. 2023 Nov 10;39(12):btad680. doi: 10.1093/bioinformatics/btad680

Table 2.

Isotonic calibration.^a

\begin{matrix} (i) & x_{°, 1}, z_{1, k} & x_{°, 2}, z_{2, k} & \dots & x_{°, q - 1}, z_{q - 1, k} & x_{°, q}, z_{q, k} \\ (ii) & x_{°, (1)} & x_{°, (2)} & \dots & x_{°, (q - 1)} & x_{°, (q)} \\ (iii) & \begin{matrix} w_{°, 1} = \\ x_{°, (1)} \end{matrix} & \begin{matrix} w_{°, 2} = \\ x_{°, (1)} + x_{°, (2)} \end{matrix} & \dots & \begin{matrix} w_{°, q - 1} = \\ x_{°, (1)} + \dots + x_{°, (q - 1)} \end{matrix} & \begin{matrix} w_{°, q} = \\ x_{°, (1)} + \dots + x_{°, (q)} \end{matrix} \\ {(iii)}^{*} & {\hat{δ}}_{1, k} & {\hat{δ}}_{2, k} & \dots & {\hat{δ}}_{q - 1, k} & {\hat{δ}}_{q, k} \\ {(ii)}^{*} & \begin{matrix} {\hat{γ}}_{(1), k} = \\ {\hat{δ}}_{1, k} + \dots + {\hat{δ}}_{q, k} \end{matrix} & \begin{matrix} {\hat{γ}}_{(2), k} = \\ {\hat{δ}}_{2, k} + \dots + {\hat{δ}}_{q, k} \end{matrix} & \dots & \begin{matrix} {\hat{γ}}_{(q - 1), k} = \\ {\hat{δ}}_{q - 1, k} + {\hat{δ}}_{q, k} \end{matrix} & \begin{matrix} {\hat{γ}}_{(q), k} = \\ {\hat{δ}}_{q, k} \end{matrix} \\ {(i)}^{*} & {\hat{γ}}_{1, k} & {\hat{γ}}_{2, k} & \dots & {\hat{γ}}_{q - 1, k} & {\hat{γ}}_{q, k} \end{matrix}

\begin{matrix} (i) & x_{°, q + 1}, z_{q + 1, k} & x_{°, q + 2}, z_{q + 2, k} & \dots & x_{°, p - 1}, z_{p - 1, k} & x_{°, p}, z_{p, k} \\ (ii) & x_{°, (q + 1)} & x_{°, (q + 2)} & \dots & x_{°, (p - 1)} & x_{°, (p)} \\ (iii) & \begin{matrix} w_{°, q + 1} = \\ x_{°, (q + 1)} + \dots + x_{°, (p)} \end{matrix} & \begin{matrix} w_{°, q + 2} = \\ x_{°, (q + 2)} + \dots + x_{°, (p)} \end{matrix} & \dots & \begin{matrix} w_{°, p - 1} = \\ x_{°, (p - 1)} + x_{°, (p)} \end{matrix} & \begin{matrix} w_{°, p} = \\ x_{°, (p)} \end{matrix} \\ {(iii)}^{*} & {\hat{δ}}_{q + 1, k} & {\hat{δ}}_{q + 2, k} & \dots & {\hat{δ}}_{p - 1, k} & {\hat{δ}}_{p, k} \\ {(ii)}^{*} & \begin{matrix} {\hat{γ}}_{(q + 1), k} = \\ {\hat{δ}}_{q + 1, k} \end{matrix} & \begin{matrix} {\hat{γ}}_{(q + 2), k} = \\ {\hat{δ}}_{q + 1, k} + {\hat{δ}}_{q + 2, k} \end{matrix} & \dots & \begin{matrix} {\hat{γ}}_{(p - 1), k} = \\ {\hat{δ}}_{q + 1, k} + \dots + {\hat{δ}}_{p - 1, k} \end{matrix} & \begin{matrix} {\hat{γ}}_{(p), k} = \\ {\hat{δ}}_{q + 1, k} + \dots + {\hat{δ}}_{p, k} \end{matrix} \\ {(i)}^{*} & {\hat{γ}}_{q + 1, k} & {\hat{γ}}_{q + 2, k} & \dots & {\hat{γ}}_{p - 1, k} & {\hat{γ}}_{p, k} \end{matrix}

The aim is to (i) estimate the effects of the features under sign and order constraints determined by q negative (top) and p–q non-negative (bottom) prior effects, i.e. estimate $γ_{1 k}, \dots, γ_{p k}$ for $x_{1}, \dots, x_{p}$ under ${\hat{γ}}_{j k} = 0 | z_{j k} = 0$ , ${\hat{γ}}_{j k} \geq 0 | z_{j k} > 0, {\hat{γ}}_{j k} \leq 0 | z_{j k} < 0, {\hat{γ}}_{j k} \geq {\hat{γ}}_{l k} | z_{j k} \geq z_{l k}$ , and ${\hat{γ}}_{j k} \leq {\hat{γ}}_{l k} | z_{j k} \leq z_{l k}$ . This can be solved by (ii) estimating the effects of the features ordered by the co-data under sign and order constraints, i.e. estimate $γ_{(1), k}, \dots, γ_{(p), k}$ for $x_{(1)}, \dots, x_{(p)}$ under ${\hat{γ}}_{(j), k} \leq 0 | j \leq p, {\hat{γ}}_{(j), k} \geq 0 | j > p$ , and ${\hat{γ}}_{(1), k} \leq \dots \leq {\hat{γ}}_{(p), k}$ . This in turn can be solved by (iii) estimating the effects of the combined features under sign constraints, i.e. estimate $δ_{1}, \dots, δ_{p}$ for $w_{1}, \dots, w_{p}$ under ${\hat{δ}}_{j} \leq 0 | j \leq p$ and ${\hat{δ}}_{j} \geq 0 | j > p$ . Our algorithm receives the original features and the prior effects (i), orders the features by the prior effects (ii), combines the features (iii), estimates the effects of the combined features (iii) $^{*}$ , calculates the estimated effects of the ordered features (ii) $^{*}$ , and returns the estimated effects of the original features (i) $^{*}$ . $^{*}$ A row with an asterisk contains the estimates for the features in the row with the same number but without an asterisk.