. Author manuscript; available in PMC: 2018 May 2.

Published in final edited form as: Stat Med. 2017 Jun 19;36(21):3334–3360. doi: 10.1002/sim.7333

Table 3.

Variance formulas in non-degenerate case, unadjusted for estimated parameters.

\hat{σ^{2}}

, ignoring the adjustment for estimated parameters

Requires Adjustment?

{\hat{σ}}_{Δ AUC}^{2}

no tied ranks

\frac{Var ({rank}_{e}^{*} (α^{* T} x_{i}) - {rank}_{e} (α^{T} x_{i}))}{n_{0}} + \frac{Var ({rank}_{ne}^{*} (α^{* T} y_{j}) - {rank}_{ne} (α^{T} y_{j}))}{n_{1}}

{\hat{σ}}_{Δ AUC}^{2}

tied ranks

Use DeLong formula[11]

{\hat{σ}}_{{NRI}_{> 0}}^{2}

\frac{{\hat{p}}_{ne}^{up} (1 - {\hat{p}}_{ne}^{up})}{n_{0}} + \frac{{\hat{p}}_{e}^{up} (1 - {\hat{p}}_{e}^{up})}{n_{1}}

{\hat{σ}}_{NRI (r)}^{2}

\frac{{\hat{p}}_{ne}^{up} + {\hat{p}}_{ne}^{down} - {({\hat{p}}_{ne}^{up} - {\hat{p}}_{ne}^{down})}^{2}}{n_{0}} + \frac{{\hat{p}}_{ev}^{up} + {\hat{p}}_{ev}^{down} - {({\hat{p}}_{ev}^{up} - {\hat{p}}_{ev}^{down})}^{2}}{n_{1}}

{\hat{σ}}_{3 c NRI}^{2}

\frac{4 ({\hat{p}}_{ne}^{2 up} + {\hat{p}}_{ne}^{2 down}) + {\hat{p}}_{ne}^{1 up} + {\hat{p}}_{ne}^{1 down} - {(2 ({\hat{p}}_{ne}^{2 up} - {\hat{p}}_{ne}^{2 down}) + {\hat{p}}_{ne}^{1 up} - {\hat{p}}_{ne}^{1 down})}^{2}}{n_{0}} + \frac{4 ({\hat{p}}_{ev}^{2 up} + {\hat{p}}_{ev}^{2 down}) + {\hat{p}}_{ev}^{1 up} + {\hat{p}}_{ev}^{1 down} - {(2 ({\hat{p}}_{ev}^{2 up} - {\hat{p}}_{ev}^{2 down}) + {\hat{p}}_{ev}^{1 up} - {\hat{p}}_{ev}^{1 down})}^{2}}{n_{1}}

Yes

{\hat{σ}}_{IDI}^{2}

\frac{Var (Δ predp (x_{i}))}{n_{0}} + \frac{Var (Δ predp (y_{j}))}{n_{1}}

Yes