Because symbiotic rhizobia reproduce clonally and are ultimately released from legume nodules into the soil, nodule number and mass or projected area have been treated as multiplicative components of rhizobial fitness [ 67, 129, 132- 134].
In this way, the between-study variance is modelled, not as a multiplicative component, but, as an additive component of between-study variance to the within-study variance (se i ): (1c) y i = α + β × s e i + μ i + ε i weighted by 1 s e i 2 + τ 2 where μ i ~ N (0, τ 2 ) & ε i ~ N (0, s e i 2 ).
The omission of the multiplicative component (ie, if a pure additive model was used) led to an estimation of the additive component that might be biased, overestimating the effect of the covariate on excess mortality.
Although alternative additive-multiplicative models of risk could be developed by considering other functions for the multiplicative component (e.g. exp), we have focused on the expit function because of its mathematical advantages.
We can explain this result by the fact that the value of the additive component was very small relative to the multiplicative component.
The ε ij terms account for residual variation (interaction) not explained by the multiplicative component.
Particularly for age, the additive component became higher when age increased, whereas the inverse phenomenon was observed for the multiplicative component (even if it was nonsignificant).
Alternatively, a model combining both excess and relative mortality gives the possibility of testing the additive and multiplicative components and then of reducing the model to the most appropriate one.
The study designs a methodology to separate the observed change in ratings into two multiplicative components: one associated with the deterioration of the banks' solvency itself and another associated with the change in the agencies' valuation criteria.
This model has the advantage of giving the possibility of testing the additive and multiplicative components and then to reduce the model to the most appropriate one.
