ = x ₂.

The conditional array distribution of y ₂ given y ₁ is an S _J distribution defined by:

$$ \begin{gathered} \gamma_{2} {\text{ replaced by [}}\gamma_{2} - \rho \{ \gamma_{1} + \delta_{1} f_{I} (y_{1} )\} ] (1 - \rho^{2} )^{{ - {\frac{1}{2}}}} \hfill \\ {\text{and}} \hfill \\ \delta_{2} {\text{ replaced by }}\delta_{2} (1 - \rho^{2} )^{{ - {\frac{1}{2}}}} \hfill \\ \end{gathered} $$

(44)

For S _I = S _B we have:

$$ \begin{gathered} \gamma_{2} \,{\text{replaced by }}\left[ {\gamma_{2} - \rho \left\{ {\gamma_{1} + \delta_{1} \,{ \ln }\left( {{\frac{{y_{1} }}{{1 - y_{1} }}}} \right)} \right\}} \right](1 - \rho^{2} )^{{ - {\frac{1}{2}}}} \hfill \\ {\text{and}} \hfill \\ \delta_{2} \,{\text{replaced by}}\,\delta_{2} (1 - \rho^{2} )^{{ - {\frac{1}{2}}}} \hfill \\ \end{gathered} $$

(45)

An example of the calculation of the conditional mean and variance using the S _BB model is given here. Consider a pipefitter exposed to 2.0 mg/m³ of total particulate; what is the estimated mean and variance of the manganese exposure using the S _BB model. This requires solution of Eq. 31 for the mean and Eq. 36 for the variance. These equations in turn depend upon expressions (32–40) that require the conditional values of γ ₂ and δ ₂ which are given by Eq. 45. This calculation requires the results from the fitting of the univariate distributions for pipefitters in Table 3 where the 1 subscript refers to the total particulate, and the 2 subscript is for manganese:

ε2 = 0.0	ε1 = 0.0235
λ2 = 1.0114	λ1 = 73.6627
γ2 = 2.3265	γ1 = 4.344
δ2 = 0.729	δ1 = 1.254

From the median regression shown in Fig. 5 the correlation coefficient is the square root of the R ² or ρ = 0.712. At this point the information is available to evaluate the Eq. 45 as follows:

$$ \gamma_{2} {\text{ replaced by }}\left[ {\gamma_{2} - \rho \left\{ {\gamma_{1} + \delta_{1} \,{ \ln }\left( {{\frac{{y_{1} }}{{1 - y_{1} }}}} \right)} \right\}} \right](1 - \rho^{2} )^{{ - {\frac{1}{2}}}} $$

$$ \begin{gathered} \gamma_{2} = \left[ {2.3265 - 0.712\left\{ {4.344 + 1.254\,{ \ln }\left( {{\frac{0.0268}{0.9732}}} \right)} \right\}} \right](1.424) \hfill \\ \gamma_{2} = 3.4758 \hfill \\ \end{gathered} $$

$$ \begin{gathered} \delta_{2} {\text{ replaced by }}\delta_{2} (1 - \rho^{2} )^{{ - {\frac{1}{2}}}} \hfill \\ \delta_{2} = 0.729(1.424) = 1.038 \hfill \\ \end{gathered} $$

Equations 35 and 40 are now solved with a computer code using the values:

$$ \begin{aligned} \gamma_{2} = & 3.4758 \\ \delta_{2} = & 1.038 \\ \end{aligned} $$

The mean and standard deviation of y are returned as 0.04972 and 0.05 respectively which are converted to exposures (C) by noting: C = λy + ε or:

$$ \begin{aligned} &\bar{C}_{M} = 1.0114 \times 0.04972 + 0 = 0.050 \\ &\sigma = 1.0114 \times 0.05 + 0 = 0.051 \\ \end{aligned} $$

Thus the conditional mean and standard deviation of manganese exposure for a pipefitter exposed to 2.0 mg/m³ of total; particulate is 0.05 mg/m³ for both (i.e., a 100% coefficient of variation). Note that the conditional median exposure for this case using Eq. 19 is 0.035 mg/m³.