. Author manuscript; available in PMC: 2016 May 1.

Published in final edited form as: Genet Epidemiol. 2015 Apr 6;39(4):306–316. doi: 10.1002/gepi.21899

Table 1.

A summary of the PRS test and its modifications based on sample splitting.


Data	D ={(Y_i, X_i)\|i = 1, …, n} = D₁ υ D₂ split to two parts D₁ and D₂.

Model	Logit [P_r(Y_i = 1)] = β_M,0 + X_ijβ_M,j, i ∈ D₁.

Output	$w_{j} = w_{j} (D_{1}) = {\hat{β}}_{M, j} (D_{1}) I (p_{j} (D_{1}) < P_{T}), w = {(w_{1}, \dots, w_{k})}^{'}$ ,
	$X_{w, i j} = w_{j} X_{i j}, X_{w, i} = {(X_{w, i 1}, \dots, X_{w, i k})}^{'}$ .

Model	$Logit [P_{r} (Y_{i} = 1)] = β_{0} + Σ_{j = 1}^{k} X_{w, i j} β_{j}, i ∊ D_{2}$ .

Output	$U_{w} = {(U_{w, 1}, \dots, U_{w, k})}^{'} = U_{w} (D_{2}) = Σ_{i ∊ D_{2}} X_{w, i} (Y_{i} - {\overset{‒}{Y}}_{(2)}), {\overset{‒}{Y}}_{(2)} = Σ_{i ∊ D_{2}} Y_{i} ∕ ∣ D_{2} ∣$ ,
	$V_{w} = Cov (U_{w}) = {\overset{‒}{Y}}_{(2)} (1 - {\overset{‒}{Y}}_{(2)}) Σ_{i ∊ D_{2}} (X_{w, i} - {\overset{‒}{X}}_{w}) {(X_{w, i} - {\overset{‒}{X}}_{w})}^{'}, {\overset{‒}{X}}_{w} = Σ_{i ∊ D_{2}} X_{w, i} ∕ ∣ D_{2} ∣$ .

Tests	$T_{P R S} \approx T_{P o l y - S u m} = 1^{'} U_{w} ∕ \sqrt{1^{'} V_{w} 1}$ .
	$T_{P o l y - S S U} = U_{w}^{'} U_{w} = Σ_{j = 1}^{k} U_{w, j}^{2}$ .
	$T_{P o l y - S S U w} = Σ_{j = 1}^{k} U_{w, j}^{2} ∕ V_{w, j j}$ .
	$T_{P o l y - U m i n P} = {max}_{j = 1}^{k} U_{w, j}^{2} ∕ V_{w, j j}$ .