Modeling Exposure Mixtures

Hey Jérôme, me again :slight_smile:

I posted this else where but didn’t get a strong answer one way or the other. Keen on your thoughts.

ACGIH suggests summing the concentration / OEL ratios for additive mixtures:

Σ Cn/OELn > 1 = Unacceptable
Σ Cn/OELn < 1 = Acceptable

I wondered if there is any statistical issue with modelling the summed ratio, in addition to individual contaminate analysis. I had two initial thoughts (remember, I’m still self teaching stats - doesn’t come naturally yet!).

  1. Calculate the sum ratio and treat it like any other analysis with an OEL of 1.

To trial this, I simulated a dataset of 25 samples, each with two chemicals. Both chemical’s were sampled from a log-normal distribution. Sigma of the Chemical 2 distribution was x0.2 chemical 1. I think you would expect exposures to be highly correlated like that - say the chemical source is the same process.

The chemical 1, chemical 2, and summed ratios samples were all log-normal (pretty much).

Correlation chem1 ~ chem 2 = .90 as expected.

If you were using exceedance fraction as your measure for acceptability - it would go from good to bad.

Does that make sense? Is there a better way of attacking this problem? Seems like this would be very important for larger and larger mixtures.

  1. Model chem1, chem2 … chemN is an GLM

y = ß° + ß1X1 + ß2X2 … + Bi + e

Kind of treat each chemical like a ‘exposure determinant’.

But it feels weird because I think the chemicals would be fixed effects given you would know exactly what the results are, unlikely say the effect of an LEV system being off or on.

Also because you’d expect X1 (chemical 1) ~ X2 (chemical 2) to be highly correlated in most cases, this LM does work so well?

As you can tell, I’m a bit confused. Any help would be greatly appreciated as always.

John :smiley:

Hi john,

Always good questions, giving me an excuse to escape admin work.

Here some reflections :

Let’s call the sum of concentration/OEL ratios HI for hazard index, and HQi each individual ratio of concentation/OEL.

The theory is that HI is basically a sum of lognormal distributions : the concentrations are lognormal, the OELs are constants, so the HQs are lognormal. So the distribution of HI is the sum of lognormal distributions that might or might not be correlated depending on the process.

I do not have the stats to show that this sum is supposed to be lognormal, but empirical evaluation suggests so ( Kumagai, S, & Matsunaga, I. (1992). Fluctuation of occupational exposures indices to mixtures. Annals of occupational hygiene, 36(2), 131‑143.)

The fact that HI suggests overexposure but not the individual analyses is OK : it is the purpose of this index, showing potential risk due to addition of toxic effects.

The fact that the chemical levels are correlated does not pose any challenge in my opinion : the values of HIs are independent within your sample, and the values for each chemical are also independant within each chemical. I see nothing that eg would cause underestimation of GSDs in your table.

In such an analysis I would look at exceedance for the HIs, and also evaluate to what extent one of the individual chemicals majorly drives the value of the HI.

A devious simulation geek could maybe find a particular exemple where exceedance would be too high for the individuall chemicals, but OK for the HI, which would seem strange ( weird correlation pattern which would reduce drastically the HI variability), but I am not sure it would serve any useful purpose.

For your statistical model, I am not sure I follow, what is Y in this model ?

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So because HIs are expected to be log-normal (and can be tested for log normality on case by case basis, you can assess HIs just like HQs or chemicals directly (e.g using ExpoStats). Right?

In terms of how much each individual chemical drives the HI, isn’t that obvious by assessing the individual chemicals? If not, what method would you use to assess that? I thought the ExpoStats Tool3 but seems like that can only be used for categorical determinants.

The paper you referenced calculating distributions for a “Specific Factors” and “Common Factor”. But in truth, I’m not 100% sure how to interpret that.

The Y in the “statistical model” was meant to represent the distributions of HIs.

Just like a one-way mixed linear model to explain between / within- worker variance is:

Y = μ + b + e
Y = logged concentrations
mu = AM
b = random effect of between workers
e = error (within worker)

I thought you could someone incorporate the HQi’s to create a similar model (X1, X2 etc.). But really don’t know what I’m talking about.

Is there anything wrong with putting HI’s into ANOVA for between worker variance analysis? I feel like no, but multiple chemical distributions underlying the HI makes me feel … unsure about things.

Thanks. Hope what I wrote makes sense. :rofl:


I don’t have anything to add to this, but I was wondering this same thing and I am very interested in this discussion, thank you!

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Hey Mike.

I’ve been given a bunch of articles to read through from Peter K (who visits these forums) on this topic. It’ll take me a few days to chunk through them but will let you know what I find!