Chronic exposure assessment, daily consumed foods
Model based usual intake
Foods are consumed on a daily basis.
For individual
with
Note that
The mean
where
The usual intake
Model based usual intake on the transformed scale
For the model based usual intake first note that the conditional distribution
It follows that the usual intake
Model based using a logarithmic transformation
For the logarithmic transform the usual intake
In this case
Model based using a power transformation
For the power transformation the integral can be approximated by means of N-point Gauss-Hermite integration. This results in the following usual intake
with
An alternative method for obtaining model based usual intakes for the power transformation employs a Taylor series expansion for the power, see e.g. Kipnis et al. (2009). This is however less accurate than Gauss-Hermite integration. For the power transformation simulation is required to derive the usual intake distribution: simulate a random effect
Model assisted usual intake on the transformed scale
The model assisted approach employs a prediction for the usual intakes of every individual in the study. This requires a prediction of the individual random effect
In the one-way random effects model the Best Linear Unbiased Prediction for
in which
and thus
which is smaller than the between individual variance
which has the correct variance
Model assisted using a logarithmic transformation
For the logarithmic transformation the usual intake
So the model assisted individual intake
Kipnis et al. (2009) employs the conditional distribution of
Since all distributions in the one-way random effects model are normal it follows that:
where the last parameter represents the covariance between
with
and
A prediction for the usual intake
For the logarithmic transform
which is a function of
It follows that the expectation of the prediction equals
which equals the mean of the usual intake. However the variance of the prediction equals
Which is less than the variance of the usual intake. The approach of Kipnis et al. (2009) will therefor result in too much shrinkage of the model assisted usual intake.
Model assisted using a power transformation
For the power transformation a model assisted BLUP with optimal properties, as derived above, cannot be constructed. The approach of Kipnis et al. (2009) can however be used to obtain a prediction in the following way. First approximate
Secondly again use Gauss-Hermite to approximate the expectation of the conditional distribution giving the prediction
which is a function of
The proposed prediction then equals