Table 1 Summary of the proposed score test \(S_s\) and its special cases

General method

Score Test \(S_s\)

Maximum of the weighted sum of Z scores

\(\varvec{W }=\varvec{ Z}^{\varvec{'}} \varvec{R}^{\varvec{-1}}\)

\(S_{s} = \varvec{Z}^{\varvec{'}}\varvec{R}^{\varvec{-1}}\varvec{Z}\)

Special cases of the score test \(S_s\)

Special case 1: Sum of Squared Score Statistic (SSU)

Weighted sum of Z scores, weights are Z scores

\(\varvec{W =R^{-1}Z, R=I}\)

\(S_{Q} = \varvec{Z^{'}Z}\)

Special case 2: SNP-set (Sequence) Kernel Association Test (SKAT)

Weighted sum of Z scores, weights are weighted Z scores

\(\varvec{W =R^{-1}Z}, \varvec{R}=diag(a_{1}, \ldots , a_{m}), a_{m} \sim beta(1,25)\)

\(S_{SKAT} = \varvec{Z^{'}R^{-1}Z}\)

Special case 3: PathSPU(2)

Weighted sum of Z scores, weights are eQTL weighted Z scores

\(\varvec{W =R^{-1}Z}, \varvec{R}=diag(a_{1}, \ldots , a_{m}), a_{m}\) are gene derived weights

\(S_{pathSPU(2)} = \varvec{Z^{'}R^{-1}Z}\)

Special case 4: Sum of Powered Score (SPU): Data-adaptive weighted combination test.

Weighted sum of Z scores, weights are function of Z scores

\(\varvec{W} = \varvec{Z}^{\gamma - 1}\)

\(SPU(\gamma ) = \sum _{m=1}^{M}\varvec{Z}_m^{\gamma }, \gamma = 1,2,\ldots ,8,\infty\)

Special case 5: Burden test

Weighted sum of Z scores, weights are all 1s

\(\varvec{W} = (1, \ldots , 1)^{'}\)

\(L_B = L(1, \ldots , 1) = \sum _{m=1}^{M}{\textbf{Z}}_{\textbf{m}}\)

Special case 6: Weighted Sum Statistic

Weighted sum of Z scores, weights are related to MAFs

\(\varvec{W} = (\frac{1}{\sqrt{p_1(1-p_1)}}, \ldots , \frac{1}{\sqrt{p_m(1-p_m)}})^{'}\), where \(p_m\) is the MAF

\(L_W = L(\frac{1}{\sqrt{p_1(1-p_1)}}, \ldots , \frac{1}{\sqrt{p_m(1-p_m)}}) = \sum _{m=1}^{M}\frac{1}{\sqrt{p_m(1-p_m)}}\cdot {\textbf{Z}}_{\textbf{m}}\)