(x_{i}, d_{i})

contributes

f (x_{i}; η)^{d_{i}} S (x_{i}; η)^{1 - d_{i}} = h (x_{i}; η)^{d_{i}} S (x_{i}; η),

where

d_{i}

indicates an event. Multiply these contributions, take the log-likelihood, and maximize it over the parametric survival-model parameter

η

, numerically if the score equations have no closed form.

b

Solution

For constant hazard

h_{i} (t) = λ

S_{i} (t) = e^{- λ t}

. With

D = \sum_{i} d_{i}

T = \sum_{i} x_{i}

ℓ (λ) = D lo g λ - λ T, λ = \frac{D}{T} .

Thus the maximum-likelihood estimate is events divided by person-time at risk.

Put

D_{k} = \sum_{i : z_{i} = k} d_{i}

and

T_{k} = \sum_{i : z_{i} = k} x_{i}

. Since the cumulative hazard is

θ e^{z_{i} β} x_{i}

ℓ (θ, β) = (D_{0} + D_{1}) lo g θ + D_{1} β - θ (T_{0} + e^{β} T_{1}) .

The score equations imply

D_{1} = θ e^{β} T_{1}

and

D_{0} = θ T_{0}

. Therefore

θ = \frac{D _{0}}{T _{0}}, β = l o g \frac{D _{1} / T _{1}}{D _{0} / T _{0}},

when both event counts are positive, with the usual infinite boundary estimates otherwise.

λ_{0} = θ

and

λ_{1} = θ e^{β}

. Separate constant-hazard likelihoods give

λ_{0} = \frac{D _{0}}{T _{0}}, λ_{1} = \frac{D _{1}}{T _{1}} .

Part (c) then verifies exactly that

λ_{0} = θ

and

λ_{1} = θ e^{β}

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