Chronic exposure as a function of covariates

The intake frequency and transformed intake amounts may be modelled as a function of covariates. MCRA allows one covariable and/or one cofactor.

Table 159 Intake frequencies and amounts, modelled as a function of covariates.

	Frequencies	Amounts
cofactor	$\mathit{logit}(\pi )={\beta }_{0l}$	$\mathit{transf}(y_{ij})={\beta }_{0l}+c_i+u_{ij}$
covariable	$\mathit{logit}(\pi )={\beta }_0+{\beta }_{1}f(x_1;\mathit{df})$	$\mathit{transf}(y_{ij})={\beta }_0+{\beta }_{l}f(x_1;\mathit{df})+c_i+u_{ij}$
both	$\mathit{logit}(\pi )={\beta }_{0l}+{\beta }_{1}f(x_1;\mathit{df})$	$\mathit{transf}(y_{ij})={\beta }_{0l}+{\beta }_{l}f(x_1;\mathit{df})+c_i+u_{ij}$
interaction	$\mathit{logit}(\pi )={\beta }_{0l}+{\beta }_{1l}f(x_1;\mathit{df})$	$\mathit{transf}(y_{ij})={\beta }_{0l}+{\beta }_{1l}f(x_1;\mathit{df})+c_i+u_{ij}$

Here $l=1\cdots L$ and $L$ is the number of levels of the cofactor, $y_{ij}$ , the intake amount, $x_1$ is the covariable, $f$ is a polynomial function with the degrees of freedom $\mathit{df}$, $c_i$ and $u_{ij}$ are the individual effect and interaction effect, respectively. These effects are assumed to be normally distributed $N(0,{\sigma }_{between}^2)$ resp. $N(0,{\sigma }_{within}^2)$. The degree of the function is determined by backward or forward selection. In the output, the usual intake is displayed for a specified number of values of the covariable and/or the levels of the cofactor.