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This analysis employed a cross-basis function that describes a bidimensional association between temperature and number of lag days (Gasparrini et al. 2010).
We extended these models to time-varying DLNMs by including a linear interaction between time and the cross-basis variables.
The cross-basis is specified by the choice of two basis, one for each dimension, among a set of possible options such as splines, polynomials, or step functions.
The simpler lag-basis for DLMs in (1) is a special case of the more complex cross-basis for DLNMs in (2).
These models may be fitted through common regression techniques with the inclusion of cross-basis matrix W in the design matrix.
The definition of identical cross-basis functions in all the locations can be problematic in the presence of substantially different exposure ranges.
The set of v x × v ℓ = 20 coefficients of the cross-basis variables with associated (co variance matrices, estimated in each region, are then reduced.
This modelling class is based on the definition of a cross-basis, a bi-dimensional space of functions describing the association along the spaces of predictor and lags.
The temperature-mortality relationship is modelled through the same cross-basis used for the full two-stage analysis illustrated later, composed by two B-spline bases.
These models are defined by 16 cross-basis variables obtained by the combination of the exposure response and lag response functions, with 4 df each.
Also, alternative parameterization of the cross-basis functions may reduce the correlation pattern in the first stage and avoid estimation problems in the second-stage multivariate model.
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