46
Step 4, => Keep updating βim+1'> and ^m+1^ until they converge respectively
Step 5, ≠> Let m = m + l, repeat steps 2, 3 and 4 until β(m+1'> and β^l+1^i converge
respectively.
Consequently, we could get estimated hazard functions Λ(∙) and Ψ(∙), and survival
functions S(∙) and R(β.
Specifically, at Step 1,
Assuming independent censoring, we fit two Cox proportional hazards models to
get initial estimators β^ and βc°'1 for β and βc, respectively. Then we use the Breslow
(1972) method to obtain estimators for baseline cumulative hazard functions, which
give estimates of baseline survival functions.
• Baseline survival functions
For an event T and i < k, we have
^obsii1 -)
£ exp(Z'kβ^γ'
obsh>obsi
(3.13)
S,o°∖θ = eτp{-A^(t)} = exp{-
obsi<t
For dependent censoring C and i < k, we have
7⅜⅜) =
exp{-Φ^(t)} = exp{-
obsi<t
^obsi,2 ɪ
£ exp(Wkβ^γ,
obδk'>obsi
(3.14)
• Marginal cumulative distribution functions