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 89 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.