Infant mortality at the Kigali University Teaching Hospital: Application of Aalen additive hazards model and comparison with other classical survival models.

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Paul Gatabazi, Sileshi Fanta Melesse & Shaun Ramroop

School of Mathematics, Statistics and Computer Sciences,

University of KwaZulu-Natal, Pietermaritzburg Campus South Africa

Email: [email protected]

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α

β

β β β β

ψ = 𝑒^𝛽^𝑘

β ψ

β β β

β

β β β β

β

𝐵_𝑘(𝑡) = ∫ 𝛽₀^𝑡 _𝑘(𝑣)𝑑𝑣

β

β β 𝐵_𝑘(𝑡) = ∫ 𝛽₀^𝑡 𝑘𝑑𝑣 = 𝛽_𝑘𝑡

𝑌_𝑖(𝑡) = {1, if individual 𝑖 is at risk at time 𝑡0, Otherwise.

𝑑𝑁_𝑖(𝑡) = ∑^𝑝_𝑘=0𝑌_𝑖(𝑡)𝑥_𝑖𝑘(𝑡)𝑑𝐵_𝑘(𝑡) + 𝑑𝑀_𝑖

d𝐁̂(t) = [(𝐗(t))^′𝐗(t)]⁻¹(𝐗(t))^′d𝐍(t).

p-value (P) Interpretation

P > 0.1 No evidence to reject the null hypothesis

0.05 < P ≤ 0.1 Slight evidence against the null hypothesis 0.01 < P ≤ 0.05 Moderate evidence against the null hypothesis 0.001 < P ≤ 0.01 Strong evidence against the null hypothesis

P ≤ 0.001 Overwhelming evidence against the null hypothesis

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𝐁̂(t) = ∫ [(𝐗(t))^𝑡 ^′𝐗(t)]⁻¹(𝐗(t))^′d𝐍(t)

0

= ∑_𝑡_𝑗_≤𝑡[(𝐗(𝑡_𝑗))^′𝐗(𝑡_𝑗)]⁻¹(𝐗(𝑡_𝑗))^′𝑦_𝑗

𝐁̂(t)

Var[𝐁̂(t)] =

∑𝑡_𝑗≤𝑡[(𝐗(𝑡𝑗))^′𝐗(𝑡𝑗)]⁻¹(𝐗(𝑡𝑗))^′𝐃(𝑡𝑗)𝐗(𝑡𝑗)[(𝐗(𝑡𝑗))^′𝐗(𝑡𝑗)]⁻¹

𝜷̂(𝑡𝑗) = [(𝐗(𝑡𝑗))^′𝐗(𝑡𝑗)]⁻¹(𝐗(𝑡𝑗))^′𝑦𝑗

Var[𝜷̂(𝑡𝑗)] =

[(𝐗(𝑡_𝑗))^′𝐗(𝑡_𝑗)]⁻¹(𝐗(𝑡_𝑗))^′𝐃(𝑡_𝑗)𝐗(𝑡_𝑗)[(𝐗(𝑡_𝑗))^′𝐗(𝑡_𝑗)]⁻¹

α

𝐵̂_𝑘(𝑡) = ±𝑧^𝛼

2√𝜎̂_𝑘𝑘(𝑡)

β ∀ 𝜖

β β

∆𝐵̂_𝑘(𝑡) β

𝐵̂_𝑘(𝑡) β

β

𝐮̂ = ∑ 𝐊_𝑡_𝑗 _𝑗𝜷̂(𝑡_𝑗) 𝜷̂(𝑡_𝑗)

𝑆̂_𝐾𝑀(𝑡_𝑗−1) 𝑆̂_𝐾𝑀(𝑡_𝑗−1)

𝑡_𝑗−1 𝑆̂_𝐾𝑀(𝑡₀) =

1

𝑆̂_𝐾𝑀(𝑡_𝑗−1)/se 𝛽̂_𝑘𝑘(𝑡𝑗) 𝛽̂_𝑘𝑘(𝑡𝑗)

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𝐮̂

Var̂ (𝐮̂) = ∑ 𝐊_𝑗Var[𝜷̂(𝑡𝑗)𝐊^′_𝑗]

𝑡𝑗

= ∑ 𝐊_𝑡_𝑗 _𝑗[(𝐗(𝑡_𝑗))^′𝐗(𝑡_𝑗)]⁻¹(𝐗(𝑡_𝑗))^′𝐃(𝑡_𝑗)𝐗(𝑡_𝑗)[(𝐗(𝑡_𝑗))^′𝐗(𝑡_𝑗)]⁻¹𝐊^′_𝑗

𝑧_𝑢_𝑘 = ^𝑢̂^𝑘

se(𝑢̂𝑘)

𝑢̂_𝑘 𝐮̂

se(𝑢̂_𝑘) Var̂ (𝐮̂)

uk

z

α

β α

𝐵(𝑡) = ∫ 𝛽(𝑣)𝑑𝑣₀^𝑡 α

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β

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(a) APGAR=4/10 to APGAR=6/10 (b) APGAR=7/10 and above .

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(

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≤

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