Parameter Types

Splines

B-splines play the role of basis functions, so that any function in the space can be expressed as a linear combination of B-splines.

\[ f(x)=\sum_i \alpha_i B_i(\hat{x}) \] where \(\alpha_i\) is the height coefficient for spline \(i\), \(\hat{x}\) is the rescaled position \(\hat{x}=\frac{x-k_i-\frac{1}{2}\omega}{\omega}\) where \(k_i\) is the position of knot \(i\) and \(\omega\) is the spline width. The basis spline function is given by

\[ B(x) = \left. \begin{cases} \dfrac{x^2}{2} & 0 < x \leq 1 \\ \dfrac{-2x^2+6x-3}{2} & 1 < x \leq 2 \\ \dfrac{x^2-6x+9}{2} & 2 < x \leq 3 \\ 0 & x \leq 0 \text{ and } x>3 \end{cases} \right\} \]

Trend Modifiers

Some model parameters include time trends (e.g. with a year variable) or age trends using logistic (or sometimes linear) regression models: \(\beta_0+\beta_1*x\). To account for the potential of non-linear trends we include an exponent \(\alpha\) to modify the trend variable: \(\beta_0+\beta_1*x^\alpha\) which is fit via calibration. We constrain \(\alpha \le 1\) to ensure that the trend relationship is either linear (\(\alpha=1\)) or concave (\(\alpha<1\)) to model potential attenuation of trends over time (or by age). This approach maintains a monotonic trend relationship while helping to guard against extrapolating to unreasonable levels.