Algorithm 2.

For a pathway consisting of p proteins measured on n samples with p > n − 1, we independently sample B subsets, each contains n − 2 proteins.

For each subset X_b, b = 1, …, B, compute ${\mathrm{\lambda ̂}}_{sel}^b$; then set ${\mathrm{\lambda ̂}}_{sel}=\frac{1}{B}{\sum }_b{\mathrm{\lambda ̂}}_{sel}^b$.

Calculate the observed pathway statistic as $\mathrm{R}\mathrm{H}{\mathrm{T}}_{\mathit{\text{Bootstrap}}}({\mathrm{\lambda ̂}}_{sel})=\frac{1}{B}{\sum }_b\mathrm{R}\mathrm{H}{\mathrm{T}}_b({\mathrm{\lambda ̂}}_{sel})$.

Resample observed log ratios M times, preserving the missing data pattern of X, and generate M null pathways ${\mathbf{X}}_1^0,\dots ,{\mathbf{X}}_M^0$.

For each ${\mathbf{X}}_j^0$, j = 1, …, M, make B subsets of proteins as outlined in Step 1, and calculate $\mathrm{R}\mathrm{H}{\mathrm{T}}_{\mathit{\text{Bootstrap}},j}^0({\mathrm{\lambda ̂}}_{sel})$ as in Step 3.

Calculate p-value by comparing RHT_Bootstrap(λ̂_sel) with ${\{\mathrm{R}\mathrm{H}{\mathrm{T}}_{\mathit{\text{Bootstrap}},j}^0({\mathrm{\lambda ̂}}_{sel})\}}_j$.