Chronic exposure as a function of covariates

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

Table 76 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.