A Novel Variable Selection Method for Classification with Application to Single Nucleotide Polymorphism Data

𝑋 𝑌 𝑝 𝑛 𝑛_𝑔 𝑋_{𝑖 𝑖 𝑖 = 1,2, … , 𝑝} 𝑋_{𝑖𝑗 𝑖 𝑗 𝑗 = 1,2, … , 𝑛} 𝐷′_/𝑟2 𝛽 𝜎2 𝜇 Ω 𝑘 𝑑 𝐺 _{= 0,1, … , 𝐺 − 1} 𝑅 _{𝑟 = 1,2, … , 𝑅} 𝑇 _{𝑡𝑟 = {𝑡𝑟,1, … , 𝑡𝑟,𝑛} 𝑟} 𝑌𝑟 𝑦_𝑟 = {𝑦_𝑟,1, … , 𝑦_{𝑟,𝑛𝑟}} 𝑌𝑆𝑁𝑃𝑖 𝑖 𝑖 = 1,2, … , 𝑝 𝑋𝑆𝑁𝑃𝑖 𝑖 𝑖 = 1,2, … , 𝑝 𝑏 _{𝑏 = (𝑏1, … , 𝑏𝑅)} ∅𝑟 ℎ_𝑟−1 _𝑟

𝑥_{𝑟,𝑗 𝑟} 𝑥_{𝑆𝑁𝑃𝑖,𝑗 𝑖} 𝑧_𝑟,𝑗 𝛼_𝑟 𝑟 ℳ𝒱𝒩 𝔻 𝜔_{𝑞 𝑞 = 1,2, … , 𝑄} 𝜑 𝜓 𝜃 𝑓 𝜋_𝑔 𝕔 ℵ _{𝔫 =} 1,2, … , ℵ 𝐿1 𝐿₂ 𝜆

𝑋_𝑖 𝑡 𝑖 𝑡 𝐹𝑠𝑡 𝑖 𝐹𝑠𝑡 𝐹 𝐾 = 3, 5

𝑋₁, 𝑋₂ 𝑋₃

𝑝

𝑛 (𝑝 ≫ 𝑛)

𝑡

𝑡

𝑡

𝜒2

𝑚11 𝑚12 𝑚13

𝑝 (𝑝 ≫ 𝑛)

𝑃𝑟(𝑌 = 1|𝑋) = 𝑒 𝛽0+𝛽𝑖𝑋𝑖 1 + 𝑒𝛽0+𝛽𝑖𝑋𝑖 𝛽₀ 𝛽₁ 𝑋_𝑖 𝑋_𝑖 𝑋_𝑖, 𝑖 = 1,2, … , 𝑝

𝑋_𝑖

𝑡 𝑖 𝑡 𝑔 = 0,1 𝑖 𝑡_𝑖 = 𝑚𝑎𝑥 {|𝑋⃗̅𝑖𝑔− 𝑋⃗̅𝑖| 𝑀_𝑔𝑆_𝑖 , 𝑔 = 0,1, … , 𝐺 − 1} 𝑋⃗̅𝑖𝑔 𝑋⃗̅𝑖 𝑖 𝑔 |𝑋⃗̅𝑖𝑔− 𝑋⃗̅𝑖| 0 ⇒ 𝑋⃗_𝑖(1)= {1,0,0}, ⇒ 𝑋⃗_𝑖(2) = {0,1,0}, 0 ⇒ 𝑋⃗_𝑖(3) = {0,0,1}. 𝑀_𝑔 = √1 𝑛_𝑔+ 1 𝑁 𝑆_𝑖2 = 1 𝑁 − 𝐺∑ ∑ (𝑋⃗𝑖𝑗− 𝑋⃗̅𝑖𝑔) (𝑋⃗𝑖𝑗− 𝑋⃗̅𝑖𝑔) 𝑇 𝑗∈𝑔 𝐺−1 𝑔=0 𝑋⃗𝑖𝑗 𝑖 𝑗 𝑁 𝑛_𝑔 𝑔 𝑆_𝑖 𝑡

𝑝 𝑡 𝑡 𝑡 𝑖 𝑗 𝑛1= 1000 𝑋⃗̅𝑖1 𝑛0= 1000 𝑋⃗̅_𝑖0 𝑁 = 2000 _{𝑋⃗̅𝑖} 𝑆_𝑖 𝑡 𝑡 𝑡

𝑡 𝑡 𝐹 𝑡 𝐹 (𝐹_𝑠𝑡) 𝐹𝑠𝑡 = 𝑉𝑎𝑟_𝑎 𝑎̅ ⋅ 𝑡̅ 𝑎̅ = ∑ 𝑎𝑔 1 𝑔=0 𝑉𝑎𝑟_𝑎 = ∑(𝑎𝑔 − 𝑎̅) 2 2 1 𝑔=0 𝑎 𝑡 𝑎̅ 𝑡̅ 𝑉𝑎𝑟𝑎 𝑎𝑔 𝑔 𝐹𝑠𝑡

𝐹𝑠𝑡 𝑖 𝑗 𝑖 𝑎1 = 14 𝑡1= 543 𝑎0= 2 𝑡0 = 67 𝑁 = 2000 𝑎̅ = 16 𝑡̅ = 610 𝑉𝑎𝑟_𝑎 𝑡 𝐹 𝐹

𝐿2 𝐿(𝛽₀, 𝛽, 𝜆) = −𝑙(𝛽₀, 𝛽) +𝜆 2 ‖𝛽‖2 2 𝑙 𝜆 𝜆 𝛽 glmnet 𝜆 𝜆 ‖𝛽‖₂ 𝐿₂ ‖𝛽‖₂2 _{= ∑ 𝛽} 𝑖2 𝑝 𝑖=1 𝛽 𝐿2 𝐿2 𝐿1

𝐿1

𝑌 = 𝑔 𝑋 = 𝑥 𝑃𝑟(𝑌 = 𝑔|𝑋 = 𝑥) =𝑃𝑟(𝑌 = 𝑔) × 𝑃𝑟(𝑋 = 𝑥|𝑌 = 𝑔) 𝑃𝑟(𝑋 = 𝑥) 𝑃𝑟(𝑌 = 𝑔) 𝑗 𝑔 𝑃𝑟(𝑋 = 𝑥) 𝑥 𝑃𝑟(𝑋 = 𝑥|𝑌 = 𝑔) 𝑋 𝑔 naïvebayes

𝑡 𝐹𝑠𝑡 𝑌 𝑋 𝑌 𝑋 𝐾 𝑥₀ 𝐾 𝑥₀ 𝑁₀ 𝑔 𝑔 𝑃𝑟(𝑌 = 𝑔|𝑋 = 𝑥₀) = 1 𝐾∑𝑗∈𝑁0𝐼(𝑦𝑗 = 𝑔)

𝑥0 𝐾 = 1, 3, 5 class 𝐾 = 1 𝐾 = 3 𝐾 = 5 𝐾 𝐾 = 5 𝐾 = 3, 5 𝐾 = 3 𝐾 = 5 𝑡 𝐹𝑠𝑡

𝛽₀+ 𝛽₁𝑋₁+ 𝛽₂𝑋₂ = 0 𝛽0, 𝛽1 𝛽2 𝑋 = (𝑋1, 𝑋2)𝑇 𝑝 𝛽₀+ 𝛽₁𝑋₁+ 𝛽₂𝑋₂+ ⋯ + 𝛽_𝑝𝑋_𝑝= 0 𝑦_𝑗 ∈ {0,1} 𝛽₀+ 𝛽₁𝑋_𝑗1+ 𝛽₂𝑋_𝑗2+ ⋯ + 𝛽_𝑝𝑋_𝑝 > 𝑀 𝑦_𝑗 = 1

𝛽0+ 𝛽1𝑋𝑗1+ 𝛽2𝑋𝑗2+ ⋯ + 𝛽𝑝𝑋𝑝 < 𝑀 𝑦𝑗 = 0 𝑗 = 1,2, … , 𝑛. 𝑀 e1071 = 𝑀 = 𝑡 𝐹𝑠𝑡

𝑔 𝑔 = 0,1 𝑝 𝑋_𝑖 𝑝 𝑋1, 𝑋2, … , 𝑋𝑝 𝑗, 𝑗 = 1,2, … , 𝑛. 𝑋₁ Ω Ω_𝐿 Ω_𝑅 Ω 𝑋₁ (1) = {Ω𝐿 𝑖𝑓 𝑋𝑖 ∈ (0) Ω_𝑅 𝑖𝑓 𝑋_𝑖 ∈ (1,2) (2) = {Ω𝐿 𝑖𝑓 𝑋𝑖 ∈ (0,1) Ω𝑅 𝑖𝑓 𝑋𝑖 ∈ (2) (3) = {Ω𝐿 𝑖𝑓 𝑋𝑖 ∈ (0,2) Ω𝑅 𝑖𝑓 𝑋𝑖 ∈ (1) 𝑌 = 𝑔 𝑋₂ Ω_𝐿 𝑋₃

Ω𝑅

𝑖(Ω) = 2𝑃𝑟_Ω(1 − 𝑃𝑟_Ω)

𝑃𝑟Ω Ω

rpart

(𝑌 = 0)

𝑡 𝐹𝑠𝑡

𝑲 = 𝟓 𝑡 𝐹𝑠𝑡 𝑲 = 𝟓 𝑡 𝐹𝑠𝑡 𝑲 = 𝟓 𝑡 𝐹𝑠𝑡 𝑲 = 𝟓 𝑡 𝐹𝑠𝑡

𝑡 𝐹

𝑡 𝐹

         

𝑆𝑁𝑅 =𝜎𝑠𝑖𝑔𝑛𝑎𝑙 2 𝜎_{𝑛𝑜𝑖𝑠𝑒}2 𝜎_{𝑠𝑖𝑔𝑛𝑎𝑙}2 𝜎_{𝑛𝑜𝑖𝑠𝑒}2 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑝 𝐺 𝑡 𝑋_𝑖 (𝑖 = 1,2, … , 𝑝) 𝑡

𝑡_𝑖 = 𝑚𝑎𝑥 {|𝑋⃗̅𝑖𝑔− 𝑋⃗̅𝑖| 𝑀_𝑔𝑆_𝑖 , 𝑔 = 0,1, … , 𝐺 − 1} 𝑋⃗̅𝑖𝑔 𝑋⃗̅𝑖 𝑖 𝑔 𝑖 |𝑋⃗̅_𝑖𝑔− 𝑋⃗̅_𝑖| (0 ⇒ 𝑋⃗_𝑖(1) = {1,0,0}, ⇒ 𝑋⃗_𝑖(2) = {0,1,0}, 0 ⇒ 𝑋⃗_𝑖(3)= {0,0,1}). 𝑀_𝑔 = √1 𝑛_𝑔+ 1 𝑁 𝑆_𝑖2 = 1 𝑁 − 𝐺∑ ∑ (𝑋⃗𝑖𝑗− 𝑋⃗̅𝑖𝑔) (𝑋⃗𝑖𝑗− 𝑋⃗̅𝑖𝑔) 𝑇 𝑗∈𝑔 𝐺−1 𝑔=0 𝑋⃗𝑖𝑗 𝑖 𝑗 𝑁 𝑛𝑔 𝑔 𝑆_𝑖 𝑡 𝑝 𝑡 𝑡

𝑡 𝑔( ) 𝑌 𝑋 𝑔(𝐸(𝑌|𝑋)) = 𝑋₁𝛽₁+ 𝑋₂𝛽₂ 𝑋₁𝛽₁ 𝑋₂𝛽₂ 𝑋 = [𝑋1, 𝑋2] 𝛽 = (𝛽1, 𝛽2)𝑇 𝑋1 𝑛 × 𝑝1 𝑋2 𝑛 × 𝑝₂ 𝛽₁ 𝑝₁× 1 𝛽₂ 𝑝₂× 1 𝑝₁+ 𝑝₂ = 𝑝 𝛽̂_𝑚𝑎𝑥 𝛽̂_𝑚𝑎𝑥 ℒ(𝛽̂_𝑚𝑎𝑥|𝑦) 𝛽̂ 𝛽 ℒ(𝛽̂|𝑦)

𝐷𝑒𝑣(𝑦, 𝑋, 𝛽̂) = −2 𝑙𝑜𝑔 ℒ(𝛽̂|𝑦) ℒ(𝛽̂𝑚𝑎𝑥|𝑦) 𝑆𝑁𝑅̂ =𝐷𝑒𝑣(𝑦, 𝑋1, 𝛽̂1) − 𝐷𝑒𝑣(𝑦, 𝑋, 𝛽̂) 𝐷𝑒𝑣(𝑦, 𝑋, 𝛽̂) 𝑋₂ 𝛽̂₂ 𝑋₁ 𝛽̂₁ 𝛽̂₁ 𝛽̂ 𝑌 𝑌 𝛽₀ 𝑋₁ 𝛽₁ 𝑋₁𝛽₁ 𝑋_𝑖 (𝑖 = 1,2, … , 𝑝)

𝑡𝑆𝑁𝑅̂𝑖 = 𝐷𝑒𝑣(𝑦, 𝛽̂₀) − 𝐷𝑒𝑣(𝑦, 𝑋_𝑖, 𝛽̂_𝑖) 𝐷𝑒𝑣(𝑦, 𝑋𝑖, 𝛽̂𝑖) 𝐷𝑒𝑣(𝑦, 𝛽̂₀) 𝛽̂₀ 𝐷𝑒𝑣(𝑦, 𝑋𝑖, 𝛽̂𝑖) 𝛽̂0 𝛽̂𝑖 𝑋𝑖 (𝑖 = 1,2, … , 𝑝) 𝑡𝑆𝑁𝑅̂ 𝑎𝑑𝑗𝑢𝑠𝑡𝑒𝑑 =𝐷𝑒𝑣(𝑦, 𝛽̂0) − 𝐷𝑒𝑣(𝑦, 𝑋𝑖, 𝛽̂𝑖) + 𝑑0− 𝑑𝑖 𝐷𝑒𝑣(𝑦, 𝑋𝑖, 𝛽̂𝑖) + 𝑑𝑖 𝑑₀ 𝑑_𝑖

𝑌 𝑝 𝑋_𝑖 (𝑖 = 1, 2, … , 𝑝) 𝐺 𝑔 = 0,1, … , 𝐺 − 1 𝐺 = 2, 𝑔 = 1 𝑔 = 0 𝑗 (𝑗 = 1,2, … , 𝑛) 𝑔 = 1 𝑖 𝑃𝑟(𝑌_𝑗 = 1|𝑋_𝑖) 𝑃𝑟(𝑌𝑗 = 1|𝑋𝑖) = 𝜋𝑗 = 𝑒𝛽0+𝑋𝑖𝑗𝑇𝛽𝑖 1 + 𝑒𝛽0+𝑋𝑖𝑗𝑇𝛽𝑖 𝑃𝑟(𝑌𝑗 = 0|𝑋𝑖) = 1 − 𝜋𝑗 = 1 1 + 𝑒𝛽0+𝑋𝑖𝑗𝑇𝛽𝑖 𝑋_𝑖𝑗 𝑖 𝑗 𝜃 = (𝛽0, 𝛽1, … , 𝛽𝑝) 𝑇

𝐿(𝜃) = ∑[𝑌_𝑗 𝑙𝑜𝑔 𝜋_𝑗+ (1 − 𝑌_𝑗)𝑙𝑜𝑔(1 − 𝜋_𝑗)] 𝑛 𝑗=1 𝑔(𝜃) = 𝐿(𝜃) − 𝜆 ∑|𝛽_𝑖| 𝑝 𝑖=1 𝛽₀ 𝜆 ∑𝑝_𝑖=1|𝛽_𝑖| 𝜆 𝛽𝑖 glmnet 𝑘 𝑘

𝑘 𝑘 𝑘 𝜔𝑖 𝑘 𝜔𝑖 = ∑ 𝑡𝑆𝑁𝑅̂ 100 𝑘=1 𝑘 𝑘 𝛽_𝑖

𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 = 𝑇𝑃 𝑇𝑃+𝐹𝑁 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑖𝑡𝑦 = 𝑇𝑁 𝑇𝑁+𝐹𝑃 𝑃𝐶𝐶 = 𝑇𝑃 + 𝑇𝑁 𝑇𝑃 + 𝐹𝑁 + 𝑇𝑁 + 𝐹𝑃

𝑃𝑃𝑉 = 𝑇𝑃

𝑇𝑃+𝐹𝑃 𝑁𝑃𝑉 =

𝑇𝑁 𝑇𝑁+𝐹𝑁

𝐴𝐼𝐶 = 𝐷𝑒𝑣(𝑦, 𝑋_𝑖, 𝛽̂_𝑖) + 2𝑑_𝑖 𝑑𝑖 𝑡𝑆𝑁𝑅̂ = 𝐷𝑒𝑣(𝑦, 𝛽̂0) 𝐷𝑒𝑣(𝑦, 𝑋𝑖, 𝛽̂𝑖) − 1 𝑡𝑆𝑁𝑅̂ =𝐷𝑒𝑣(𝑦, 𝛽̂0) 𝐴𝐼𝐶 − 2𝑑_𝑖 − 1 𝑛

𝐵𝐼𝐶 = 𝐷𝑒𝑣(𝑦, 𝑋_𝑖, 𝛽̂_𝑖) + 𝑑 𝑙𝑜𝑔(𝑛)

𝑡𝑆𝑁𝑅̂ = 𝐷𝑒𝑣(𝑦, 𝛽̂0)

𝐵𝐼𝐶 − 𝑑 𝑙𝑜𝑔(𝑛)− 1

𝑅2 = 1 −𝐷𝑒𝑣(𝑦, 𝑋𝑖, 𝛽̂𝑖) 𝐷𝑒𝑣(𝑦, 𝛽̂0)

𝐺 𝐺 = 2, 𝑔 = 1 𝑔 = 0 𝑇. 𝑅 ≥ 1 𝑡𝑟 = (𝑡_𝑟,1, … , 𝑡_{𝑟,𝑛𝑟}) 𝑡_𝑟,1< ⋯ < 𝑡_{𝑟,𝑛𝑟} 𝑟 = 1, … , 𝑅 𝑅 𝑌𝑟 = (𝑌𝑟,1, … , 𝑌𝑟,𝑛𝑟) 𝑡 < 𝑇 𝑔 𝑡 𝑔 = 0 𝑔 = 1 𝑗 (𝑗 = 1, … , 𝑛_𝑟) 𝑟 (𝑟 = 1, … , 𝑅) 𝑌_𝑟,𝑗 ∅_𝑟𝑔 {ℎ_𝑟−1_{𝐸(𝑌 𝑟,𝑗|𝑏, 𝑔)} = 𝑥𝑟,𝑗 𝑔𝑇 𝛼_𝑟𝑔 + 𝑧_𝑟,𝑗𝑔𝑇𝑏_𝑟, 𝑟 = 1, … , 𝑅, 𝑗 = 1, … , 𝑛_𝑟.

ℎ𝑟−1 𝑟 𝑥_𝑟,𝑗𝑔 𝑧_𝑟,𝑗𝑔 𝑔 𝛼_𝑟𝑔 𝑟 𝑔 𝑏 = (𝑏1, … , 𝑏𝑅) 𝑏, 𝑔~ ∑ 𝜔_𝑞𝑔 𝑄𝑔 𝑞=1 ℳ𝒱𝒩(𝜇_𝑞𝑔, 𝔻_𝑞𝑔), ℳ𝒱𝒩(𝜇, 𝔻) 𝜇 𝔻 𝜔_𝑞, (𝑞 = 1, … , 𝑄) 𝑄𝑔 𝜑(∙; 𝜇, 𝔻) 𝜓𝑔: = (𝛼₁𝑔, … , 𝛼_𝑅𝑔, ∅₁𝑔, … , ∅_𝑅𝑔) 𝜃𝑔: =

(𝜔𝑔, 𝜇₁𝑔, … , 𝜇_𝑄𝑔𝑔, 𝔻₁𝑔, … , 𝔻_𝑄𝑔𝑔) 𝑌_𝑟 𝑌_{𝑆𝑁𝑃𝑖}, (𝑖 = 1,2, … 𝑝) 𝑌_{𝑆𝑁𝑃𝑖} 𝑌_{𝑆𝑁𝑃𝑖1} 𝑌_{𝑆𝑁𝑃𝑖2} 𝑌_{𝑆𝑁𝑃1} 𝑌𝑆𝑁𝑃₁ 𝑌𝑆𝑁𝑃₁₁ 𝑌𝑆𝑁𝑃₁₂

𝑥_𝑟,𝑗𝑔 {ℎ𝑟 −1_{𝐸(𝑌 𝑟,𝑗|𝑏, 𝑔)} = 𝑥𝑟,𝑗 𝑔𝑇 𝛼_𝑟𝑔+ 𝑧_𝑟,𝑗𝑔𝑇𝑏𝑟, 𝑟 = 1, … , 𝑅, 𝑗 = 1, … , 𝑛𝑟 ℎ_𝑆𝑁𝑃−1 _𝑖{𝐸(𝑌𝑆𝑁𝑃𝑖,𝑗|𝑔)} = 𝑥𝑆𝑁𝑃𝑖,𝑗 𝑔𝑇 𝛼_𝑆𝑁𝑃 𝑖 𝑔 , 𝑖 = 1, … , 𝑝, 𝑗 = 1, … , 𝑛𝑆𝑁𝑃𝑖 𝑅 𝑥_𝑟,𝑗𝑔𝑇 = (𝑥_𝑟,𝑗𝑔𝑇, 𝑋_𝑆𝑁𝑃 𝑖,𝑗 𝑔𝑇 , … , 𝑋_𝑆𝑁𝑃 𝑝,𝑗 𝑔𝑇 ).

𝑋_𝑆𝑁𝑃 11,𝑗 𝑔 𝑋_𝑆𝑁𝑃 12,𝑗 𝑔 𝜓𝑔 𝜃𝑔 𝑔 𝑃𝑟_{𝑔,𝑛𝑒𝑤} = 𝜋𝑔𝑓̂𝑔,𝑛𝑒𝑤 ∑𝐺−1_𝑔̌=0𝜋_𝑔̃𝑓̂_{𝑔̃,𝑛𝑒𝑤} 𝑔 = 0, … , 𝐺 − 1, 𝑓̂ 𝜋𝑔 = 𝑃𝑟(𝑔), 𝑔 = 0, … , 𝐺 − 1 𝑓𝑔,𝑛𝑒𝑤

ℵ 𝑓𝑔,𝑛𝑒𝑤 𝑓_{𝑔,𝑛𝑒𝑤} 𝑌_𝑛𝑒𝑤 = (𝑦_{𝑛𝑒𝑤,1}, … , 𝑦_{𝑛𝑒𝑤,𝑅}) 𝑓_{𝑔,𝑛𝑒𝑤} 𝑌𝑛𝑒𝑤 𝑓_{𝑔,𝑛𝑒𝑤} 𝑌_𝑛𝑒𝑤 𝑓𝑔,𝑛𝑒𝑤

𝑔 = 0 𝑔 = 1

glm

glmnet

0 50 100 150 200 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0

Number of SNPs in the model based on univariate tSNR ranking

C la s s if ic a ti o n p e rf o rm a n c e PCC AUC NPV PPV Sensitivity Specificity

0 50 100 150 200 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0

Number of SNPs in the model based on cumulative tSNR ranking

C la s s if ic a ti o n p e rf o rm a n c e PCC AUC NPV PPV Sensitivity Specificity

𝑋𝑖 , 𝑖 = 1,2, … , 𝑝 𝑡𝑆𝑁𝑅̂𝑖 𝑡𝑆𝑁𝑅̂𝑖 = 𝐷𝑒𝑣(𝑦, 𝛽̂0) − 𝐷𝑒𝑣(𝑦, 𝑋𝑖, 𝛽̂𝑖) 𝐷𝑒𝑣(𝑦, 𝑋𝑖, 𝛽̂𝑖) 𝐷𝑒𝑣(𝑦, 𝛽̂₀) 𝛽̂₀ 𝐷𝑒𝑣(𝑦, 𝑋_𝑖, 𝛽̂_𝑖) 𝛽̂0 𝛽̂𝑖 𝑖 𝑋𝑖 (𝑖 = 1,2, … , 𝑝)

n = 1,212

n = 303 𝑘

𝜔𝑖 𝑘 𝜔𝑖 = ∑ 𝑡𝑆𝑁𝑅̂ 100 𝑘=1 𝑋1, 𝑋2 𝑋3 𝑘 𝑋1 𝑋2 𝑋3 𝑋1+ 𝑋2+ 𝑋3 𝑋1 𝑋1+ 𝑋2 𝑋2+ 𝑋3 𝜔𝑖 𝜔1 𝜔2 𝜔3

𝑡𝑆𝑁𝑅̂ 𝑎𝑑𝑗𝑢𝑠𝑡𝑒𝑑 =𝐷𝑒𝑣(𝑦, 𝛽̂0) − 𝐷𝑒𝑣(𝑦, 𝑋𝑖, 𝛽̂𝑖) + 𝑑0− 𝑑𝑖 𝐷𝑒𝑣(𝑦, 𝑋𝑖, 𝛽̂𝑖) + 𝑑𝑖 𝑑₀ 𝑑_𝑖 𝑘 𝑘 𝛽_𝑖

0 10 20 30 40 50 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0

Number of SNPs in the model based on univariate tSNR ranking

C la s s if ic a ti o n p e rf o rm a n c e PCC AUC NPV PPV Sensitivity Specificity 0 10 20 30 40 50 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0

Number of SNPs in the model based on univariate tSNR ranking

C la s s if ic a ti o n p e rf o rm a n c e PCC AUC NPV PPV Sensitivity Specificity

glmnet

𝜆

0 10 20 30 40 50 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0

Number of SNPs in the model based on cumulative tSNR ranking

C la s s ifi c a tio n p e rf o rm a n c e PCC AUC NPV PPV Sensitivity Specificity

0 50 100 150 200 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0

Number of SNPs in the model based on cumulative tSNR ranking

C la s s if ic a ti o n p e rf o rm a n c e PCC AUC NPV PPV Sensitivity Specificity 0 50 100 150 200 0 1 2 3 4 5

Number of SNPs in the model based on cumulative tSNR ranking

A d ju s te d t S N R

(𝑛 = 639) (𝑛 = 1,515)

           

0 500 1000 1500 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 Time (days) S e iz u re s Remission 0 500 1000 1500 0 2 4 6 8 10 Time (days) lo g (1 + T o ta l S e iz u re s ) Remission 0 500 1000 1500 0 1 2 3 4 5 6 Time (days) N u m b e r o f A d v e rs e E v e n ts Remission 0 500 1000 1500 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 Time (days) S e iz u re s Refractory 0 500 1000 1500 0 2 4 6 8 10 Time (days) lo g (1 + T o ta l S e iz u re s ) Refractory 0 500 1000 1500 0 1 2 3 4 5 Time (days) N u m b e r o f A d v e rs e E v e n ts Refractory

mixAK 𝑗 (𝑗 = 1, … , 𝑛𝑟) 𝑟 (𝑟 = 1, … , 𝑅) 𝑌_𝑟,𝑗 𝑔 𝑏 ∅_𝑟𝑔 ℎ_𝑟−1{𝐸(𝑌_𝑟,𝑗|𝑏, 𝑈 = 𝑔)} = 𝑥_𝑟,𝑗𝑔𝑇𝛼_𝑟𝑔 + 𝑧_𝑟,𝑗𝑔𝑇𝑏_𝑟 𝑟 = 1, … , 𝑅, 𝑗 = 1, … , 𝑛_𝑟 𝑅 = 3 𝐸(𝑌1,𝑗|𝑏, 𝑈 = 1) = ℎ1−1(𝑥1,𝑗 𝑔𝑇 𝛼₁𝑔+ 𝑧_1,𝑗𝑔𝑇𝑏1) 𝐸(𝑌_2,𝑗|𝑏, 𝑈 = 1) = ℎ₂−1(𝑥_2,𝑗𝑔𝑇𝛼₂𝑔+ 𝑧_2,𝑗𝑔𝑇𝑏₂) 𝐸(𝑌_3,𝑗|𝑏, 𝑈 = 1) = ℎ₃−1_(𝑥 3,𝑗 𝑔𝑇 𝛼₃𝑔+ 𝑧_3,𝑗𝑔𝑇𝑏₃) 𝐸(𝑌_1,𝑗|𝑏, 𝑈 = 0) = ℎ₁−1(𝑥_1,𝑗𝑔𝑇𝛼₁𝑔+ 𝑧_1,𝑗𝑔𝑇𝑏₁) 𝐸(𝑌_2,𝑗|𝑏, 𝑈 = 0) = ℎ₂−1_(𝑥 2,𝑗 𝑔𝑇 𝛼₂𝑔+ 𝑧_2,𝑗𝑔𝑇𝑏₂) 𝐸(𝑌_3,𝑗|𝑏, 𝑈 = 0) = ℎ₃−1(𝑥_3,𝑗𝑔𝑇𝛼₃𝑔+ 𝑧_3,𝑗𝑔𝑇𝑏₃) 𝑥_𝑟,𝑗𝑔

𝑧_𝑟,𝑗𝑔 = 1 𝑏 = (𝑏1, 𝑏2, 𝑏3)𝑇 𝑔 𝑃𝑟_{𝑔,𝑛𝑒𝑤} = 𝜋𝑔𝑓̂𝑔,𝑛𝑒𝑤 ∑𝐺−1_𝑔̌=0𝜋_𝑔̃𝑓̂_{𝑔̃,𝑛𝑒𝑤} 𝑔 = 0, … , 𝐺 − 1, 𝑓̂ 𝜋𝑔 = 𝑃𝑟(𝑔), 𝑔 = 0, … , 𝐺 − 1 𝑓𝑔,𝑛𝑒𝑤 ℵ coda 𝛼_𝑟𝑔 ∅_𝑟𝑔 𝑓𝑔,𝑛𝑒𝑤

𝑌𝑛𝑒𝑤 = (𝑦𝑛𝑒𝑤,1, … , 𝑦𝑛𝑒𝑤,𝑅) 𝑓𝑔,𝑛𝑒𝑤 𝑌𝑛𝑒𝑤 𝑏_𝑛𝑒𝑤 𝕔 𝕔 𝑌𝑟

𝑋𝑟 𝑌_{𝑆𝑁𝑃1} 𝑌_{𝑆𝑁𝑃2} {ℎ𝑟 −1_{𝐸(𝑌 𝑟,𝑗|𝑏, 𝑔)} = 𝑥𝑟,𝑗 𝑔𝑇 𝛼_𝑟𝑔+ 𝑧_𝑟,𝑗𝑔𝑇𝑏_𝑟, 𝑟 = 1, … , 𝑅, 𝑗 = 1, … , 𝑛_𝑟 ℎ_𝑆𝑁𝑃−1 _𝑖{𝐸(𝑌_{𝑆𝑁𝑃𝑖,𝑗}|𝑔)} = 𝑥_𝑆𝑁𝑃 𝑖,𝑗 𝑔𝑇 𝛼_𝑆𝑁𝑃 𝑖 𝑔 , 𝑖 = 1, … , 𝑝, 𝑗 = 1, … , 𝑛_{𝑆𝑁𝑃𝑖} 𝑋_{𝑆𝑁𝑃1} 𝑋_{𝑆𝑁𝑃2} 𝑅 𝑥_𝑟,𝑗𝑔𝑇 = (𝑥_𝑟,𝑗𝑔𝑇, 𝑋_𝑆𝑁𝑃 𝑖,𝑗 𝑔𝑇 , … , 𝑋_𝑆𝑁𝑃 𝑝,𝑗 𝑔𝑇 ).

𝑌𝑆𝑁𝑃𝑖 𝑋𝑆𝑁𝑃𝑖

0 50 100 150 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0

Number of SNPs in the model based on weightage ranking

C la s s ifi c a tio n p e rf o rm a n c e PCC AUC NPV PPV Sensitivity Specificity

0 50 100 150 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0

Number of SNPs in the model based on weightage ranking

C la s s ifi c a tio n p e rf o rm a n c e 0 1 2 3 4 5 6 A d ju s te d t S N R 0 50 100 150 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0

Number of SNPs in the model based on weightage ranking

C la s s if ic a ti o n p e rf o rm a n c e PCC AUC NPV PPV Sensitivity Specificity

𝑌𝑆𝑁𝑃1 𝑌𝑆𝑁𝑃2 𝑋𝑆𝑁𝑃1 𝑋𝑆𝑁𝑃2 𝑋𝑆𝑁𝑃1 + 𝑋𝑆𝑁𝑃₂ 𝑌𝑆𝑁𝑃1 𝑌𝑆𝑁𝑃2 𝑌𝑆𝑁𝑃1 + 𝑌𝑆𝑁𝑃₂ 𝑌_{𝑆𝑁𝑃1} 𝑌_{𝑆𝑁𝑃1}+ 𝑌_{𝑆𝑁𝑃2} 𝑋𝑆𝑁𝑃1+ 𝑋𝑆𝑁𝑃2

𝑥𝑆𝑁𝑃₁ 𝑥𝑆𝑁𝑃_{2 + 𝑥}𝑥𝑆𝑁𝑃1

𝑆𝑁𝑃2

𝑌𝑆𝑁𝑃₁ 𝑌𝑆𝑁𝑃_{2 + 𝑌}𝑌𝑆𝑁𝑃1

𝑆𝑁𝑃2

𝑌𝑆𝑁𝑃1 𝑌𝑆𝑁𝑃2

𝑌𝑆𝑁𝑃1 + 𝑌𝑆𝑁𝑃2