Evaluating methods for Lasso selective inference in biomedical research: a comparative simulation study

Table 3 Primary performance measures investigated in this study

Measure	Definition	Approximation by simulation
Coverage	\({\mathbb{P}}\left[{\beta }_{.,\widehat{M}}\in C{I}_{.,\widehat{M}}\right]\)	\(\frac{{\sum }_{j\in {M}_{F}}{\sum }_{M\subseteq {M}_{F}}{\sum }_{s\in S}{\mathbb{I}}\left[{\widehat{M}}_{s}=M\wedge {\beta }_{j,M}\in C{I}_{j,M}\right]}{{\sum }_{j\in {M}_{F}}{\sum }_{s\in S}{\mathbb{I}}\left[j\in {\widehat{M}}_{s}\right]}\)
Power	\({\mathbb{P}}\left[{\beta }_{.,\widehat{M}}\in C{I}_{.,\widehat{M}}\|{\beta }_{.,\widehat{M}}\ne 0\right]\)	\(\frac{\sum_{j\in {M}_{F}}{\sum }_{M\subseteq {M}_{F}}{\sum }_{s\in S}{\mathbb{I}}\left[{\widehat{M}}_{s}=M\wedge 0\notin C{I}_{j,M}\right]}{{\sum }_{j\in {M}_{F}}{\sum }_{s\in S}{\mathbb{I}}\left[j\in {\widehat{M}}_{s}\wedge {\beta }_{j,{\widehat{M}}_{s}}\ne 0\right]}\)
Type 1 error	\({\mathbb{P}}\left[{\beta }_{.,\widehat{M}}\in C{I}_{.,\widehat{M}}\|{\beta }_{.,\widehat{M}}=0\right]\)	\(\frac{\sum_{j\in {M}_{F}}{\sum }_{M\subseteq {M}_{F}}{\sum }_{s\in S}{\mathbb{I}}\left[{\widehat{M}}_{s}=M\wedge 0\notin C{I}_{j,M}\right]}{{\sum }_{j\in {M}_{F}}{\sum }_{s\in S}{\mathbb{I}}\left[j\in {\widehat{M}}_{s}\wedge {\beta }_{j,{\widehat{M}}_{s}}=0\right]}\)

We denote the set of all iterations of a simulation scenario by \(S=\{1,\dots ,{n}_{sim}\}\). The full model using all predictors is written as \({M}_{F}=\{1,\dots ,p\}\), the selected model in a specific iteration \(s\) is written as \({\widehat{M}}_{s}\). By the use of \({\mathbb{I}}[.]\) we denote the indicator function for the event specified between square brackets. Note that for methods without variable selection, the estimands reduce to the usual definitions of frequentist properties. More details on the derivation of the approximation in the simulation are given in the Supplementary Material Sect. 3.1

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