Exposure biomarker conversions from data

In table Exposure biomarker conversions, conversion factors are specified to convert measured biomarkers to the biomarkers of interest. Conversion factors may be dependent on individual properties like age and gender. Specify in table Exposure biomarker conversion subgroups age and/or gender specific conversion factors.

For age, specify the lower bound of the age interval (in years) of the exposure biomarker conversion subgroup. Individuals belong to a subgroup when the age of the individual is equal or greater than the specified lower bound and smaller than the specified lower age of the next subgroup.

Check option Use exposure biomarker conversion factors subgroup to use age and/or gender specific conversion factors in your assessment.

Column VariabilityDistributionType in table Exposure biomarker conversions is used to specify the variability distribution to sample from. When the column is left empty, the conversion factor itself is used. Currently, four distributions are available.

Uniform distribution

The uniform distribution is defined on the interval [$a$, $b$] where $a$ and $b$ are the minimum and maximum value and $a>0$, $b>0$.

The conversion factor is the nominal value of the uniform distribution, and

$$factor=\frac{a+b}{2}$$

For the variability upper value, $b$ = $upper$ and $a$ is calculated as:

$$a=2\cdot factor-b$$

Lognormal distribution

The lognormal distribution is characterised by parameters $\mu$ and $\sigma$, which are the mean and standard deviation on log-scale.

The conversion factor is the nominal value of the lognormal distribution, $\mu =ln(factor)$. The variability upper value specifies the p95 of the standard lognormal. Parameter $\sigma$ is calculated as:

$$\sigma =\frac{ln(upper)-\mu }{1.645}$$

Beta distribution

The standard beta distribution is defined on the interval (0, 1) and is usually characterised by two parameters $a$ and $b$, with $a>0$, $b>0$ (see e.g. Mood et al. (1974)). Alternatively, it can be parameterised by the mean (conversion factor)

$$\mu =a/(a+b)\in (0,1)$$

and the variance

$${\sigma }^2=ab/(a+b+1{)}^{-1}(a+b{)}^{-2}$$

Note that:

$$\frac{ab}{(a+b{)}^2(a+b+1)}=\frac{\mu (1-\mu )}{(a+b{)}^2(a+b+1)}<\frac{\mu (1-\mu )}{1}=\mu (1-\mu )\in (0.{0.5}^2)$$

The conversion factor is the nominal value of the beta distribution, here the mean. The variability upper value is the variance of the beta distribution. The equations solved for $a$ and $b$ show that:

$$a=(\frac{1-\mu }{{\sigma }^2}-\frac{1}{\mu })\cdot {\mu }^2$$

and

$$b=a\cdot (\frac{1}{\mu }-1)$$

InverseUniform distribution

The inverse uniform distribution is defined on the interval [$a$, $b$] where $a$ and $b$ are the minimum and maximum value and $a>0$, $b>0$.

The conversion factor is the median (nominal) value of the inverse uniform distribution, and the median is equal to

$$factor=\frac{2}{a+b}$$

The variability upper value is equal to $\frac{1}{a}$, so

$$a=\frac{1}{upper}$$

and

$$b=\frac{2}{factor}-a$$

Uniform distributed values are drawn from the interval [$a$, $b$] and inverted. See e.g. Invers distribution and Inverse transform.