Chronic exposure assessment, daily consumed foods

Model based usual intake

Foods are consumed on a daily basis.

For individual i on day j let Yij denote the 24 hour recall of a food (i=1n;j=1ni). In most cases within-individual random variation is dependent on the individual mean and has a skewed distribution. It is therefore customary to define a one-way random effects model for Yij on some transformed scale

Yij=g(Yij)=μi+bi+wij

with biN(0,σb2) and wijN(0,σw2)

Note that bi represents variation between individuals and wij represents variation within individuals between days.

The mean μi may depend on a set of covariate Zi=(Zi1,,Zip):

μi=β0+β1tZi

where β0 and β1 are regression coefficients.

The usual intake Ti for an individual i is defined as the mean consumption over many many days. This assumes that the untransformed intakes Yij are unbiased for true usual intake rather than the transformed intakes Yij. In mathematical terms Ti is the expectation of the intake for this individual where the expectation is taken over the random day effect:

Ti=Ew[g1(μi+bi+wij)|bi]=F(bi)

Model based usual intake on the transformed scale

For the model based usual intake first note that the conditional distribution

(μi+bi+wij|bi)N(μi+bi,σw2)

It follows that the usual intake Ti is given by

Ti=Ew[g1(μi+bi+wij|bi)]=g1(μi+bi+wij)12πσw2exp(w22σw2)dw

Model based using a logarithmic transformation

For the logarithmic transform the usual intake Ti can be written in closed form using the formula for the mean of the lognormal distribution:

Ti=exp(μi+bi+σw2/2)

In this case Ti follows a log-normal distribution with mean μi+σw2/2 and variance σb2. This fully specifies the usual intake distribution, e.g. the mean and variance of the usual intake are given by

μiT=E[Ti]=exp(μi+σw2/2+σb2/2)
σiT2=Var[Ti]=[exp(σb2)1]exp(2μi+σw2+σb2)

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

Ti1πj=1Nwj(μi+bi+2σwxj)p

with p the inverse of the power transformation. A similar approximation can be used for the Box-Cox transformation. There can be a small problem with Gauss-Hermite integration. The summation term (μi+bi+2σwxj)p can not be calculated when the factor between round brackets is negative and the power p is not an integer. This can happen when (μi+bi) is small relative to the between day standard error σw. In that case the corresponding term is set to zero. This is not a flaw in the numerical method but in the statistical model since the model allows negative intakes on the transformed scale which cannot be transformed back to the natural scale. The mean and variance of Ti can be approximated again by using Gauss-Hermite integration:

μiT=E[Ti]=1πk=1Nwk1πj=1Nwj(μi+2σwxj+2σbxk)
σiT=Var[Ti]=1πk=1Nwk[1πj=1Nwj(μi+2σwxj+2σbxk)]2μT2

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 bi for many individuals and then approximate Ti for these individuals. The Ti values then form a sample form the usual intake distribution.

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 bi for every individual.

In the one-way random effects model the Best Linear Unbiased Prediction for (μi+bi) is given by

BLUPi=μi+(Y¯iμi)(σb2σb2+σw2/ni)

in which Y¯i is the mean of the transformed intakes for individual i. BLUPs have optimal properties for some purposes, but not for the purpose of representing the variation σb2 between individuals. This can be seen by noting that

Var(Y¯i)=σb2+σw2/ni

and thus

Var(BLUPi)=(σb4σb2+σw2/ni)

which is smaller than the between individual variance σb2. As an alternative a modified BLUP can be defined by means of

modifiedBLUPi=μi+(Y¯iμi)(σb2σb2+σw2/ni)

which has the correct variance σb2 and also the correct mean μi. However these optimal properties disappear when modified BLUPs are directly backtransformed to the original scale.

Model assisted using a logarithmic transformation

For the logarithmic transformation the usual intake Ti follows a log-normal distribution with mean μi+σw2/2 and variance σb2. If we can construct a BLUP like stochastic variable with the same mean and variance, then this variable be an unbiased predictor with the correct variance. It is easy to see that the following variable has the same distribution as Ti

modelassistedBLUPi=μi+σw22+(Y¯iμi)(σb2σb2+σw2/ni)

So the model assisted individual intake exp(modelassistedBLUPi) has the same distribution as the usual intake and is thus the best predictor for usual intake.

Kipnis et al. (2009) employs the conditional distribution of bi given the observations Yi1,,Yini to obtain a prediction. First note that

(bi|Yi1,,Yini)=(bi|Yi1,,Yini)=(bi|Y¯i)

Since all distributions in the one-way random effects model are normal it follows that:

(bi,Y¯i)BivariateNormal(0,μi,σb2,σb2+σw2/ni,σb2)

where the last parameter represents the covariance between bi and Y¯i. It follows that the conditional distribution

(bi|Y¯i)N(μc,σc2)

with

μc=σb2σb2+σw2/ni(Y¯iμi)

and

σc2=σb2σw2/niσb2+σw2/ni

A prediction for the usual intake Ti=F(bi) is then obtained by the expectation

E[F(bi)|Y¯i]=F(b)ϕ(b;μc,σc2)db

For the logarithmic transform F(bi)=exp(μi+bi+σw2/2) and the expectation reduces to

E[F(bi)|Y¯i]=exp(μi+μc+σc2/2+σw2/2)

which is a function of Y¯i through μc. To obtain the mean and variance of the prediction note that

μi+μc+σc2/2+σw2/2N(μi+σb2σw2/ni2(σb2+σw2/ni)+σw22,σb4σb2+σw2/ni)

It follows that the expectation of the prediction equals

E[E[F(bi)|Y¯i]]=exp(μi+σb2σw2/ni2(σb2+σw2/ni)+σw22+σb42(σb2+σw2/ni))
=exp(μi+σb22+σw22)

which equals the mean of the usual intake. However the variance of the prediction equals

Var[E[F(bi|Y¯i]]=[exp(σb4(σb2+σw2/ni))1]exp(2μi+σb2+σw2)

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 Ti=F(bi) by Gauss-Hermite integration:

F(bi)=Ti1πj=1Nwi(μi+bi+2σwxi)p

Secondly again use Gauss-Hermite to approximate the expectation of the conditional distribution giving the prediction Pi.

Pi=E[F(bi)|Y¯i]=F(bi)ϕ(b;μc,σc2)db1πk=1Nwkj=1Nwj(μi+μc+2σwxj+2σcxk)p

which is a function of Y¯i through μc. It is likely that the thus obtained predictions Pi have a variance that is too small. If we would know the mean μiP and variance σiP2 of the predictions, the predictions could be linearly rescaled to have the correct mean μiT and variance iT2. The mean and variance of the prediction can be calculated using Gauss-Hermite integration.

μiP=1πl=1Nwl1πk=1Nwkj=1Nwj(μi+2σb2σb2+σw2/nixl+2σwxj+2σcxk)p
σiP2=1πl=1Nwl[1πk=1Nwkj=1Nwj(μi+2σb2σb2+σw2/nixl+2σwxj+2σcxk)p]2μiP2

The proposed prediction then equals

Pi=μiT+σiTσiP(PiμiP)